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0 {\displaystyle \lambda =0} corresponds to the trivial case of no collision. Substituting the non trivial value of λ {\displaystyle \lambda } in (3) we get the desired result (1). Since all equations are in vectorial form, this derivation is valid also for three dimensions with spheres. == See also == Collision Inelastic collision Coefficient of restitution == References == === General references === Landau, L. D.; Lifshitz, E. M. (1976). Mechanics (3rd ed.). Pergamon Press. ISBN 0-08-021022-8. Raymond, David J. "10.4.1 Elastic collisions". A Radically Modern Approach to Introductory Physics. Vol. 1: Fundamental Principles. Socorro, New Mexico: New Mexico Tech Press. ISBN 978-0-9830394-5-7. Serway, Raymond A.; Jewett, John W. (2014). "9: Linear Momentum and Collisions". Physics for scientists and engineers with modern physics (9th ed.). Boston. ISBN 978-1-133-95405-7.{{cite book}}: CS1 maint: location missing publisher (link) == External links == Rigid Body Collision Resolution in three dimensions including a derivation using the conservation laws
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"page_id": 65907,
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"title": "Elastic collision"
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An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed. The molecules of a gas or liquid rarely experience perfectly elastic collisions because kinetic energy is exchanged between the molecules' translational motion and their internal degrees of freedom with each collision. At any one instant, half the collisions are – to a varying extent – inelastic (the pair possesses less kinetic energy after the collision than before), and half could be described as “super-elastic” (possessing more kinetic energy after the collision than before). Averaged across an entire sample, molecular collisions are elastic. Although inelastic collisions do not conserve kinetic energy, they do obey conservation of momentum. Simple ballistic pendulum problems obey the conservation of kinetic energy only when the block swings to its largest angle. In nuclear physics, an inelastic collision is one in which the incoming particle causes the nucleus it strikes to become excited or to break up. Deep inelastic scattering is a method of probing the structure of subatomic particles in much the same way as Rutherford probed the inside of the atom (see Rutherford scattering). Such experiments were performed on protons in the late 1960s using high-energy electrons at the Stanford Linear Accelerator (SLAC). As in Rutherford scattering, deep inelastic scattering of electrons by proton targets revealed that most of the incident electrons interact very little and pass straight through, with only a small number bouncing back. This indicates that the charge in the proton is concentrated in small lumps, reminiscent of Rutherford's discovery that the positive charge in an atom is concentrated at the
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{
"page_id": 65908,
"source": null,
"title": "Inelastic collision"
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nucleus. However, in the case of the proton, the evidence suggested three distinct concentrations of charge (quarks) and not one. == Formula == The formula for the velocities after a one-dimensional collision is: v a = C R m b ( u b − u a ) + m a u a + m b u b m a + m b v b = C R m a ( u a − u b ) + m a u a + m b u b m a + m b {\displaystyle {\begin{aligned}v_{a}&={\frac {C_{R}m_{b}(u_{b}-u_{a})+m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}}\\v_{b}&={\frac {C_{R}m_{a}(u_{a}-u_{b})+m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}}\end{aligned}}} where va is the final velocity of the first object after impact vb is the final velocity of the second object after impact ua is the initial velocity of the first object before impact ub is the initial velocity of the second object before impact ma is the mass of the first object mb is the mass of the second object CR is the coefficient of restitution; if it is 1 we have an elastic collision; if it is 0 we have a perfectly inelastic collision, see below. In a center of momentum frame the formulas reduce to: v a = − C R u a v b = − C R u b {\displaystyle {\begin{aligned}v_{a}&=-C_{R}u_{a}\\v_{b}&=-C_{R}u_{b}\end{aligned}}} For two- and three-dimensional collisions the velocities in these formulas are the components perpendicular to the tangent line/plane at the point of contact. If assuming the objects are not rotating before or after the collision, the normal impulse is: J n = m a m b m a + m b ( 1 + C R ) ( u b → − u a → ) ⋅ n → {\displaystyle J_{n}={\frac {m_{a}m_{b}}{m_{a}+m_{b}}}(1+C_{R})({\vec {u_{b}}}-{\vec {u_{a}}})\cdot {\vec {n}}} where n → {\displaystyle {\vec {n}}} is the normal vector. Assuming no friction,
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{
"page_id": 65908,
"source": null,
"title": "Inelastic collision"
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this gives the velocity updates: Δ v a → = J n m a n → Δ v b → = − J n m b n → {\displaystyle {\begin{aligned}\Delta {\vec {v_{a}}}&={\frac {J_{n}}{m_{a}}}{\vec {n}}\\\Delta {\vec {v_{b}}}&=-{\frac {J_{n}}{m_{b}}}{\vec {n}}\end{aligned}}} == Perfectly inelastic collision == A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost. In a perfectly inelastic collision, i.e., a zero coefficient of restitution, the colliding particles stick together. In such a collision, kinetic energy is lost by bonding the two bodies together. This bonding energy usually results in a maximum kinetic energy loss of the system. It is necessary to consider conservation of momentum: (Note: In the sliding block example above, momentum of the two body system is only conserved if the surface has zero friction. With friction, momentum of the two bodies is transferred to the surface that the two bodies are sliding upon. Similarly, if there is air resistance, the momentum of the bodies can be transferred to the air.) The equation below holds true for the two-body (Body A, Body B) system collision in the example above. In this example, momentum of the system is conserved because there is no friction between the sliding bodies and the surface. m a u a + m b u b = ( m a + m b ) v {\displaystyle m_{a}u_{a}+m_{b}u_{b}=\left(m_{a}+m_{b}\right)v} where v is the final velocity, which is hence given by v = m a u a + m b u b m a + m b {\displaystyle v={\frac {m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}}} The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision
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{
"page_id": 65908,
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"title": "Inelastic collision"
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is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy from one particle to the other; the fact that this depends on the frame shows how relative this is. The change in kinetic energy is hence: Δ K E = 1 2 μ u r e l 2 = 1 2 m a m b m a + m b | u a − u b | 2 {\displaystyle \Delta KE={1 \over 2}\mu u_{\rm {rel}}^{2}={\frac {1}{2}}{\frac {m_{a}m_{b}}{m_{a}+m_{b}}}|u_{a}-u_{b}|^{2}} where μ is the reduced mass and urel is the relative velocity of the bodies before collision. With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation). == Partially inelastic collisions == Partially inelastic collisions are the most common form of collisions in the real world. In this type of collision, the objects involved in the collisions do not stick, but some kinetic energy is still lost. Friction, sound and heat are some ways the kinetic energy can be lost through partial inelastic collisions. == See also == Collision Elastic collision Coefficient of restitution == References ==
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{
"page_id": 65908,
"source": null,
"title": "Inelastic collision"
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|
In physics and in particular in the theory of magnetism, an antidynamo theorem is one of several results that restrict the type of magnetic fields that may be produced by dynamo action. One notable example is Thomas Cowling's antidynamo theorem which states that no axisymmetric magnetic field can be maintained through a self-sustaining dynamo action by an axially symmetric current. Similarly, the Zeldovich's antidynamo theorem states that a two-dimensional, planar flow cannot maintain the dynamo action. == Consequences == Apart from the Earth's magnetic field, some other bodies such as Jupiter and Saturn, and the Sun have significant magnetic fields whose major component is a dipole, an axisymmetric magnetic field. These magnetic fields are self-sustained through fluid motion in the Sun or planets, with the necessary non-symmetry for the planets deriving from the Coriolis force caused by their rapid rotation, and one cause of non-symmetry for the Sun being its differential rotation. The magnetic fields of planets with slow rotation periods and/or solid cores, such as Mercury, Venus, and Mars, have dissipated to almost nothing by comparison. The impact of the known anti-dynamo theorems is that successful dynamos do not possess a high degree of symmetry. == See also == Dynamo theory Magnetosphere of Jupiter Magnetosphere of Saturn == References ==
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{
"page_id": 32244084,
"source": null,
"title": "Antidynamo theorem"
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Maleic hydrazide, often known by the brand name Fazor, is a plant growth regulator that reduces growth through preventing cell division but not cell enlargement. It is applied to the foliage of potato, onion, garlic and carrot crops to prevent sprouting during storage. It can also be used to control volunteer potatoes that are left in the field during harvesting. It was first identified in the 1940s but was not used commercially in the United Kingdom until 1984. The banning of chlorpropham as a sprout suppressant in 2019 has led renewed interest in how maleic hydrazide can be used in potatoes. == References ==
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"page_id": 67895670,
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"title": "Maleic hydrazide"
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Swallowing, also called deglutition or inglutition in scientific and medical contexts, is a physical process of an animal's digestive tract (e.g. that of a human body) that allows for an ingested substance (typically food) to pass from the mouth to the pharynx and then into the esophagus. In colloquial English, the term "swallowing" is also used to describe the action of gulping, i.e. taking in a large mouthful of food without any biting. Swallowing is performed by an initial push from back part of the tongue (with the tongue tip contacting the hard palate for mechanical anchorage) and subsequent coordinated contractions of the pharyngeal muscles. The portion of food, drink and/or other material (e.g. mucus, secretions and medications) that moves into the gullet in one swallow is called a bolus, which is then propelled through to the stomach for further digestion by autonomic peristalsis of the esophagus. Swallowing is an important part of eating and drinking. If the process fails and the bolus to be swallowed mistakenly goes into the trachea, then choking or pulmonary aspiration can occur. In the human body, such incidents are prevented by an automatic trapdoor-like inversion of the epiglottis to temporarily cover the larynx and close off the upper airway, controlled by a complex reflex that facilitates the elevation of the hyoid bone and thyroid cartilage at the same time. The body will also initiate a cough reflex to expel any unwanted material that have accidentally entered the airway. A separate gag reflex, which involves the elevation of the uvula and tightening of the soft palate, prevents food from wrongly entering the nasal cavity above during swallowing. == In humans == Swallowing comes so easily to most people that the process rarely prompts much thought. However, from the viewpoints of physiology, of speech–language pathology, and
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of health care for people with difficulty in swallowing (dysphagia), it is an interesting topic with extensive scientific literature. === Coordination and control === Eating and swallowing are complex neuromuscular activities consisting essentially of three phases, an oral, pharyngeal and esophageal phase. Each phase is controlled by a different neurological mechanism. The oral phase, which is entirely voluntary, is mainly controlled by the medial temporal lobes and limbic system of the cerebral cortex with contributions from the motor cortex and other cortical areas. The pharyngeal swallow is started by the oral phase and subsequently is coordinated by the swallowing center on the medulla oblongata and pons. The reflex is initiated by touch receptors in the pharynx as a bolus of food is pushed to the back of the mouth by the tongue, or by stimulation of the palate (palatal reflex). Swallowing is a complex mechanism using both skeletal muscle (tongue) and smooth muscles of the pharynx and esophagus. The autonomic nervous system (ANS) coordinates this process in the pharyngeal and esophageal phases. === Phases === ==== Oral phase ==== Prior to the following stages of the oral phase, the mandible depresses and the lips abduct to allow food or liquid to enter the oral cavity. Upon entering the oral cavity, the mandible elevates and the lips adduct to assist in oral containment of the food and liquid. The following stages describe the normal and necessary actions to form the bolus, which is defined as the state of the food in which it is ready to be swallowed. 1) Moistening Food is moistened by saliva from the salivary glands (parasympathetic). 2) Mastication Food is mechanically broken down by the action of the teeth controlled by the muscles of mastication (V3) acting on the temporomandibular joint. This results in a bolus which
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"page_id": 196983,
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is moved from one side of the oral cavity to the other by the tongue. Buccinator (VII) helps to contain the food against the occlusal surfaces of the teeth. The bolus is ready for swallowing when it is held together by saliva (largely mucus), sensed by the lingual nerve of the tongue (VII—chorda tympani and IX—lesser petrosal) (V3). Any food that is too dry to form a bolus will not be swallowed. 3) Trough formation A trough is then formed at the back of the tongue by the intrinsic muscles (XII). The trough obliterates against the hard palate from front to back, forcing the bolus to the back of the tongue. The intrinsic muscles of the tongue (XII) contract to make a trough (a longitudinal concave fold) at the back of the tongue. The tongue is then elevated to the roof of the mouth (by the mylohyoid (mylohyoid nerve—V3), genioglossus, styloglossus and hyoglossus (the rest XII)) such that the tongue slopes downwards posteriorly. The contraction of the genioglossus and styloglossus (both XII) also contributes to the formation of the central trough. 4) Movement of the bolus posteriorly At the end of the oral preparatory phase, the food bolus has been formed and is ready to be propelled posteriorly into the pharynx. In order for anterior to posterior transit of the bolus to occur, orbicularis oris contracts and adducts the lips to form a tight seal of the oral cavity. Next, the superior longitudinal muscle elevates the apex of the tongue to make contact with the hard palate and the bolus is propelled to the posterior portion of the oral cavity. Once the bolus reaches the palatoglossal arch of the oropharynx, the pharyngeal phase, which is reflex and involuntary, then begins. Receptors initiating this reflex are proprioceptive (afferent limb of reflex
