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the "requirement niche". The requirement niche is bounded by both the availability of resources as well as the effects of coexisting consumers (e.g. competitors and predators). === Coexistence under contemporary niche theory === Contemporary niche theory provides three requirements that must be met in order for two species (consumers) to coexist: The requirement niches of both consumers must overlap. Each consumer must outcompete the other for the resource that it needs most. For example, if two plants (P1 and P2) are competing for nitrogen and phosphorus in a given ecosystem, they will only coexist if they are limited by different resources (P1 is limited by nitrogen and P2 is limited by phosphorus, perhaps) and each species must outcompete the other species to get that resource (P1 needs to be better at obtaining nitrogen and P2 needs to be better at obtaining phosphorus). Intuitively, this makes sense from an inverse perspective: If both consumers are limited by the same resource, one of the species will ultimately be the better competitor, and only that species will survive. Furthermore, if P1 was outcompeted for the nitrogen (the resource it needed most) it would not survive. Likewise, if P2 was outcompeted for phosphorus, it would not survive. The availability of the limiting resources (nitrogen and phosphorus in the above example) in the environment are equivalent. These requirements are interesting and controversial because they require any two species to share a certain environment (have overlapping requirement niches) but fundamentally differ the ways that they use (or "impact") that environment. These requirements have repeatedly been violated by nonnative (i.e. introduced and invasive) species, which often coexist with new species in their nonnative ranges, but do not appear to be constricted these requirements. In other words, contemporary niche theory predicts that species will be unable to invade
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new environments outside of their requirement (i.e. realized) niche, yet many examples of this are well-documented. Additionally, contemporary niche theory predicts that species will be unable to establish in environments where other species already consume resources in the same ways as the incoming species, however examples of this are also numerous. == Niche differentiation == In ecology, niche differentiation (also known as niche segregation, niche separation and niche partitioning) refers to the process by which competing species use the environment differently in a way that helps them to coexist. The competitive exclusion principle states that if two species with identical niches (ecological roles) compete, then one will inevitably drive the other to extinction. This rule also states that two species cannot occupy the same exact niche in a habitat and coexist together, at least in a stable manner. When two species differentiate their niches, they tend to compete less strongly, and are thus more likely to coexist. Species can differentiate their niches in many ways, such as by consuming different foods, or using different areas of the environment. As an example of niche partitioning, several anole lizards in the Caribbean islands share common diets—mainly insects. They avoid competition by occupying different physical locations. Although these lizards might occupy different locations, some species can be found inhabiting the same range, with up to 15 in certain areas. For example, some live on the ground while others are arboreal. Species who live in different areas compete less for food and other resources, which minimizes competition between species. However, species who live in similar areas typically compete with each other. === Detection and quantification === The Lotka–Volterra equation states that two competing species can coexist when intra-specific (within species) competition is greater than inter-specific (between species) competition. Since niche differentiation concentrates competition within-species,
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due to a decrease in between-species competition, the Lotka-Volterra model predicts that niche differentiation of any degree will result in coexistence. In reality, this still leaves the question of how much differentiation is needed for coexistence. A vague answer to this question is that the more similar two species are, the more finely balanced the suitability of their environment must be in order to allow coexistence. There are limits to the amount of niche differentiation required for coexistence, and this can vary with the type of resource, the nature of the environment, and the amount of variation both within and between the species. To answer questions about niche differentiation, it is necessary for ecologists to be able to detect, measure, and quantify the niches of different coexisting and competing species. This is often done through a combination of detailed ecological studies, controlled experiments (to determine the strength of competition), and mathematical models. To understand the mechanisms of niche differentiation and competition, much data must be gathered on how the two species interact, how they use their resources, and the type of ecosystem in which they exist, among other factors. In addition, several mathematical models exist to quantify niche breadth, competition, and coexistence (Bastolla et al. 2005). However, regardless of methods used, niches and competition can be distinctly difficult to measure quantitatively, and this makes detection and demonstration of niche differentiation difficult and complex. === Development === Over time, two competing species can either coexist, through niche differentiation or other means, or compete until one species becomes locally extinct. Several theories exist for how niche differentiation arises or evolves given these two possible outcomes. ==== Current competition (The Ghost of Competition Present) ==== Niche differentiation can arise from current competition. For instance, species X has a fundamental niche of the entire
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slope of a hillside, but its realized niche is only the top portion of the slope because species Y, which is a better competitor but cannot survive on the top portion of the slope, has excluded it from the lower portion of the slope. With this scenario, competition will continue indefinitely in the middle of the slope between these two species. Because of this, detection of the presence of niche differentiation (through competition) will be relatively easy. Importantly, there is no evolutionary change of the individual species in this case; rather this is an ecological effect of species Y out-competing species X within the bounds of species Y's fundamental niche. ==== Via past extinctions (The Ghost of Competition Past) ==== Another way by which niche differentiation can arise is via the previous elimination of species without realized niches. This asserts that at some point in the past, several species inhabited an area, and all of these species had overlapping fundamental niches. However, through competitive exclusion, the less competitive species were eliminated, leaving only the species that were able to coexist (i.e. the most competitive species whose realized niches did not overlap). Again, this process does not include any evolutionary change of individual species, but it is merely the product of the competitive exclusion principle. Also, because no species is out-competing any other species in the final community, the presence of niche differentiation will be difficult or impossible to detect. ==== Evolving differences ==== Finally, niche differentiation can arise as an evolutionary effect of competition. In this case, two competing species will evolve different patterns of resource use so as to avoid competition. Here too, current competition is absent or low, and therefore detection of niche differentiation is difficult or impossible. === Types === Below is a list of ways that
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species can partition their niche. This list is not exhaustive, but illustrates several classic examples. ==== Resource partitioning ==== Resource partitioning is the phenomenon where two or more species divides out resources like food, space, resting sites etc. to coexist. For example, some lizard species appear to coexist because they consume insects of differing sizes. Alternatively, species can coexist on the same resources if each species is limited by different resources, or differently able to capture resources. Different types of phytoplankton can coexist when different species are differently limited by nitrogen, phosphorus, silicon, and light. In the Galapagos Islands, finches with small beaks are more able to consume small seeds, and finches with large beaks are more able to consume large seeds. If a species' density declines, then the food it most depends on will become more abundant (since there are so few individuals to consume it). As a result, the remaining individuals will experience less competition for food. Although "resource" generally refers to food, species can partition other non-consumable objects, such as parts of the habitat. For example, warblers are thought to coexist because they nest in different parts of trees. Species can also partition habitat in a way that gives them access to different types of resources. As stated in the introduction, anole lizards appear to coexist because each uses different parts of the forests as perch locations. This likely gives them access to different species of insects. Research has determined that plants can recognize each other's root systems and differentiate between a clone, a plant grown from the same mother plants seeds, and other species. Based on the root secretions, also called exudates, plants can make this determination. The communication between plants starts with the secretions from plant roots into the rhizosphere. If another plant that is
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kin is entering this area the plant will take up exudates. The exudate, being several different compounds, will enter the plants root cell and attach to a receptor for that chemical halting growth of the root meristem in that direction, if the interaction is kin. Simonsen discusses how plants accomplish root communication with the addition of beneficial rhizobia and fungal networks and the potential for different genotypes of the kin plants, such as the legume M. Lupulina, and specific strains of nitrogen fixing bacteria and rhizomes can alter relationships between kin and non-kin competition. This means there could be specific subsets of genotypes in kin plants that selects well with specific strains that could outcompete other kin. What might seem like an instance in kin competition could just be different genotypes of organisms at play in the soil that increase the symbiotic efficiency. ==== Predator partitioning ==== Predator partitioning occurs when species are attacked differently by different predators (or natural enemies more generally). For example, trees could differentiate their niche if they are consumed by different species of specialist herbivores, such as herbivorous insects. If a species density declines, so too will the density of its natural enemies, giving it an advantage. Thus, if each species is constrained by different natural enemies, they will be able to coexist. Early work focused on specialist predators; however, more recent studies have shown that predators do not need to be pure specialists, they simply need to affect each prey species differently. The Janzen–Connell hypothesis represents a form of predator partitioning. ==== Conditional differentiation ==== Conditional differentiation (sometimes called temporal niche partitioning) occurs when species differ in their competitive abilities based on varying environmental conditions. For example, in the Sonoran Desert, some annual plants are more successful during wet years, while others are more
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successful during dry years. As a result, each species will have an advantage in some years, but not others. When environmental conditions are most favorable, individuals will tend to compete most strongly with member of the same species. For example, in a dry year, dry-adapted plants will tend to be most limited by other dry-adapted plants. This can help them to coexist through a storage effect. ==== Competition-predation trade-off ==== Species can differentiate their niche via a competition-predation trade-off if one species is a better competitor when predators are absent, and the other is better when predators are present. Defenses against predators, such as toxic compounds or hard shells, are often metabolically costly. As a result, species that produce such defenses are often poor competitors when predators are absent. Species can coexist through a competition-predation trade-off if predators are more abundant when the less defended species is common, and less abundant if the well-defended species is common. This effect has been criticized as being weak, because theoretical models suggest that only two species within a community can coexist because of this mechanism. == Segregation versus restriction == Two ecological paradigms deal with the problem. The first paradigm predominates in what may be called "classical" ecology. It assumes that niche space is largely saturated with individuals and species, leading to strong competition. Niches are restricted because "neighbouring" species, i.e., species with similar ecological characteristics such as similar habitats or food preferences, prevent expansion into other niches or even narrow niches down. This continual struggle for existence is an important assumption of natural selection introduced by Darwin as an explanation for evolution. The other paradigm assumes that niche space is to a large degree vacant, i.e., that there are many vacant niches. It is based on many empirical studies and theoretical investigations
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especially of Kauffman 1993. Causes of vacant niches may be evolutionary contingencies or brief or long-lasting environmental disturbances. Both paradigms agree that species are never "universal" in the sense that they occupy all possible niches; they are always specialized, although the degree of specialization varies. For example, there is no universal parasite which infects all host species and microhabitats within or on them. However, the degree of host specificity varies strongly. Thus, Toxoplasma (Protista) infects numerous vertebrates including humans, Enterobius vermicularis infects only humans. The following mechanisms for niche restriction and segregation have been proposed: Niche restriction: Species must be specialized in order to survive. They may survive for a while in less optimal habitats under favourable conditions, but they will be extinguished when conditions become less favourable, for example due to changed weather conditions (this aspect was especially emphasized by Price 1983). Niches remain narrow or become narrower as the result of natural selection in order to enhance the chances of mating. This "mating theory of niche restriction" is supported by the observation that niches of asexual stages are often wider than those of sexually mature stages; that niches become narrower at the time of mating; and that microhabitats of sessile species and of species with small population sizes often are narrower than those of non-sessile species and of species with large population sizes. Niche segregation: The random selection of niches in largely empty niche space will often automatically lead to segregation (this mechanism is of particular importance in the second paradigm). Niches are segregated due to interspecific competition (this mechanism is of particular importance in the first paradigm). Niches of similar species are segregated (as the result of natural selection) in order to prevent interspecific hybridisation, because hybrids are less fit. (Many cases of niche segregation explained by
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interspecific competition are better explained by this mechanism, i.e., "reinforcement of reproductive barriers") (e.g., Rohde 2005b). === Relative significance of the mechanisms === Both paradigms acknowledge a role for all mechanisms (except possibly for that of random selection of niches in the first paradigm), but emphasis on the various mechanisms varies. The first paradigm stresses the paramount importance of interspecific competition, whereas the second paradigm tries to explain many cases which are thought to be due to competition in the first paradigm, by reinforcement of reproductive barriers and/or random selection of niches. – Many authors believe in the overriding importance of interspecific competition. Intuitively, one would expect that interspecific competition is of particular importance in all those cases in which sympatric species (i.e., species occurring together in the same area) with large population densities use the same resources and largely exhaust them. However, Andrewartha and Birch (1954,1984) and others have pointed out that most natural populations usually don't even approach exhaustion of resources, and too much emphasis on interspecific competition is therefore wrong. Concerning the possibility that competition has led to segregation in the evolutionary past, Wiens (1974, 1984) concluded that such assumptions cannot be proven, and Connell (1980) found that interspecific competition as a mechanism of niche segregation has been proven only for some pest insects. Barker (1983), in his review of competition in Drosophila and related genera, which are among the best known animal groups, concluded that the idea of niche segregation by interspecific competition is attractive, but that no study has yet been able to show a mechanism responsible for segregation. Without specific evidence, the possibility of random segregation can never be excluded, and assumption of such randomness can indeed serve as a null-model. – Many physiological and morphological differences between species can prevent hybridization. Evidence for
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niche segregation as the result of reinforcement of reproductive barriers is especially convincing in those cases in which such differences are not found in allopatric but only in sympatric locations. For example, Kawano (2002) has shown this for giant rhinoceros beetles in Southeast Asia. Two closely related species occur in 12 allopatric (i.e., in different areas) and 7 sympatric (i.e., in the same area) locations. In the former, body length and length of genitalia are practically identical, in the latter, they are significantly different, and much more so for the genitalia than the body, convincing evidence that reinforcement is an important factor (and possibly the only one) responsible for niche segregation. - The very detailed studies of communities of Monogenea parasitic on the gills of marine and freshwater fishes by several authors have shown the same. Species use strictly defined microhabitats and have very complex copulatory organs. This and the fact that fish replicas are available in almost unlimited numbers, makes them ideal ecological models. Many congeners (species belonging to the same genus) and non-congeners were found on single host species. The maximum number of congeners was nine species. The only limiting factor is space for attachment, since food (blood, mucus, fast regenerating epithelial cells) is in unlimited supply as long as the fish is alive. Various authors, using a variety of statistical methods, have consistently found that species with different copulatory organs may co-occur in the same microhabitat, whereas congeners with identical or very similar copulatory organs are spatially segregated, convincing evidence that reinforcement and not competition is responsible for niche segregation. For a detailed discussion, especially of competition and reinforcement of reproductive barriers, see == Coexistence without niche differentiation: exceptions to the rule == Some competing species have been shown to coexist on the same resource with no
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observable evidence of niche differentiation and in "violation" of the competitive exclusion principle. One instance is in a group of hispine beetle species. These beetle species, which eat the same food and occupy the same habitat, coexist without any evidence of segregation or exclusion. The beetles show no aggression either intra- or inter-specifically. Coexistence may be possible through a combination of non-limiting food and habitat resources and high rates of predation and parasitism, though this has not been demonstrated. This example illustrates that the evidence for niche differentiation is by no means universal. Niche differentiation is also not the only means by which coexistence is possible between two competing species. However, niche differentiation is a critically important ecological idea which explains species coexistence, thus promoting the high biodiversity often seen in many of the world's biomes. Research using mathematical modelling is indeed demonstrating that predation can indeed stabilize lumps of very similar species. Willow warbler and chiffchaff and other very similar warblers can serve as an example. The idea is that it is also a good strategy to be very similar to a successful species or have enough dissimilarity. Other examples of nearly identical species clusters occupying the same niche were water beetles, prairie birds and algae. The basic idea is that there can be clusters of very similar species all applying the same successful strategy and between them open spaces. Here the species cluster takes the place of a single species in the classical ecological models. == Niche and Geographic Range == The geographic range of a species can be viewed as a spatial reflection of its niche, along with characteristics of the geographic template and the species that influence its potential to colonize. The fundamental geographic range of a species is the area it occupies in which environmental
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conditions are favorable, without restriction from barriers to disperse or colonize. A species will be confined to its realized geographic range when confronting biotic interactions or abiotic barriers that limit dispersal, a more narrow subset of its larger fundamental geographic range. An early study on ecological niches conducted by Joseph H. Connell analyzed the environmental factors that limit the range of a barnacle (Chthamalus stellatus) on Scotland's Isle of Cumbrae. In his experiments, Connell described the dominant features of C. stellatus niches and provided explanation for their distribution on intertidal zone of the rocky coast of the Isle. Connell described the upper portion of C. stellatus's range is limited by the barnacle's ability to resist dehydration during periods of low tide. The lower portion of the range was limited by interspecific interactions, namely competition with a cohabiting barnacle species and predation by a snail. By removing the competing B. balanoides, Connell showed that C. stellatus was able to extend the lower edge of its realized niche in the absence of competitive exclusion. These experiments demonstrate how biotic and abiotic factors limit the distribution of an organism. == Parameters == The different dimensions, or plot axes, of a niche represent different biotic and abiotic variables. These factors may include descriptions of the organism's life history, habitat, trophic position (place in the food chain), and geographic range. According to the competitive exclusion principle, no two species can occupy the same niche in the same environment for a long time. The parameters of a realized niche are described by the realized niche width of that species. Some plants and animals, called specialists, need specific habitats and surroundings to survive, such as the spotted owl, which lives specifically in old growth forests. Other plants and animals, called generalists, are not as particular and can
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survive in a range of conditions, for example the dandelion. == See also == Ontogenetic niche shift Marginal distribution (biology) Fitness landscape Niche differentiation Overpopulation Phylogenetic niche conservatism Unified neutral theory of biodiversity Character displacement == References == == Further reading == == External links == Concept of ecological niche Ontology of the niche Niche restriction and segregation Vacant niche Latitude-niche width hypothesis Walter, G.H. (May 1991). "What is resource partitioning?". J. Theor. Biol. 150 (2): 137–43. Bibcode:1991JThBi.150..137W. doi:10.1016/S0022-5193(05)80327-3. PMID 1890851.
