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Inside a rotating spherical shell the acceleration due to the Lense–Thirring effect would be |
for "MG" ≪ "Rc"2 or more precisely, |
The spacetime inside the rotating spherical shell will not be flat. A flat spacetime inside a rotating mass shell is possible if the shell is allowed to deviate from a precisely spherical shape and the mass density inside the shell is allowed to vary. |
A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an anti-node, a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes. |
Standing waves result when two sinusoidal wave trains of the same frequency are moving in opposite directions in the same space and interfere with each other. They occur when waves are reflected at a boundary, such as sound waves reflected from a wall or electromagnetic waves reflected from the end of a transmission line, and particularly when waves are confined in a resonator at resonance, bouncing back and forth between two boundaries, such as in an organ pipe or guitar string. |
In a standing wave the nodes are a series of locations at equally spaced intervals where the wave amplitude (motion) is zero (see animation above). At these points the two waves add with opposite phase and cancel each other out. They occur at intervals of half a wavelength (λ/2). Midway between each pair of nodes are locations where the amplitude is maximum. These are called the antinodes. At these points the two waves add with the same phase and reinforce each other. |
In cases where the two opposite wave trains are not the same amplitude, they do not cancel perfectly, so the amplitude of the standing wave at the nodes is not zero but merely a minimum. This occurs when the reflection at the boundary is imperfect. This is indicated by a finite standing wave ratio (SWR), the ratio of the amplitude of the wave at the antinode to the amplitude at the node. |
In resonance of a two dimensional surface or membrane, such as a drumhead or vibrating metal plate, the nodes become nodal lines, lines on the surface where the surface is motionless, dividing the surface into separate regions vibrating with opposite phase. These can be made visible by sprinkling sand on the surface, and the intricate patterns of lines resulting are called Chladni figures. |
In transmission lines a voltage node is a current antinode, and a voltage antinode is a current node. |
Nodes are the points of zero displacement, not the points where two constituent waves intersect. |
Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection: |
A sound wave consists of alternating cycles of compression and expansion of the wave medium. During compression, the molecules of the medium are forced together, resulting in the increased pressure and density. During expansion the molecules are forced apart, resulting in the decreased pressure and density. |
The number of nodes in a specified length is directly proportional to the frequency of the wave. |
The characteristic sound that allows the listener to identify a particular instrument is largely due to the relative magnitude of the harmonics created by the instrument. |
In quantum field theory, a contact term is a radiatively induced point-like interaction. |
These typically occur when the vertex for the emission of a massless particle such as a photon, a graviton, or a gluon, is proportional to formula_1 (the invariant momentum of the radiated particle). |
This factor cancels the formula_2 of the Feynman propagator, and causes the exchange of the massless particle to produce a point-like |
formula_3-function effective interaction, rather than the usual |
formula_4 long-range potential. A notable example occurs in the weak interactions where a W-boson radiative correction to a gluon vertex produces a formula_1 term, leading to |
what is known as a "penguin" interaction. The contact term then generates a correction to the full action of the theory. |
Contact terms occur in gravity when there are non-minimal interactions, |
The non-minimal couplings are quantum equivalent to an "Einstein frame," with a pure Einstein-Hilbert action, |
owing to gravitational contact terms. These arise classically from graviton exchange interactions. |
The contact terms are an essential, yet hidden, part of the action and, if they are ignored, the Feynman diagram loops in different frames yield different results. At the leading order in formula_9 including the contact terms is equivalent to performing a Weyl Transformation to remove the non-minimal couplings and taking the theory to the Einstein-Hilbert form. In this sense, the Einstein-Hilbert form of the action is unique and "frame ambiguities" in loop calculations do not exist. |
Unification of the observable fundamental phenomena of nature is one of the primary goals of physics. The two great unifications to date are Isaac Newton’s unification of gravity and astronomy, and James Clerk Maxwell’s unification of electromagnetism; the latter has been further unified with the concept of electroweak interaction. This process of "unifying" forces continues today, with the ultimate goal of finding a theory of everything. |
The "first great unification" was Isaac Newton's 17th century unification of gravity, which brought together the understandings of the observable phenomena of gravity on Earth with the observable behaviour of celestial bodies in space. |
Unification of Magnetism, Electricity, Light and related radiation. |
The "second great unification" was James Clerk Maxwell's 19th century unification of electromagnetism. It brought together the understandings of the observable phenomena of magnetism, electricity and light (and more broadly, the spectrum of electromagnetic radiation). This was followed in the 20th century by Albert Einstein's unification of space and time, and of mass and energy. Later, quantum field theory unified quantum mechanics and special relativity. |
The ancient Chinese observed that certain rocks (lodestone and magnetite) were attracted to one another by an invisible force. This effect was later called magnetism, which was first rigorously studied in the 17th century. But even before the Chinese discovered magnetism, the ancient Greeks knew of other objects such as amber, that when rubbed with fur would cause a similar invisible attraction between the two. This was also first studied rigorously in the 17th century and came to be called electricity. Thus, physics had come to understand two observations of nature in terms of some root cause (electricity and magnetism). However, further work in the 19th century revealed that these two forces were just two different aspects of one force—electromagnetism. |
This process of "unifying" forces continues today, and electromagnetism and the weak nuclear force are now considered to be two aspects of the electroweak interaction. |
Unification of the remaining fundamental forces: Theory of Everything. |