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is IX and efferent limb is the pharyngeal plexus- IX and X). They are scattered over the base of the tongue, the palatoglossal and palatopharyngeal arches, the tonsillar fossa, uvula and posterior pharyngeal wall. Stimuli from the receptors of this phase then provoke the pharyngeal phase. In fact, it has been shown that the swallowing reflex can be initiated entirely by peripheral stimulation of the internal branch of the superior laryngeal nerve. This phase is voluntary and involves important cranial nerves: V (trigeminal), VII (facial) and XII (hypoglossal). ==== Pharyngeal phase ==== For the pharyngeal phase to work properly all other egress from the pharynx must be occluded—this includes the nasopharynx and the larynx. When the pharyngeal phase begins, other activities such as chewing, breathing, coughing and vomiting are concomitantly inhibited. 5) Closure of the nasopharynx The soft palate is tensed by tensor palatini (Vc), and then elevated by levator palatini (pharyngeal plexus—IX, X) to close the nasopharynx. There is also the simultaneous approximation of the walls of the pharynx to the posterior free border of the soft palate, which is carried out by the palatopharyngeus (pharyngeal plexus—IX, X) and the upper part of the superior constrictor (pharyngeal plexus—IX, X). 6) The pharynx prepares to receive the bolus The pharynx is pulled upwards and forwards by the suprahyoid and longitudinal pharyngeal muscles – stylopharyngeus (IX), salpingopharyngeus (pharyngeal plexus—IX, X) and palatopharyngeus (pharyngeal plexus—IX, X) to receive the bolus. The palatopharyngeal folds on each side of the pharynx are brought close together through the superior constrictor muscles, so that only a small bolus can pass. 7) Opening of the auditory tube The actions of the levator palatini (pharyngeal plexus—IX, X), tensor palatini (Vc) and salpingopharyngeus (pharyngeal plexus—IX, X) in the closure of the nasopharynx and elevation of the pharynx opens the
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auditory tube, which equalises the pressure between the nasopharynx and the middle ear. This does not contribute to swallowing, but happens as a consequence of it. 8) Closure of the oropharynx The oropharynx is kept closed by palatoglossus (pharyngeal plexus—IX, X), the intrinsic muscles of tongue (XII) and styloglossus (XII). 9) Laryngeal closure The primary laryngopharyngeal protective mechanism to prevent aspiration during swallowing is via the closure of the true vocal folds. The adduction of the vocal cords is affected by the contraction of the lateral cricoarytenoids and the oblique and transverse arytenoids (all recurrent laryngeal nerve of vagus). Since the true vocal folds adduct during the swallow, a finite period of apnea (swallowing apnea) must necessarily take place with each swallow. When relating swallowing to respiration, it has been demonstrated that swallowing occurs most often during expiration, even at full expiration a fine air jet is expired probably to clear the upper larynx from food remnants or liquid. The clinical significance of this finding is that patients with a baseline of compromised lung function will, over a period of time, develop respiratory distress as a meal progresses. Subsequently, false vocal fold adduction, adduction of the aryepiglottic folds and retroversion of the epiglottis take place. The aryepiglotticus (recurrent laryngeal nerve of vagus) contracts, causing the arytenoids to appose each other (closes the laryngeal aditus by bringing the aryepiglottic folds together), and draws the epiglottis down to bring its lower half into contact with arytenoids, thus closing the aditus. Retroversion of the epiglottis, while not the primary mechanism of protecting the airway from laryngeal penetration and aspiration, acts to anatomically direct the food bolus laterally towards the piriform fossa. Additionally, the larynx is pulled up with the pharynx under the tongue by stylopharyngeus (IX), salpingopharyngeus (pharyngeal plexus—IX, X), palatopharyngeus (pharyngeal plexus—IX,
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X) and inferior constrictor (pharyngeal plexus—IX, X). This phase is passively controlled reflexively and involves cranial nerves V, X (vagus), XI (accessory) and XII (hypoglossal). The respiratory center of the medulla is directly inhibited by the swallowing center for the very brief time that it takes to swallow. This means that it is briefly impossible to breathe during this phase of swallowing and the moment where breathing is prevented is known as deglutition apnea. 10) Hyoid elevation The hyoid is elevated by digastric (V & VII) and stylohyoid (VII), lifting the pharynx and larynx up even further. 11) Bolus transits pharynx The bolus moves down towards the esophagus by pharyngeal peristalsis which takes place by sequential contraction of the superior, middle and inferior pharyngeal constrictor muscles (pharyngeal plexus—IX, X). The lower part of the inferior constrictor (cricopharyngeus) is normally closed and only opens for the advancing bolus. Gravity plays only a small part in the upright position—in fact, it is possible to swallow solid food even when standing on one's head. The velocity through the pharynx depends on a number of factors such as viscosity and volume of the bolus. In one study, bolus velocity in healthy adults was measured to be approximately 30–40 cm/s. ==== Esophageal phase ==== 12) Esophageal peristalsis Like the pharyngeal phase of swallowing, the esophageal phase of swallowing is under involuntary neuromuscular control. However, propagation of the food bolus is significantly slower than in the pharynx. The bolus enters the esophagus and is propelled downwards first by striated muscle (recurrent laryngeal, X) then by the smooth muscle (X) at a rate of 3–5 cm/s. The upper esophageal sphincter relaxes to let food pass, after which various striated constrictor muscles of the pharynx as well as peristalsis and relaxation of the lower esophageal sphincter sequentially push
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the bolus of food through the esophagus into the stomach. 13) Relaxation phase Finally the larynx and pharynx move down with the hyoid mostly by elastic recoil. Then the larynx and pharynx move down from the hyoid to their relaxed positions by elastic recoil. Swallowing therefore depends on coordinated interplay between many various muscles, and although the initial part of swallowing is under voluntary control, once the deglutition process is started, it is quite hard to stop it. ==== Clinical significance ==== Swallowing becomes a great concern for the elderly since strokes and Alzheimer's disease can interfere with the autonomic nervous system. Speech pathologists commonly diagnose and treat this condition since the speech process uses the same neuromuscular structures as swallowing. Diagnostic procedures commonly performed by a speech pathologist to evaluate dysphagia include Fiberoptic Endoscopic Evaluation of Swallowing and Modified Barium Swallow Study. Occupational Therapists may also offer swallowing rehabilitation services as well as prescribing modified feeding techniques and utensils. Consultation with a dietician is essential, in order to ensure that the individual with dysphagia is able to consume sufficient calories and nutrients to maintain health. In terminally ill patients, a failure of the reflex to swallow leads to a build-up of mucus or saliva in the throat and airways, producing a noise known as a death rattle (not to be confused with agonal respiration, which is an abnormal pattern of breathing due to cerebral ischemia or hypoxia). Abnormalities of the pharynx and/or oral cavity may lead to oropharyngeal dysphagia. Abnormalities of the esophagus may lead to esophageal dysphagia. The failure of the lower esophagus sphincter to respond properly to swallowing is called achalasia. M-Type Swallowing With practice, people can learn to swallow fluidly without closing the mouth by merely manipulating the tongue and jaw to drive fluids or foods
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down the esophagus. With a continuous motion, an individual forges breathing and priorities the swallowed matter. This intermediate level of muscle manipulation is similar to the techniques used by sword swallowers. == In non-mammal animals == In many birds, the esophagus is largely a mere gravity chute, and in such events as a seagull swallowing a fish or a stork swallowing a frog, swallowing consists largely of the bird lifting its head with its beak pointing up and guiding the prey with tongue and jaws so that the prey slides inside and down. In fish, the tongue is largely bony and much less mobile and getting the food to the back of the pharynx is helped by pumping water in its mouth and out of its gills. In snakes, the work of swallowing is done by raking with the lower jaw until the prey is far enough back to be helped down by body undulations. == See also == Dysphagia Occlusion Speech and language pathology == References == == External links == Nosek, Thomas M. "Section 6/6ch3/s6ch3_15". Essentials of Human Physiology. Archived from the original on 2016-03-24. Overview at nature.com Anatomy and physiology of swallowing at dysphagia.com Swallowing animation (flash) at hopkins-gi.org [Article on French Wikipedia] See : "déglutition atypique" = unfunctional or pathological swallowing. Normal Swallowing and Dysphagia: Pediatric Population
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"page_id": 196983,
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Nuclear Science and Techniques is a monthly peer-reviewed, scientific journal that is published by Science Press and Springer. This journal was established in 1990. The editor-in-chief is Yu-Gang Ma. The journal covers all theoretical and experimental aspects of nuclear physics and technology, including synchrotron radiation applications, beam line technology, accelerator, ray technology and applications, nuclear chemistry, radiochemistry, and radiopharmaceuticals and nuclear medicine, nuclear electronics and instrumentation, nuclear energy science and engineering. == Abstracing and indexing == The journal is indexed in the Science Citation Index Expanded. According to the Journal Citation Reports, the journal has a 2017 impact factor of 1.085, ranking it 18th out of 33 journals in the category "Nuclear Science and Technology" and 18th out of 20 journals in the category "Physics, Nuclear". == References == == External links == Official website
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"page_id": 40173942,
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"title": "Nuclear Science and Techniques"
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In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics. == Types == There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations. However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations, arising from the definitions of kinematic quantities: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). A differential equation of motion, usually identified as some physical law (for example, F = ma), and applying definitions of physical quantities, is used to set up an equation to solve a kinematics problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a set of solutions. A particular solution can be obtained by setting the
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"page_id": 65913,
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"title": "Equations of motion"
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initial values, which fixes the values of the constants. Stated formally, in general, an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr/dt), and its acceleration (the second derivative of r, a = d2r/dt2), and time t. Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in r is a second-order ordinary differential equation (ODE) in r, M [ r ( t ) , r ˙ ( t ) , r ¨ ( t ) , t ] = 0 , {\displaystyle M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,} where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t = 0, r ( 0 ) , r ˙ ( 0 ) . {\displaystyle \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,.} The solution r(t) to the equation of motion, with specified initial values, describes the system for all times t after t = 0. Other dynamical variables like the momentum p of the object, or quantities derived from r and p like angular momentum, can be used in place of r as the quantity to solve for from some equation of motion, although the position of the object at time t is by far the most sought-after quantity. Sometimes, the equation will be linear and is more likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions. == History == Kinematics, dynamics and the mathematical models of
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{
"page_id": 65913,
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"title": "Equations of motion"
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the universe developed incrementally over three millennia, thanks to many thinkers, only some of whose names we know. In antiquity, priests, astrologers and astronomers predicted solar and lunar eclipses, the solstices and the equinoxes of the Sun and the period of the Moon. But they had nothing other than a set of algorithms to guide them. Equations of motion were not written down for another thousand years. Medieval scholars in the thirteenth century — for example at the relatively new universities in Oxford and Paris — drew on ancient mathematicians (Euclid and Archimedes) and philosophers (Aristotle) to develop a new body of knowledge, now called physics. At Oxford, Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, who were of similar stature to the intellectuals at the University of Paris. Thomas Bradwardine extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time. Nicholas Oresme further extended Bradwardine's arguments. The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion. For writers on kinematics before Galileo, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, greater velocity as a result of greater elevation. Only Domingo de Soto, a Spanish theologian, in his commentary on Aristotle's Physics published in 1545, after defining "uniform difform" motion (which is uniformly accelerated motion) – the word velocity was not used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies
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and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. De Soto's comments are remarkably correct regarding the definitions of acceleration (acceleration was a rate of change of motion (velocity) in time) and the observation that acceleration would be negative during ascent. Discourses such as these spread throughout Europe, shaping the work of Galileo Galilei and others, and helped in laying the foundation of kinematics. Galileo deduced the equation s = 1/2gt2 in his work geometrically, using the Merton rule, now known as a special case of one of the equations of kinematics. Galileo was the first to show that the path of a projectile is a parabola. Galileo had an understanding of centrifugal force and gave a correct definition of momentum. This emphasis of momentum as a fundamental quantity in dynamics is of prime importance. He measured momentum by the product of velocity and weight; mass is a later concept, developed by Huygens and Newton. In the swinging of a simple pendulum, Galileo says in Discourses that "every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc." His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion. He did not generalize and make them applicable to bodies not subject to the earth's gravitation. That step was Newton's contribution. The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.) Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics.