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In astrophysics and statistical mechanics, Jeans's theorem, named after James Jeans, states that any steady-state solution of the collisionless Boltzmann equation depends on the phase space coordinates only through integrals of motion in the given potential, and conversely any function of the integrals is a steady-state solution. Jeans's theorem is most often discussed in the context of potentials characterized by three, global integrals. In such potentials, all of the orbits are regular, i.e. non-chaotic; the Kepler potential is one example. In generic potentials, some orbits respect only one or two integrals and the corresponding motion is chaotic. Jeans's theorem can be generalized to such potentials as follows: The phase-space density of a stationary stellar system is constant within every well-connected region. A well-connected region is one that cannot be decomposed into two finite regions such that all trajectories lie, for all time, in either one or the other. Invariant tori of regular orbits are such regions, but so are the more complex parts of phase space associated with chaotic trajectories. Integrability of the motion is therefore not required for a steady state. == Mathematical description == Consider the collisionless Boltzmann equation for the distribution function f ( x , v , t ) {\displaystyle f(\mathbf {x} ,\mathbf {v} ,t)} ∂ f ∂ t + v ⋅ ∇ f + 1 m F ⋅ ∇ v f = 0. {\displaystyle {\frac {\partial f}{\partial t}}+\mathbf {v} \cdot \nabla f+{\frac {1}{m}}\mathbf {F} \cdot \nabla _{v}f=0.} Consider the Lagrangian approach to the particle's motion in which case, the required equations are d x d t = v {\displaystyle {\frac {d\mathbf {x} }{dt}}=\mathbf {v} } d v d t = F m . {\displaystyle {\frac {d\mathbf {v} }{dt}}={\frac {\mathbf {F} }{m}}.} Let the solutions of these equations be x = x ( α 1 ,
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… , α 6 , t ) {\displaystyle \mathbf {x} =\mathbf {x} (\alpha _{1},\dots ,\alpha _{6},t)} v = v ( α 1 , … , α 6 , t ) {\displaystyle \mathbf {v} =\mathbf {v} (\alpha _{1},\dots ,\alpha _{6},t)} where α i {\displaystyle \alpha _{i}} s are the integration constants. Let us assume that from the above set, we are able to solve α i {\displaystyle \alpha _{i}} , that is to say, we are able to find α i = α i ( x , v , t ) . {\displaystyle \alpha _{i}=\alpha _{i}(\mathbf {x} ,\mathbf {v} ,t).} Now consider an arbitrary function of α i {\displaystyle \alpha _{i}} 's, f = f ( α 1 , … , α 6 ) . {\displaystyle f=f(\alpha _{1},\dots ,\alpha _{6}).} Then this function is the solution of the collisionless Boltzmann equation, as can be verified by substituting this function into the collisionless Boltzmann equation to find ∑ i = 1 6 ∂ f ∂ α i [ ∂ α i ∂ t + v ⋅ ∇ α i + 1 m F ⋅ ∇ v α i ] = ∑ i = 1 6 ∂ f ∂ α i d α i d t = 0. {\displaystyle \sum _{i=1}^{6}{\frac {\partial f}{\partial \alpha _{i}}}\left[{\frac {\partial \alpha _{i}}{\partial t}}+\mathbf {v} \cdot \nabla \alpha _{i}+{\frac {1}{m}}\mathbf {F} \cdot \nabla _{v}\alpha _{i}\right]=\sum _{i=1}^{6}{\frac {\partial f}{\partial \alpha _{i}}}{\frac {d\alpha _{i}}{dt}}=0.} This proves the theorem. A trivial set of integration constants are the initial location x 0 {\displaystyle \mathbf {x} _{0}} and the initial velocities v 0 {\displaystyle \mathbf {v} _{0}} of the particle. In this case, any function f = f ( x 0 ( x , v , t ) , v 0 ( x , v , t ) ) {\displaystyle f=f(\mathbf {x} _{0}(\mathbf
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{x} ,\mathbf {v} ,t),\mathbf {v} _{0}(\mathbf {x} ,\mathbf {v} ,t))} is a solution of the collisionless Boltzmann equation. == See also == Jeans equations == References ==
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"page_id": 35784363,
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In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. == Overview == === Phase space coordinates (p, q) and Hamiltonian H === Let ( M , L ) {\displaystyle (M,{\mathcal {L}})} be a mechanical system with configuration space M {\displaystyle M} and smooth Lagrangian L . {\displaystyle {\mathcal {L}}.} Select a standard coordinate system ( q , q ˙ ) {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} on M . {\displaystyle M.} The quantities p i ( q , q ˙ , t ) = def ∂ L / ∂ q ˙ i {\displaystyle \textstyle p_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)~{\stackrel {\text{def}}{=}}~{\partial {\mathcal {L}}}/{\partial {\dot {q}}^{i}}} are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant t , {\displaystyle t,} the Legendre transformation of L {\displaystyle {\mathcal {L}}} is defined as the map ( q , q ˙ ) → ( p , q ) {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\to \left({\boldsymbol {p}},{\boldsymbol {q}}\right)} which is assumed to have a smooth inverse ( p , q ) → ( q , q ˙ ) . {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})\to ({\boldsymbol {q}},{\boldsymbol {\dot {q}}}).} For a system with n {\displaystyle n} degrees of freedom, the Lagrangian mechanics defines the energy function E L ( q , q ˙ , t ) = def ∑ i = 1 n q ˙ i ∂ L ∂ q ˙ i − L . {\displaystyle E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot
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"page_id": 198319,
"source": null,
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{q}}},t)\,{\stackrel {\text{def}}{=}}\,\sum _{i=1}^{n}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}.} The Legendre transform of L {\displaystyle {\mathcal {L}}} turns E L {\displaystyle E_{\mathcal {L}}} into a function H ( p , q , t ) {\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)} known as the Hamiltonian. The Hamiltonian satisfies H ( ∂ L ∂ q ˙ , q , t ) = E L ( q , q ˙ , t ) {\displaystyle {\mathcal {H}}\left({\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}},{\boldsymbol {q}},t\right)=E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)} which implies that H ( p , q , t ) = ∑ i = 1 n p i q ˙ i − L ( q , q ˙ , t ) , {\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)=\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t),} where the velocities q ˙ = ( q ˙ 1 , … , q ˙ n ) {\displaystyle {\boldsymbol {\dot {q}}}=({\dot {q}}^{1},\ldots ,{\dot {q}}^{n})} are found from the ( n {\displaystyle n} -dimensional) equation p = ∂ L / ∂ q ˙ {\displaystyle \textstyle {\boldsymbol {p}}={\partial {\mathcal {L}}}/{\partial {\boldsymbol {\dot {q}}}}} which, by assumption, is uniquely solvable for q ˙ {\displaystyle {\boldsymbol {\dot {q}}}} . The ( 2 n {\displaystyle 2n} -dimensional) pair ( p , q ) {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})} is called phase space coordinates. (Also canonical coordinates). === From Euler–Lagrange equation to Hamilton's equations === In phase space coordinates ( p , q ) {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})} , the ( n {\displaystyle n} -dimensional) Euler–Lagrange equation ∂ L ∂ q − d d t ∂ L ∂ q ˙ = 0 {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}-{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\boldsymbol {q}}}}}=0} becomes Hamilton's equations in 2 n {\displaystyle 2n} dimensions === From stationary action principle to Hamilton's equations === Let P (
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"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
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a , b , x a , x b ) {\displaystyle {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} be the set of smooth paths q : [ a , b ] → M {\displaystyle {\boldsymbol {q}}:[a,b]\to M} for which q ( a ) = x a {\displaystyle {\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}} and q ( b ) = x b . {\displaystyle {\boldsymbol {q}}(b)={\boldsymbol {x}}_{b}.} The action functional S : P ( a , b , x a , x b ) → R {\displaystyle {\mathcal {S}}:{\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})\to \mathbb {R} } is defined via S [ q ] = ∫ a b L ( t , q ( t ) , q ˙ ( t ) ) d t = ∫ a b ( ∑ i = 1 n p i q ˙ i − H ( p , q , t ) ) d t , {\displaystyle {\mathcal {S}}[{\boldsymbol {q}}]=\int _{a}^{b}{\mathcal {L}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))\,dt=\int _{a}^{b}\left(\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)\right)\,dt,} where q = q ( t ) {\displaystyle {\boldsymbol {q}}={\boldsymbol {q}}(t)} , and p = ∂ L / ∂ q ˙ {\displaystyle {\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\boldsymbol {\dot {q}}}} (see above). A path q ∈ P ( a , b , x a , x b ) {\displaystyle {\boldsymbol {q}}\in {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} is a stationary point of S {\displaystyle {\mathcal {S}}} (and hence is an equation of motion) if and only if the path ( p ( t ) , q ( t ) ) {\displaystyle ({\boldsymbol {p}}(t),{\boldsymbol {q}}(t))} in phase space coordinates obeys the Hamilton equations. === Basic physical interpretation === A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one nonrelativistic particle of mass m. The value H ( p , q ) {\displaystyle H(p,q)} of the Hamiltonian is
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"page_id": 198319,
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}
|
the total energy of the system, in this case the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here p is the momentum mv and q is the space coordinate. Then H = T + V , T = p 2 2 m , V = V ( q ) {\displaystyle {\mathcal {H}}=T+V,\qquad T={\frac {p^{2}}{2m}},\qquad V=V(q)} T is a function of p alone, while V is a function of q alone (i.e., T and V are scleronomic). In this example, the time derivative of q is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum. The time derivative of the momentum p equals the Newtonian force, and so the second Hamilton equation means that the force equals the negative gradient of potential energy. == Example == A spherical pendulum consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity. Spherical coordinates are used to describe the position of the mass in terms of (r, θ, φ), where r is fixed, r = ℓ. The Lagrangian for this system is L = 1 2 m ℓ 2 ( θ ˙ 2 + sin 2 θ φ ˙ 2 ) + m g ℓ cos θ . {\displaystyle L={\frac {1}{2}}m\ell ^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \ {\dot {\varphi }}^{2}\right)+mg\ell \cos \theta .} Thus the Hamiltonian is H = P θ θ ˙ + P φ φ ˙ − L {\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\varphi }{\dot {\varphi }}-L} where P θ = ∂ L ∂ θ ˙ = m ℓ 2 θ ˙ {\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=m\ell ^{2}{\dot {\theta
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
}}} and P φ = ∂ L ∂ φ ˙ = m ℓ 2 sin 2 θ φ ˙ . {\displaystyle P_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}.} In terms of coordinates and momenta, the Hamiltonian reads H = [ 1 2 m ℓ 2 θ ˙ 2 + 1 2 m ℓ 2 sin 2 θ φ ˙ 2 ] ⏟ T + [ − m g ℓ cos θ ] ⏟ V = P θ 2 2 m ℓ 2 + P φ 2 2 m ℓ 2 sin 2 θ − m g ℓ cos θ . {\displaystyle H=\underbrace {\left[{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}^{2}\right]} _{T}+\underbrace {{\Big [}-mg\ell \cos \theta {\Big ]}} _{V}={\frac {P_{\theta }^{2}}{2m\ell ^{2}}}+{\frac {P_{\varphi }^{2}}{2m\ell ^{2}\sin ^{2}\theta }}-mg\ell \cos \theta .} Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations, θ ˙ = P θ m ℓ 2 φ ˙ = P φ m ℓ 2 sin 2 θ P θ ˙ = P φ 2 m ℓ 2 sin 3 θ cos θ − m g ℓ sin θ P φ ˙ = 0. {\displaystyle {\begin{aligned}{\dot {\theta }}&={P_{\theta } \over m\ell ^{2}}\\[6pt]{\dot {\varphi }}&={P_{\varphi } \over m\ell ^{2}\sin ^{2}\theta }\\[6pt]{\dot {P_{\theta }}}&={P_{\varphi }^{2} \over m\ell ^{2}\sin ^{3}\theta }\cos \theta -mg\ell \sin \theta \\[6pt]{\dot {P_{\varphi }}}&=0.\end{aligned}}} Momentum P φ {\displaystyle P_{\varphi }} , which corresponds to the vertical component of angular momentum L z = ℓ sin θ × m ℓ sin θ φ ˙ {\displaystyle L_{z}=\ell \sin \theta \times m\ell \sin \theta \,{\dot {\varphi }}} , is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
axis. Being absent from the Hamiltonian, azimuth φ {\displaystyle \varphi } is a cyclic coordinate, which implies conservation of its conjugate momentum. == Deriving Hamilton's equations == Hamilton's equations can be derived by a calculation with the Lagrangian L {\displaystyle {\mathcal {L}}} , generalized positions qi, and generalized velocities ⋅qi, where i = 1 , … , n {\displaystyle i=1,\ldots ,n} . Here we work off-shell, meaning q i {\displaystyle q^{i}} , q ˙ i {\displaystyle {\dot {q}}^{i}} , t {\displaystyle t} are independent coordinates in phase space, not constrained to follow any equations of motion (in particular, q ˙ i {\displaystyle {\dot {q}}^{i}} is not a derivative of q i {\displaystyle q^{i}} ). The total differential of the Lagrangian is: d L = ∑ i ( ∂ L ∂ q i d q i + ∂ L ∂ q ˙ i d q ˙ i ) + ∂ L ∂ t d t . {\displaystyle \mathrm {d} {\mathcal {L}}=\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}\,\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .} The generalized momentum coordinates were defined as p i = ∂ L / ∂ q ˙ i {\displaystyle p_{i}=\partial {\mathcal {L}}/\partial {\dot {q}}^{i}} , so we may rewrite the equation as: d L = ∑ i ( ∂ L ∂ q i d q i + p i d q ˙ i ) + ∂ L ∂ t d t = ∑ i ( ∂ L ∂ q i d q i + d ( p i q ˙ i ) − q ˙ i d p i ) + ∂ L ∂ t d t . {\displaystyle {\begin{aligned}\mathrm {d} {\mathcal {L}}=&\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+p_{i}\mathrm {d}
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
{\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t\\=&\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+\mathrm {d} (p_{i}{\dot {q}}^{i})-{\dot {q}}^{i}\,\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\,.\end{aligned}}} After rearranging, one obtains: d ( ∑ i p i q ˙ i − L ) = ∑ i ( − ∂ L ∂ q i d q i + q ˙ i d p i ) − ∂ L ∂ t d t . {\displaystyle \mathrm {d} \!\left(\sum _{i}p_{i}{\dot {q}}^{i}-{\mathcal {L}}\right)=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .} The term in parentheses on the left-hand side is just the Hamiltonian H = ∑ p i q ˙ i − L {\textstyle {\mathcal {H}}=\sum p_{i}{\dot {q}}^{i}-{\mathcal {L}}} defined previously, therefore: d H = ∑ i ( − ∂ L ∂ q i d q i + q ˙ i d p i ) − ∂ L ∂ t d t . {\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\,\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .} One may also calculate the total differential of the Hamiltonian H {\displaystyle {\mathcal {H}}} with respect to coordinates q i {\displaystyle q^{i}} , p i {\displaystyle p_{i}} , t {\displaystyle t} instead of q i {\displaystyle q^{i}} , q ˙ i {\displaystyle {\dot {q}}^{i}} , t {\displaystyle t} , yielding: d H = ∑ i ( ∂ H ∂ q i d q i + ∂ H ∂ p i d p i ) + ∂ H ∂ t d t . {\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .} One may now equate these two
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
expressions for d H {\displaystyle d{\mathcal {H}}} , one in terms of L {\displaystyle {\mathcal {L}}} , the other in terms of H {\displaystyle {\mathcal {H}}} : ∑ i ( − ∂ L ∂ q i d q i + q ˙ i d p i ) − ∂ L ∂ t d t = ∑ i ( ∂ H ∂ q i d q i + ∂ H ∂ p i d p i ) + ∂ H ∂ t d t . {\displaystyle \sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ =\ \sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .} Since these calculations are off-shell, one can equate the respective coefficients of d q i {\displaystyle \mathrm {d} q^{i}} , d p i {\displaystyle \mathrm {d} p_{i}} , d t {\displaystyle \mathrm {d} t} on the two sides: ∂ H ∂ q i = − ∂ L ∂ q i , ∂ H ∂ p i = q ˙ i , ∂ H ∂ t = − ∂ L ∂ t . {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\partial {\mathcal {L}} \over \partial t}\ .} On-shell, one substitutes parametric functions q i = q i ( t ) {\displaystyle q^{i}=q^{i}(t)} which define a trajectory in phase space with velocities q ˙ i = d d t q i ( t ) {\displaystyle {\dot {q}}^{i}={\tfrac {d}{dt}}q^{i}(t)} , obeying Lagrange's equations: d d t ∂ L ∂ q ˙ i − ∂ L ∂ q i = 0 . {\displaystyle {\frac {\mathrm {d} }{\mathrm
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
{d} t}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0\ .} Rearranging and writing in terms of the on-shell p i = p i ( t ) {\displaystyle p_{i}=p_{i}(t)} gives: ∂ L ∂ q i = p ˙ i . {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial q^{i}}}={\dot {p}}_{i}\ .} Thus Lagrange's equations are equivalent to Hamilton's equations: ∂ H ∂ q i = − p ˙ i , ∂ H ∂ p i = q ˙ i , ∂ H ∂ t = − ∂ L ∂ t . {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\dot {p}}_{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}\,.} In the case of time-independent H {\displaystyle {\mathcal {H}}} and L {\displaystyle {\mathcal {L}}} , i.e. ∂ H / ∂ t = − ∂ L / ∂ t = 0 {\displaystyle \partial {\mathcal {H}}/\partial t=-\partial {\mathcal {L}}/\partial t=0} , Hamilton's equations consist of 2n first-order differential equations, while Lagrange's equations consist of n second-order equations. Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but important theoretical results can be derived from them, because coordinates and momenta are independent variables with nearly symmetric roles. Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate q i {\displaystyle q_{i}} does not occur in the Hamiltonian (i.e. a cyclic coordinate), the corresponding momentum coordinate p i {\displaystyle p_{i}} is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set. This effectively reduces the problem from n coordinates to (n − 1) coordinates: this is the basis of symplectic reduction in geometry. In the Lagrangian framework, the conservation of momentum also follows immediately, however all the generalized
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
velocities q ˙ i {\displaystyle {\dot {q}}_{i}} still occur in the Lagrangian, and a system of equations in n coordinates still has to be solved. The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in classical mechanics, and suggest analogous formulations in quantum mechanics: the path integral formulation and the Schrödinger equation. == Properties of the Hamiltonian == The value of the Hamiltonian H {\displaystyle {\mathcal {H}}} is the total energy of the system if and only if the energy function E L {\displaystyle E_{\mathcal {L}}} has the same property. (See definition of H {\displaystyle {\mathcal {H}}} ). d H d t = ∂ H ∂ t {\displaystyle {\frac {d{\mathcal {H}}}{dt}}={\frac {\partial {\mathcal {H}}}{\partial t}}} when p ( t ) {\displaystyle \mathbf {p} (t)} , q ( t ) {\displaystyle \mathbf {q} (t)} form a solution of Hamilton's equations. Indeed, d H d t = ∂ H ∂ p ⋅ p ˙ + ∂ H ∂ q ⋅ q ˙ + ∂ H ∂ t , {\textstyle {\frac {d{\mathcal {H}}}{dt}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}\cdot {\dot {\boldsymbol {p}}}+{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}\cdot {\dot {\boldsymbol {q}}}+{\frac {\partial {\mathcal {H}}}{\partial t}},} and everything but the final term cancels out. H {\displaystyle {\mathcal {H}}} does not change under point transformations, i.e. smooth changes q ↔ q ′ {\displaystyle {\boldsymbol {q}}\leftrightarrow {\boldsymbol {q'}}} of space coordinates. (Follows from the invariance of the energy function E L {\displaystyle E_{\mathcal {L}}} under point transformations. The invariance of E L {\displaystyle E_{\mathcal {L}}} can be established directly). ∂ H ∂ t = − ∂ L ∂ t . {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}.} (See § Deriving Hamilton's equations). − ∂ H ∂ q i = p ˙ i = ∂ L ∂ q i