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called "canonical transformations", which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by formula_1 and formula_2, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself formula_3 as one of the new canonical momentum coordinates. |
In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups. |
All of these objects are named in honor of Siméon Denis Poisson. |
Given two functions f and g that depend on phase space and time, their Poisson bracket formula_4 is another function that depends on phase space and time. The following rules hold for any three functions formula_5 of phase space and time: |
Also, if a function formula_10 is constant over phase space (but may depend on time), then formula_11 for any formula_12. |
In canonical coordinates (also known as Darboux coordinates) formula_13 on the phase space, given two functions formula_14 and formula_15, the Poisson bracket takes the form |
The Poisson brackets of the canonical coordinates are |
Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that formula_19 is a function on the solution’s trajectory-manifold. Then from the multivariable chain rule, |
Further, one may take formula_21 and formula_22 to be solutions to Hamilton's equations; that is, |
Thus, the time evolution of a function formula_12 on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time formula_26 being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that "any time formula_26" in the solution to Hamilton's equations, |
can serve as the bracket coordinates. "Poisson brackets are canonical invariants". |
The operator in the convective part of the derivative, formula_30, is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)). |
An integrable dynamical system will have constants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function formula_31 is a constant of motion. This implies that if formula_32 is a trajectory or solution to Hamilton's equations of motion, then |
where, as above, the intermediate step follows by applying the equations of motion and we supposed that formula_12 does not explicitly depend on time. This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure (or "distribution function" on the phase space) is given by the above. |
If the Poisson bracket of formula_12 and formula_37 vanishes (formula_38), then formula_12 and formula_37 are said to be in involution. In order for a Hamiltonian system to be completely integrable, formula_41 independent constants of motion must be in mutual involution, where formula_41 is the number of degrees of freedom. |
Furthermore, according to Poisson's Theorem, if two quantities formula_43 and formula_44 are explicitly time independent (formula_45) constants of motion, so is their Poisson bracket formula_46. This does not always supply a useful result, however, since the number of possible constants of motion is limited (formula_47 for a system with formula_41 degrees of freedom), and so the result may be trivial (a constant, or a function of formula_43 and formula_44.) |
Let formula_51 be a symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form formula_52 which is both closed (i.e., its exterior derivative formula_53 vanishes) and non-degenerate. For example, in the treatment above, take formula_51 to be formula_55 and take |
If formula_57 is the interior product or contraction operation defined by formula_58, then non-degeneracy is equivalent to saying that for every one-form formula_59 there is a unique vector field formula_60 such that formula_61. Alternatively, formula_62. Then if formula_63 is a smooth function on formula_51, the Hamiltonian vector field formula_65 can be defined to be formula_66. It is easy to see that |
The Poisson bracket formula_68 on ("M", ω) is a bilinear operation on differentiable functions, defined by formula_69; the Poisson bracket of two functions on "M" is itself a function on "M". The Poisson bracket is antisymmetric because: |
Here "Xgf" denotes the vector field "Xg" applied to the function "f" as a directional derivative, and formula_71 denotes the (entirely equivalent) Lie derivative of the function "f". |
If α is an arbitrary one-form on "M", the vector field Ωα generates (at least locally) a flow formula_72 satisfying the boundary condition formula_73 and the first-order differential equation |
The formula_72 will be symplectomorphisms (canonical transformations) for every "t" as a function of "x" if and only if formula_76; when this is true, Ωα is called a symplectic vector field. Recalling Cartan's identity formula_77 and "d"ω = 0, it follows that formula_78. Therefore, Ωα is a symplectic vector field if and only if α is a closed form. Since formula_79, it follows that every Hamiltonian vector field "Xf" is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From above, under the Hamiltonian flow "XH", |
This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when "{f,H} = 0", "f" is a constant of motion of the system. In addition, in canonical coordinates (with formula_81 and formula_82), Hamilton's equations for the time evolution of the system follow immediately from this formula. |
It also follows from that the Poisson bracket is a derivation; that is, it satisfies a non-commutative version of Leibniz's product rule: |
The Poisson bracket is intimately connected to the Lie bracket of the Hamiltonian vector fields. Because the Lie derivative is a derivation, |
Thus if "v" and "w" are symplectic, using formula_84, Cartan's identity, and the fact that formula_85 is a closed form, |
Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of abstract algebra, the symplectic vector fields form a subalgebra of the Lie algebra of smooth vector fields on "M", and the Hamiltonian vector fields form an ideal of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional) Lie group of symplectomorphisms of "M". |
It is widely asserted that the Jacobi identity for the Poisson bracket, |
follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is sufficient to show that: |
where the operator formula_90 on smooth functions on "M" is defined by formula_91 and the bracket on the right-hand side is the commutator of operators, formula_92. By , the operator formula_90 is equal to the operator "Xg". The proof of the Jacobi identity follows from because the Lie bracket of vector fields is just their commutator as differential operators. |
The algebra of smooth functions on M, together with the Poisson bracket forms a Poisson algebra, because it is a Lie algebra under the Poisson bracket, which additionally satisfies Leibniz's rule . We have shown that every symplectic manifold is a Poisson manifold, that is a manifold with a "curly-bracket" operator on smooth functions such that the smooth functions form a Poisson algebra. However, not every Poisson manifold arises in this way, because Poisson manifolds allow for degeneracy which cannot arise in the symplectic case. |