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"page_id": 65913,
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He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope. Galileo also was interested by the laws of the pendulum, his first observations of which were as a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum. More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum. Thus we arrive at René Descartes, Isaac Newton, Gottfried Leibniz, et al.; and the evolved forms of the equations of motion that begin to be recognized as the modern ones. Later the equations of motion also appeared in electrodynamics, when describing the motion of charged particles in electric and magnetic fields, the Lorentz force is the general equation which serves as the definition of what is meant by an electric field and magnetic field. With the advent of special relativity and general relativity, the theoretical modifications to spacetime meant the classical equations of motion were also modified to account for the finite speed of light, and curvature of spacetime. In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations. However, the equations of quantum mechanics can also be considered "equations of motion", since they are differential equations of the wavefunction, which describes how a quantum state behaves analogously using the space and time coordinates of the
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particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields. == Kinematic equations for one particle == === Kinematic quantities === From the instantaneous position r = r(t), instantaneous meaning at an instant value of time t, the instantaneous velocity v = v(t) and acceleration a = a(t) have the general, coordinate-independent definitions; v = d r d t , a = d v d t = d 2 r d t 2 {\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}} Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature. The rotational analogues are the "angular vector" (angle the particle rotates about some axis) θ = θ(t), angular velocity ω = ω(t), and angular acceleration α = α(t): θ = θ n ^ , ω = d θ d t , α = d ω d t , {\displaystyle {\boldsymbol {\theta }}=\theta {\hat {\mathbf {n} }}\,,\quad {\boldsymbol {\omega }}={\frac {d{\boldsymbol {\theta }}}{dt}}\,,\quad {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}\,,} where n̂ is a unit vector in the direction of the axis of rotation, and θ is the angle the object turns through about the axis. The following relation holds for a point-like particle, orbiting about some axis with angular velocity ω: v = ω × r {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} } where r is the position vector of the particle (radial from the
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rotation axis) and v the tangential velocity of the particle. For a rotating continuum rigid body, these relations hold for each point in the rigid body. === Uniform acceleration === The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below. ==== Constant translational acceleration in a straight line ==== These equations apply to a particle moving linearly, in three dimensions in a straight line with constant acceleration. Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. v = v 0 + a t [ 1 ] r = r 0 + v 0 t + 1 2 a t 2 [ 2 ] r = r 0 + 1 2 ( v + v 0 ) t [ 3 ] v 2 = v 0 2 + 2 a ( r − r 0 ) [ 4 ] r = r 0 + v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}v&=v_{0}+at&[1]\\r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}&[2]\\r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t&[3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)&[4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}&[5]\\\end{aligned}}} where: r0 is the particle's initial position r is the particle's final position v0 is the particle's initial velocity v is the particle's final velocity a is the particle's acceleration t is the time interval Here a is constant acceleration, or in the case of bodies moving under the influence of gravity, the standard gravity g is used. Note that each of the equations contains four of the five variables, so in this situation it
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is sufficient to know three out of the five variables to calculate the remaining two. In some programs, such as the IGCSE Physics and IB DP Physics programs (international programs but especially popular in the UK and Europe), the same formulae would be written with a different set of preferred variables. There u replaces v0 and s replaces r - r0. They are often referred to as the SUVAT equations, where "SUVAT" is an acronym from the variables: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. In these variables, the equations of motion would be written v = u + a t [ 1 ] s = u t + 1 2 a t 2 [ 2 ] s = 1 2 ( u + v ) t [ 3 ] v 2 = u 2 + 2 a s [ 4 ] s = v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}v&=u+at&[1]\\s&=ut+{\tfrac {1}{2}}at^{2}&[2]\\s&={\tfrac {1}{2}}(u+v)t&[3]\\v^{2}&=u^{2}+2as&[4]\\s&=vt-{\tfrac {1}{2}}at^{2}&[5]\\\end{aligned}}} ==== Constant linear acceleration in any direction ==== The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion take an almost identical form. The only difference is that the square magnitudes of the velocities require the dot product. The derivations are essentially the same as in the collinear case, v = a t + v 0 [ 1 ] r = r 0 + v 0 t + 1 2 a t 2 [ 2 ] r = r 0 + 1 2 ( v + v 0 ) t [ 3 ] v 2 = v 0 2 + 2 a ⋅ ( r − r 0 ) [ 4 ] r = r 0 + v t − 1 2 a
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t 2 [ 5 ] {\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {a} t+\mathbf {v} _{0}&[1]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}&[2]\\\mathbf {r} &=\mathbf {r} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} +\mathbf {v} _{0}\right)t&[3]\\\mathbf {v} ^{2}&=\mathbf {v} _{0}^{2}+2\mathbf {a} \cdot \left(\mathbf {r} -\mathbf {r} _{0}\right)&[4]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} t-{\tfrac {1}{2}}\mathbf {a} t^{2}&[5]\\\end{aligned}}} although the Torricelli equation [4] can be derived using the distributive property of the dot product as follows: v 2 = v ⋅ v = ( v 0 + a t ) ⋅ ( v 0 + a t ) = v 0 2 + 2 t ( a ⋅ v 0 ) + a 2 t 2 {\displaystyle v^{2}=\mathbf {v} \cdot \mathbf {v} =(\mathbf {v} _{0}+\mathbf {a} t)\cdot (\mathbf {v} _{0}+\mathbf {a} t)=v_{0}^{2}+2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}} ( 2 a ) ⋅ ( r − r 0 ) = ( 2 a ) ⋅ ( v 0 t + 1 2 a t 2 ) = 2 t ( a ⋅ v 0 ) + a 2 t 2 = v 2 − v 0 2 {\displaystyle (2\mathbf {a} )\cdot (\mathbf {r} -\mathbf {r} _{0})=(2\mathbf {a} )\cdot \left(\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\right)=2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}=v^{2}-v_{0}^{2}} ∴ v 2 = v 0 2 + 2 ( a ⋅ ( r − r 0 ) ) {\displaystyle \therefore v^{2}=v_{0}^{2}+2(\mathbf {a} \cdot (\mathbf {r} -\mathbf {r} _{0}))} ==== Applications ==== Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwards into the air. Given initial velocity u, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity g. While these quantities appear to be scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as unidirectional
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vectors. Choosing s to measure up from the ground, the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also the acceleration on the ball due to it. At the highest point, the ball will be at rest: therefore v = 0. Using equation [4] in the set above, we have: s = v 2 − u 2 − 2 g . {\displaystyle s={\frac {v^{2}-u^{2}}{-2g}}.} Substituting and cancelling minus signs gives: s = u 2 2 g . {\displaystyle s={\frac {u^{2}}{2g}}.} ==== Constant circular acceleration ==== The analogues of the above equations can be written for rotation. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary, ω = ω 0 + α t θ = θ 0 + ω 0 t + 1 2 α t 2 θ = θ 0 + 1 2 ( ω 0 + ω ) t ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) θ = θ 0 + ω t − 1 2 α t 2 {\displaystyle {\begin{aligned}\omega &=\omega _{0}+\alpha t\\\theta &=\theta _{0}+\omega _{0}t+{\tfrac {1}{2}}\alpha t^{2}\\\theta &=\theta _{0}+{\tfrac {1}{2}}(\omega _{0}+\omega )t\\\omega ^{2}&=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\\\theta &=\theta _{0}+\omega t-{\tfrac {1}{2}}\alpha t^{2}\\\end{aligned}}} where α is the constant angular acceleration, ω is the angular velocity, ω0 is the initial angular velocity, θ is the angle turned through (angular displacement), θ0 is the initial angle, and t is the time taken to rotate from the initial state to the final state. === General planar motion === These are the kinematic equations for a particle traversing a path in a plane, described by position r = r(t). They are simply the time derivatives of the position vector in plane polar coordinates
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using the definitions of physical quantities above for angular velocity ω and angular acceleration α. These are instantaneous quantities which change with time. The position of the particle is r = r ( r ( t ) , θ ( t ) ) = r e ^ r {\displaystyle \mathbf {r} =\mathbf {r} \left(r(t),\theta (t)\right)=r\mathbf {\hat {e}} _{r}} where êr and êθ are the polar unit vectors. Differentiating with respect to time gives the velocity v = e ^ r d r d t + r ω e ^ θ {\displaystyle \mathbf {v} =\mathbf {\hat {e}} _{r}{\frac {dr}{dt}}+r\omega \mathbf {\hat {e}} _{\theta }} with radial component dr/dt and an additional component rω due to the rotation. Differentiating with respect to time again obtains the acceleration a = ( d 2 r d t 2 − r ω 2 ) e ^ r + ( r α + 2 ω d r d t ) e ^ θ {\displaystyle \mathbf {a} =\left({\frac {d^{2}r}{dt^{2}}}-r\omega ^{2}\right)\mathbf {\hat {e}} _{r}+\left(r\alpha +2\omega {\frac {dr}{dt}}\right)\mathbf {\hat {e}} _{\theta }} which breaks into the radial acceleration d2r/dt2, centripetal acceleration –rω2, Coriolis acceleration 2ωdr/dt, and angular acceleration rα. Special cases of motion described by these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration. === General 3D motions === In 3D space, the equations in spherical coordinates (r, θ, φ) with corresponding unit vectors êr, êθ and êφ, the position, velocity, and acceleration generalize respectively to r = r ( t ) = r e ^ r v = v e ^ r + r d θ d t e ^ θ + r d φ d t sin
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θ e ^ φ a = ( a − r ( d θ d t ) 2 − r ( d φ d t ) 2 sin 2 θ ) e ^ r + ( r d 2 θ d t 2 + 2 v d θ d t − r ( d φ d t ) 2 sin θ cos θ ) e ^ θ + ( r d 2 φ d t 2 sin θ + 2 v d φ d t sin θ + 2 r d θ d t d φ d t cos θ ) e ^ φ {\displaystyle {\begin{aligned}\mathbf {r} &=\mathbf {r} \left(t\right)=r\mathbf {\hat {e}} _{r}\\\mathbf {v} &=v\mathbf {\hat {e}} _{r}+r\,{\frac {d\theta }{dt}}\mathbf {\hat {e}} _{\theta }+r\,{\frac {d\varphi }{dt}}\,\sin \theta \mathbf {\hat {e}} _{\varphi }\\\mathbf {a} &=\left(a-r\left({\frac {d\theta }{dt}}\right)^{2}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin ^{2}\theta \right)\mathbf {\hat {e}} _{r}\\&+\left(r{\frac {d^{2}\theta }{dt^{2}}}+2v{\frac {d\theta }{dt}}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin \theta \cos \theta \right)\mathbf {\hat {e}} _{\theta }\\&+\left(r{\frac {d^{2}\varphi }{dt^{2}}}\,\sin \theta +2v\,{\frac {d\varphi }{dt}}\,\sin \theta +2r\,{\frac {d\theta }{dt}}\,{\frac {d\varphi }{dt}}\,\cos \theta \right)\mathbf {\hat {e}} _{\varphi }\end{aligned}}\,\!} In the case of a constant φ this reduces to the planar equations above. == Dynamic equations of motion == === Newtonian mechanics === The first general equation of motion developed was Newton's second law of motion. In its most general form it states the rate of change of momentum p = p(t) = mv(t) of an object equals the force F = F(x(t), v(t), t) acting on it,: 1112 F = d p d t {\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}} The force in the equation is not the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } since
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m is a constant in Newtonian mechanics. Newton's second law applies to point-like particles, and to all points in a rigid body. They also apply to each point in a mass continuum, like deformable solids or fluids, but the motion of the system must be accounted for; see material derivative. In the case the mass is not constant, it is not sufficient to use the product rule for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with conservation of momentum; see variable-mass system. It may be simple to write down the equations of motion in vector form using Newton's laws of motion, but the components may vary in complicated ways with spatial coordinates and time, and solving them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex. The momentum form is preferable since this is readily generalized to more complex systems, such as special and general relativity (see four-momentum).: 112 It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces. For a number of particles (see many body problem), the equation of motion for one particle i influenced by other particles is d p i d t = F E + ∑ i
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≠ j F i j {\displaystyle {\frac {d\mathbf {p} _{i}}{dt}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}} where pi is the momentum of particle i, Fij is the force on particle i by particle j, and FE is the resultant external force due to any agent not part of system. Particle i does not exert a force on itself. Euler's laws of motion are similar to Newton's laws, but they are applied specifically to the motion of rigid bodies. The Newton–Euler equations combine the forces and torques acting on a rigid body into a single equation. Newton's second law for rotation takes a similar form to the translational case, τ = d L d t , {\displaystyle {\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}\,,} by equating the torque acting on the body to the rate of change of its angular momentum L. Analogous to mass times acceleration, the moment of inertia tensor I depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity, τ = I α . {\displaystyle {\boldsymbol {\tau }}=\mathbf {I} {\boldsymbol {\alpha }}.} Again, these equations apply to point like particles, or at each point of a rigid body. Likewise, for a number of particles, the equation of motion for one particle i is d L i d t = τ E + ∑ i ≠ j τ i j , {\displaystyle {\frac {d\mathbf {L} _{i}}{dt}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,,} where Li is the angular momentum of particle i, τij the torque on particle i by particle j, and τE is resultant external torque (due to any agent not part of system). Particle i does not exert a torque on itself. === Applications === Some examples of Newton's law include describing the motion of
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a simple pendulum, − m g sin θ = m d 2 ( ℓ θ ) d t 2 ⇒ d 2 θ d t 2 = − g ℓ sin θ , {\displaystyle -mg\sin \theta =m{\frac {d^{2}(\ell \theta )}{dt^{2}}}\quad \Rightarrow \quad {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{\ell }}\sin \theta \,,} and a damped, sinusoidally driven harmonic oscillator, F 0 sin ( ω t ) = m ( d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x ) . {\displaystyle F_{0}\sin(\omega t)=m\left({\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}x\right)\,.} For describing the motion of masses due to gravity, Newton's law of gravity can be combined with Newton's second law. For two examples, a ball of mass m thrown in the air, in air currents (such as wind) described by a vector field of resistive forces R = R(r, t), − G m M | r | 2 e ^ r + R = m d 2 r d t 2 + 0 ⇒ d 2 r d t 2 = − G M | r | 2 e ^ r + A {\displaystyle -{\frac {GmM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {R} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}+0\quad \Rightarrow \quad {\frac {d^{2}\mathbf {r} }{dt^{2}}}=-{\frac {GM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {A} } where G is the gravitational constant, M the mass of the Earth, and A = R/m is the acceleration of the projectile due to the air currents at position r and time t. The classical N-body problem for N particles each interacting with each other due to gravity is a set of N nonlinear coupled second order ODEs, d 2 r i d t 2 = G ∑ i ≠ j m j | r j − r i | 3
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( r j − r i ) {\displaystyle {\frac {d^{2}\mathbf {r} _{i}}{dt^{2}}}=G\sum _{i\neq j}{\frac {m_{j}}{|\mathbf {r} _{j}-\mathbf {r} _{i}|^{3}}}(\mathbf {r} _{j}-\mathbf {r} _{i})} where i = 1, 2, ..., N labels the quantities (mass, position, etc.) associated with each particle. == Analytical mechanics == Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system has N degrees of freedom, then one can use a set of N generalized coordinates q(t) = [q1(t), q2(t) ... qN(t)], to define the configuration of the system. They can be in the form of arc lengths or angles. They are a considerable simplification to describe motion, since they take advantage of the intrinsic constraints that limit the system's motion, and the number of coordinates is reduced to a minimum. The time derivatives of the generalized coordinates are the generalized velocities q ˙ = d q d t . {\displaystyle \mathbf {\dot {q}} ={\frac {d\mathbf {q} }{dt}}\,.} The Euler–Lagrange equations are d d t ( ∂ L ∂ q ˙ ) = ∂ L ∂ q , {\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,} where the Lagrangian is a function of the configuration q and its time rate of change dq/dt (and possibly time t) L = L [ q ( t ) , q ˙ ( t ) , t ] . {\displaystyle L=L\left[\mathbf {q} (t),\mathbf {\dot {q}} (t),t\right]\,.} Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled N second order ODEs in the coordinates are obtained. Hamilton's equations are p ˙ = − ∂ H ∂ q , q ˙ = + ∂ H ∂ p , {\displaystyle \mathbf {\dot {p}} =-{\frac {\partial H}{\partial