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
{\displaystyle -{\frac {\partial {\mathcal {H}}}{\partial q^{i}}}={\dot {p}}_{i}={\frac {\partial {\mathcal {L}}}{\partial q^{i}}}} . (Compare Hamilton's and Euler-Lagrange equations or see § Deriving Hamilton's equations). ∂ H ∂ q i = 0 {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=0} if and only if ∂ L ∂ q i = 0 {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0} .A coordinate for which the last equation holds is called cyclic (or ignorable). Every cyclic coordinate q i {\displaystyle q^{i}} reduces the number of degrees of freedom by 1 {\displaystyle 1} , causes the corresponding momentum p i {\displaystyle p_{i}} to be conserved, and makes Hamilton's equations easier to solve. == Hamiltonian as the total system energy == In its application to a given system, the Hamiltonian is often taken to be H = T + V {\displaystyle {\mathcal {H}}=T+V} where T {\displaystyle T} is the kinetic energy and V {\displaystyle V} is the potential energy. Using this relation can be simpler than first calculating the Lagrangian, and then deriving the Hamiltonian from the Lagrangian. However, the relation is not true for all systems. The relation holds true for nonrelativistic systems when all of the following conditions are satisfied ∂ V ( q , q ˙ , t ) ∂ q ˙ i = 0 , ∀ i {\displaystyle {\frac {\partial V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}=0\;,\quad \forall i} ∂ T ( q , q ˙ , t ) ∂ t = 0 {\displaystyle {\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial t}}=0} T ( q , q ˙ ) = ∑ i = 1 n ∑ j = 1 n ( c i j ( q ) q ˙ i q ˙ j ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})=\sum _{i=1}^{n}\sum _{j=1}^{n}{\biggl (}c_{ij}({\boldsymbol {q}}){\dot {q}}_{i}{\dot {q}}_{j}{\biggr )}} where t {\displaystyle t} is time, n {\displaystyle n} is
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{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
the number of degrees of freedom of the system, and each c i j ( q ) {\displaystyle c_{ij}({\boldsymbol {q}})} is an arbitrary scalar function of q {\displaystyle {\boldsymbol {q}}} . In words, this means that the relation H = T + V {\displaystyle {\mathcal {H}}=T+V} holds true if T {\displaystyle T} does not contain time as an explicit variable (it is scleronomic), V {\displaystyle V} does not contain generalised velocity as an explicit variable, and each term of T {\displaystyle T} is quadratic in generalised velocity. === Proof === Preliminary to this proof, it is important to address an ambiguity in the related mathematical notation. While a change of variables can be used to equate L ( p , q , t ) = L ( q , q ˙ , t ) {\displaystyle {\mathcal {L}}({\boldsymbol {p}},{\boldsymbol {q}},t)={\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)} , it is important to note that ∂ L ( q , q ˙ , t ) ∂ q ˙ i ≠ ∂ L ( p , q , t ) ∂ q ˙ i {\displaystyle {\frac {\partial {\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}\neq {\frac {\partial {\mathcal {L}}({\boldsymbol {p}},{\boldsymbol {q}},t)}{\partial {\dot {q}}_{i}}}} . In this case, the right hand side always evaluates to 0. To perform a change of variables inside of a partial derivative, the multivariable chain rule should be used. Hence, to avoid ambiguity, the function arguments of any term inside of a partial derivative should be stated. Additionally, this proof uses the notation f ( a , b , c ) = f ( a , b ) {\displaystyle f(a,b,c)=f(a,b)} to imply that ∂ f ( a , b , c ) ∂ c = 0 {\displaystyle {\frac {\partial f(a,b,c)}{\partial c}}=0} . === Application to systems of point masses === For a system
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{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
of point masses, the requirement for T {\displaystyle T} to be quadratic in generalised velocity is always satisfied for the case where T ( q , q ˙ , t ) = T ( q , q ˙ ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)=T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} , which is a requirement for H = T + V {\displaystyle {\mathcal {H}}=T+V} anyway. === Conservation of energy === If the conditions for H = T + V {\displaystyle {\mathcal {H}}=T+V} are satisfied, then conservation of the Hamiltonian implies conservation of energy. This requires the additional condition that V {\displaystyle V} does not contain time as an explicit variable. ∂ V ( q , q ˙ , t ) ∂ t = 0 {\displaystyle {\frac {\partial V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial t}}=0} In summary, the requirements for H = T + V = constant of time {\displaystyle {\mathcal {H}}=T+V={\text{constant of time}}} to be satisfied for a nonrelativistic system are V = V ( q ) {\displaystyle V=V({\boldsymbol {q}})} T = T ( q , q ˙ ) {\displaystyle T=T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} T {\displaystyle T} is a homogeneous quadratic function in q ˙ {\displaystyle {\boldsymbol {\dot {q}}}} Regarding extensions to the Euler-Lagrange formulation which use dissipation functions (See Lagrangian mechanics § Extensions to include non-conservative forces), e.g. the Rayleigh dissipation function, energy is not conserved when a dissipation function has effect. It is possible to explain the link between this and the former requirements by relating the extended and conventional Euler-Lagrange equations: grouping the extended terms into the potential function produces a velocity dependent potential. Hence, the requirements are not satisfied when a dissipation function has effect. == Hamiltonian of a charged particle in an electromagnetic field == A sufficient illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle
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{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
in an electromagnetic field. In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units): L = ∑ i 1 2 m x ˙ i 2 + ∑ i q x ˙ i A i − q φ , {\displaystyle {\mathcal {L}}=\sum _{i}{\tfrac {1}{2}}m{\dot {x}}_{i}^{2}+\sum _{i}q{\dot {x}}_{i}A_{i}-q\varphi ,} where q is the electric charge of the particle, φ is the electric scalar potential, and the Ai are the components of the magnetic vector potential that may all explicitly depend on x i {\displaystyle x_{i}} and t {\displaystyle t} . This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law m x ¨ = q E + q x ˙ × B , {\displaystyle m{\ddot {\mathbf {x} }}=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \,,} and is called minimal coupling. The canonical momenta are given by: p i = ∂ L ∂ x ˙ i = m x ˙ i + q A i . {\displaystyle p_{i}={\frac {\partial {\mathcal {L}}}{\partial {\dot {x}}_{i}}}=m{\dot {x}}_{i}+qA_{i}.} The Hamiltonian, as the Legendre transformation of the Lagrangian, is therefore: H = ∑ i x ˙ i p i − L = ∑ i ( p i − q A i ) 2 2 m + q φ . {\displaystyle {\mathcal {H}}=\sum _{i}{\dot {x}}_{i}p_{i}-{\mathcal {L}}=\sum _{i}{\frac {\left(p_{i}-qA_{i}\right)^{2}}{2m}}+q\varphi .} This equation is used frequently in quantum mechanics. Under gauge transformation: A → A + ∇ f , φ → φ − f ˙ , {\displaystyle \mathbf {A} \rightarrow \mathbf {A} +\nabla f\,,\quad \varphi \rightarrow \varphi -{\dot {f}}\,,} where f(r, t) is any scalar function of space and time. The aforementioned Lagrangian, the canonical momenta, and the Hamiltonian transform like: L → L ′ = L + q d f d t , p → p ′ = p
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
+ q ∇ f , H → H ′ = H − q ∂ f ∂ t , {\displaystyle L\rightarrow L'=L+q{\frac {df}{dt}}\,,\quad \mathbf {p} \rightarrow \mathbf {p'} =\mathbf {p} +q\nabla f\,,\quad H\rightarrow H'=H-q{\frac {\partial f}{\partial t}}\,,} which still produces the same Hamilton's equation: ∂ H ′ ∂ x i | p i ′ = ∂ ∂ x i | p i ′ ( x ˙ i p i ′ − L ′ ) = − ∂ L ′ ∂ x i | p i ′ = − ∂ L ∂ x i | p i ′ − q ∂ ∂ x i | p i ′ d f d t = − d d t ( ∂ L ∂ x ˙ i | p i ′ + q ∂ f ∂ x i | p i ′ ) = − p ˙ i ′ {\displaystyle {\begin{aligned}\left.{\frac {\partial H'}{\partial {x_{i}}}}\right|_{p'_{i}}&=\left.{\frac {\partial }{\partial {x_{i}}}}\right|_{p'_{i}}({\dot {x}}_{i}p'_{i}-L')=-\left.{\frac {\partial L'}{\partial {x_{i}}}}\right|_{p'_{i}}\\&=-\left.{\frac {\partial L}{\partial {x_{i}}}}\right|_{p'_{i}}-q\left.{\frac {\partial }{\partial {x_{i}}}}\right|_{p'_{i}}{\frac {df}{dt}}\\&=-{\frac {d}{dt}}\left(\left.{\frac {\partial L}{\partial {{\dot {x}}_{i}}}}\right|_{p'_{i}}+q\left.{\frac {\partial f}{\partial {x_{i}}}}\right|_{p'_{i}}\right)\\&=-{\dot {p}}'_{i}\end{aligned}}} In quantum mechanics, the wave function will also undergo a local U(1) group transformation during the Gauge Transformation, which implies that all physical results must be invariant under local U(1) transformations. === Relativistic charged particle in an electromagnetic field === The relativistic Lagrangian for a particle (rest mass m {\displaystyle m} and charge q {\displaystyle q} ) is given by: L ( t ) = − m c 2 1 − x ˙ ( t ) 2 c 2 + q x ˙ ( t ) ⋅ A ( x ( t ) , t ) − q φ ( x ( t ) , t ) {\displaystyle {\mathcal {L}}(t)=-mc^{2}{\sqrt {1-{\frac {{{\dot {\mathbf {x} }}(t)}^{2}}{c^{2}}}}}+q{\dot {\mathbf {x} }}(t)\cdot \mathbf {A} \left(\mathbf {x} (t),t\right)-q\varphi \left(\mathbf {x} (t),t\right)} Thus
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
the particle's canonical momentum is p ( t ) = ∂ L ∂ x ˙ = m x ˙ 1 − x ˙ 2 c 2 + q A {\displaystyle \mathbf {p} (t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {x} }}}}={\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}+q\mathbf {A} } that is, the sum of the kinetic momentum and the potential momentum. Solving for the velocity, we get x ˙ ( t ) = p − q A m 2 + 1 c 2 ( p − q A ) 2 {\displaystyle {\dot {\mathbf {x} }}(t)={\frac {\mathbf {p} -q\mathbf {A} }{\sqrt {m^{2}+{\frac {1}{c^{2}}}{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}}} So the Hamiltonian is H ( t ) = x ˙ ⋅ p − L = c m 2 c 2 + ( p − q A ) 2 + q φ {\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}\cdot \mathbf {p} -{\mathcal {L}}=c{\sqrt {m^{2}c^{2}+{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}+q\varphi } This results in the force equation (equivalent to the Euler–Lagrange equation) p ˙ = − ∂ H ∂ x = q x ˙ ⋅ ( ∇ A ) − q ∇ φ = q ∇ ( x ˙ ⋅ A ) − q ∇ φ {\displaystyle {\dot {\mathbf {p} }}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {x} }}=q{\dot {\mathbf {x} }}\cdot ({\boldsymbol {\nabla }}\mathbf {A} )-q{\boldsymbol {\nabla }}\varphi =q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi } from which one can derive d d t ( m x ˙ 1 − x ˙ 2 c 2 ) = d d t ( p − q A ) = p ˙ − q ∂ A ∂ t − q ( x ˙ ⋅ ∇ ) A = q ∇ ( x ˙ ⋅ A ) − q ∇ φ − q ∂ A ∂
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
t − q ( x ˙ ⋅ ∇ ) A = q E + q x ˙ × B {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}\right)&={\frac {\mathrm {d} }{\mathrm {d} t}}(\mathbf {p} -q\mathbf {A} )={\dot {\mathbf {p} }}-q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi -q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \end{aligned}}} The above derivation makes use of the vector calculus identity: 1 2 ∇ ( A ⋅ A ) = A ⋅ J A = A ⋅ ( ∇ A ) = ( A ⋅ ∇ ) A + A × ( ∇ × A ) . {\displaystyle {\tfrac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)=\mathbf {A} \cdot \mathbf {J} _{\mathbf {A} }=\mathbf {A} \cdot (\nabla \mathbf {A} )=(\mathbf {A} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {A} ).} An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, P = γ m x ˙ ( t ) = p − q A {\displaystyle \mathbf {P} =\gamma m{\dot {\mathbf {x} }}(t)=\mathbf {p} -q\mathbf {A} } , is H ( t ) = x ˙ ( t ) ⋅ P ( t ) + m c 2 γ + q φ ( x ( t ) , t ) = γ m c 2 + q φ ( x ( t ) , t ) = E + V {\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}(t)\cdot \mathbf {P} (t)+{\frac {mc^{2}}{\gamma }}+q\varphi (\mathbf {x} (t),t)=\gamma mc^{2}+q\varphi (\mathbf {x} (t),t)=E+V} This has the advantage that kinetic momentum P {\displaystyle \mathbf {P} } can be measured experimentally whereas canonical
|
{
"page_id": 198319,
"source": null,
"title": "Hamiltonian mechanics"
}
|
momentum p {\displaystyle \mathbf {p} } cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), E = γ m c 2 {\displaystyle E=\gamma mc^{2}} , plus the potential energy, V = q φ {\displaystyle V=q\varphi } . == From symplectic geometry to Hamilton's equations == === Geometry of Hamiltonian systems === The Hamiltonian can induce a symplectic structure on a smooth even-dimensional manifold M2n in several equivalent ways, the best known being the following: As a closed nondegenerate symplectic 2-form ω. According to Darboux's theorem, in a small neighbourhood around any point on M there exist suitable local coordinates p 1 , ⋯ , p n , q 1 , ⋯ , q n {\displaystyle p_{1},\cdots ,p_{n},\ q_{1},\cdots ,q_{n}} (canonical or symplectic coordinates) in which the symplectic form becomes: ω = ∑ i = 1 n d p i ∧ d q i . {\displaystyle \omega =\sum _{i=1}^{n}dp_{i}\wedge dq_{i}\,.} The form ω {\displaystyle \omega } induces a natural isomorphism of the tangent space with the cotangent space: T x M ≅ T x ∗ M {\displaystyle T_{x}M\cong T_{x}^{*}M} . This is done by mapping a vector ξ ∈ T x M {\displaystyle \xi \in T_{x}M} to the 1-form ω ξ ∈ T x ∗ M {\displaystyle \omega _{\xi }\in T_{x}^{*}M} , where ω ξ ( η ) = ω ( η , ξ ) {\displaystyle \omega _{\xi }(\eta )=\omega (\eta ,\xi )} for all η ∈ T x M {\displaystyle \eta \in T_{x}M} . Due to the bilinearity and non-degeneracy of ω {\displaystyle \omega } , and the fact that dim T x M = dim T x ∗ M {\displaystyle \dim T_{x}M=\dim T_{x}^{*}M} , the mapping ξ → ω ξ {\displaystyle
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\xi \to \omega _{\xi }} is indeed a linear isomorphism. This isomorphism is natural in that it does not change with change of coordinates on M . {\displaystyle M.} Repeating over all x ∈ M {\displaystyle x\in M} , we end up with an isomorphism J − 1 : Vect ( M ) → Ω 1 ( M ) {\displaystyle J^{-1}:{\text{Vect}}(M)\to \Omega ^{1}(M)} between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every f , g ∈ C ∞ ( M , R ) {\displaystyle f,g\in C^{\infty }(M,\mathbb {R} )} and ξ , η ∈ Vect ( M ) {\displaystyle \xi ,\eta \in {\text{Vect}}(M)} , J − 1 ( f ξ + g η ) = f J − 1 ( ξ ) + g J − 1 ( η ) . {\displaystyle J^{-1}(f\xi +g\eta )=fJ^{-1}(\xi )+gJ^{-1}(\eta ).} (In algebraic terms, one would say that the C ∞ ( M , R ) {\displaystyle C^{\infty }(M,\mathbb {R} )} -modules Vect ( M ) {\displaystyle {\text{Vect}}(M)} and Ω 1 ( M ) {\displaystyle \Omega ^{1}(M)} are isomorphic). If H ∈ C ∞ ( M × R t , R ) {\displaystyle H\in C^{\infty }(M\times \mathbb {R} _{t},\mathbb {R} )} , then, for every fixed t ∈ R t {\displaystyle t\in \mathbb {R} _{t}} , d H ∈ Ω 1 ( M ) {\displaystyle dH\in \Omega ^{1}(M)} , and J ( d H ) ∈ Vect ( M ) {\displaystyle J(dH)\in {\text{Vect}}(M)} . J ( d H ) {\displaystyle J(dH)} is known as a Hamiltonian vector field. The respective differential equation on M {\displaystyle M} x ˙ = J ( d H ) ( x ) {\displaystyle {\dot {x}}=J(dH)(x)} is called Hamilton's equation. Here x = x ( t ) {\displaystyle
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x=x(t)} and J ( d H ) ( x ) ∈ T x M {\displaystyle J(dH)(x)\in T_{x}M} is the (time-dependent) value of the vector field J ( d H ) {\displaystyle J(dH)} at x ∈ M {\displaystyle x\in M} . A Hamiltonian system may be understood as a fiber bundle E over time R, with the fiber Et being the position space at time t ∈ R. The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T∗Et, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form. Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the phase space. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field. The Hamiltonian vector field induces a Hamiltonian flow on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, an isotopy of symplectomorphisms, starting with the identity. By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called "the Hamiltonian mechanics" of the Hamiltonian system. The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra. If F and G are smooth functions on M then the smooth
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"page_id": 198319,
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function ω(J(dF), J(dG)) is properly defined; it is called a Poisson bracket of functions F and G and is denoted {F, G}. The Poisson bracket has the following properties: bilinearity antisymmetry Leibniz rule: { F 1 ⋅ F 2 , G } = F 1 { F 2 , G } + F 2 { F 1 , G } {\displaystyle \{F_{1}\cdot F_{2},G\}=F_{1}\{F_{2},G\}+F_{2}\{F_{1},G\}} Jacobi identity: { { H , F } , G } + { { F , G } , H } + { { G , H } , F } ≡ 0 {\displaystyle \{\{H,F\},G\}+\{\{F,G\},H\}+\{\{G,H\},F\}\equiv 0} non-degeneracy: if the point x on M is not critical for F then a smooth function G exists such that { F , G } ( x ) ≠ 0 {\displaystyle \{F,G\}(x)\neq 0} . Given a function f d d t f = ∂ ∂ t f + { f , H } , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}f={\frac {\partial }{\partial t}}f+\left\{f,{\mathcal {H}}\right\},} if there is a probability distribution ρ, then (since the phase space velocity ( p ˙ i , q ˙ i ) {\displaystyle ({\dot {p}}_{i},{\dot {q}}_{i})} has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so ∂ ∂ t ρ = − { ρ , H } {\displaystyle {\frac {\partial }{\partial t}}\rho =-\left\{\rho ,{\mathcal {H}}\right\}} This is called Liouville's theorem. Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if {G, H} = 0, then G is conserved and the symplectomorphisms are symmetry transformations. A Hamiltonian may have multiple conserved quantities Gi. If the symplectic manifold has dimension 2n and there are n functionally independent conserved quantities Gi which are in involution (i.e., {Gi, Gj} = 0), then the Hamiltonian is
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"page_id": 198319,