Given a smooth vector field formula_94 on the configuration space, let formula_95 be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket: |
This important result is worth a short proof. Write a vector field formula_94 at point formula_98 in the configuration space as |
where the formula_100 is the local coordinate frame. The conjugate momentum to formula_94 has the expression |
where the formula_2 are the momentum functions conjugate to the coordinates. One then has, for a point formula_104 in the phase space, |
The above holds for all formula_106, giving the desired result. |
Poisson brackets deform to Moyal brackets upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra, or, equivalently in Hilbert space, quantum commutators. The Wigner-İnönü group contraction of these (the classical limit, ) yields the above Lie algebra. |
To state this more explicitly and precisely, the universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit). The Moyal product is then a special case of the star product on the algebra of symbols. An explicit definition of the algebra of symbols, and the star product is given in the article on the universal enveloping algebra. |
In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time. |
Liouville's theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does "not" imply that the ergodic hypothesis holds for all Hamiltonian systems. |
The ergodic hypothesis is often assumed in the statistical analysis of computational physics. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. This assumption—that it is as good to simulate a system over a long time as it is to make many independent realizations of the same system—is not always correct. (See, for example, the Fermi–Pasta–Ulam–Tsingou experiment of 1953.) |
Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible. |
Systems that are ergodic are said to have the property of ergodicity; a broad range of systems in geometry, physics and stochastic probability theory are ergodic. Ergodic systems are studied in ergodic theory. |
In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase space can be sufficiently large that the thermodynamic equilibrium state exhibits some form of ergodicity breaking. A common example is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero. The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking. |
However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments. Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner. In practice this means that on sufficiently short time scales (e.g. those of parts of seconds, minutes, or a few hours) the systems may behave as "solids", i.e. with a positive shear modulus, but on extremely long scales, e.g. over millennia or eons, as "liquids", or with two or more time scales and "plateaux" in between. |
The "commutative property" (or "commutative law") is a property generally associated with binary operations and functions. If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to "commute" under that operation. |
The term "commutative" is used in several related senses. |
Two well-known examples of commutative binary operations: |
Subtraction is noncommutative, since formula_10. However it is classified more precisely as anti-commutative, since formula_11. |
Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for and are |
Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative. For example, let formula_12 and formula_13. Then |
This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation). |
Matrix multiplication of square matrices is almost always noncommutative, for example: |
The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., "b" × "a" = −("a" × "b"). |
Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products. Euclid is known to have assumed the commutative property of multiplication in his book "Elements". Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics. |
The first recorded use of the term "commutative" was in a memoir by François Servois in 1814, which used the word "commutatives" when describing functions that have what is now called the commutative property. The word is a combination of the French word "commuter" meaning "to substitute or switch" and the suffix "-ative" meaning "tending to" so the word literally means "tending to substitute or switch". The term then appeared in English in 1838 in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh. |
In truth-functional propositional logic, "commutation", or "commutativity" refer to two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are: |
where "formula_19" is a metalogical symbol representing "can be replaced in a proof with". |
"Commutativity" is a property of some logical connectives of truth functional propositional logic. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies. |
In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs. |
The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result. |
Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function |
which is clearly commutative (interchanging "x" and "y" does not affect the result), but it is not associative (since, for example, formula_25 but formula_26). More such examples may be found in commutative non-associative magmas. |
Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line formula_27. As an example, if we let a function "f" represent addition (a commutative operation) so that formula_28 then formula_29 is a symmetric function, which can be seen in the adjacent image. |
For relations, a symmetric relation is analogous to a commutative operation, in that if a relation "R" is symmetric, then formula_30. |
In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as formula_31 (meaning multiply by formula_31), and formula_33. These two operators do not commute as may be seen by considering the effect of their compositions formula_34 and formula_35 (also called products of operators) on a one-dimensional wave function formula_36: |
According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the formula_31-direction of a particle are represented by the operators formula_31 and formula_40, respectively (where formula_41 is the reduced Planck constant). This is the same example except for the constant formula_42, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary. |
In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature "C" as the intersection point of two infinitely close normal lines to the curve. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. This term is generally used in physics regarding to study of lenses and mirrors. |
It can also be defined as the spherical distance between the point at which all the rays falling on a lens or mirror either seems to converge to (in the case of convex lenses and concave mirrors) or diverge from (in the case of concave lenses or convex mirrors) and the lens/mirror itself. |
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