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\mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,} where the Hamiltonian H = H [ q ( t ) , p ( t ) , t ] , {\displaystyle H=H\left[\mathbf {q} (t),\mathbf {p} (t),t\right]\,,} is a function of the configuration q and conjugate "generalized" momenta p = ∂ L ∂ q ˙ , {\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}\,,} in which ∂/∂q = (∂/∂q1, ∂/∂q2, …, ∂/∂qN) is a shorthand notation for a vector of partial derivatives with respect to the indicated variables (see for example matrix calculus for this denominator notation), and possibly time t, Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled 2N first order ODEs in the coordinates qi and momenta pi are obtained. The Hamilton–Jacobi equation is − ∂ S ( q , t ) ∂ t = H ( q , p , t ) . {\displaystyle -{\frac {\partial S(\mathbf {q} ,t)}{\partial t}}=H\left(\mathbf {q} ,\mathbf {p} ,t\right)\,.} where S [ q , t ] = ∫ t 1 t 2 L ( q , q ˙ , t ) d t , {\displaystyle S[\mathbf {q} ,t]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,dt\,,} is Hamilton's principal function, also called the classical action is a functional of L. In this case, the momenta are given by p = ∂ S ∂ q . {\displaystyle \mathbf {p} ={\frac {\partial S}{\partial \mathbf {q} }}\,.} Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order non-linear PDE, in N + 1 variables. The action S allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any differentiable symmetry of
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the action of a physical system has a corresponding conservation law, a theorem due to Emmy Noether. All classical equations of motion can be derived from the variational principle known as Hamilton's principle of least action δ S = 0 , {\displaystyle \delta S=0\,,} stating the path the system takes through the configuration space is the one with the least action S. == Electrodynamics == In electrodynamics, the force on a charged particle of charge q is the Lorentz force: F = q ( E + v × B ) {\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)} Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle: m d 2 r d t 2 = q ( E + d r d t × B ) {\displaystyle m{\frac {d^{2}\mathbf {r} }{dt^{2}}}=q\left(\mathbf {E} +{\frac {d\mathbf {r} }{dt}}\times \mathbf {B} \right)} or its momentum: d p d t = q ( E + p × B m ) {\displaystyle {\frac {d\mathbf {p} }{dt}}=q\left(\mathbf {E} +{\frac {\mathbf {p} \times \mathbf {B} }{m}}\right)} The same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass m and charge q: L = 1 2 m r ˙ ⋅ r ˙ + q A ⋅ r ˙ − q ϕ {\displaystyle L={\tfrac {1}{2}}m\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot {\dot {\mathbf {r} }}-q\phi } where A and ϕ are the electromagnetic scalar and vector potential fields. The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by: P = ∂ L ∂ r ˙ = m r ˙ + q A {\displaystyle \mathbf {P} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} } instead
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of just mv, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation. Alternatively the Hamiltonian (and substituting into the equations): H = ( P − q A ) 2 2 m + q ϕ {\displaystyle H={\frac {\left(\mathbf {P} -q\mathbf {A} \right)^{2}}{2m}}+q\phi } can derive the Lorentz force equation. == General relativity == === Geodesic equation of motion === The above equations are valid in flat spacetime. In curved spacetime, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a geodesic of the curved spacetime (the shortest length of curve between two points). For curved manifolds with a metric tensor g, the metric provides the notion of arc length (see line element for details). The differential arc length is given by:: 1199 d s = g α β d x α d x β {\displaystyle ds={\sqrt {g_{\alpha \beta }dx^{\alpha }dx^{\beta }}}} and the geodesic equation is a second-order differential equation in the coordinates. The general solution is a family of geodesics:: 1200 d 2 x μ d s 2 = − Γ μ α β d x α d s d x β d s {\displaystyle {\frac {d^{2}x^{\mu }}{ds^{2}}}=-\Gamma ^{\mu }{}_{\alpha \beta }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}} where Γ μαβ is a Christoffel symbol of the second kind, which contains the metric (with respect to the coordinate system). Given the mass-energy distribution provided by the stress–energy tensor T αβ, the Einstein field equations are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of spacetime is equivalent to a gravitational field (see equivalence principle). Mass falling in curved spacetime is equivalent to a mass falling in a
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gravitational field - because gravity is a fictitious force. The relative acceleration of one geodesic to another in curved spacetime is given by the geodesic deviation equation: D 2 ξ α d s 2 = − R α β γ δ d x α d s ξ γ d x δ d s {\displaystyle {\frac {D^{2}\xi ^{\alpha }}{ds^{2}}}=-R^{\alpha }{}_{\beta \gamma \delta }{\frac {dx^{\alpha }}{ds}}\xi ^{\gamma }{\frac {dx^{\delta }}{ds}}} where ξα = x2α − x1α is the separation vector between two geodesics, D/ds (not just d/ds) is the covariant derivative, and Rαβγδ is the Riemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field.: 34–35 For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to Newton's law of gravity. === Spinning objects === In general relativity, rotational motion is described by the relativistic angular momentum tensor, including the spin tensor, which enter the equations of motion under covariant derivatives with respect to proper time. The Mathisson–Papapetrou–Dixon equations describe the motion of spinning objects moving in a gravitational field. == Analogues for waves and fields == Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of waves and fields are always partial differential equations, since the waves or fields are functions of space and time. For a particular solution, boundary conditions along with initial conditions need to be specified. Sometimes in the following contexts, the wave or field equations are also called "equations of motion". === Field
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equations === Equations that describe the spatial dependence and time evolution of fields are called field equations. These include Maxwell's equations for the electromagnetic field, Poisson's equation for Newtonian gravitational or electrostatic field potentials, the Einstein field equation for gravitation (Newton's law of gravity is a special case for weak gravitational fields and low velocities of particles). This terminology is not universal: for example although the Navier–Stokes equations govern the velocity field of a fluid, they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead. === Wave equations === Equations of wave motion are called wave equations. The solutions to a wave equation give the time-evolution and spatial dependence of the amplitude. Boundary conditions determine if the solutions describe traveling waves or standing waves. From classical equations of motion and field equations; mechanical, gravitational wave, and electromagnetic wave equations can be derived. The general linear wave equation in 3D is: 1 v 2 ∂ 2 X ∂ t 2 = ∇ 2 X {\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}X}{\partial t^{2}}}=\nabla ^{2}X} where X = X(r, t) is any mechanical or electromagnetic field amplitude, say: the transverse or longitudinal displacement of a vibrating rod, wire, cable, membrane etc., the fluctuating pressure of a medium, sound pressure, the electric fields E or D, or the magnetic fields B or H, the voltage V or current I in an alternating current circuit, and v is the phase velocity. Nonlinear equations model the dependence of phase velocity on amplitude, replacing v by v(X). There are other linear and nonlinear wave equations for very specific applications, see for example the Korteweg–de Vries equation. === Quantum theory === In quantum theory, the wave and field concepts both appear. In quantum mechanics the
|
{
"page_id": 65913,
"source": null,
"title": "Equations of motion"
}
|
analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the Schrödinger equation in its most general form: i ℏ ∂ Ψ ∂ t = H ^ Ψ , {\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi \,,} where Ψ is the wavefunction of the system, Ĥ is the quantum Hamiltonian operator, rather than a function as in classical mechanics, and ħ is the Planck constant divided by 2π. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the correspondence principle, in the limit that ħ becomes zero. To compare to measurements, operators for observables must be applied the quantum wavefunction according to the experiment performed, leading to either wave-like or particle-like results. Throughout all aspects of quantum theory, relativistic or non-relativistic, there are various formulations alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance: the Heisenberg equation of motion resembles the time evolution of classical observables as functions of position, momentum, and time, if one replaces dynamical observables by their quantum operators and the classical Poisson bracket by the commutator, the phase space formulation closely follows classical Hamiltonian mechanics, placing position and momentum on equal footing, the Feynman path integral formulation extends the principle of least action to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians. == See also == == References ==
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{
"page_id": 65913,
"source": null,
"title": "Equations of motion"
}
|
In vitro compartmentalization (IVC) is an emulsion-based technology that generates cell-like compartments in vitro. These compartments are designed such that each contains no more than one gene. When the gene is transcribed and/or translated, its products (RNAs and/or proteins) become 'trapped' with the encoding gene inside the compartment. By coupling the genotype (DNA) and phenotype (RNA, protein), compartmentalization allows the selection and evolution of phenotype. == History == In vitro compartmentalization method was first developed by Dan Tawfik and Andrew Griffiths. Based on the idea that Darwinian evolution relies on the linkage of genotype to phenotype, Tawfik and Griffiths designed aqueous compartments of water-in-oil (w/o) emulsions to mimic cellular compartments that can link genotype and phenotype. Emulsions of cell-like compartments were formed by adding in vitro transcription/translation reaction mixture to stirred mineral oil containing surfactants. The mean droplet diameter was measured to be 2.6 μm by laser diffraction. As a proof of concept, Tawfik and Griffiths designed a selection experiment using a pool of DNA sequences, including the gene encoding HaeIII DNA methyltransferase (M.HaeIII) in the presence of 107-fold excess of genes encoding a different enzyme folA. The 3’ of each DNA sequences was purposely designed to contain a HaeIII recognition site which, in the presence of expressed methyltransferase, would be methylated and, thus, resistant to restriction enzyme digestion. By selecting for DNA sequences that survive the endonuclease digestion, Tawfik and Griffiths found that the M.HaeIII genes were enriched by at least 1000-fold over the folA genes within the first round of selection. == Method == === Emulsion technology === Water-in-oil (w/o) emulsions are created by mixing aqueous and oil phases with the help of surfactants. A typical IVC emulsion is formed by first generating oil-surfactant mixture by stirring, and then gradually adding the aqueous phase to the oil-surfactant mixture.
|
{
"page_id": 15991162,
"source": null,
"title": "In vitro compartmentalization"
}
|
For stable emulsion formation, a mixture of HLB (hydrophile-lipophile balance) and low HLB surfactants are needed. Some combinations of surfactants used to generate oil-surfactant mixture are mineral oil / 0.5% Tween 80 / 4.5% Span 80 / sodium deoxycholate and a more heat stable version, light mineral oil / 0.4% Tween 80 / 4.5% Span 80 / 0.05% Triton X-100. The aqueous phase containing transcription and/or translation components is slowly added to the oil surfactants, and the formation of w/o is facilitated by homogenizing, stirring or using hand extruding device. The emulsion quality can be determined by light microscopy and/or dynamic light scattering techniques. The emulsion is quite diverse, and greater homogenization speeds helps to produce smaller droplets with narrower size distribution. However, homogenization speeds has to be controlled, since speed over 13,500 r.p.m tends to result in a significant loss of enzyme activity on the level of transcription. The most widely used emulsion formation gives droplets with a mean diameter of 2-3μm, and an average volume of ~5 femtoliters, or 1010 aqueous droplet per ml of emulsions. The ratio of genes to droplets is designed such that most of the droplets contains no more than a single gene statistically. === In vitro transcription/translation === IVC enables the miniaturization of large-scale techniques that can now be done on the micro scale including coupled in vitro transcription and translation (IVTT) experiments. Streamlining and integrating transcription and translation allows for fast and highly controllable experimental designs. IVTT can be done both in bulk emulsions and in microdroplets by utilizing droplet-based microfluidics. Microdroplets, droplets on the scale of pico to femtoliters, have been successfully used as single DNA molecule vessels. This droplet technology allows high throughput analysis with many different selection pressures in a single experimental setup. IVTT in microdroplets is preferred when
|
{
"page_id": 15991162,
"source": null,
"title": "In vitro compartmentalization"
}
|
overexpression of a desired protein would be toxic to a host cell minimizing the utility of the transcription and translation mechanisms. IVC has used bacterial cell, wheat germ and rabbit reticulocyte (RRL) extracts for transcription and translation. It is also possible to use bacterial reconstituted translation system such as PURE in which translation components are individually purified and later combined. When expressing eukaryote or complex proteins, it is desirable to use eukaryotic translation systems such as wheat germ extract or more superior alternative, RRL extract. In order to use RRL for transcription and translation, traditional emulsion formulation cannot be used as it abolishes translation. Instead, a novel emulsion formulation: 4% Abil EM90 / light mineral oil was developed and demonstrated to be functional in expressing luciferase and human telomerase. === Breaking emulsion and coupling of genotype and phenotype === Once transcription and/or translation has completed in the droplets, emulsion will be broken by successive steps of removing mineral oil and surfactants to allow for subsequent selection. At this stage, it is crucial to have a method to ‘track’ each gene products to the encoding gene as they become free floating in a heterogeneous population of molecules. There are three major approaches to track down each phenotype to its genotype. The first method is to attach each DNA molecule with a biotin group and an additional coding sequence for streptavidin (STABLE display). All the newly formed proteins/peptides will be in fusion with streptavidin molecules and bind to their biotinylated coding sequence. An improved version attached two biotin molecules to the ends of a DNA molecule to increase the avidity between DNA molecule and streptavidin-fused peptides, and used a low GC content synthetic streptavidin gene to increase efficiency and specificity during PCR amplification. The second method is to covalently link DNA and
|
{
"page_id": 15991162,
"source": null,
"title": "In vitro compartmentalization"
}
|
protein. Two strategies have been demonstrated. The first is to form M.HaeIII fusion proteins. Each expressed protein/polypeptide will be in fusion with Hae III DNA methyltransferase domain, which is able to bind covalently to DNA fragments containing the sequence 5′-GGC*-3′, where C* is 5-fluoro-2 deoxycytidine. The second strategy is to use monomeric mutant of VirD2 enzyme. When a protein/peptide is expressed in fusion with Agrobacterium protein VirD2, it will bind to its DNA coding sequence that has a single-stranded overhang comprising VirD2 T-border recognition sequences. The third method is to link phenotype and genotype via beads. The beads used will be coated with streptavidin to allow for the binding of biotinylated DNA, in addition, the beads will also display cognate binding partner to the affinity tag that will be expressed in fusion with the protein/peptide. == Selection == Depending on the phenotype to be selected, difference selection strategies will be used. Selection strategy can be divided into three major categories: selection for binding, selection for catalysis and selection for regulation. The phenotype to be selected can range from RNA to peptide to protein. By selecting for binding, the most commonly evolved phenotypes are peptide/proteins that have selective affinity to a specific antibody or DNA molecule. An example is the selection of proteins that have affinity to zinc finger DNA by Sepp et al. By selecting for catalytic proteins/RNAs, new variants with novel or improved enzymatic property are usually isolated. For example, new ribozyme variants with trans-ligase activity were selected and exhibited multiple turnovers. By selecting for regulation, inhibitors of DNA nucleases can be selected, such as protein inhibitors of the Colicin E7 DNase. == Advantages == Comparing to other in vitro display technologies, IVC has two major advantages. The first advantage is its ability to control reactions within the droplets.