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Liouville integrable. The Liouville–Arnold theorem says that, locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism into a new Hamiltonian with the conserved quantities Gi as coordinates; the new coordinates are called action–angle coordinates. The transformed Hamiltonian depends only on the Gi, and hence the equations of motion have the simple form G ˙ i = 0 , φ ˙ i = F i ( G ) {\displaystyle {\dot {G}}_{i}=0\quad ,\quad {\dot {\varphi }}_{i}=F_{i}(G)} for some function F. There is an entire field focusing on small deviations from integrable systems governed by the KAM theorem. The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined. === Riemannian manifolds === An important special case consists of those Hamiltonians that are quadratic forms, that is, Hamiltonians that can be written as H ( q , p ) = 1 2 ⟨ p , p ⟩ q {\displaystyle {\mathcal {H}}(q,p)={\tfrac {1}{2}}\langle p,p\rangle _{q}} where ⟨ , ⟩q is a smoothly varying inner product on the fibers T∗qQ, the cotangent space to the point q in the configuration space, sometimes called a cometric. This Hamiltonian consists entirely of the kinetic term. If one considers a Riemannian manifold or a pseudo-Riemannian manifold, the Riemannian metric induces a linear isomorphism between the tangent and cotangent bundles. (See Musical isomorphism). Using this isomorphism, one can define a cometric. (In coordinates, the matrix defining the cometric is the inverse of the matrix defining the metric.) The solutions to the Hamilton–Jacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. The existence of such solutions, and the completeness of the
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"page_id": 198319,
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set of solutions, are discussed in detail in the article on geodesics. See also Geodesics as Hamiltonian flows. === Sub-Riemannian manifolds === When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point q of the configuration space manifold Q, so that the rank of the cometric is less than the dimension of the manifold Q, one has a sub-Riemannian manifold. The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that every sub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by the Chow–Rashevskii theorem. The continuous, real-valued Heisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by H ( x , y , z , p x , p y , p z ) = 1 2 ( p x 2 + p y 2 ) . {\displaystyle {\mathcal {H}}\left(x,y,z,p_{x},p_{y},p_{z}\right)={\tfrac {1}{2}}\left(p_{x}^{2}+p_{y}^{2}\right).} pz is not involved in the Hamiltonian. === Poisson algebras === Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A2 maps to a nonnegative real number. A further generalization is given by Nambu dynamics. === Generalization to quantum mechanics through Poisson
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bracket === Hamilton's equations above work well for classical mechanics, but not for quantum mechanics, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time. However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of the Poisson algebra over p and q to the algebra of Moyal brackets. Specifically, the more general form of the Hamilton's equation reads d f d t = { f , H } + ∂ f ∂ t , {\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\left\{f,{\mathcal {H}}\right\}+{\frac {\partial f}{\partial t}},} where f is some function of p and q, and H is the Hamiltonian. To find out the rules for evaluating a Poisson bracket without resorting to differential equations, see Lie algebra; a Poisson bracket is the name for the Lie bracket in a Poisson algebra. These Poisson brackets can then be extended to Moyal brackets comporting to an inequivalent Lie algebra, as proven by Hilbrand J. Groenewold, and thereby describe quantum mechanical diffusion in phase space (See Phase space formulation and Wigner–Weyl transform). This more algebraic approach not only permits ultimately extending probability distributions in phase space to Wigner quasi-probability distributions, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevant conserved quantities in a system. == See also == == References == == Further reading == == External links == Binney, James J., Classical Mechanics (lecture notes) (PDF), University of Oxford, retrieved 27 October 2010 Tong, David, Classical Dynamics (Cambridge lecture notes), University of Cambridge, retrieved 27 October 2010 Hamilton, William Rowan, On a General Method in Dynamics, Trinity College Dublin Malham, Simon J.A. (2016), An
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introduction to Lagrangian and Hamiltonian mechanics (lecture notes) (PDF) Morin, David (2008), Introduction to Classical Mechanics (Additional material: The Hamiltonian method) (PDF)
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"page_id": 198319,
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The Ferrier rearrangement is an organic reaction that involves a nucleophilic substitution reaction combined with an allylic shift in a glycal (a 2,3-unsaturated glycoside). It was discovered by the carbohydrate chemist Robert J. Ferrier. == Mechanism == In the first step, a delocalized allyloxocarbenium ion (2) is formed, typically with the aid of a Lewis acid like indium(III) chloride or boron trifluoride. This ion reacts in situ with an alcohol, yielding a mixture of the α (3) and β (4) anomers of the 2-glycoside, with the double bond shifted to position 3,4. == Examples == == Modifications == === Forming of C-glycosides === By replacing the alcohol with a silane, C-glycosides can be formed. With triethylsilane (R'=H), the reaction yields a 2,3-unsaturated deoxy sugar. === Nitrogen analogue === An analogous reaction with nitrogen as the heteroatom was described in 1984 for the synthesis of the antibiotic substance streptazolin. == References ==
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{
"page_id": 1836722,
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"title": "Ferrier rearrangement"
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An alchemist is a person versed in the art of alchemy. Western alchemy flourished in Greco-Roman Egypt, the Islamic world during the Middle Ages, and then in Europe from the 13th to the 18th centuries. Indian alchemists and Chinese alchemists made contributions to Eastern varieties of the art. Alchemy is still practiced today by a few, and alchemist characters still appear in recent fictional works and video games. Many alchemists are known from the thousands of surviving alchemical manuscripts and books. Some of their names are listed below. Due to the tradition of pseudepigraphy, the true author of some alchemical writings may differ from the name most often associated with that work. Some well-known historical figures such as Albertus Magnus and Aristotle are often incorrectly named amongst the alchemists as a result. == Legendary == Anqi Sheng Hermes Trismegistus Ostanes Nicolas Flamel Perenelle Flamel Christian Rosenkreuz Abraham Eleazar == Greco-Roman Egypt == == India == Kanada, sage and philosopher (6th century BC) Nagarjuna (born 931) Yogi Vemana Siddhar Tamil sage and philosophers Nayanmars Tamil sage and philosophers Alvars Tamil sage and philosophers Vallalar, Tamil 18th Century sage and philosopher Arunagirinathar Tamil 15th Century sage and philosopher Agastiyar Tamil Sage Korakkar Tamil Sage Thirumoolar Tamil Sage Bogar Tamil Sage Kagapujandar Tamil Sage Vaalmiki Tamil Sage Pattinathar Tamil Sage Kalangi Nathar Tamil Sage Pathanjali Tamil Sage Naradhar == China == Kong Anguo (ca. 156 – ca. 74 BC) Keng Hsien-Seng (circa A.D. 975) Ge Hong/ Ko Hong (283–343) Jiajing Emperor (1507 – 1567) Liu Yiming (1734–1821) Pao Ku Ko (third century A.D.) Shen Yu Hsiu (15th century) Sun Pu-Eh (12th century) Sun Simiao (died 682) Tao Hongjing (456–536) Wei Boyang Xu Fu (255 BC- 210 BC) Zhang Guo the Elder (c. 600) Zou Yan (305 BC – 240 BC) == Arabic-Islamic world
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{
"page_id": 657083,
"source": null,
"title": "List of alchemists"
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== Khalid ibn Yazid, known in Latin as Calid (died 704) Jabir ibn Hayyan, known in Latin as Geber (died c. 806–816) Dhu al-Nun al-Misri (born 796) Abu Bakr al-Razi (c. 865–925 or 935) Ibn Umayl, known in Latin as Senior Zadith (c. 900–960) al-Tughrai (1061–1121) Artephius (c. 1150) al-Jildaki, also written al-Jaldaki (died 1342) == Europe == == Revival and modern == == Scholars of alchemy == == Indirectly involved with alchemy == Rudolf II, Holy Roman Emperor (1552–1612) Qin Shi Huang (259–210 BC) Yongzheng Emperor (1678–1735) == In fiction == == See also == List of alchemical substances List of astrologers List of occultists List of occult symbols
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"page_id": 657083,
"source": null,
"title": "List of alchemists"
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Albert of Saxony (Latin: Albertus de Saxonia; c. 1320 – 8 July 1390) was a German philosopher and mathematician known for his contributions to logic and physics. He was bishop of Halberstadt from 1366 until his death. == Life == Albert was born at Rickensdorf near Helmstedt, the son of a farmer in a small village. Due to his talent, he was sent to study at the University of Prague and the University of Paris. At Paris, he became a Master of Arts (a professor), and held this post from 1351 until 1362. He also studied theology at the College of Sorbonne, although without receiving a degree. In 1353, he was rector of the University of Paris. After 1362, Albert went to the court of Pope Urban V in Avignon as an envoy of Rudolf IV, Duke of Austria to negotiate the founding of the University of Vienna. The negotiations were successful, and Albert became the first rector in 1365. In 1366, Albert was elected bishop of Halberstadt (counted as Albert III), the diocese in which he was born. As bishop, he allied himself with Magnus with the Necklace, Duke of Brunswick-Lüneburg, against Gerhard of Berg, Bishop of Hildesheim, and was taken prisoner by Gerhard in the battle of Dinckler in 1367. He died at Halberstadt in 1390. == Philosophy == Albert was a pupil of Jean Buridan and his thoughts were shaped by Buridan's teachings on physics and logic. As a natural philosopher, he contributed to the spread of Parisian natural philosophy throughout Italy and central Europe. Similar to Buridan, Albert combined critical analysis of language with epistemological pragmatism. Albert distinguishes, as his teacher did, between what is absolutely impossible or contradictory and what is impossible “in the common course of nature” and considers hypotheses under circumstances that are
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not naturally possible but imaginable given God's absolute power. He was later regarded as one of the principal adherents of nominalism, along with his near contemporaries at Paris, namely Buridan and Marsilius of Inghen. The subsequent wide circulation of Albert's work made him a better-known figure in some areas than contemporaries like Buridan and Nicole Oresme. Albert's work in logic was in part an extension of William of Ockham's commentaries on the logica vetus (i. e. on Porphyry, and Aristotle's Categoriae and De interpretatione) which were made the subject of a series of works called Quaestiones by Albert. Albert of Saxony's teachings on logic and metaphysics were extremely influential. The theory of impetus introduced a third stage to the two stage theory of John Philoponus. Initial stage. Motion is in a straight line in direction of impetus which is dominant while gravity is insignificant Intermediate stage. Path begins to deviate downwards from straight line as part of a great circle as air resistance slows projectile and gravity recovers. Last stage. Gravity alone draws projectile downwards vertically as all impetus is spent. This theory was a precursor to the modern theory of inertia. Although Buridan remained the predominant figure in logic, Albert's Perutilis logica (c. 1360) was destined to serve as a popular text because of its systematic nature and also because it takes up and develops essential aspects of the Ockhamist position. Albert accepted Ockham's conception of the nature of a sign. Albert believed that signification rests on a referential relation of the sign to the individual thing, and that the spoken sign depends for its signification on the conceptual sign. Albert followed Ockham in his conception of universals and in his theory of supposition. Specifically, Albert preserved Ockham's notion of simple supposition, understood as the direct reference of a
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"page_id": 853697,
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term to the concept on which it depends when it signifies an extra-mental thing. Albert followed Ockham in his theory of categories and contrary to Buridan, refused to treat quantity as a feature of reality in its own right, but rather reduced it to a disposition of substance and quality. Albert established signification through a referential relation to a singular thing defining the relation of the spoken to conceptual signs as a relation of subordination. Albert's treatment of relation was highly original. Although, like Ockham, he refused to construe relations as things distinct from absolute entities, he clearly ascribed them to an act of the soul by which absolute entities are compared and placed in relation to each other. He therefore completely rejected certain propositions Ockham had admitted reasonable, even if he did not construe them in the same way. Albert's voluminous collection of Sophismata (c. 1359) examined various sentences that raise difficulties of interpretation due to the presence of syncategorematic terms such as quantifiers and certain prepositions, which, according to medieval logicians, do not have a proper and determinate signification but rather modify the signification of the other terms in the propositions in which they occur. In his Sophismata, he followed William Heytesbury. In his analysis of epistemic verbs or of infinity, Albert admitted that a proposition has its own signification, which is not that of its terms: just like a syncategorematic term, a proposition signifies a "mode of a thing". Albert made use of the idea of the distinguishable signification of the proposition in defining truth and in dealing with “insolubles” or paradoxes of self-reference. In this work he shows that since every proposition, by its very form, signifies that it is true, an insoluble proposition will turn out to be false because it will signify at once
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both that it is true and that it is false. Albert also authored commentaries on Ars Vetus, a set of twenty-five Quaestiones logicales (c. 1356) that involved semantical problems and the status of logic, and Quaestiones on the Posterior Analytics. Albert explored in a series of disputed questions the status of logic and semantics, as well as the theory of reference and truth. Albert was influenced by English logicians and was influential in the diffusion of terminist logic in central Europe. Albert is considered a major contributor in his theory of consequences, found in his Perutilis Logica. Albert took a major step forward in the medieval theory of logical deduction. However, it was his commentary on Aristotle's Physics that was especially widely read. Many manuscripts of it can be found in France and Italy, in Erfurt and Prague. Albert's Physics basically guaranteed the transmission of the Parisian tradition to Italy, where it was authoritative along with the works of Heytesbury and John Dumbleton. His commentary on Aristotle's De caelo was also influential, eventually eclipsing Buridan's commentary on this text. Blasius of Parma read it in Bologna between 1379 and 1382. A little later, it enjoyed a wide audience at Vienna. His Treatise on Proportions was often quoted in Italy where, in addition to the texts of Thomas Bradwardine and Oresme, it influenced the application of the theory of proportions to motion. Albert's commentaries on the Nicomachean Ethics and the Economics also survive (both unedited), as well as several short mathematical texts, most notably Tractatus proportionum (c. 1353). Although Albert studied theology in Paris, no theological writing survived. Albert played an essential role in the diffusion throughout Italy and central Europe of Parisian ideas which bore the mark of Buridan's teachings, but which were also clearly shaped by Albert's own grasp
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"page_id": 853697,
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of English innovations. At the same time, Albert was not merely a compiler of the work of others. Rather, he constructed proofs of undeniable originality on many topics in logic and physics. == Works == Perutilis Logica Magistri Alberti de Saxonia (Very Useful Logic), Venice 1522 and Hildesheim 1974 (reproduction) Albert of Saxony's Twenty-Five Disputed Questions on Logic. A Critical Edition of His Quaestiones circa logicam, by Michael J. Fitzgerald, Leiden: Brill, 2002 Quaestiones in artem veterem critical edition by Angel Muñoz Garcia, Maracaibo, Venezuela: Universidad del Zulia,1988 Quaestiones on the Posterior Analytics Quaestiones logicales (Logical Questions) De consequentiis (On Consequences) - attributed De locis dialecticis (On Dialectical Topics) - attributed Sophismata et Insolubilia et Obligationes, Paris 1489 and Hildesheim 1975 (reproduction) Expositio et quaestiones in Aristotelis Physicam ad Albertum de Saxonia attributae critical edition by Benoit Patar, Leuven, Peeters Publishers, 1999 Questiones subtilissime in libros Aristotelis de caelo et mundo, Venetiis, 1492. Questiones subtilissime super libros posteriorum, Venetiis 1497 Hildesheim 1986 (reproduction) Alberti de Saxonia Quæstiones in Aristotelis De cælo critical edition by Benoit Patar, Leuven, Peeters Publishers, 2008 De latudinibus, Padua 1505 De latitudinibus formarum De maximo et minimo De quadratura circuli - Question on the Squaring of the Circle Tractatus proportionum, Venice 1496 and Vienna 1971: editor Hubertus L. Busard == Modern editions and English translations == Tractatus proportionum: Der Tractatus proportionum von Albert von Sachsen, Osterreichische Akademie der Wissenschaften, math.-nat. Klasse, Denkschriften 116(2):44–72. Springer, Vienna, 1971. Perutilis logica, Latin text and Spanish translation by A. Muñoz-Garcia, Universidad Nacional Autonoma de Mexico, 1988. Quaestiones in Artem Veterem, Latin text and Spanish translation by A. Muñoz-Garcia, Maracaibo, Universidad del Zulia, 1988. De proprietates terminorum (second tract of the Perutilis logica), edited by C. Kann, Die Eigenschaften der Termini, Brill, Leiden, 1993. Quaestiones super libros Physicorum, edited by B.