|
{
"page_id": 15991162,
"source": null,
"title": "In vitro compartmentalization"
}
|
Hydrophobic and hydrophilic components can be delivered to each droplet in a step-wise fashion without compromising the chemical integrity of the droplet, and thus by controlling what to be added and when to be added, the reaction in each droplet is controlled. In addition, depending on the nature of the reaction to be carried out, the pH of each droplet can also be changed. More recently, photocaged substrates were used and their participation in a reaction was regulated by photo-activation. The second advantage is that IVC allows the selection of catalytic molecules. As an example, Griffiths et al. was able to select for phosphotriesterase variants with higher Kcat by detecting product formation and amount using anti-product antibody and flow cytometry respectively. == Related technologies == CIS display Phage display Bacterial display Yeast display Ribosome display mRNA display == References ==
|
{
"page_id": 15991162,
"source": null,
"title": "In vitro compartmentalization"
}
|
Integrative and conjugative elements (ICEs) are mobile genetic elements present in both Gram-positive and Gram-negative bacteria. In a donor cell, ICEs are located primarily on the chromosome, but have the ability to excise themselves from the genome and transfer to recipient cells via bacterial conjugation. Due to their physical association with chromosomes, identifying integrative and conjugative elements has proven challenging, but in silico analysis of bacterial genomes indicate these elements are widespread among many microorganisms. ICEs have been detected in Pseudomonadota (e.g., Pseudomonas spp., Aeromonas spp., E. coli, Haemophilus spp.), Actinomycetota and Bacillota. Among many other virulence determinants, ICEs may spread antibiotic and metal ion resistance genes across prokaryotic phyla. In addition, ICE elements may also facilitate the mobilisation of other DNA modules such as genomic islands. == Characteristics == Although ICEs exhibit various mechanisms promoting their integration, transfer and regulation, they share many common characteristics. ICEs comprise all mobile genetic elements with self-replication, integration, and conjugation abilities, including conjugative transposons, regardless of the particular conjugation and integration mechanism by which they act. Some immobile genomic pathogenicity islands are also believed to be defective ICEs that have lost their ability to conjugate. ICEs combine certain features of the following mobile genetic elements: Bacteriophages that have the ability to insert into and excise from bacterial chromosomes. Transposons that, besides their inherent transposable activity, can additionally be subject to horizontal gene transfer via conjugation. Conjugative plasmids that transfer from donor to recipient bacteria via conjugation. In contrast to plasmids and phages, integrative and conjugative elements cannot remain in an extrachromosomal form in the cytoplasm of bacterial cells and replicate only with the chromosome they reside in. ICEs possess the structure organized into three gene modules that are responsible for their integration with the chromosome, excision from the genome and conjugation, as well
|
{
"page_id": 66978170,
"source": null,
"title": "Integrative and conjugative element"
}
|
as regulatory genes. All integrative and conjugative elements encode integrases that are essential for controlling the excision, transfer and integration of an ICE. The representative example of ICE integrases is the integrase encoded by lambda phage. The transfer of an integrated ICE element from the donor to recipient bacterium must be preceded by its excision from the chromosome that is co-promoted by small DNA-binding proteins, the so-called recombination directionality factors. The dynamics of the integration and excision processes are specific to each integrative and conjugative element. == References ==
|
{
"page_id": 66978170,
"source": null,
"title": "Integrative and conjugative element"
}
|
The Vg1 ribozyme is a manganese dependent RNA enzyme or ribozyme which is the smallest ribozyme to be identified. It was identified in the 3′ UTR of Xenopus Vg1 mRNA transcripts and mouse beta-actin mRNA. This ribozyme was identified from in vitro studies that showed that the Vg1 mRNA was cleaved within the 3′ UTR in the absence of protein. Studying the Vg1 mRNA 3′UTR a manganese-dependent ribozyme was predicted to exist. This ribozyme was shown to be located adjacent to the polyadenylation site and in vitro studies showed that it catalyzes a first-order reaction where its mechanism of cleavage is similar to the manganese ribozyme present in Tetrahymena group I introns. In vivo studies showed that this ribozyme is not functional with the cell. == References == == External links == Page for Manganese dependent ribozyme in Vg1 mRNA at Rfam
|
{
"page_id": 24117627,
"source": null,
"title": "Vg1 ribozyme"
}
|
Since the first printing of Carl Linnaeus's Species Plantarum in 1753, plants have been assigned one epithet or name for their species and one name for their genus, a grouping of related species. Many of these plants are listed in Stearn's Dictionary of Plant Names for Gardeners. William Stearn (1911–2001) was one of the pre-eminent British botanists of the 20th century: a Librarian of the Royal Horticultural Society, a president of the Linnean Society and the original drafter of the International Code of Nomenclature for Cultivated Plants. The first column below contains seed-bearing genera from Stearn and other sources as listed, excluding those names that no longer appear in more modern works, such as Plants of the World by Maarten J. M. Christenhusz (lead author), Michael F. Fay and Mark W. Chase. Plants of the World is also used for the family and order classification for each genus. The second column gives a meaning or derivation of the word, such as a language of origin. The last two columns indicate additional citations. == Key == Latin: = derived from Latin (otherwise Greek, except as noted) Ba = listed in Ross Bayton's The Gardener's Botanical Bu = listed in Lotte Burkhardt's Index of Eponymic Plant Names CS = listed in both Allen Coombes's The A to Z of Plant Names and Stearn's Dictionary of Plant Names for Gardeners G = listed in David Gledhill's The Names of Plants St = listed in Stearn's Dictionary of Plant Names for Gardeners == Genera == == See also == Glossary of botanical terms List of Greek and Latin roots in English List of Latin and Greek words commonly used in systematic names List of plant genera named for people: A–C, D–J, K–P, Q–Z List of plant family names with etymologies == Notes == ==
|
{
"page_id": 65667453,
"source": null,
"title": "List of plant genus names with etymologies (A–C)"
}
|
Citations == == References == Bayton, Ross (2020). The Gardener's Botanical: An Encyclopedia of Latin Plant Names. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-20017-0. Burkhardt, Lotte (2018). Verzeichnis eponymischer Pflanzennamen – Erweiterte Edition [Index of Eponymic Plant Names – Extended Edition] (pdf) (in German). Berlin: Botanic Garden and Botanical Museum, Freie Universität Berlin. doi:10.3372/epolist2018. ISBN 978-3-946292-26-5. S2CID 187926901. Retrieved January 1, 2021. See http://creativecommons.org/licenses/by/4.0/ for license. Christenhusz, Maarten; Fay, Michael Francis; Chase, Mark Wayne (2017). Plants of the World: An Illustrated Encyclopedia of Vascular Plants. Chicago, Illinois: Kew Publishing and The University of Chicago Press. ISBN 978-0-226-52292-0. Coombes, Allen (2012). The A to Z of Plant Names: A Quick Reference Guide to 4000 Garden Plants. Portland, Oregon: Timber Press. ISBN 978-1-60469-196-2. Cullen, Katherine E. (2006). Biology: The People Behind the Science. New York, New York: Infobase Publishing. ISBN 978-0-8160-7221-7. Gledhill, David (2008). The Names of Plants. New York, New York: Cambridge University Press. ISBN 978-0-521-86645-3. The Linnean Society (August 1992). "Publications by William T. Stearn on bibliographical, botanical and horticultural subjects, 1977–1991; a chronological list". Botanical Journal of the Linnean Society. 109 (4): 443–451. doi:10.1111/j.1095-8339.1992.tb01443.x. ISSN 0024-4074. Stearn, William (2002). Stearn's Dictionary of Plant Names for Gardeners. London: Cassell. ISBN 978-0-304-36469-5. == Further reading == Brown, Roland (1956). Composition of Scientific Words. Washington, DC: Smithsonian Institution Press. ISBN 978-1-56098-848-9. {{cite book}}: ISBN / Date incompatibility (help) Lewis, Charlton (1891). An Elementary Latin Dictionary. Oxford: Oxford University Press. ISBN 978-0-19-910205-1. {{cite book}}: ISBN / Date incompatibility (help) Available online at the Perseus Digital Library. Liddell, Henry George; Scott, Robert (2013) [1888/1889]. An Intermediate Greek–English Lexicon. Mansfield Centre, Connecticut: Martino Fine Books. ISBN 978-1-61427-397-4. Available online at the Perseus Digital Library. Quattrocchi, Umberto (2000). CRC World Dictionary of Plant Names, Volume I, A–C. Boca Raton, Florida: CRC Press. ISBN 978-0-8493-2675-2.
|
{
"page_id": 65667453,
"source": null,
"title": "List of plant genus names with etymologies (A–C)"
}
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The molecular formula C5H5NS (molar mass: 111.16 g/mol, exact mass: 111.0143 u) may refer to: 2-Mercaptopyridine Thiazepines 1,2-Thiazepine 1,3-Thiazepine 1,4-Thiazepine
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{
"page_id": 23921022,
"source": null,
"title": "C5H5NS"
}
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In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems. These systems may be rectangular like cartesian, Curvilinear coordinates like polar coordinates or other systems. The object trajectories may be specified with respect to other objects which may themselve be in motion relative to a standard reference. Rotating systems may also be used. Numerous practical problems in kinematics involve constraints, such as mechanical linkages, ropes, or rolling disks. == Overview == Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. The study of how forces act on bodies falls within kinetics or dynamics (including analytical dynamics), not kinematics. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure mathematics since it can be studied without considering the mass of a body or the forces acting upon it. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. Another way to describe kinematics is as the specification of the possible states of a physical system. Dynamics then describes the evolution of a system through such states. Robert Spekkens argues that this division cannot be empirically tests and
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{
"page_id": 65914,
"source": null,
"title": "Kinematics"
}
|
thus has no physical basis. Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering, robotics, and biomechanics, kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine, a robotic arm or the human skeleton. Geometric transformations, including called rigid transformations, are used to describe the movement of components in a mechanical system, simplifying the derivation of the equations of motion. They are also central to dynamic analysis. Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design a mechanism for a desired range of motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism. Relativistic_kinematics applies the special theory of relativity to the geometry of object motion. The topics include time dilation, length contraction and the Lorentz transformation.: 12.8 The kinematics of relativity operates in a spacetime geometry where spatial points are augmented with a time coordinate to form 4-vectors.: 221 Werner Heisenberg reinterpreted classical kinetics for quantum systems in his 1925 paper "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships". Dirac noted the similarity in structure between Heisenberg's formulations and classical Poisson brackets.: 143 In a follow up paper in 1927 Heisenberg showed that classical kinematic notions like velocity and energy are valid in quantum mechanics, but pairs of conjugate kinematic and dynamic quantities cannot be simultaneously measure, a result he called indeterminacy but which became known as the uncertainty principle. == Etymology == The term kinematic is the English version of A.M. Ampère's cinématique, which
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{
"page_id": 65914,
"source": null,
"title": "Kinematics"
}
|
he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν kinein ("to move"). Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek γρᾰ́φω grapho ("to write"). == Kinematics of a particle trajectory in a non-rotating frame of reference == Particle kinematics is the study of the trajectory of particles. The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the x-axis and north is in the direction of the y-axis, then the coordinate vector to the base of the tower is r = (0 m, −50 m, 0 m). If the tower is 50 m high, and this height is measured along the z-axis, then the coordinate vector to the top of the tower is r = (0 m, −50 m, 50 m). In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to a reference frame. The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position
|
{
"page_id": 65914,
"source": null,
"title": "Kinematics"
}
|
vector r {\displaystyle {\bf {r}}} can be expressed as r = ( x , y , z ) = x x ^ + y y ^ + z z ^ , {\displaystyle \mathbf {r} =(x,y,z)=x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }},} where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} are the Cartesian coordinates and x ^ {\displaystyle {\hat {\mathbf {x} }}} , y ^ {\displaystyle {\hat {\mathbf {y} }}} and z ^ {\displaystyle {\hat {\mathbf {z} }}} are the unit vectors along the x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} coordinate axes, respectively. The magnitude of the position vector | r | {\displaystyle \left|\mathbf {r} \right|} gives the distance between the point r {\displaystyle \mathbf {r} } and the origin. | r | = x 2 + y 2 + z 2 . {\displaystyle |\mathbf {r} |={\sqrt {x^{2}+y^{2}+z^{2}}}.} The direction cosines of the position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector. The trajectory of a particle is a vector function of time, r ( t ) {\displaystyle \mathbf {r} (t)} , which defines the curve traced by the moving particle, given by r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x ( t ) {\displaystyle x(t)} , y ( t ) {\displaystyle y(t)} , and z ( t ) {\displaystyle z(t)} describe each coordinate of the particle's position as a function of time. === Velocity and speed === The velocity of
|
{
"page_id": 65914,
"source": null,
"title": "Kinematics"
}
|
a particle is a vector quantity that describes the direction as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle (displacement) by the time interval. This ratio is called the average velocity over that time interval and is defined as v ¯ = Δ r Δ t = Δ x Δ t x ^ + Δ y Δ t y ^ + Δ z Δ t z ^ = v ¯ x x ^ + v ¯ y y ^ + v ¯ z z ^ {\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta z}{\Delta t}}{\hat {\mathbf {z} }}={\bar {v}}_{x}{\hat {\mathbf {x} }}+{\bar {v}}_{y}{\hat {\mathbf {y} }}+{\bar {v}}_{z}{\hat {\mathbf {z} }}\,} where Δ r {\displaystyle \Delta \mathbf {r} } is the displacement vector during the time interval Δ t {\displaystyle \Delta t} . In the limit that the time interval Δ t {\displaystyle \Delta t} approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, v = lim Δ t → 0 Δ r Δ t = d r d t = v x x ^ + v y y ^ + v z z ^ . {\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {{\text{d}}\mathbf {r} }{{\text{d}}t}}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}.} Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is tangent to the particle's trajectory at every position along its path. In
|
{
"page_id": 65914,
"source": null,
"title": "Kinematics"
}
|
a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The speed of an object is the magnitude of its velocity. It is a scalar quantity: v = | v | = d s d t , {\displaystyle v=|\mathbf {v} |={\frac {{\text{d}}s}{{\text{d}}t}},} where s {\displaystyle s} is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, d s d t {\displaystyle {\frac {{\text{d}}s}{{\text{d}}t}}} is non-negative, which implies that speed is also non-negative. === Acceleration === The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over a time interval is defined as the ratio. a ¯ = Δ v ¯ Δ t = Δ v ¯ x Δ t x ^ + Δ v ¯ y Δ t y ^ + Δ v ¯ z Δ t z ^ = a ¯ x x ^ + a ¯ y y ^ + a ¯ z z ^ {\displaystyle \mathbf {\bar {a}} ={\frac {\Delta \mathbf {\bar {v}} }{\Delta t}}={\frac {\Delta {\bar {v}}_{x}}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta {\bar {v}}_{y}}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta {\bar {v}}_{z}}{\Delta t}}{\hat {\mathbf {z} }}={\bar {a}}_{x}{\hat {\mathbf {x} }}+{\bar {a}}_{y}{\hat {\mathbf {y} }}+{\bar {a}}_{z}{\hat {\mathbf {z} }}\,} where Δv is the average velocity and Δt is