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Patar, Expositio et Quaestiones in Aristotelis Physicam ad Albertum de Saxonia attributae, Louvain, Peeters, 1999 (3 volumes). Quaestiones circa Logicam: Twenty-Five Disputed Questions on Logic, trans. Michael J. Fitzgerald, Dallas Medieval Texts and Translations 9, Louvain and Paris: Peeters, 2010. == See also == John Buridan List of Roman Catholic scientist-clerics == References == == Further reading == Joel Biard (ed.), Itinéraires d’Albert de Saxe. Paris Vienne au XIVe siècle, Paris, Vrin, 1991. Grant, Edward, A Companion to Philosophy in the Middle Ages, In Gracia, J., J., E. & Noone, T. B. (Eds.), Blackwell Companions to Philosophy, Malden, MA: Blackwell, 2003. Moody, Ernest A. (1970). "Albert of Saxony". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 93–95. ISBN 0-684-10114-9. Pasnau, Robert, The Cambridge History of Medieval Philosophy, Cambridge: Cambridge University Press, 2010. Thijssen, Johannes M. M. H. (2007). "Albert of Saxony". New Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 34–36. ISBN 978-0-684-31320-7. J.M.M.H. Thijssen, The Buridan School Reassessed. John Buridan and Albert of Saxony, Vivarium 42, 2004, pp. 18–42. == External links == Biard, Joël. "Albert of Saxony". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. O'Connor, John J.; Robertson, Edmund F., "Albert of Saxony (philosopher)", MacTutor History of Mathematics Archive, University of St Andrews Friedrich Wilhelm Bautz (1975). "Albert von Sachsen (eigentlich: Albert von Rickmersdorf; auch: Albert von Helmstedt; Albertus de Saxonia)". In Bautz, Friedrich Wilhelm (ed.). Biographisch-Bibliographisches Kirchenlexikon (BBKL) (in German). Vol. 1. Hamm: Bautz. cols. 83–84. ISBN 3-88309-013-1. Zedlers Universal-Lexicon, vol. 1, p. 542 Rochus von Liliencron (1875). "Albert, Bischof von Halberstadt". Allgemeine Deutsche Biographie (in German). Vol. 1. Leipzig: Duncker & Humblot. pp. 182–183.
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Corals of the World Online, or simply Corals of the World, is an open access website aiming to become a fully comprehensive and up-to-date reference on the subjects of coral taxonomy, biogeography and identification. It is focused only on the group of zooxanthellate Scleractinia, also called stony corals, the order of Cnidaria that is the main builder of coral reefs. This website was created by J. E. N. Veron (also known as "Charlie" Veron), an Australian biologist expert, pioneer in the study of corals, and Mary Stafford-Smith, with the collaboration of Lyndon DeVantier, Emre Turak, and the contributions of many others from around the world. Its beta version was put online in 2016. It supersedes, expands and makes globally available the three-volume book Corals of the World by Veron and Stafford-Smith, itself an important and well established reference on the topic, both in academia and in the reef aquarism industry. The governmental research center Australian Institute of Marine Science includes the website in its Data Repository as a resource for research, conservation, education, and policy-making. The development of the site is sponsored by the Corals of the World Foundation, a non-profit organization. Given that the website is freely accessible to all, it can be considered not only very useful for scientists, but also for serious reef aquarists who wish to deepen their knowledge of stony corals. == See also == World Register of Marine Species (WoRMS) Census of Marine Life (CoML) Ocean Biogeographic Information System (OBIS) International Commission on Zoological Nomenclature (ICZN) == References == == External links == Corals of the World
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"page_id": 79496899,
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"title": "Corals of the World (website)"
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Sodium sesquicarbonate (systematic name: trisodium hydrogendicarbonate) Na3H(CO3)2 is a double salt of sodium bicarbonate and sodium carbonate (NaHCO3 · Na2CO3), and has a needle-like crystal structure. However, the term is also applied to an equimolar mixture of those two salts, with whatever water of hydration the sodium carbonate includes, supplied as a powder. The dihydrate, Na3H(CO3)2 · 2H2O, occurs in nature as the evaporite mineral trona. Due to concerns about the toxicity of borax which was withdrawn as a cleaning and laundry product, sodium sesquicarbonate is sold in the European Union (EU) as "Borax substitute". It is also known as one of the E number food additives E500(iii). == Uses == Sodium sesquicarbonate is used in bath salts, swimming pools, as an alkalinity source for water treatment, and as a phosphate-free product replacing the trisodium phosphate for heavy duty cleaning. Sodium sesquicarbonate is used in the conservation of copper and copper alloy artifacts that corrode due to contact with salt (called "bronze disease" due to its effect on bronze). The chloride from salt forms copper(I) chloride. In the presence of oxygen and water, even the small amount of moisture in the atmosphere, the cuprous chloride forms copper(II) chloride and hydrochloric acid, the latter of which dissolves the metal and forms more cuprous chloride in a self-sustaining reaction that leads to the entire destruction of the object. Treatment with sodium sesquicarbonate removes copper(II) chlorides from the corroded layer. It is also used as a precipitating water softener, which combines with hard water minerals (calcium- and magnesium-based minerals) to form an insoluble precipitate, removing these hardness minerals from the water. It is the carbonate moiety which forms the precipitate, the bicarbonate being included to moderate the material's alkalinity. In Chinese cuisine, it is known as mǎyájiǎn (simplified:马牙碱 traditional: 馬牙鹼) which roughly translates
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{
"page_id": 12912325,
"source": null,
"title": "Sodium sesquicarbonate"
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|
to “horse tooth alkaline” and traditionally used as an ingredient in the marinade for century eggs, a dish generally made from duck eggs preserved whole in an highly alkaline mixture. == References ==
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{
"page_id": 12912325,
"source": null,
"title": "Sodium sesquicarbonate"
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|
Le Vaillant (French: The Valiant) (died 4 June 1916) was a pigeon used by the French Army in the First World War. The bird was the last held at Fort Vaux before it was overrun in the Battle of Verdun. Le Vaillant carried a message from the fort's commander Sylvain Reynal to his senior officers requesting reinforcements but was mortally wounded in flight. The bird was posthumously appointed to the Legion of Honour and is commemorated by a plaque at the fort. == Background == Fort Vaux was a fortification guarding the north-east approach to the city of Verdun. The fort was besieged by German forces during the 1916 Battle of Verdun and by early June the remaining French garrison was under the command of Commandant Sylvain Raynal. Telephone connection between the fort and the Verdun citadel had been severed by German troops and Raynal's only means of communication was by messenger pigeon, of which he had four. With German attacks continuing to gain ground Raynal sent the first of his pigeons on 2 June. The message requested that artillery fire be directed upon the fort against German troops that had occupied its upperworks. The pigeon arrived at the citadel, despite injury but had lost the ring containing the message. Raynal's penultimate bird was received and was awarded the Croix de Guerre for its flight. == Flight of 4 June == On 4 June Raynal released his last pigeon, number 787.15, named Le Vaillant. The message he bore included the text "we are holding. But ... relief is imperative ... This is my last pigeon". Le Vaillant had been affected by gas released from German shells and was revived by a number of trips to a loophole in Raynal's command post. He set off at 11:30 a.m. Le Vaillant delivered
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{
"page_id": 72156872,
"source": null,
"title": "Le Vaillant"
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|
the message to the dovecot at the citadel. The bird was grievously wounded and died in the hands of the citadel's pigeon master. Because of the message, five relief parties were sent to reinforce Raynal, arriving on 5 June. The garrison lacked water and ammunition and Raynal was forced to surrender his position and 600 surviving troops on 7 June. == Legacy == Le Vaillant was posthumously appointed to the Legion of Honour, the only pigeon to be so rewarded during the war. The diploma of the award hung in the headquarters of the French army signals units. Le Vaillant was stuffed and preserved and is now in the Mont Valérien Military Pigeon Museum in Suresne. Le Vaillant was commemorated by a series of postcards issued after the war. He was formally recorded as Mort pour la France (died for France). A plaque memoralising the bird, with a depiction of him, is in the courtyard of Fort Vaux, being erected by the pigeon fanciers societies of France on 24 June 1929. The French Army perpetuates the history of messenger pigeons and the 8th Signal Regiment maintains a dovecote of 200 pigeons for ceremonial use and in case of all other communications being lost. == See also == Valiant (film) - about a Second World War British messenger pigeon named Valiant. == References ==
|
{
"page_id": 72156872,
"source": null,
"title": "Le Vaillant"
}
|
Venom optimization hypothesis, also known as venom metering, is a biological hypothesis which postulates that venomous animals have physiological control over their production and use of venoms. It explains the economic use of venom because venom is a metabolically expensive product, and that there is a biological mechanism for controlling their specific use. The hypothetical concept was proposed by Esther Wigger, Lucia Kuhn-Nentwig, and Wolfgang Nentwig of the Zoological Institute at the University of Bern, Switzerland, in 2002. A number of venomous animals have been experimentally found to regulate the amount of venom they use during predation or defensive situations. Species of anemones, jellyfish, ants, scorpions, spiders, and snakes are found to use their venoms frugally depending on the situation and size of their preys or predators. == Development == Venom optimization hypothesis was postulated by Wigger, Kuhn-Nentwig, and Nentwig from their studies of the amount of venom used by a wandering spider Cupiennius salei. This spider produces a neurotoxic peptide called CsTx-1 for paralysing its prey. It does not weave webs for trapping preys, and therefore, entirely depends on its venom for predation. It is known to prey on a variety of insects including butterflies, moths, earwigs, cockroaches, flies and grasshoppers. Its venom glands store only about 10 μl of crude venom. Refilling of the glands takes 2–3 days and the lethal efficacy of the venom is, initially, very low for several days, requiring 8 to 18 days for full effect. It was found that the amount of venom released differed for each specific prey. For example, for bigger and stronger insects like beetles, the spider uses the entire amount of its venom; while for small ones, it uses only a small amount, thus economizing its costly venom. In fact, experiments show that the amount of venom released is
|
{
"page_id": 40502985,
"source": null,
"title": "Venom optimization hypothesis"
}
|
just sufficient (at the lethal dose) to paralyze the target organism depending on the size or strength, and is not more than what is necessary. == Concept == Animal venoms are complex biomolecules and hence, their biological synthesis require high metabolic activity. A particular venom itself is a complex chemical mixture composed of hundreds of proteins and non-proteinaceous compounds, resulting in a potent weapon for prey immobilization and predator deterrence. The metabolic cost of venom is sufficiently high to result in secondary loss of venom whenever its use becomes non-essential to survival of the animal. This suggests that venomous animals may have evolved strategies for minimizing venom expenditure, that they should use them only as and when required, and that too in optimal amount. == References ==
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{
"page_id": 40502985,
"source": null,
"title": "Venom optimization hypothesis"
}
|
Joseph Michael Forshaw is an Australian ornithologist, and expert on parrots. He was the former head of wildlife conservation for the Australian National Parks and Wildlife Service. == Bibliography == Forshaw, Joseph M. (2006). Parrots of the World; an Identification Guide. Illustrated by Frank Knight. Princeton University Press. ISBN 0-691-09251-6. == References ==
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{
"page_id": 6948556,
"source": null,
"title": "Joseph Forshaw"
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|
The following is a list of notable proteins that are produced from recombinant DNA, using biomolecular engineering. In many cases, recombinant human proteins have replaced the original animal-derived version used in medicine. The prefix "rh" for "recombinant human" appears less and less in the literature. A much larger number of recombinant proteins is used in the research laboratory. These include both commercially available proteins (for example most of the enzymes used in the molecular biology laboratory), and those that are generated in the course specific research projects. == Human recombinants that largely replaced animal or harvested from human types == === Medicinal applications === Human growth hormone (rHGH): Humatrope from Lilly and Serostim from Serono replaced cadaver harvested human growth hormone human insulin (BHI): Humulin from Lilly and Novolin from Novo Nordisk among others largely replaced bovine and porcine insulin for human therapy. Some prefer to continue using the animal-sourced preparations, as there is some evidence that synthetic insulin varieties are more likely to induce hypoglycemia unawareness. Remaining manufacturers of highly purified animal-sourced insulin include the U.K.'s Wockhardt Ltd. (headquartered in India), Argentina's Laboratorios Beta S.A., and China's Wanbang Biopharma Co. Follicle-stimulating hormone (FSH) as a recombinant gonadotropin preparation replaced Serono's Pergonal which was previously isolated from post-menopausal female urine Factor VIII: Kogenate from Bayer replaced blood harvested factor VIII === Research applications === Ribosomal proteins: For the studies of individual ribosomal proteins, the use of proteins that are produced and purified from recombinant sources has largely replaced those that are obtained through isolation. However, isolation is still required for the studies of the whole ribosome. Lysosomal proteins: Lysosomal proteins are difficult to produce recombinantly due to the number and type of post-translational modifications that they have (e.g. glycosylation). As a result, recombinant lysosomal proteins are usually produced in mammalian
|
{
"page_id": 3016399,
"source": null,
"title": "List of recombinant proteins"
}
|
cells. Plant cell culture was used to produce FDA-approved glycosylated lysosomal protein-drug, and additional drug candidates. Recent studies have shown that it may be possible to produce recombinant lysosomal proteins with microorganisms such as Escherichia coli and Saccharomyces cerevisiae. Recombinant lysosomal proteins are used for both research and medical applications, such as enzyme replacement therapy. == Human recombinants with recombination as only source == === Medicinal applications === Erythropoietin (EPO): Epogen from Amgen Granulocyte colony-stimulating factor (G-CSF): filgrastim sold as Neupogen from Amgen; pegfilgrastim sold as Neulasta alpha-galactosidase A: Fabrazyme by Genzyme alpha-L-iduronidase: (rhIDU; laronidase) Aldurazyme by BioMarin Pharmaceutical and Genzyme N-acetylgalactosamine-4-sulfatase (rhASB; galsulfase): Naglazyme by BioMarin Pharmaceutical Dornase alfa, a DNase sold under the trade name Pulmozyme by Genentech Tissue plasminogen activator (TPA) Activase by Genentech Glucocerebrosidase: Ceredase by Genzyme Interferon (IF) Interferon-beta-1a: Avonex from Biogen Idec; Rebif from Serono; Interferon beta-1b as Betaseron from Schering. It is being investigated for the treatments of diseases including Guillain-Barré syndrome and multiple sclerosis. Insulin-like growth factor 1 (IGF-1) Rasburicase, a Urate Oxidase analog sold as Elitek from Sanofi == Animal recombinants == === Medicinal applications === Bovine somatotropin (bST) Porcine somatotropin (pST) Bovine Chymosin == Bacterial recombinants == === Industrial applications === Xylanases Proteases, which have found applications in both the industrial (such as the food industry) and domestic settings. == Viral recombinants == === Medicinal applications === Envelope protein of the hepatitis B virus marketed as Engerix-B by SmithKline Beecham HPV Vaccine proteins == Plant recombinants == === Research applications === Polyphenol oxidases (PPOs): These include both catechol oxidases and tyrosinases. In additional to research, PPOs have also found applications as biocatalysts. Cystatins are proteins that inhibit cysteine proteases. Research are ongoing to evaluate the potential of using cystatins in crop protection to control herbivorous pests and pathogens. === Industrial
|
{
"page_id": 3016399,
"source": null,
"title": "List of recombinant proteins"
}
|
applications === Laccases have found a wide range of application, from food additive and beverage processing to biomedical diagnosis, and as cross‐linking agents for furniture construction or in the production of biofuels. The tyrosinase‐induced polymerization of peptides offers facile access to artificial mussel foot protein analogues. Next generation universal glues can be envisioned that perform effectively even under rigorous seawater conditions and adapt to a broad range of difficult surfaces. == See also == Protein production Gene expression Protein purification Host cell protein == References == == External links == Laboratorios Beta S.A website CP Pharma/Wockhardt UK website
|
{
"page_id": 3016399,
"source": null,
"title": "List of recombinant proteins"
}
|
In chromatography, endcapping refers to the replacement of accessible silanol groups in a bonded stationary phase by trimethylsilyl groups. End-capped columns have much lower residual silanol group activity compared to non-endcapped columns. Endcapped columns show decreased retention for hydrogen bond acceptors, such as ionized bases, and increased retention for protonated bases. == References ==
|
{
"page_id": 20514516,
"source": null,
"title": "Endcapping"
}
|
Francesca Ferlaino (born 1977) is an Italian-Austrian experimental physicist known for her research on quantum matter. She is a professor of physics at the University of Innsbruck. == Biography == Francesca Ferlaino was born in Naples, Italy. She studied physics at the University of Naples Federico II (1995–2000) and was an undergraduate research fellow at the International School for Advanced Studies (SISSA) in Trieste (1999–2000). She did a PhD in physics at the University of Florence and the European Laboratory for Non-Linear Spectroscopy (LENS) (2001–2004). In 2007 she moved to the University of Innsbruck, Austria, where she was a research and teaching associate and started her own research group. In 2014 she became a professor of physics at the University of Innsbruck and research director at the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences. == Work == Her research activity explores quantum phenomena in atomic gases at ultralow temperatures with contributions spanning topics including quantum matter of atoms and molecules and few-body and scattering physics. Over the last years, she focuses specifically on the strongly magnetic, and rather unexplored, Erbium and Dysprosium atomic species, realizing in 2012 world's first Bose-Einstein condensation of Erbium, and in 2018 the first dipolar quantum mixture of Erbium and Dysprosium. In 2019, she was able to prepare the first long-lived supersolid state, an elusive and paradoxical state where superfluid flow and crystal rigidity coexist. With these systems, she has explored a variety of many-body quantum phenomena dictated by the long-range and anisotropic dipolar interaction among the atoms. In 2021 she created supersolid states along two dimensions. In 2024 her team reported the observation of quantum vortices in the supersolid phase == Awards and fellowships == Her work has earned her multiple awards, including the Grand Prix de Physique
|
{
"page_id": 67503829,
"source": null,
"title": "Francesca Ferlaino"
}
|
"Cécile-DeWitt Morette/École de Physique des Houches" from the French Academy of Sciences (2019), the Junior BEC Award (2019), the Feltrinelli Prize (2017) and the Erwin Schrödinger Prize (2017), the highest award of the Austrian Academy of Sciences. In addition, she is the recipient of an Alexander von Humboldt Professorship (2013), a START-Prize (2009) and three ERC Grants (Starting 2010, Consolidator 2016 and Advanced 2022) She was elected as a Fellow of the American Physical Society (APS) in 2019, after a nomination from the APS Division of Atomic, Molecular and Optical Physics, "for ground-breaking experiments on dipolar quantum gases of erbium atoms, including the attainment of quantum degeneracy of bosons and fermions, studies on quantum-chaotical scattering, the formation of quantum droplets, and investigations on the roton spectrum". == References == == External links == Dipolar Quantum Gas Group Research group of Francesca Ferlaino
|
{
"page_id": 67503829,
"source": null,
"title": "Francesca Ferlaino"
}
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Molecular gastronomy is the scientific approach of cuisine from primarily the perspective of chemistry. The composition (molecular structure), properties (mass, viscosity, etc) and transformations (chemical reactions, reactant products) of an ingredient are addressed and utilized in the preparation and appreciation of the ingested products. It is a branch of food science that approaches the preparation and enjoyment of nutrition from the perspective of a scientist at the scale of atoms, molecules, and mixtures. Nicholas Kurti, Hungarian physicist, and Hervé This, at the INRA in France, coined "Molecular and Physical Gastronomy" in 1988. == Examples == === Eponymous recipes === New dishes named after famous scientists include: Gibbs – infusing vanilla pods in egg white with sugar, adding olive oil and then microwave cooking. Named after physicist Josiah Willard Gibbs (1839–1903). Vauquelin – using orange juice or cranberry juice with added sugar when whipping eggs to increase the viscosity and to stabilize the foam, and then microwave cooking. Named after Nicolas Vauquelin (1763–1829), one of Lavoisier's teachers. Baumé – soaking a whole egg for a month in alcohol to create a coagulated egg. Named after the French chemist Antoine Baumé (1728–1804). == History == There are many branches of food science that study different aspects of food, such as safety, microbiology, preservation, chemistry, engineering, and physics. Until the advent of molecular gastronomy, there was no branch dedicated to studying the chemical processes of cooking in the home and in restaurants. Food science has primarily been concerned with industrial food production and, while the disciplines may overlap, they are considered separate areas of investigation. The creation of the discipline of molecular gastronomy was intended to bring together what had previously been fragmented and isolated investigations into the chemical and physical processes of cooking into an organized discipline within food science, to address
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{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
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what the other disciplines within food science either do not cover, or cover in a manner intended for scientists rather than cooks. The term "molecular and physical gastronomy" was coined in 1988 by Hungarian physicist Nicholas Kurti and French physical chemist Hervé This. In 1992, it became the title for a set of workshops held in Erice, Italy (originally titled "Science and Gastronomy") that brought together scientists and professional cooks for discussions about the science behind traditional cooking preparations. Eventually, the shortened term "molecular gastronomy" became the name of the approach, based on exploring the science behind traditional cooking methods. Kurti and This considered the creation of a formal discipline around the subjects discussed in the meetings. After Kurti's death in 1998, the name of the Erice workshops were changed by This to "The International Workshop on Molecular Gastronomy 'N. Kurti'". This remained the sole director of the subsequent workshops from 1999, and continued his research in the field of molecular gastronomy at the Inra-AgroParisTech International Centre for Molecular Gastronomy, remaining in charge of organizing the international meetings. === Precursors === The idea of using techniques developed in chemistry to study food is not a new one, for instance the discipline of food science has existed for many years. Kurti and This acknowledged this fact and though they decided that a new, organized and specific discipline should be created within food science that investigated the processes in regular cooking (as food science was primarily concerned with the nutritional properties of food and developing methods to process food on an industrial scale), there are several notable examples throughout history of investigations into the science of everyday cooking recorded as far as back to 18th century. ==== Marie-Antoine Carême (1784–1833) ==== The concept of molecular gastronomy was perhaps presaged by Marie-Antoine Carême,
|
{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
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|
one of the most famous French chefs, who said in the early 19th century that when making a food stock "the broth must come to a boil very slowly, otherwise the albumin coagulates, hardens; the water, not having time to penetrate the meat, prevents the gelatinous part of the osmazome from detaching itself." ==== Raymond Roussel (1877-1933) ==== French writer Raymond Roussel, in his 1914 story "L'Allée aux lucioles" ("The Alley of Fireflies"), introduces a fictionalized version of French chemist Antoine de Lavoisier who, in the story, creates an apparently edible semi-permeable coating ("invol ...") that he uses to encase a tiny frozen sculpture made from one type of wine, which is immersed in another type of wine. The story cites the fictional event as significant "in both the annals of science and the history of improved gastronomy." ==== Evelyn G. Halliday and Isabel T. Noble ==== In 1943 the University of Chicago Press published a book titled Food Chemistry and Cookery by the then University of Chicago Associate Professor of Home Economics Evelyn G. Halliday and University of Minnesota Associate Professor of Home Economics Isabel T. Noble. In the foreword of the 346-page book, the authors state that, "The main purpose of this book is to give an understanding of the chemical principles upon which good practices in food preparation and preservation are based." The book includes chapters such as "The Chemistry of Milk", "The Chemistry of Baking Powders and Their Use in Baking", "The Chemistry of Vegetable Cookery" and "Determination of Hydrogen Ion Concentration" and contains numerous illustrations of lab experiments including a Distillation Apparatus for Vegetable Samples and a Pipette for Determining the Relative Viscosity of Pectin Solutions. The professors had previously published The Hows and Whys of Cooking in 1928. ==== Belle Lowe ==== In 1932,
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{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
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|
Belle Lowe, then the professor of Food and Nutrition at Iowa State College, published a book titled Experimental Cookery: From The Chemical And Physical Standpoint which became a standard textbook for home economics courses across the United States. The book is an exhaustively researched look into the science of everyday cooking referencing hundreds of sources and including many experiments. At a length of over 600 pages with section titles such as "The Relation Of Cookery To Colloidal Chemistry", "Coagulation Of Proteins", "The Factors Affecting The Viscosity Of Cream And Ice Cream", "Syneresis", "Hydrolysis Of Collagen" and "Changes In Cooked Meat And The Cooking Of Meat", the volume rivals or exceeds the scope of many other books on the subject, at a much earlier date. ==== Elizabeth Cawdry Thomas ==== Though rarely credited, the origins of the Erice workshops (originally entitled "Science and Gastronomy") can be traced back to cooking teacher Elizabeth Cawdry Thomas, who studied at Le Cordon Bleu in London and ran a cooking school in Berkeley, California. The one-time wife of a physicist, Thomas had many friends in the scientific community and an interest in the science of cooking. In 1988, while attending a meeting at the Ettore Majorana Center for Scientific Culture in Erice, Thomas had a conversation with Professor Ugo Valdrè of the University of Bologna, who agreed with her that the science of cooking was an undervalued subject, and encouraged Kurti to organize a workshop at the Ettore Majorana Center. However nothing happened until Kurti met Hervé This: both approached the director of the Ettore Majorana center, physicist Antonino Zichichi, who liked the idea. They invited the food science writer Harold McGee to join them as invited co-director of the first workshops in 1992. === Nicholas Kurti === University of Oxford physicist Nicholas Kurti advocated
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{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
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|
applying scientific knowledge to culinary problems. He was one of the first television cooks in the UK, hosting a black-and-white television show in 1969 entitled The Physicist in the Kitchen, where he demonstrated techniques such as using a syringe to inject hot mince pies with brandy in order to avoid disturbing the crust. That same year, he held a presentation for the Royal Society of London (also entitled "The Physicist in the Kitchen") in which he stated: I think it is a sad reflection on our civilization that while we can and do measure the temperature in the atmosphere of Venus we do not know what goes on inside our soufflés. Kurti demonstrated making meringue in a vacuum chamber, the cooking of sausages by connecting them across a car battery, the digestion of protein by fresh pineapple juice, and a reverse baked alaska—hot inside, cold outside—cooked in a microwave oven. Kurti was also an advocate of low temperature cooking, repeating 18th century experiments by British scientist Benjamin Thompson by leaving a 2 kg (4.4 lb) lamb joint in an oven at 80 °C (176 °F). After 8.5 hours, both the inside and outside temperature of the lamb joint were around 75 °C (167 °F), and the meat was tender and juicy. With his wife Giana, Kurti edited an anthology on food and science by fellows and foreign members of the Royal Society. === Hervé This === Hervé This started collecting "culinary precisions" (old kitchen wives' tales and cooking tricks) the 24th of March 1980, and started testing these precisions to see which held up; his collection eventually numbered some 25,000. In 1995, he received a PhD in Physical Chemistry of Materials, for which he wrote his thesis on "La gastronomie moléculaire et physique" (molecular and physical gastronomy). He served as
|
{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
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|
an adviser to the French minister of education, lectured internationally, and was invited to join the lab of Nobel-winning molecular chemist Jean-Marie Lehn. This has published several books in French, four of which have been translated into English, including Molecular Gastronomy: Exploring the Science of Flavor, Kitchen Mysteries: Revealing the Science of Cooking, Cooking: The Quintessential Art, and Building a Meal: From Molecular Gastronomy to Culinary Constructivism. He currently publishes a series of essays in French, and hosts free monthly seminars on molecular gastronomy at the INRA in France. He gives free and public seminars on molecular gastronomy every month, and annually gives a public and free course on molecular gastronomy. Hervé This also authors a website and a pair of blogs on the subject in French, and publishes monthly collaborations with French chef Pierre Gagnaire on Gagnaire's website. == Objectives == The objectives of molecular gastronomy, as defined by Hervé This, are seeking for the mechanisms of culinary transformations and processes (from a chemical and physical point of view) in three areas: the social phenomena linked to culinary activity the artistic component of culinary activity the technical component of culinary activity The original fundamental objectives of molecular gastronomy were defined by This in his doctoral dissertation as: Investigating culinary and gastronomical proverbs, sayings and old wives' tales Exploring existing recipes Introducing new tools, ingredients and methods into the kitchen Inventing new dishes Using molecular gastronomy to help the general public understand the contribution of science to society Hervé This later recognized points 3, 4, and 5 as being not entirely scientific endeavors (more application of technology and educational), and has revised the list. === Areas of investigation === Prime topics for study include How ingredients are changed by different cooking methods How all the senses play their own roles
|
{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
}
|
in our appreciation of food The mechanisms of aroma release and the perception of taste and flavor How and why we evolved our particular taste and flavor sense organs and our general food likes and dislikes How cooking methods affect the eventual flavor and texture of food ingredients How new cooking methods might produce improved results of texture and flavor How our brains interpret the signals from all our senses to tell us the "flavor" of food How our enjoyment of food is affected by other influences, our environment, our mood, how it is presented, who prepares it, etc. == Chefs == In the late 1990s and early 2000s, the term started to be used to describe a new style of cooking in which some chefs began to explore new possibilities in the kitchen by embracing science, research, technological advances in equipment and various natural gums and hydrocolloids produced by the commercial food processing industry. It has since been used to describe the food and cooking of a number of famous chefs, though many of them do not accept the term as a description of their style of cooking. Chefs who are often associated with molecular gastronomy because of their embrace of science include Heston Blumenthal, Grant Achatz, Ferran Adrià, José Andrés, Marcel Vigneron, Homaro Cantu, Michael Carlson, Wylie Dufresne, and Adam Melonas. Despite their central role in the popularisation of science-based cuisine, both Adria and Blumenthal have expressed their frustration with the common mis-classification of their food and cooking as "molecular gastronomy", On 10 December 2006 Blumenthal and Harold McGee published a 'Statement on the "New Cookery" in the Observer in order to summarise what they saw as the central tenets of modern cuisine. Ferran Adria of El Bulli and Thomas Keller of the French Laundry and Per Se
|
{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
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|
signed up to this and together released a joint statement in 2006 clarifying their approach to cooking, stating that the term "molecular gastronomy" was coined in 1992 for a single workshop that did not influence them, and that the term does not describe any style of cooking. In February 2011, Nathan Myhrvold published Modernist Cuisine, which led many chefs to further classify molecular gastronomy versus modernist cuisine. Myhrvold believes that his cooking style should not be called molecular gastronomy. == Techniques, tools and ingredients == Carbon dioxide source, for adding bubbles and making foams Foams can also be made with an immersion blender Liquid nitrogen, for flash freezing and shattering Ice cream maker, often used to make unusual flavors, including savory Anti-griddle, for cooling and freezing Thermal immersion circulator for sous-vide (low temperature cooking) Food dehydrator Centrifuge Maltodextrin – can turn a high-fat liquid into a powder Sugar substitutes Enzymes Lecithin – an emulsifier and non-stick agent Hydrocolloids such as starch, gelatin, pectin and natural gums – used as thickening agents, gelling agents, emulsifying agents and stabilizers, sometimes needed for foams Transglutaminase – a protein binder, called meat glue Spherification – a caviar-like effect Syringe, for injecting unexpected fillings Edible paper made from soybeans and potato starch, for use with edible fruit inks and an inkjet printer Aromatic accompaniment: gases trapped in a bag, a serving device, or the food itself; an aromatic substance presented as a garnish or creative serveware; or a smell produced by burning Presentation style is often whimsical or avant-garde, and may include unusual serviceware Unusual flavor combinations (food pairings) are favored, such as combining savory and sweet Using ultrasound to achieve more precise cooking times == Alternative names and related pursuits == The term molecular gastronomy was originally intended to refer only to the scientific
|
{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
}
|
investigation of cooking, though it has been adopted by a number of people and applied to cooking itself or to describe a style of cuisine. Other names for the style of cuisine practiced by these chefs include: Avant-garde cuisine Culinary constructivism Cocina de vanguardia – term used by Ferran Adrià Emotional cuisine Experimental cuisine Forward-thinking movement – term used at Grant Achatz's Alinea Kitchen science Modern cuisine Modernist cuisine, which shares its name with a cookbook by Nathan Myhrvold, and which is endorsed by Ferran Adrià of El Bulli and David Chang Molecular cuisine Molecular cooking New cuisine New cookery Nueva cocina Progressive cuisine Techno-emotional cuisine—term preferred by elBulli research and development chef Ferran Adrià Technologically forward cuisine Vanguard cuisine Techno-cuisine No singular name has ever been applied in consensus, and the term "molecular gastronomy" continues to be used often as a blanket term to refer to any and all of these things—particularly in the media. Ferran Adrià hates the term "molecular gastronomy" and prefers 'deconstructivist' to describe his style of cooking. A 2006 open letter by Ferran Adria, Heston Blumenthal, Thomas Keller and Harold McGee published in The Times used no specific term, referring only to "a new approach to cooking" and "our cooking". == See also == === People === === Restaurants === === Subjects === == References == == Further reading == Caporaso, Nicola, Diego Formisano (2016). Developments, applications, and trends of molecular gastronomy among food scientists and innovative chefs. Food Reviews International, 32(4), 417–435. Hoelscher, Dietmar, Molecular kitchen and moleculare mixology: you can do what you imagine (2008 DVD) ISBN 978-3-00-022641-0 Kurti, Nicholas, But the Crackling Is Superb, Institute of Physics Publishing, 1998 ISBN 978-0-85274-301-0 McGee, Harold, The Curious Cook. North Point Press, Berkeley, 1990. McGee, Harold, On Food and Cooking: The Science and Lore of
|
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"source": null,
"title": "Molecular gastronomy"
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|
the Kitchen. Scribner, New York, 2004. ISBN 0-684-80001-2. This, Hervé, Building a Meal: From Molecular Gastronomy to Culinary Constructivism, Columbia University Press 2009 ISBN 978-0-231-14466-7 This, Hervé, Pierre Gagnaire: Cooking: The Quintessential Art, University of California Press 2008 ISBN 978-0-520-25295-0 This, Hervé, Kitchen Mysteries: Revealing the Science of Cooking. Columbia University Press, New York, 2007 ISBN 978-0-231-14170-3 This, Hervé, Molecular Gastronomy: Exploring the Science of Flavor. Columbia University Press, New York, 2006. ISBN 978-0-231-13312-8 Wolke, Robert L., "What Einstein Told His Cook: Kitchen Science Explained" (2002, 350p) ISBN 0-393-01183-6 == External links == John Mariani, Decline of Modernist Molecular Cuisine July 24, 2013 esquire.com Grubstreet, Rebuttal of John Marianis esquire article July 24, 2013 grubstreet
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{
"page_id": 2623191,
"source": null,
"title": "Molecular gastronomy"