|
{
"page_id": 65914,
"source": null,
"title": "Kinematics"
}
|
the time interval. The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative, a = lim Δ t → 0 Δ v Δ t = d v d t = a x x ^ + a y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {{\text{d}}\mathbf {v} }{{\text{d}}t}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Alternatively, a = lim ( Δ t ) 2 → 0 Δ r ( Δ t ) 2 = d 2 r d t 2 = a x x ^ + a y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{(\Delta t)^{2}\to 0}{\frac {\Delta \mathbf {r} }{(\Delta t)^{2}}}={\frac {{\text{d}}^{2}\mathbf {r} }{{\text{d}}t^{2}}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Thus, acceleration is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The magnitude of the acceleration of an object is the magnitude |a| of its acceleration vector. It is a scalar quantity: | a | = | v ˙ | = d v d t . {\displaystyle |\mathbf {a} |=|{\dot {\mathbf {v} }}|={\frac {{\text{d}}v}{{\text{d}}t}}.} === Relative position vector === A relative position vector is a vector that defines the position of one point relative to another. It is the difference in position of the two points. The position of one point A relative to another point B is simply the difference between their positions r A / B = r A − r B {\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r}
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"page_id": 65914,
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_{B}} which is the difference between the components of their position vectors. If point A has position components r A = ( x A , y A , z A ) {\displaystyle \mathbf {r} _{A}=\left(x_{A},y_{A},z_{A}\right)} and point B has position components r B = ( x B , y B , z B ) {\displaystyle \mathbf {r} _{B}=\left(x_{B},y_{B},z_{B}\right)} then the position of point A relative to point B is the difference between their components: r A / B = r A − r B = ( x A − x B , y A − y B , z A − z B ) {\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r} _{B}=\left(x_{A}-x_{B},y_{A}-y_{B},z_{A}-z_{B}\right)} === Relative velocity === The velocity of one point relative to another is simply the difference between their velocities v A / B = v A − v B {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}} which is the difference between the components of their velocities. If point A has velocity components v A = ( v A x , v A y , v A z ) {\displaystyle \mathbf {v} _{A}=\left(v_{A_{x}},v_{A_{y}},v_{A_{z}}\right)} and point B has velocity components v B = ( v B x , v B y , v B z ) {\displaystyle \mathbf {v} _{B}=\left(v_{B_{x}},v_{B_{y}},v_{B_{z}}\right)} then the velocity of point A relative to point B is the difference between their components: v A / B = v A − v B = ( v A x − v B x , v A y − v B y , v A z − v B z ) {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}=\left(v_{A_{x}}-v_{B_{x}},v_{A_{y}}-v_{B_{y}},v_{A_{z}}-v_{B_{z}}\right)} Alternatively, this same result could be obtained by computing the time derivative of the relative position vector rB/A. === Relative acceleration === The acceleration of one point C relative
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"page_id": 65914,
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to another point B is simply the difference between their accelerations. a C / B = a C − a B {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}} which is the difference between the components of their accelerations. If point C has acceleration components a C = ( a C x , a C y , a C z ) {\displaystyle \mathbf {a} _{C}=\left(a_{C_{x}},a_{C_{y}},a_{C_{z}}\right)} and point B has acceleration components a B = ( a B x , a B y , a B z ) {\displaystyle \mathbf {a} _{B}=\left(a_{B_{x}},a_{B_{y}},a_{B_{z}}\right)} then the acceleration of point C relative to point B is the difference between their components: a C / B = a C − a B = ( a C x − a B x , a C y − a B y , a C z − a B z ) {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}=\left(a_{C_{x}}-a_{B_{x}},a_{C_{y}}-a_{B_{y}},a_{C_{z}}-a_{B_{z}}\right)} Assuming that the initial conditions of the position, r 0 {\displaystyle \mathbf {r} _{0}} , and velocity v 0 {\displaystyle \mathbf {v} _{0}} at time t = 0 {\displaystyle t=0} are known, the first integration yields the velocity of the particle as a function of time. v ( t ) = v 0 + ∫ 0 t a ( τ ) d τ {\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\int _{0}^{t}\mathbf {a} (\tau )\,{\text{d}}\tau } Additional relations between displacement, velocity, acceleration, and time can be derived. If the acceleration is constant, a = Δ v Δ t = v − v 0 t {\displaystyle \mathbf {a} ={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v} -\mathbf {v} _{0}}{t}}} can be substituted into the above equation to give: r ( t ) = r 0 + ( v + v 0 2 ) t . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\left({\frac {\mathbf
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"page_id": 65914,
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{v} +\mathbf {v} _{0}}{2}}\right)t.} A relationship between velocity, position and acceleration without explicit time dependence can be obtained by solving the average acceleration for time and substituting and simplifying t = v − v 0 a {\displaystyle t={\frac {\mathbf {v} -\mathbf {v} _{0}}{\mathbf {a} }}} ( r − r 0 ) ⋅ a = ( v − v 0 ) ⋅ v + v 0 2 , {\displaystyle \left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =\left(\mathbf {v} -\mathbf {v} _{0}\right)\cdot {\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\ ,} where ⋅ {\displaystyle \cdot } denotes the dot product, which is appropriate as the products are scalars rather than vectors. 2 ( r − r 0 ) ⋅ a = | v | 2 − | v 0 | 2 . {\displaystyle 2\left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} The dot product can be replaced by the cosine of the angle α between the vectors (see Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in which case: 2 | r − r 0 | | a | cos α = | v | 2 − | v 0 | 2 . {\displaystyle 2\left|\mathbf {r} -\mathbf {r} _{0}\right|\left|\mathbf {a} \right|\cos \alpha =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} In the case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors (α) is 0, so cos 0 = 1 {\displaystyle \cos 0=1} , and | v | 2 = | v 0 | 2 + 2 | a | | r − r 0 | . {\displaystyle |\mathbf {v} |^{2}=|\mathbf {v} _{0}|^{2}+2\left|\mathbf {a} \right|\left|\mathbf {r} -\mathbf {r} _{0}\right|.} This can be simplified using the notation for the
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magnitudes of the vectors | a | = a , | v | = v , | r − r 0 | = Δ r {\displaystyle |\mathbf {a} |=a,|\mathbf {v} |=v,|\mathbf {r} -\mathbf {r} _{0}|=\Delta r} where Δ r {\displaystyle \Delta r} can be any curvaceous path taken as the constant tangential acceleration is applied along that path, so v 2 = v 0 2 + 2 a Δ r . {\displaystyle v^{2}=v_{0}^{2}+2a\Delta r.} This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that Δ r = ∫ v d t {\textstyle \Delta r=\int v\,{\text{d}}t} or Δ r {\displaystyle \Delta r} is the area under a velocity–time graph. We can take Δ r {\displaystyle \Delta r} by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the A ⋅ B {\displaystyle A\cdot B} where A {\displaystyle A} is the width and B {\displaystyle B} is the height. In this case A = t {\displaystyle A=t} and B = v 0 {\displaystyle B=v_{0}} (the A {\displaystyle A} here is different from the acceleration a {\displaystyle a} ). This means that the bottom area is t v 0 {\displaystyle tv_{0}} . Now let's find the top area (a triangle). The area of a triangle is 1 2 B H {\textstyle {\frac {1}{2}}BH} where B {\displaystyle B} is the base and H {\displaystyle H} is the height. In this case, B = t {\displaystyle B=t} and H = a t {\displaystyle H=at} or A = 1 2 B H = 1 2 a t t = 1 2 a t 2 = a t 2 2 {\textstyle A={\frac {1}{2}}BH={\frac {1}{2}}att={\frac {1}{2}}at^{2}={\frac
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"page_id": 65914,
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{at^{2}}{2}}} . Adding v 0 t {\displaystyle v_{0}t} and a t 2 2 {\textstyle {\frac {at^{2}}{2}}} results in the equation Δ r {\displaystyle \Delta r} results in the equation Δ r = v 0 t + a t 2 2 {\textstyle \Delta r=v_{0}t+{\frac {at^{2}}{2}}} . This equation is applicable when the final velocity v is unknown. == Particle trajectories in cylindrical-polar coordinates == It is often convenient to formulate the trajectory of a particle r(t) = (x(t), y(t), z(t)) using polar coordinates in the X–Y plane. In this case, its velocity and acceleration take a convenient form. Recall that the trajectory of a particle P is defined by its coordinate vector r measured in a fixed reference frame F. As the particle moves, its coordinate vector r(t) traces its trajectory, which is a curve in space, given by: r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x̂, ŷ, and ẑ are the unit vectors along the x, y and z axes of the reference frame F, respectively. Consider a particle P that moves only on the surface of a circular cylinder r(t) = constant, it is possible to align the z axis of the fixed frame F with the axis of the cylinder. Then, the angle θ around this axis in the x–y plane can be used to define the trajectory as, r ( t ) = r cos ( θ ( t ) ) x ^ + r sin ( θ ( t ) ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=r\cos(\theta (t)){\hat {\mathbf {x} }}+r\sin(\theta (t)){\hat {\mathbf
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"page_id": 65914,
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{y} }}+z(t){\hat {\mathbf {z} }},} where the constant distance from the center is denoted as r, and θ(t) is a function of time. The cylindrical coordinates for r(t) can be simplified by introducing the radial and tangential unit vectors, r ^ = cos ( θ ( t ) ) x ^ + sin ( θ ( t ) ) y ^ , θ ^ = − sin ( θ ( t ) ) x ^ + cos ( θ ( t ) ) y ^ . {\displaystyle {\hat {\mathbf {r} }}=\cos(\theta (t)){\hat {\mathbf {x} }}+\sin(\theta (t)){\hat {\mathbf {y} }},\quad {\hat {\mathbf {\theta } }}=-\sin(\theta (t)){\hat {\mathbf {x} }}+\cos(\theta (t)){\hat {\mathbf {y} }}.} and their time derivatives from elementary calculus: d r ^ d t = ω θ ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {r} }}}{{\text{d}}t}}=\omega {\hat {\mathbf {\theta } }}.} d 2 r ^ d t 2 = d ( ω θ ^ ) d t = α θ ^ − ω 2 r ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {r} }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(\omega {\hat {\mathbf {\theta } }})}{{\text{d}}t}}=\alpha {\hat {\mathbf {\theta } }}-\omega ^{2}{\hat {\mathbf {r} }}.} d θ ^ d t = − ω r ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {\theta } }}}{{\text{d}}t}}=-\omega {\hat {\mathbf {r} }}.} d 2 θ ^ d t 2 = d ( − ω r ^ ) d t = − α r ^ − ω 2 θ ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {\theta } }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(-\omega {\hat {\mathbf {r} }})}{{\text{d}}t}}=-\alpha {\hat {\mathbf {r} }}-\omega ^{2}{\hat {\mathbf {\theta } }}.} Using this notation, r(t) takes the form, r ( t ) = r r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} In general, the trajectory r(t)
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"page_id": 65914,
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is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes: r ( t ) = r ( t ) r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r(t){\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} Where r, θ, and z might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector vP is the time derivative of the trajectory r(t), which yields: v P = d d t ( r r ^ + z z ^ ) = v r ^ + r ω θ ^ + v z z ^ = v ( r ^ + θ ^ ) + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=v{\hat {\mathbf {r} }}+r\mathbf {\omega } {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v({\hat {\mathbf {r} }}+{\hat {\mathbf {\theta } }})+v_{z}{\hat {\mathbf {z} }}.} Similarly, the acceleration aP, which is the time derivative of the velocity vP, is given by: a P = d d t ( v r ^ + v θ ^ + v z z ^ ) = ( a − v ω ) r ^ + ( a + v ω ) θ ^ + a z z ^ . {\displaystyle \mathbf {a} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(v{\hat {\mathbf {r} }}+v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}\right)=(a-v\omega ){\hat {\mathbf {r} }}+(a+v\omega ){\hat {\mathbf {\theta } }}+a_{z}{\hat {\mathbf {z} }}.} The term − v ω r ^ {\displaystyle -v\omega {\hat {\mathbf {r} }}} acts toward the center of curvature of the path at that point on the path, is commonly called the centripetal acceleration. The term v ω θ ^ {\displaystyle v\omega {\hat {\mathbf {\theta }
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"page_id": 65914,
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}}} is called the Coriolis acceleration. === Constant radius === If the trajectory of the particle is constrained to lie on a cylinder, then the radius r is constant and the velocity and acceleration vectors simplify. The velocity of vP is the time derivative of the trajectory r(t), v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ + v z z ^ = v θ ^ + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}.} === Planar circular trajectories === A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the z axis: r ( t ) = r r ^ + z z ^ , {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }},} where r and z0 are constants. In this case, the velocity vP is given by: v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ = v θ ^ , {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}=v{\hat {\mathbf {\theta } }},} where ω {\displaystyle \omega } is the angular velocity of the unit vector θ^ around the z axis of the cylinder. The acceleration aP of the particle P is now given by: a P = d ( v θ ^ ) d t = a θ ^ − v θ r ^ . {\displaystyle \mathbf {a} _{P}={\frac {{\text{d}}(v{\hat {\mathbf {\theta } }})}{{\text{d}}t}}=a{\hat {\mathbf {\theta } }}-v\theta {\hat {\mathbf {r} }}.} The components a r = − v θ , a θ =
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"page_id": 65914,
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a , {\displaystyle a_{r}=-v\theta ,\quad a_{\theta }=a,} are called, respectively, the radial and tangential components of acceleration. The notation for angular velocity and angular acceleration is often defined as ω = θ ˙ , α = θ ¨ , {\displaystyle \omega ={\dot {\theta }},\quad \alpha ={\ddot {\theta }},} so the radial and tangential acceleration components for circular trajectories are also written as a r = − r ω 2 , a θ = r α . {\displaystyle a_{r}=-r\omega ^{2},\quad a_{\theta }=r\alpha .} == Point trajectories in a body moving in the plane == The movement of components of a mechanical system are analyzed by attaching a reference frame to each part and determining how the various reference frames move relative to each other. If the structural stiffness of the parts are sufficient, then their deformation can be neglected and rigid transformations can be used to define this relative movement. This reduces the description of the motion of the various parts of a complicated mechanical system to a problem of describing the geometry of each part and geometric association of each part relative to other parts. Geometry is the study of the properties of figures that remain the same while the space is transformed in various ways—more technically, it is the study of invariants under a set of transformations. These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry. The coordinates of points in a plane are two-dimensional vectors in R2 (two dimensional space). Rigid transformations are those that preserve the distance between any two points. The set of rigid transformations in an n-dimensional space
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"page_id": 65914,
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is called the special Euclidean group on Rn, and denoted SE(n). === Displacements and motion === The position of one component of a mechanical system relative to another is defined by introducing a reference frame, say M, on one that moves relative to a fixed frame, F, on the other. The rigid transformation, or displacement, of M relative to F defines the relative position of the two components. A displacement consists of the combination of a rotation and a translation. The set of all displacements of M relative to F is called the configuration space of M. A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the motion of M relative to F. The motion of a body consists of a continuous set of rotations and translations. === Matrix representation === The combination of a rotation and translation in the plane R2 can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2 rotation matrix A(φ) and the 2×1 translation vector d = (dx, dy), as: [ T ( ϕ , d ) ] = [ A ( ϕ ) d 0 1 ] = [ cos ϕ − sin ϕ d x sin ϕ cos ϕ d y 0 0 1 ] . {\displaystyle [T(\phi ,\mathbf {d} )]={\begin{bmatrix}A(\phi )&\mathbf {d} \\\mathbf {0} &1\end{bmatrix}}={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}.} These homogeneous transforms perform rigid transformations on the points in the plane z = 1, that is, on points with coordinates r = (x, y, 1). In particular, let r define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of
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"page_id": 65914,
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M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by: P = [ T ( ϕ , d ) ] r = [ cos ϕ − sin ϕ d x sin ϕ cos ϕ d y 0 0 1 ] [ x y 1 ] . {\displaystyle \mathbf {P} =[T(\phi ,\mathbf {d} )]\mathbf {r} ={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\1\end{bmatrix}}.} Homogeneous transforms represent affine transformations. This formulation is necessary because a translation is not a linear transformation of R2. However, using projective geometry, so that R2 is considered a subset of R3, translations become affine linear transformations. == Pure translation == If a rigid body moves so that its reference frame M does not rotate (θ = 0) relative to the fixed frame F, the motion is called pure translation. In this case, the trajectory of every point in the body is an offset of the trajectory d(t) of the origin of M, that is: r ( t ) = [ T ( 0 , d ( t ) ) ] p = d ( t ) + p . {\displaystyle \mathbf {r} (t)=[T(0,\mathbf {d} (t))]\mathbf {p} =\mathbf {d} (t)+\mathbf {p} .} Thus, for bodies in pure translation, the velocity and acceleration of every point P in the body are given by: v P = r ˙ ( t ) = d ˙ ( t ) = v O , a P = r ¨ ( t ) = d ¨ ( t ) = a O , {\displaystyle \mathbf {v} _{P}={\dot {\mathbf {r} }}(t)={\dot {\mathbf {d} }}(t)=\mathbf {v} _{O},\quad \mathbf {a} _{P}={\ddot {\mathbf {r}
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"page_id": 65914,
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}}(t)={\ddot {\mathbf {d} }}(t)=\mathbf {a} _{O},} where the dot denotes the derivative with respect to time and vO and aO are the velocity and acceleration, respectively, of the origin of the moving frame M. Recall the coordinate vector p in M is constant, so its derivative is zero. == Rotation of a body around a fixed axis == Objects like a playground merry-go-round, ventilation fans, or hinged doors can be modeled as rigid bodies rotating about a single fixed axis.: 37 The z-axis has been chosen by convention. === Position === This allows the description of a rotation as the angular position of a planar reference frame M relative to a fixed F about this shared z-axis. Coordinates p = (x, y) in M are related to coordinates P = (X, Y) in F by the matrix equation: P ( t ) = [ A ( t ) ] p , {\displaystyle \mathbf {P} (t)=[A(t)]\mathbf {p} ,} where [ A ( t ) ] = [ cos ( θ ( t ) ) − sin ( θ ( t ) ) sin ( θ ( t ) ) cos ( θ ( t ) ) ] , {\displaystyle [A(t)]={\begin{bmatrix}\cos(\theta (t))&-\sin(\theta (t))\\\sin(\theta (t))&\cos(\theta (t))\end{bmatrix}},} is the rotation matrix that defines the angular position of M relative to F as a function of time. === Velocity === If the point p does not move in M, its velocity in F is given by v P = P ˙ = [ A ˙ ( t ) ] p . {\displaystyle \mathbf {v} _{P}={\dot {\mathbf {P} }}=[{\dot {A}}(t)]\mathbf {p} .} It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P(t), v P = [ A ˙ ( t ) ] [ A
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"page_id": 65914,
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( t ) − 1 ] P = [ Ω ] P , {\displaystyle \mathbf {v} _{P}=[{\dot {A}}(t)][A(t)^{-1}]\mathbf {P} =[\Omega ]\mathbf {P} ,} where the matrix [ Ω ] = [ 0 − ω ω 0 ] , {\displaystyle [\Omega ]={\begin{bmatrix}0&-\omega \\\omega &0\end{bmatrix}},} is known as the angular velocity matrix of M relative to F. The parameter ω is the time derivative of the angle θ, that is: ω = d θ d t . {\displaystyle \omega ={\frac {{\text{d}}\theta }{{\text{d}}t}}.} === Acceleration === The acceleration of P(t) in F is obtained as the time derivative of the velocity, A P = P ¨ ( t ) = [ Ω ˙ ] P + [ Ω ] P ˙ , {\displaystyle \mathbf {A} _{P}={\ddot {P}}(t)=[{\dot {\Omega }}]\mathbf {P} +[\Omega ]{\dot {\mathbf {P} }},} which becomes A P = [ Ω ˙ ] P + [ Ω ] [ Ω ] P , {\displaystyle \mathbf {A} _{P}=[{\dot {\Omega }}]\mathbf {P} +[\Omega ][\Omega ]\mathbf {P} ,} where [ Ω ˙ ] = [ 0 − α α 0 ] , {\displaystyle [{\dot {\Omega }}]={\begin{bmatrix}0&-\alpha \\\alpha &0\end{bmatrix}},} is the angular acceleration matrix of M on F, and α = d 2 θ d t 2 . {\displaystyle \alpha ={\frac {{\text{d}}^{2}\theta }{{\text{d}}t^{2}}}.} The description of rotation then involves these three quantities: Angular position: the oriented distance from a selected origin on the rotational axis to a point of an object is a vector r(t) locating the point. The vector r(t) has some projection (or, equivalently, some component) r⊥(t) on a plane perpendicular to the axis of rotation. Then the angular position of that point is the angle θ from a reference axis (typically the positive x-axis) to the vector r⊥(t) in a known rotation sense (typically given by the right-hand rule). Angular velocity: the angular
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"page_id": 65914,
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velocity ω is the rate at which the angular position θ changes with respect to time t: ω = d θ d t {\displaystyle \omega ={\frac {{\text{d}}\theta }{{\text{d}}t}}} The angular velocity is represented in Figure 1 by a vector Ω pointing along the axis of rotation with magnitude ω and sense determined by the direction of rotation as given by the right-hand rule. Angular acceleration: the magnitude of the angular acceleration α is the rate at which the angular velocity ω changes with respect to time t: α = d ω d t {\displaystyle \alpha ={\frac {{\text{d}}\omega }{{\text{d}}t}}} The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges: ω f = ω i + α t {\displaystyle \omega _{\mathrm {f} }=\omega _{\mathrm {i} }+\alpha t\!} θ f − θ i = ω i t + 1 2 α t 2 {\displaystyle \theta _{\mathrm {f} }-\theta _{\mathrm {i} }=\omega _{\mathrm {i} }t+{\tfrac {1}{2}}\alpha t^{2}} θ f − θ i = 1 2 ( ω f + ω i ) t {\displaystyle \theta _{\mathrm {f} }-\theta _{\mathrm {i} }={\tfrac {1}{2}}(\omega _{\mathrm {f} }+\omega _{\mathrm {i} })t} ω f 2 = ω i 2 + 2 α ( θ f − θ i ) . {\displaystyle \omega _{\mathrm {f} }^{2}=\omega _{\mathrm {i} }^{2}+2\alpha (\theta _{\mathrm {f} }-\theta _{\mathrm {i} }).} Here θi and θf are, respectively, the initial and final angular positions, ωi and ωf are, respectively, the initial and final angular velocities, and α is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector. == Point trajectories in body moving in three dimensions == Important formulas
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in kinematics define the velocity and acceleration of points in a moving body as they trace trajectories in three-dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either Newton's second law or Lagrange's equations. === Position === In order to define these formulas, the movement of a component B of a mechanical system is defined by the set of rotations [A(t)] and translations d(t) assembled into the homogeneous transformation [T(t)]=[A(t), d(t)]. If p is the coordinates of a point P in B measured in the moving reference frame M, then the trajectory of this point traced in F is given by: P ( t ) = [ T ( t ) ] p = [ P 1 ] = [ A ( t ) d ( t ) 0 1 ] [ p 1 ] . {\displaystyle \mathbf {P} (t)=[T(t)]\mathbf {p} ={\begin{bmatrix}\mathbf {P} \\1\end{bmatrix}}={\begin{bmatrix}A(t)&\mathbf {d} (t)\\0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}.} This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), which is hopefully clear in context. This equation for the trajectory of P can be inverted to compute the coordinate vector p in M as: p = [ T ( t ) ] − 1 P ( t ) = [ p 1 ] = [ A ( t ) T − A ( t ) T d ( t ) 0 1 ] [ P ( t ) 1 ] . {\displaystyle \mathbf {p} =[T(t)]^{-1}\mathbf {P} (t)={\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}={\begin{bmatrix}A(t)^{\text{T}}&-A(t)^{\text{T}}\mathbf {d} (t)\\0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}.} This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is: [ A ( t ) ] T [ A ( t ) ] = I . {\displaystyle [A(t)]^{\text{T}}[A(t)]=I.\!}
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=== Velocity === The velocity of the point P along its trajectory P(t) is obtained as the time derivative of this position vector, v P = [ T ˙ ( t ) ] p = [ v P 0 ] = ( d d t [ A ( t ) d ( t ) 0 1 ] ) [ p 1 ] = [ A ˙ ( t ) d ˙ ( t ) 0 0 ] [ p 1 ] . {\displaystyle \mathbf {v} _{P}=[{\dot {T}}(t)]\mathbf {p} ={\begin{bmatrix}\mathbf {v} _{P}\\0\end{bmatrix}}=\left({\frac {d}{dt}}{\begin{bmatrix}A(t)&\mathbf {d} (t)\\0&1\end{bmatrix}}\right){\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}={\begin{bmatrix}{\dot {A}}(t)&{\dot {\mathbf {d} }}(t)\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}.} The dot denotes the derivative with respect to time; because p is constant, its derivative is zero. This formula can be modified to obtain the velocity of P by operating on its trajectory P(t) measured in the fixed frame F. Substituting the inverse transform for p into the velocity equation yields: v P = [ T ˙ ( t ) ] [ T ( t ) ] − 1 P ( t ) = [ v P 0 ] = [ A ˙ d ˙ 0 0 ] [ A d 0 1 ] − 1 [ P ( t ) 1 ] = [ A ˙ d ˙ 0 0 ] A − 1 [ 1 − d 0 A ] [ P ( t ) 1 ] = [ A ˙ A − 1 − A ˙ A − 1 d + d ˙ 0 0 ] [ P ( t ) 1 ] = [ A ˙ A T − A ˙ A T d + d ˙ 0 0 ] [ P ( t ) 1 ] v P = [ S ] P . {\displaystyle {\begin{aligned}\mathbf {v} _{P}&=[{\dot {T}}(t)][T(t)]^{-1}\mathbf {P} (t)\\[4pt]&={\begin{bmatrix}\mathbf {v} _{P}\\0\end{bmatrix}}={\begin{bmatrix}{\dot
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{A}}&{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}A&\mathbf {d} \\0&1\end{bmatrix}}^{-1}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[4pt]&={\begin{bmatrix}{\dot {A}}&{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}A^{-1}{\begin{bmatrix}1&-\mathbf {d} \\0&A\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[4pt]&={\begin{bmatrix}{\dot {A}}A^{-1}&-{\dot {A}}A^{-1}\mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[4pt]&={\begin{bmatrix}{\dot {A}}A^{\text{T}}&-{\dot {A}}A^{\text{T}}\mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[6pt]\mathbf {v} _{P}&=[S]\mathbf {P} .\end{aligned}}} The matrix [S] is given by: [ S ] = [ Ω − Ω d + d ˙ 0 0 ] {\displaystyle [S]={\begin{bmatrix}\Omega &-\Omega \mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}} where [ Ω ] = A ˙ A T , {\displaystyle [\Omega ]={\dot {A}}A^{\text{T}},} is the angular velocity matrix. Multiplying by the operator [S], the formula for the velocity vP takes the form: v P = [ Ω ] ( P − d ) + d ˙ = ω × R P / O + v O , {\displaystyle \mathbf {v} _{P}=[\Omega ](\mathbf {P} -\mathbf {d} )+{\dot {\mathbf {d} }}=\omega \times \mathbf {R} _{P/O}+\mathbf {v} _{O},} where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω]; the vector R P / O = P − d , {\displaystyle \mathbf {R} _{P/O}=\mathbf {P} -\mathbf {d} ,} is the position of P relative to the origin O of the moving frame M; and v O = d ˙ , {\displaystyle \mathbf {v} _{O}={\dot {\mathbf {d} }},} is the velocity of the origin O. === Acceleration === The acceleration of a point P in a moving body B is obtained as the time derivative of its velocity vector: A P = d d t v P = d d t ( [ S ] P ) = [ S ˙ ] P + [ S ] P ˙ = [ S ˙ ] P + [ S ] [ S ] P . {\displaystyle \mathbf {A} _{P}={\frac {d}{dt}}\mathbf {v} _{P}={\frac {d}{dt}}\left([S]\mathbf {P} \right)=[{\dot {S}}]\mathbf {P} +[S]{\dot {\mathbf {P}
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"page_id": 65914,
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}}=[{\dot {S}}]\mathbf {P} +[S][S]\mathbf {P} .} This equation can be expanded firstly by computing [ S ˙ ] = [ Ω ˙ − Ω ˙ d − Ω d ˙ + d ¨ 0 0 ] = [ Ω ˙ − Ω ˙ d − Ω v O + A O 0 0 ] {\displaystyle [{\dot {S}}]={\begin{bmatrix}{\dot {\Omega }}&-{\dot {\Omega }}\mathbf {d} -\Omega {\dot {\mathbf {d} }}+{\ddot {\mathbf {d} }}\\0&0\end{bmatrix}}={\begin{bmatrix}{\dot {\Omega }}&-{\dot {\Omega }}\mathbf {d} -\Omega \mathbf {v} _{O}+\mathbf {A} _{O}\\0&0\end{bmatrix}}} and [ S ] 2 = [ Ω − Ω d + v O 0 0 ] 2 = [ Ω 2 − Ω 2 d + Ω v O 0 0 ] . {\displaystyle [S]^{2}={\begin{bmatrix}\Omega &-\Omega \mathbf {d} +\mathbf {v} _{O}\\0&0\end{bmatrix}}^{2}={\begin{bmatrix}\Omega ^{2}&-\Omega ^{2}\mathbf {d} +\Omega \mathbf {v} _{O}\\0&0\end{bmatrix}}.} The formula for the acceleration AP can now be obtained as: A P = Ω ˙ ( P − d ) + A O + Ω 2 ( P − d ) , {\displaystyle \mathbf {A} _{P}={\dot {\Omega }}(\mathbf {P} -\mathbf {d} )+\mathbf {A} _{O}+\Omega ^{2}(\mathbf {P} -\mathbf {d} ),} or A P = α × R P / O + ω × ω × R P / O + A O , {\displaystyle \mathbf {A} _{P}=\alpha \times \mathbf {R} _{P/O}+\omega \times \omega \times \mathbf {R} _{P/O}+\mathbf {A} _{O},} where α is the angular acceleration vector obtained from the derivative of the angular velocity vector; R P / O = P − d , {\displaystyle \mathbf {R} _{P/O}=\mathbf {P} -\mathbf {d} ,} is the relative position vector (the position of P relative to the origin O of the moving frame M); and A O = d ¨ {\displaystyle \mathbf {A} _{O}={\ddot {\mathbf {d} }}} is the acceleration of the origin of the moving frame M. == Kinematic constraints == Kinematic
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constraints are constraints on the movement of components of a mechanical system. Kinematic constraints can be considered to have two basic forms, (i) constraints that arise from hinges, sliders and cam joints that define the construction of the system, called holonomic constraints, and (ii) constraints imposed on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane, which are called non-holonomic constraints. The following are some common examples. === Kinematic coupling === A kinematic coupling exactly constrains all 6 degrees of freedom. === Rolling without slipping === An object that rolls against a surface without slipping obeys the condition that the velocity of its center of mass is equal to the cross product of its angular velocity with a vector from the point of contact to the center of mass: v G ( t ) = Ω × r G / O . {\displaystyle {\boldsymbol {v}}_{G}(t)={\boldsymbol {\Omega }}\times {\boldsymbol {r}}_{G/O}.} For the case of an object that does not tip or turn, this reduces to v = r ω {\displaystyle v=r\omega } . === Inextensible cord === This is the case where bodies are connected by an idealized cord that remains in tension and cannot change length. The constraint is that the sum of lengths of all segments of the cord is the total length, and accordingly the time derivative of this sum is zero. A dynamic problem of this type is the pendulum. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord. An equilibrium problem (i.e. not kinematic) of this type is the catenary. === Kinematic pairs === Reuleaux called the ideal connections between components
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"page_id": 65914,
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that form a machine kinematic pairs. He distinguished between higher pairs which were said to have line contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple classification. ==== Lower pair ==== A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three-dimensional) body to a corresponding point line or plane in the fixed solid body. There are the following cases: A revolute pair, or hinged joint, requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom, which is pure rotation about the axis of the hinge. A prismatic joint, or slider, requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body maintain contact with a similar parallel plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom. This degree of freedom is the distance of the slide along the line. A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body. It is a combination of a revolute joint and a sliding joint. This joint has two degrees of freedom. The position of the moving body is defined by both the
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"page_id": 65914,