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In organic chemistry, the Arndt–Eistert reaction is the conversion of a carboxylic acid to its homologue. It is named for the German chemists Fritz Arndt (1885–1969) and Bernd Eistert (1902–1978). The method entails treating an acid chloride with diazomethane. It is a popular method of producing β-amino acids from α-amino acids. == Conditions == Aside from the acid chloride substrate, three reagents are required: diazomethane, water, and a metal catalyst. Each has been well investigated. The diazomethane is required in excess so as to react with the HCl formed previously. Not taking diazomethane in excess results in HCl reacting with the diazoketone to form chloromethyl ketone and N2. Mild conditions allow this reaction to take place while not affecting complex or reducible groups in the reactant-acid. The reaction requires the presence of a nucleophile (water). A metal catalyst is required. Usually Ag2O is chosen but other metals and even light effect the reaction. === Variants === The preparation of the beta-amino acid from phenylalanine illustrates the Arndt–Eistert synthesis carried out with the Newman–Beal modification, which involves the inclusion of triethylamine in the diazomethane solution. Either triethylamine or a second equivalent of diazomethane will scavenge HCl, avoiding the formation of α-chloromethylketone side-products. Diazomethane is the traditional reagent, but analogues can also be applied. Diazomethane is toxic and potentially violently explosive, which has led to safer alternative procedures, For example, diazo(trimethylsilyl)methane has been demonstrated. Acid anhydrides can be used in place of acid chloride. The reaction yields a 1:1 mixture of the homologated acid and the corresponding methyl ester. This method can also be used with primary diazoalkanes, to produce secondary α-diazo ketones. However, there are many limitations. Primary diazoalkanes undergo azo coupling to form azines; thus the reaction conditions must be altered such that acid chloride is added to a solution
|
{
"page_id": 2950875,
"source": null,
"title": "Arndt–Eistert reaction"
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|
of diazoalkane and triethylamine at low temperature. In addition, primary diazoalkanes are very reactive, incompatible with acidic functionalities, and will react with activated alkenes including α,β-unsaturated carbonyl compounds to give 1,3-dipolar cycloaddition products. An alternative to the Arndt–Eistert reaction is the Kowalski ester homologation, which also involves the generation of a carbene equivalent but avoids diazomethane. == Reaction mechanism == The acid chloride suffers attack by diazomethane with loss of HCl. The alpha-diazoketone (RC(O)CHN2) product undergoes the metal-catalyzed Wolff rearrangement to form a ketene, which hydrates to the acid. The rearrangement leaves untouched the stereochemistry at the carbon alpha to the acid chloride. == Historical readings == Arndt, F.; Eistert, B. (1935). "Ein Verfahren zur Überführung von Carbonsäuren in ihre höheren Homologen bzw. deren Derivate" [A process for the conversion of carboxylic acids into their higher homologs or their derivatives]. Ber. Dtsch. Chem. Ges. (in German). 68 (1): 200–208. doi:10.1002/cber.19350680142. Bachmann, W. E.; Struve, W. S. (1942). "The Arndt–Eistert Reaction". Org. React. 1: 38. == See also == Curtius rearrangement Kowalski ester homologation Lossen rearrangement Nierenstein reaction Wolff rearrangement == References ==
|
{
"page_id": 2950875,
"source": null,
"title": "Arndt–Eistert reaction"
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|
Radiosensitivity is the relative susceptibility of cells, tissues, organs or organisms to the harmful effect of ionizing radiation. == Cells types affected == Cells are least sensitive when in the S phase, then the G1 phase, then the G2 phase, and most sensitive in the M phase of the cell cycle. This is described by the 'law of Bergonié and Tribondeau', formulated in 1906: X-rays are more effective on cells which have a greater reproductive activity. From their observations, they concluded that quickly dividing tumor cells are generally more sensitive than the majority of body cells. This is not always true. Tumor cells can be hypoxic and therefore less sensitive to X-rays because most of their effects are mediated by the free radicals produced by ionizing oxygen. It has meanwhile been shown that the most sensitive cells are those that are undifferentiated, well nourished, dividing quickly and highly active metabolically. Amongst the body cells, the most sensitive are spermatogonia and erythroblasts, epidermal stem cells, gastrointestinal stem cells. The least sensitive are nerve cells and muscle fibers. Very sensitive cells are also oocytes and lymphocytes, although they are resting cells and do not meet the criteria described above. The reasons for their sensitivity are not clear. There also appears to be a genetic basis for the varied vulnerability of cells to ionizing radiation. This has been demonstrated across several cancer types and in normal tissues. == Cell damage classification == The damage to the cell can be lethal (the cell dies) or sublethal (the cell can repair itself). Cell damage can ultimately lead to health effects which can be classified as either Tissue Reactions or Stochastic Effects according to the International Commission on Radiological Protection. === Tissue reactions === Tissue reactions have a threshold of irradiation under which they do not
|
{
"page_id": 1967838,
"source": null,
"title": "Radiosensitivity"
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appear and above which they typically appear. Fractionation of dose, dose rate, the application of antioxidants and other factors may affect the precise threshold at which a tissue reaction occurs. Tissue reactions include skin reactions (epilation, erythema, moist desquamation), cataracts, circulatory disease, and other conditions. Seven proteins were discovered in a systematic review, which correlated with radiosensitivity in normal tissues: γH2AX, TP53BP1, VEGFA, CASP3, CDKN2A, IL6, and IL1B. === Stochastic effects === Stochastic effects do not have a threshold of irradiation, are coincidental, and cannot be avoided. They can be divided into somatic and genetic effects. Among the somatic effects, secondary cancer is the most important. It develops because radiation causes DNA mutations directly and indirectly. Direct effects are those caused by ionizing particles and rays themselves, while the indirect effects are those that are caused by free radicals, generated especially in water radiolysis and oxygen radiolysis. The genetic effects confer the predisposition of radiosensitivity to the offspring. The process is not well understood yet. === Target structures === For decades, the main cellular target for radiation induced damage was thought to be the DNA molecule. This view has been challenged by data indicating that in order to increase survival, the cells must protect their proteins, which in turn repair the damage in the DNA. An important part of protection of proteins (but not DNA) against the detrimental effects of reactive oxygen species (ROS), which are the main mechanism of radiation toxicity, is played by non-enzymatic complexes of manganese ions and small organic metabolites. These complexes were shown to protect the proteins from oxidation in vitro and also increased radiation survival in mice. An application of the synthetically reconstituted protective mixture with manganese was shown to preserve the immunogenicity of viral and bacterial epitopes at radiation doses far above those
|
{
"page_id": 1967838,
"source": null,
"title": "Radiosensitivity"
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necessary to kill the microorganisms, thus opening a possibility for a quick whole-organism vaccine production. The intracellular manganese content and the nature of complexes it forms (both measurable by electron paramagnetic resonance) were shown to correlate with radiosensitivity in bacteria, archaea, fungi and human cells. An association was also found between total cellular manganese contents and their variation, and clinically inferred radioresponsiveness in different tumor cells, a finding that may be useful for more precise radiodosages and improved treatment of cancer patients. == See also == Background radiation Cell death Lethal dose, LD50 LNT model, Linear no-threshold response model for ionizing radiation Radiation sensitivity, the susceptibility of a material to physical or chemical changes induced by radiation == References ==
|
{
"page_id": 1967838,
"source": null,
"title": "Radiosensitivity"
}
|
Juan Mari Arzak Arratibel (born 31 July 1942) is a Spanish chef, the owner and chef of Arzak restaurant. He is considered to be one of the great masters of New Basque cuisine. He describes his cooking as "signature cuisine, Basque cuisine that's evolutionary, investigatory, and avant-garde." == Personal life == Arzak was an only child born to Juan Ramon Arzak and Francisca Arratibel in San Sebastián, Spain. He spent much of his childhood in his grandparents' restaurant. Later, Juan Mari Arzak's parents took over control of the restaurant. Juan Mari Arzak's father died in 1951, after which time his mother continued to run the restaurant until he took over control of the restaurant. Juan Mari Arzak has two daughters, Marta and Elena, with Maite Espina. == Professional life == Arzak said that his interest in cooking began at birth, and that in his childhood he would help in his family's restaurant. However, his real interest in cuisine didn't begin until his time at a hotel management school in Madrid. After Arzak's mandatory time in the military, he returned to his family's restaurant and began training as a chef, during which time he was responsible for roasting meat. Since he took over the restaurant, the restaurant has garnered much praise, and received three Michelin stars in 1989. In 2008 Arzak received the "Universal Basque" award for "adapting gastronomy, one of the most important traditions of the Basque Country, to the new times and making of it one of the most innovative of the world". He trained his daughter Elena Arzak (1969-), who has won the "Veuve Clicquot World's Best Female Chef" at the "World's 50 Best Restaurant Awards", to take over the restaurant. == External links == Restaurant Arzak website == Notes ==
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"page_id": 24708831,
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"title": "Juan Mari Arzak"
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Sim4 is a nucleotide sequence alignment program akin to BLAST but specifically tailored to DNA to cDNA/EST (Expressed Sequence Tag) alignment (as opposed to DNA–DNA or protein–protein alignment). It was written by Florea et al. == External links == A Computer Program for Aligning a cDNA Sequence with a Genomic DNA Sequence Download
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"page_id": 2688736,
"source": null,
"title": "Sim4"
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In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and currents). Even when there is a propagating electromagnetic wave produced (e.g., by a transmitting antenna), one can still identify as an evanescent field the component of the electric or magnetic field that cannot be attributed to the propagating wave observed at a distance of many wavelengths (such as the far field of a transmitting antenna). A hallmark of an evanescent field is that there is no net energy flow in that region. Since the net flow of electromagnetic energy is given by the average Poynting vector, this means that the Poynting vector in these regions, as averaged over a complete oscillation cycle, is zero. == Use of the term == In many cases one cannot simply say that a field is or is not "evanescent" – having the Poynting vector average to zero in some direction (or all directions). In most cases where they exist, evanescent fields are simply thought of and referred to the same as all other electric or magnetic fields involved, without any special recognition of those fields' evanescence. The term's use is mostly limited to distinguishing a part of a field or solution in those cases where one might only expect the fields of a propagating wave. For instance, in the illustration at the top of the article, energy is indeed carried in the horizontal direction. However, in the vertical direction, the field strength drops off exponentially with increasing distance above the surface. This leaves most of the field concentrated in a thin boundary layer very close to the interface; for that reason, it is referred to as
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a surface wave. However, despite energy flowing horizontally, along the vertical there is no net propagation of energy away from (or toward) the surface, so that one could properly describe the field as being "evanescent in the vertical direction". This is one example of the context dependence of the term. Everyday electronic devices and electrical appliances are surrounded by large fields which are evanescent; their operation involves alternating voltages (producing an electric field between them) and alternating currents (producing a magnetic field around them) which are expected to only carry power along internal wires, but not to the outsides of the devices. Even though the term "evanescent" is not mentioned in this ordinary context, the appliances' designers still may be concerned with maintaining evanescence, in order to prevent or limit production of a propagating electromagnetic wave, which would lead to radiation loss, since a propagating wave "steals" its power from the circuitry or donates unwanted interference. The term "evanescent field" does arise in various contexts where a propagating electromagnetic wave is involved (even if confined). The term then differentiates electromagnetic field components that accompany the propagating wave, but which do not themselves propagate. In other, similar cases, where a propagating electromagnetic wave would normally be expected (such as light refracted at the interface between glass and air), the term is invoked to describe that part of the field where the wave is suppressed (such as light traveling through glass, impinging on a glass-to-air interface but beyond the critical angle). Although all electromagnetic fields are classically governed according to Maxwell's equations, different technologies or problems have certain types of expected solutions, and when the primary solutions involve wave propagation the term evanescent is frequently applied to field components or solutions which do not share that property. For instance, the propagation constant
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of a hollow metal waveguide is a strong function of frequency (a dispersion relation). Below a certain frequency (the cut-off frequency) the propagation constant becomes an imaginary number. A solution to the wave equation having an imaginary wavenumber does not propagate as a wave but falls off exponentially, so the field excited at that lower frequency is considered evanescent. It can also be simply said that propagation is "disallowed" for that frequency. The formal solution to the wave equation can describe modes having an identical form, but the change of the propagation constant from real to imaginary as the frequency drops below the cut-off frequency totally changes the physical nature of the result. The solution may be described as a "cut-off mode" or an "evanescent mode";: 360 while a different author will just state that no such mode exists. Since the evanescent field corresponding to the mode was computed as a solution to the wave equation, it is often discussed as being an "evanescent wave" even though its properties (such as carrying no energy) are inconsistent with the definition of wave. Although this article concentrates on electromagnetics, the term evanescent is used similarly in fields such as acoustics and quantum mechanics, where the wave equation arises from the physics involved. In these cases, solutions to the wave equation resulting in imaginary propagation constants are likewise called "evanescent", and have the essential property that no net energy is transferred, even though there is a non-zero field. == Evanescent wave applications == In optics and acoustics, evanescent waves are formed when waves traveling in a medium undergo total internal reflection at its boundary because they strike it at an angle greater than the critical angle. The physical explanation for the existence of the evanescent wave is that the electric and magnetic fields
|
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(or pressure gradients, in the case of acoustical waves) cannot be discontinuous at a boundary, as would be the case if there was no evanescent wave field. In quantum mechanics, the physical explanation is exactly analogous—the Schrödinger wave-function representing particle motion normal to the boundary cannot be discontinuous at the boundary. Electromagnetic evanescent waves have been used to exert optical radiation pressure on small particles to trap them for experimentation, or to cool them to very low temperatures, and to illuminate very small objects such as biological cells or single protein and DNA molecules for microscopy (as in the total internal reflection fluorescence microscope). The evanescent wave from an optical fiber can be used in a gas sensor, and evanescent waves figure in the infrared spectroscopy technique known as attenuated total reflectance. In electrical engineering, evanescent waves are found in the near-field region within one third of a wavelength of any radio antenna. During normal operation, an antenna emits electromagnetic fields into the surrounding nearfield region, and a portion of the field energy is reabsorbed, while the remainder is radiated as EM waves. Recently, a graphene-based Bragg grating (one-dimensional photonic crystal) has been fabricated and demonstrated its competence for excitation of surface electromagnetic waves in the periodic structure using a prism coupling technique. In quantum mechanics, the evanescent-wave solutions of the Schrödinger equation give rise to the phenomenon of wave-mechanical tunneling. In microscopy, systems that capture the information contained in evanescent waves can be used to create super-resolution images. Matter radiates both propagating and evanescent electromagnetic waves. Conventional optical systems capture only the information in the propagating waves and hence are subject to the diffraction limit. Systems that capture the information contained in evanescent waves, such as the superlens and near field scanning optical microscopy, can overcome the diffraction limit;
|
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however these systems are then limited by the system's ability to accurately capture the evanescent waves. The limitation on their resolution is given by k ∝ 1 d ln 1 δ , {\displaystyle k\propto {\frac {1}{d}}\ln {\frac {1}{\delta }},} where k {\displaystyle k} is the maximal wave vector that can be resolved, d {\displaystyle d} is the distance between the object and the sensor, and δ {\displaystyle \delta } is a measure of the quality of the sensor. More generally, practical applications of evanescent waves can be classified as (1) those in which the energy associated with the wave is used to excite some other phenomenon within the region of space where the original traveling wave becomes evanescent (for example, as in the total internal reflection fluorescence microscope) or (2) those in which the evanescent wave couples two media in which traveling waves are allowed, and hence permits the transfer of energy or a particle between the media (depending on the wave equation in use), even though no traveling-wave solutions are allowed in the region of space between the two media. An example of this is wave-mechanical tunnelling, and is known generally as evanescent wave coupling. == Total internal reflection of light == For example, consider total internal reflection in two dimensions, with the interface between the media lying on the x axis, the normal along y, and the polarization along z. One might expect that for angles leading to total internal reflection, the solution would consist of an incident wave and a reflected wave, with no transmitted wave at all, but there is no such solution that obeys Maxwell's equations. Maxwell's equations in a dielectric medium impose a boundary condition of continuity for the components of the fields E||, H||, Dy, and By. For the polarization considered in
|
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this example, the conditions on E|| and By are satisfied if the reflected wave has the same amplitude as the incident one, because these components of the incident and reflected waves superimpose destructively. Their Hx components, however, superimpose constructively, so there can be no solution without a non-vanishing transmitted wave. The transmitted wave cannot, however, be a sinusoidal wave, since it would then transport energy away from the boundary, but since the incident and reflected waves have equal energy, this would violate conservation of energy. We therefore conclude that the transmitted wave must be a non-vanishing solution to Maxwell's equations that is not a traveling wave, and the only such solutions in a dielectric are those that decay exponentially: evanescent waves. Mathematically, evanescent waves can be characterized by a wave vector where one or more of the vector's components has an imaginary value. Because the vector has imaginary components, it may have a magnitude that is less than its real components. For the plane of incidence as the x y {\displaystyle xy} plane at z = 0 {\displaystyle z=0} and the interface of the two mediums as the x z {\displaystyle xz} plane at y = 0 {\displaystyle y=0} , the wave vector of the transmitted wave has the form k t = k y y ^ + k x x ^ {\displaystyle \mathbf {k_{t}} \ =\ k_{y}{\hat {\mathbf {y} }}+k_{x}{\hat {\mathbf {x} }}} with k x = k t sin θ t {\displaystyle k_{x}=k_{t}\sin \theta _{t}} and k y = k t cos θ t {\displaystyle k_{y}=k_{t}\cos \theta _{t}} , where k t {\displaystyle k_{t}} is the magnitude of the wave vector of the transmitted wave (so the wavenumber), θ t {\displaystyle \theta _{t}} is the angle of refraction, and x ^ {\displaystyle {\hat {\mathbf {x} }}}
|
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and y ^ {\displaystyle {\hat {\mathbf {y} }}} are the unit vectors along the x {\displaystyle x} axis direction and the y {\displaystyle y} axis direction respectively. By using the Snell's law n i sin θ i = n t sin θ t {\displaystyle n_{i}\sin \theta _{i}=n_{t}\sin \theta _{t}} where n i {\displaystyle n_{i}} , n t {\displaystyle n_{t}} , and θ i {\displaystyle \theta _{i}} are the refractive index of the medium where the incident wave and the reflected wave exist, the refractive index of the medium where the transmitted wave exists, and the angle of incidence respectively, k y = k t cos θ t = ± k t ( 1 − sin 2 θ i n t i 2 ) 1 / 2 {\displaystyle k_{y}=k_{t}\cos \theta _{t}=\pm k_{t}\left(1-{\frac {\sin ^{2}\theta _{i}}{n_{ti}^{2}}}\right)^{1/2}} . with n t i = n t n i {\textstyle n_{ti}={\frac {n_{t}}{n_{i}}}} . If a part of the condition of the total internal reflection as sin θ i > sin θ c = n t i {\displaystyle \sin \theta _{i}>\sin \theta _{c}=n_{ti}} , is satisfied, then k y = ± i k t ( sin 2 θ i n t i 2 − 1 ) 1 / 2 = ± i α {\displaystyle k_{y}=\pm ik_{t}\left({\frac {\sin ^{2}\theta _{i}}{n_{ti}^{2}}}-1\right)^{1/2}=\pm i\alpha } . If the polarization is perpendicular to the plane of incidence (along the z {\displaystyle z} direction), then the electric field of any of the waves (incident, reflected, or transmitted) can be expressed as E ( r , t ) = Re { E ( r ) e i ω t } z ^ {\displaystyle \mathbf {E} (\mathbf {r} ,t)=\operatorname {Re} \left\{E(\mathbf {r} )e^{i\omega t}\right\}\mathbf {\hat {z}} } where z ^ {\displaystyle \mathbf {\hat {z}} } is
|
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the unit vector in the z {\displaystyle z} axis direction. By assuming plane waves as E ( r ) = E 0 e − i k ⋅ r {\displaystyle E(\mathbf {r} )=E_{0}e^{-i\mathbf {k} \cdot \mathbf {r} }} , and substituting the transmitted wave vector k t {\displaystyle \mathbf {k_{t}} } into k {\displaystyle \mathbf {k} } , we find for the transmitted wave: E ( r ) = E o e − i ( − i α y + β x ) = E o e − α y − i β x {\displaystyle E(\mathbf {r} )=E_{o}e^{-i(-i\alpha y+\beta x)}=E_{o}e^{-\alpha y-i\beta x}} where α = k t ( sin 2 θ i n t i 2 − 1 ) 1 / 2 {\textstyle \alpha =k_{t}\left({\frac {\sin ^{2}\theta _{i}}{n_{ti}^{2}}}-1\right)^{1/2}} is the attenuation constant, and β = k x {\displaystyle \beta =k_{x}} is the phase constant. + i α {\displaystyle +i\alpha } is ignored since it does not physically make sense (the wave amplification along y the direction in this case). == Evanescent-wave coupling == Especially in optics, evanescent-wave coupling refers to the coupling between two waves due to physical overlap of what would otherwise be described as the evanescent fields corresponding to the propagating waves. One classical example is frustrated total internal reflection (FTIR) in which the evanescent field very close (see graph) to the surface of a dense medium at which a wave normally undergoes total internal reflection overlaps another dense medium in the vicinity. This disrupts the totality of the reflection, diverting some power into the second medium. Coupling between two optical waveguides may be effected by placing the fiber cores close together so that the evanescent field generated by one element excites a wave in the other fiber. This is used to produce fiber-optic splitters and in fiber
|
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tapping. At radio (and even optical) frequencies, such a device is called a directional coupler. The device is usually called a power divider in the case of microwave transmission and modulation. Evanescent-wave coupling is synonymous with near field interaction in electromagnetic field theory. Depending on the nature of the source element, the evanescent field involved is either predominantly electric (capacitive) or magnetic (inductive), unlike (propagating) waves in the far field where these components are connected (identical phase, in the ratio of the impedance of free space). The evanescent wave coupling takes place in the non-radiative field near each medium and as such is always associated with matter; i.e., with the induced currents and charges within a partially reflecting surface. In quantum mechanics the wave function interaction may be discussed in terms of particles and described as quantum tunneling. === Applications === Evanescent wave coupling is commonly used in photonic and nanophotonic devices as waveguide sensors or couplers (see e.g., prism coupler). Evanescent wave coupling is used to excite, for example, dielectric microsphere resonators. Evanescent coupling, as near field interaction, is one of the concerns in electromagnetic compatibility. Coupling of optical fibers without loss for fiber tapping. Evanescent wave coupling plays a major role in the theoretical explanation of extraordinary optical transmission. Evanescent wave coupling is used in powering devices wirelessly. A total internal reflection fluorescence microscope uses the evanescent wave produced by total internal reflection to excite fluorophores close to a surface. This is useful when surface properties of biological samples need to be studied. == See also == == Notes == == References == == External links == Evanescent wave s Evanescent and propagating waves animation on Youtube.com
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The molecular formula C12H15N3O2 (molar mass: 233.266 g/mol) may refer to: 5-Nitro-DMT Pardoprunox Phenylpiracetam hydrazide
|
{
"page_id": 61540071,
"source": null,
"title": "C12H15N3O2"
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Frank Roberts (22 January 1882–26 June 1963) was a New Zealand pioneer in building model railways. His models were extremely accurate and reflected the history of the New Zealand railways. == Early career == Roberts spent his early career working for New Zealand Railways Department (NZR) as a cleaner, fireman and driver before becoming a partner in an electrical firm with his brother Jack. In 1903 Roberts built his first model which was of the WA class steam locomotive using a scale of 1:19. In 1926 Roberts joined the Auckland Model Engineering Society and began building his first garden railway. Using 1¾ inch gauge track, Roberts, his brother George and W. W. Stewart built a large railway (known as the "RSR Railway") over the next 50 years at Roberts home in Epsom, Auckland. Roberts built his models from then on to a scale of 1:24 (G scale). His meticulously accurate working models of examples of the locomotives and rolling stock found on New Zealand railways became a local attraction. The popularity of this layout led to Roberts being commissioned in 1938 by NZR to build and operate a working model train layout for the New Zealand Centennial Exhibition. == New Zealand Centennial Exhibition == Roberts' work for the New Zealand Centennial Exhibition (1939-1940) showed him at the peak of his expertise. The NZR section of the exhibition was extremely popular. == Preservation == Roberts sold his models to the Railways Department in 1950 and was employed by them to maintain them as working models. In June 1993, just prior to privatisation, New Zealand Rail Limited gifted the collection to the Museum of New Zealand Te Papa Tongarewa. == External links == Models by Frank Roberts from the collection of the Museum of New Zealand Te Papa Tongarewa == References ==
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"source": null,
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=== Citations === === Bibliography === Roberts, Joyce (1976). Steam in Miniature - Frank Roberts and his garden railway. A H Reed. ISBN 0589009486. Vintage Steam - Stories by Frank Roberts, Edited by Frank Roberts and Gordon Troup, 1967 Film Letting off Steam Working models of railway locomotives at a model engineering society's field-day and children at an exhibition of N.Z.R. models. Held by Archives New Zealand. Ref: R.V.132 Film Weekly Review No. 379 (1948) Auckland Garden Railway. Held by Archives New Zealand. Ref: R.V. 689
|
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"page_id": 17696487,
"source": null,
"title": "Frank Roberts (model maker)"
}
|
The Hartle–Thorne metric is an approximate solution of the vacuum Einstein field equations of general relativity that describes the exterior of a slowly and rigidly rotating, stationary and axially symmetric body. The metric was found by James Hartle and Kip Thorne in the 1960s to study the spacetime outside neutron stars, white dwarfs and supermassive stars. It can be shown that it is an approximation to the Kerr metric (which describes a rotating black hole) when the quadrupole moment is set as q = − a 2 a M 3 {\displaystyle q=-a^{2}aM^{3}} , which is the correct value for a black hole but not, in general, for other astrophysical objects. == Metric == Up to second order in the angular momentum J {\displaystyle J} , mass M {\displaystyle M} and quadrupole moment q {\displaystyle q} , the metric in spherical coordinates is given by g t t = − ( 1 − 2 M r + 2 q r 3 P 2 + 2 M q r 4 P 2 + 2 q 2 r 6 P 2 2 − 2 3 J 2 r 4 ( 2 P 2 + 1 ) ) , g t ϕ = − 2 J r sin 2 θ , g r r = 1 + 2 M r + 4 M 2 r 2 − 2 q P 2 r 3 − 10 M q P 2 r 4 + 1 12 q 2 ( 8 P 2 2 − 16 P 2 + 77 ) r 6 + 2 J 2 ( 8 P 2 − 1 ) r 4 , g θ θ = r 2 ( 1 − 2 q P 2 r 3 − 5 M q P 2 r 4 + 1 36 q 2 (
|
{
"page_id": 60360425,
"source": null,
"title": "Hartle–Thorne metric"
}
|
44 P 2 2 + 8 P 2 − 43 ) r 6 + J 2 P 2 r 4 ) , g ϕ ϕ = r 2 sin 2 θ ( 1 − 2 q P 2 r 3 − 5 M q P 2 r 4 + 1 36 q 2 ( 44 P 2 2 + 8 P 2 − 43 ) r 6 + J 2 P 2 r 4 ) , {\displaystyle {\begin{aligned}g_{tt}&=-\left(1-{\frac {2M}{r}}+{\frac {2q}{r^{3}}}P_{2}+{\frac {2Mq}{r^{4}}}P_{2}+{\frac {2q^{2}}{r^{6}}}P_{2}^{2}-{\frac {2}{3}}{\frac {J^{2}}{r^{4}}}(2P_{2}+1)\right),\\g_{t\phi }&=-{\frac {2J}{r}}\sin ^{2}\theta ,\\g_{rr}&=1+{\frac {2M}{r}}+{\frac {4M^{2}}{r^{2}}}-{\frac {2qP_{2}}{r^{3}}}-{\frac {10MqP_{2}}{r^{4}}}+{\frac {1}{12}}{\frac {q^{2}\left(8P_{2}^{2}-16P_{2}+77\right)}{r^{6}}}+{\frac {2J^{2}(8P_{2}-1)}{r^{4}}},\\g_{\theta \theta }&=r^{2}\left(1-{\frac {2qP_{2}}{r^{3}}}-{\frac {5MqP_{2}}{r^{4}}}+{\frac {1}{36}}{\frac {q^{2}\left(44P_{2}^{2}+8P_{2}-43\right)}{r^{6}}}+{\frac {J^{2}P_{2}}{r^{4}}}\right),\\g_{\phi \phi }&=r^{2}\sin ^{2}\theta \left(1-{\frac {2qP_{2}}{r^{3}}}-{\frac {5MqP_{2}}{r^{4}}}+{\frac {1}{36}}{\frac {q^{2}\left(44P_{2}^{2}+8P_{2}-43\right)}{r^{6}}}+{\frac {J^{2}P_{2}}{r^{4}}}\right),\end{aligned}}} where P 2 = 3 cos 2 θ − 1 2 . {\displaystyle P_{2}={\frac {3\cos ^{2}\theta -1}{2}}.} == See also == Kerr metric == References ==
|
{
"page_id": 60360425,
"source": null,
"title": "Hartle–Thorne metric"
}
|
Royana (2006–2010) was Iran's and the Middle East's first successfully cloned sheep. Royana was a brown male domestic sheep and was cloned in the Royan Research Institute in Isfahan, Iran (The word Royan means embryo in Persian). He was the second cloned sheep in Royan Research Institute, but whereas the first sheep died few hours after birth, Royana lived for a few years. == Birth == On September 30, 2006, a group of scientists in Iran cloned Royana from an adult cell in a test tube in a laboratory. After the embryo proved its stability, scientists transferred it to the uterus of a female sheep. After a period of 145 days, Royana was born by caesarean section. Despite critical conditions, he survived and thrived. Royana was born on April 15, 2006, 1:30 am at Isfahan campus of Royan Institute by cesarean section in a healthy condition. == Death == Royana was euthanized after the abdominal pain was traced to his liver. It was also thought that Royana suffered premature death syndrome. Royana died at the age of three. His birth was a great step in the production of transgenic lambs containing factor IX transgenic, which is helpful in human blood clotting. == See also == List of cloned animals == References == == External links == 1host2u.com tabnak.ir theguardian.com/
|
{
"page_id": 47122154,
"source": null,
"title": "Royana"
}
|
Activity recognition aims to recognize the actions and goals of one or more agents from a series of observations on the agents' actions and the environmental conditions. Since the 1980s, this research field has captured the attention of several computer science communities due to its strength in providing personalized support for many different applications and its connection to many different fields of study such as medicine, human-computer interaction, or sociology. Due to its multifaceted nature, different fields may refer to activity recognition as plan recognition, goal recognition, intent recognition, behavior recognition, location estimation and location-based services. == Types == === Sensor-based, single-user activity recognition === Sensor-based activity recognition integrates the emerging area of sensor networks with novel data mining and machine learning techniques to model a wide range of human activities. Mobile devices (e.g. smart phones) provide sufficient sensor data and calculation power to enable physical activity recognition to provide an estimation of the energy consumption during everyday life. Sensor-based activity recognition researchers believe that by empowering ubiquitous computers and sensors to monitor the behavior of agents (under consent), these computers will be better suited to act on our behalf. Visual sensors that incorporate color and depth information, such as the Kinect, allow more accurate automatic action recognition and fuse many emerging applications such as interactive education and smart environments. Multiple views of visual sensor enable the development of machine learning for automatic view invariant action recognition. More advanced sensors used in 3D motion capture systems allow highly accurate automatic recognition, in the expenses of more complicated hardware system setup. ==== Levels of sensor-based activity recognition ==== Sensor-based activity recognition is a challenging task due to the inherent noisy nature of the input. Thus, statistical modeling has been the main thrust in this direction in layers, where the recognition at
|
{
"page_id": 15795950,
"source": null,
"title": "Activity recognition"
}
|
several intermediate levels is conducted and connected. At the lowest level where the sensor data are collected, statistical learning concerns how to find the detailed locations of agents from the received signal data. At an intermediate level, statistical inference may be concerned about how to recognize individuals' activities from the inferred location sequences and environmental conditions at the lower levels. Furthermore, at the highest level, a major concern is to find out the overall goal or subgoals of an agent from the activity sequences through a mixture of logical and statistical reasoning. === Sensor-based, multi-user activity recognition === Recognizing activities for multiple users using on-body sensors first appeared in the work by ORL using active badge systems in the early 1990s. Other sensor technology such as acceleration sensors were used for identifying group activity patterns during office scenarios. Activities of Multiple Users in intelligent environments are addressed in Gu et al. In this work, they investigate the fundamental problem of recognizing activities for multiple users from sensor readings in a home environment, and propose a novel pattern mining approach to recognize both single-user and multi-user activities in a unified solution. === Sensor-based group activity recognition === Recognition of group activities is fundamentally different from single, or multi-user activity recognition in that the goal is to recognize the behavior of the group as an entity, rather than the activities of the individual members within it. Group behavior is emergent in nature, meaning that the properties of the behavior of the group are fundamentally different than the properties of the behavior of the individuals within it, or any sum of that behavior. The main challenges are in modeling the behavior of the individual group members, as well as the roles of the individual within the group dynamic and their relationship to emergent
|
{
"page_id": 15795950,
"source": null,
"title": "Activity recognition"
}
|
behavior of the group in parallel. Challenges which must still be addressed include quantification of the behavior and roles of individuals who join the group, integration of explicit models for role description into inference algorithms, and scalability evaluations for very large groups and crowds. Group activity recognition has applications for crowd management and response in emergency situations, as well as for social networking and Quantified Self applications. == Approaches == === Activity recognition through logic and reasoning === Logic-based approaches keep track of all logically consistent explanations of the observed actions. Thus, all possible and consistent plans or goals must be considered. Kautz provided a formal theory of plan recognition. He described plan recognition as a logical inference process of circumscription. All actions and plans are uniformly referred to as goals, and a recognizer's knowledge is represented by a set of first-order statements, called event hierarchy. Event hierarchy is encoded in first-order logic, which defines abstraction, decomposition and functional relationships between types of events. Kautz's general framework for plan recognition has an exponential time complexity in worst case, measured in the size of the input hierarchy. Lesh and Etzioni went one step further and presented methods in scaling up goal recognition to scale up his work computationally. In contrast to Kautz's approach where the plan library is explicitly represented, Lesh and Etzioni's approach enables automatic plan-library construction from domain primitives. Furthermore, they introduced compact representations and efficient algorithms for goal recognition on large plan libraries. Inconsistent plans and goals are repeatedly pruned when new actions arrive. Besides, they also presented methods for adapting a goal recognizer to handle individual idiosyncratic behavior given a sample of an individual's recent behavior. Pollack et al. described a direct argumentation model that can know about the relative strength of several kinds of arguments for
|
{
"page_id": 15795950,
"source": null,
"title": "Activity recognition"
}
|
belief and intention description. A serious problem of logic-based approaches is their inability or inherent infeasibility to represent uncertainty. They offer no mechanism for preferring one consistent approach to another and are incapable of deciding whether one particular plan is more likely than another, as long as both of them can be consistent enough to explain the actions observed. There is also a lack of learning ability associated with logic based methods. Another approach to logic-based activity recognition is to use stream reasoning based on answer set programming, and has been applied to recognising activities for health-related applications, which uses weak constraints to model a degree of ambiguity/uncertainty. === Activity recognition through probabilistic reasoning === Probability theory and statistical learning models are more recently applied in activity recognition to reason about actions, plans and goals under uncertainty. In the literature, there have been several approaches which explicitly represent uncertainty in reasoning about an agent's plans and goals. Using sensor data as input, Hodges and Pollack designed machine learning-based systems for identifying individuals as they perform routine daily activities such as making coffee. Intel Research (Seattle) Lab and University of Washington at Seattle have done some important works on using sensors to detect human plans. Some of these works infer user transportation modes from readings of radio-frequency identifiers (RFID) and global positioning systems (GPS). The use of temporal probabilistic models has been shown to perform well in activity recognition and generally outperform non-temporal models. Generative models such as the Hidden Markov Model (HMM) and the more generally formulated Dynamic Bayesian Networks (DBN) are popular choices in modelling activities from sensor data. Discriminative models such as Conditional Random Fields (CRF) are also commonly applied and also give good performance in activity recognition. Generative and discriminative models both have their pros and cons
|
{
"page_id": 15795950,
"source": null,
"title": "Activity recognition"
}
|
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