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rotation about and slide along the axis. A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom. A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom. ==== Higher pairs ==== Generally speaking, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a cam joint. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints. === Kinematic chains === Rigid bodies ("links") connected by kinematic pairs ("joints") are known as kinematic chains. Mechanisms and robots are examples of kinematic chains. The degree of freedom of a kinematic chain is computed from the number of links and the number and type of joints using the mobility formula. This formula can also be used to enumerate the topologies of kinematic chains that have a given degree of freedom, which is known as type synthesis in machine design. ==== Examples ==== The planar one degree-of-freedom linkages assembled from N links and j hinges or sliding joints are: N = 2, j = 1 : a two-bar linkage that is the lever; N = 4, j = 4 : the four-bar linkage; N = 6, j = 7 : a six-bar linkage. This must have two links ("ternary links") that support three joints. There are two distinct topologies that depend on how the two ternary linkages are connected. In the Watt topology, the two ternary links
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"page_id": 65914,
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have a common joint; in the Stephenson topology, the two ternary links do not have a common joint and are connected by binary links. N = 8, j = 10 : eight-bar linkage with 16 different topologies; N = 10, j = 13 : ten-bar linkage with 230 different topologies; N = 12, j = 16 : twelve-bar linkage with 6,856 topologies. For larger chains and their linkage topologies, see R. P. Sunkari and L. C. Schmidt, "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm", Mechanism and Machine Theory #41, pp. 1021–1030 (2006). == See also == == References == == Further reading == Koetsier, Teun (1994), "§8.3 Kinematics", in Grattan-Guinness, Ivor (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol. 2, Routledge, pp. 994–1001, ISBN 0-415-09239-6 Moon, Francis C. (2007). The Machines of Leonardo Da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century. Springer. ISBN 978-1-4020-5598-0. Eduard Study (1913) D.H. Delphenich translator, "Foundations and goals of analytical kinematics". == External links == Java applet of 1D kinematics Physclips: Mechanics with animations and video clips from the University of New South Wales. Kinematic Models for Design Digital Library (KMODDL), featuring movies and photos of hundreds of working models of mechanical systems at Cornell University and an e-book library of classic texts on mechanical design and engineering. Micro-Inch Positioning with Kinematic Components
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In nuclear physics, a Borromean nucleus is an atomic nucleus comprising three bound components in which any subsystem of two components is unbound. This has the consequence that if one component is removed, the remaining two comprise an unbound resonance, so that the original nucleus is split into three parts. The name is derived from the Borromean rings, a system of three linked rings in which no pair of rings is linked. == Examples of Borromean nuclei == Many Borromean nuclei are light nuclei near the nuclear drip lines that have a nuclear halo and low nuclear binding energy. For example, the nuclei 6He, 11Li, and 22C each possess a two-neutron halo surrounding a core containing the remaining nucleons. These are Borromean nuclei because the removal of either neutron from the halo will result in a resonance unbound to one-neutron emission, whereas the dineutron (the particles in the halo) is itself an unbound system. Similarly, 17Ne is a Borromean nucleus with a two-proton halo; both the diproton and 16F are unbound. Additionally, 9Be is a Borromean nucleus comprising two alpha particles and a neutron; the removal of any one component would produce one of the unbound resonances 5He or 8Be. Several Borromean nuclei such as 9Be and the Hoyle state (an excited resonance in 12C) play an important role in nuclear astrophysics. Namely, these are three-body systems whose unbound components (formed from 4He) are intermediate steps in the triple-alpha process; this limits the rate of production of heavier elements, for three bodies must react nearly simultaneously. Borromean nuclei consisting of more than three components can also exist. These also lie along the drip lines; for instance, 8He and 14Be are five-body Borromean systems with a four-neutron halo. It is also possible that nuclides produced in the alpha process (such as
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{
"page_id": 41615741,
"source": null,
"title": "Borromean nucleus"
}
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12C and 16O) may be clusters of alpha particles, having a similar structure to Borromean nuclei. As of 2012, the heaviest known Borromean nucleus was 29F. Heavier species along the neutron drip line have since been observed; these and undiscovered heavier nuclei along the drip line are also likely to be Borromean nuclei with varying numbers (3, 5, 7, or more) of bodies. == See also == Efimov state Three-body force Halo nucleus == References ==
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"page_id": 41615741,
"source": null,
"title": "Borromean nucleus"
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Megvii (Chinese: 旷视; pinyin: Kuàngshì) is a Chinese technology company that designs image recognition and deep-learning software. Based in Beijing, the company develops artificial intelligence (AI) technology for businesses and for the public sector. Megvii is the largest provider of third-party authentication software in the world, and its product, Face++, is the world's largest computer vision platform. In 2019, the company was valued at $USD 4 billion. As of 2024, the company operates the world's largest computer vision research institute. The company has faced U.S. investment and export restrictions due to allegations of aiding the persecution of Uyghurs in China. == History == The company was founded in Beijing with Megvii standing for "mega vision." It was started by Yin Qi and two college friends.: 101 The company's core product, Face++, launched in 2012 as the first online facial recognition platform in China. In 2015 Megvii created Brain++, a deep-learning engine to help train its algorithms. Backed by GGV Capital, Megvii raised $100 million in 2016, $460 million in 2017 and $750 million in May 2019. In 2017, Megvii marketed authentication and computational photography functions to smart phone companies and mobile application developers, then smart logistics. Megvii's AI-empowered products include personal IoT, city IoT and supply chain IoT. In 2017 and 2018, Megvii beat Google, Facebook, and Microsoft in tests of image recognition at the International Conference on Computer Vision. By June 2019, Megvii had 2,349 employees, and was valued at over $4 billion, as the "world’s biggest provider of third-party authentication software", with 339 corporate clients in 112 cities in China. The Chinese government employs Megvii software. In May 2019, Human Rights Watch reported finding Face++ code in the Integrated Joint Operations Platform (IJOP), a police surveillance app used to collect data on, and track the Uyghur community in
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"page_id": 60883332,
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Xinjiang. Human Rights Watch released a correction to its report in June 2019 stating that Megvii did not appear to have collaborated on IJOP, and that the Face++ code in the app was inoperable. In March 2020, Megvii announced that it would make its deep learning framework MegEngine open-source. As of 2024, Megvii operates the world's largest computer vision research institute.: 101 === US sanctions === Megvii was sanctioned by the US government, and placed on the United States Bureau of Industry and Security's Entity List on October 9, 2019, due to the use of its technology for human rights abuses against Uyghurs in Xinjiang. In December 2021, the United States Department of the Treasury prohibited all U.S. investment in Megvii, accusing the company of complicity in aiding the persecution of Uyghurs in China. In January 2024, the United States Department of Defense named Megvii on its list of "Chinese Military Companies Operating in the United States." Following US sanctions, GGV Capital announced its intention to divest from Megvii. In 2022, Megvii laid off workers in multiple departments as a response to U.S. sanctions and ongoing tensions between the United States and China. == References == == External links == Official website
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"page_id": 60883332,
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The Frink Medal for British Zoologists is awarded by the Zoological Society of London "for significant and original contributions by a professional zoologist to the development of zoology." It consists of a bronze plaque (76 by 83 millimetres), depicting a bison and carved by British sculptor Elisabeth Frink. The Frink Medal was instituted in 1973 and first presented in 1974. == Recipients == Source ZSL == See also == List of biology awards == References == == Notes ==
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"page_id": 8192389,
"source": null,
"title": "Frink Medal"
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Mark and recapture is a method commonly used in ecology to estimate an animal population's size where it is impractical to count every individual. A portion of the population is captured, marked, and released. Later, another portion will be captured and the number of marked individuals within the sample is counted. Since the number of marked individuals within the second sample should be proportional to the number of marked individuals in the whole population, an estimate of the total population size can be obtained by dividing the number of marked individuals by the proportion of marked individuals in the second sample. The method assumes, rightly or wrongly, that the probability of capture is the same for all individuals. Other names for this method, or closely related methods, include capture-recapture, capture-mark-recapture, mark-recapture, sight-resight, mark-release-recapture, multiple systems estimation, band recovery, the Petersen method, and the Lincoln method. Another major application for these methods is in epidemiology, where they are used to estimate the completeness of ascertainment of disease registers. Typical applications include estimating the number of people needing particular services (e.g. services for children with learning disabilities, services for medically frail elderly living in the community), or with particular conditions (e.g. illegal drug addicts, people infected with HIV, etc.). == Field work related to mark-recapture == Typically a researcher visits a study area and uses traps to capture a group of individuals alive. Each of these individuals is marked with a unique identifier (e.g., a numbered tag or band), and then is released unharmed back into the environment. A mark-recapture method was first used for ecological study in 1896 by C.G. Johannes Petersen to estimate plaice, Pleuronectes platessa, populations. Sufficient time should be allowed to pass for the marked individuals to redistribute themselves among the unmarked population. Next, the researcher returns and
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"page_id": 1507717,
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captures another sample of individuals. Some individuals in this second sample will have been marked during the initial visit and are now known as recaptures. Other organisms captured during the second visit, will not have been captured during the first visit to the study area. These unmarked animals are usually given a tag or band during the second visit and then are released. Population size can be estimated from as few as two visits to the study area. Commonly, more than two visits are made, particularly if estimates of survival or movement are desired. Regardless of the total number of visits, the researcher simply records the date of each capture of each individual. The "capture histories" generated are analyzed mathematically to estimate population size, survival, or movement. When capturing and marking organisms, ecologists need to consider the welfare of the organisms. If the chosen identifier harms the organism, then its behavior might become irregular. == Notation == Let N = Number of animals in the population n = Number of animals marked on the first visit K = Number of animals captured on the second visit k = Number of recaptured animals that were marked A biologist wants to estimate the size of a population of turtles in a lake. She captures 10 turtles on her first visit to the lake, and marks their backs with paint. A week later she returns to the lake and captures 15 turtles. Five of these 15 turtles have paint on their backs, indicating that they are recaptured animals. This example is (n, K, k) = (10, 15, 5). The problem is to estimate N. == Lincoln–Petersen estimator == The Lincoln–Petersen method (also known as the Petersen–Lincoln index or Lincoln index) can be used to estimate population size if only two visits are made
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"page_id": 1507717,
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to the study area. This method assumes that the study population is "closed". In other words, the two visits to the study area are close enough in time so that no individuals die, are born, or move into or out of the study area between visits. The model also assumes that no marks fall off animals between visits to the field site by the researcher, and that the researcher correctly records all marks. Given those conditions, estimated population size is: N ^ = n K k , {\displaystyle {\hat {N}}={\frac {nK}{k}},} === Derivation === It is assumed that all individuals have the same probability of being captured in the second sample, regardless of whether they were previously captured in the first sample (with only two samples, this assumption cannot be tested directly). This implies that, in the second sample, the proportion of marked individuals that are caught ( k / K {\displaystyle k/K} ) should equal the proportion of the total population that is marked ( n / N {\displaystyle n/N} ). For example, if half of the marked individuals were recaptured, it would be assumed that half of the total population was included in the second sample. In symbols, k K = n N . {\displaystyle {\frac {k}{K}}={\frac {n}{N}}.} A rearrangement of this gives N ^ = n K k , {\displaystyle {\hat {N}}={\frac {nK}{k}},} the formula used for the Lincoln–Petersen method. === Sample calculation === In the example (n, K, k) = (10, 15, 5) the Lincoln–Petersen method estimates that there are 30 turtles in the lake. N ^ = n K k = 10 × 15 5 = 30 {\displaystyle {\hat {N}}={\frac {nK}{k}}={\frac {10\times 15}{5}}=30} == Chapman estimator == The Lincoln–Petersen estimator is asymptotically unbiased as sample size approaches infinity, but is biased at small sample sizes.
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An alternative less biased estimator of population size is given by the Chapman estimator: N ^ C = ( n + 1 ) ( K + 1 ) k + 1 − 1 {\displaystyle {\hat {N}}_{C}={\frac {(n+1)(K+1)}{k+1}}-1} === Sample calculation === The example (n, K, k) = (10, 15, 5) gives N ^ C = ( n + 1 ) ( K + 1 ) k + 1 − 1 = 11 × 16 6 − 1 = 28.3 {\displaystyle {\hat {N}}_{C}={\frac {(n+1)(K+1)}{k+1}}-1={\frac {11\times 16}{6}}-1=28.3} Note that the answer provided by this equation must be truncated not rounded. Thus, the Chapman method estimates 28 turtles in the lake. Surprisingly, Chapman's estimate was one conjecture from a range of possible estimators: "In practice, the whole number immediately less than (K+1)(n+1)/(k+1) or even Kn/(k+1) will be the estimate. The above form is more convenient for mathematical purposes."(see footnote, page 144). Chapman also found the estimator could have considerable negative bias for small Kn/N (page 146), but was unconcerned because the estimated standard deviations were large for these cases. == Confidence interval == An approximate 100 ( 1 − α ) % {\displaystyle 100(1-\alpha )\%} confidence interval for the population size N can be obtained as: K + n − k − 0.5 + ( K − k + 0.5 ) ( n − k + 0.5 ) ( k + 0.5 ) exp ( ± z α / 2 σ ^ 0.5 ) , {\displaystyle K+n-k-0.5+{\frac {(K-k+0.5)(n-k+0.5)}{(k+0.5)}}\exp(\pm z_{\alpha /2}{\hat {\sigma }}_{0.5}),} where z α / 2 {\textstyle z_{\alpha /2}} corresponds to the 1 − α / 2 {\displaystyle 1-\alpha /2} quantile of a standard normal random variable, and σ ^ 0.5 = 1 k + 0.5 + 1 K − k + 0.5 + 1 n − k + 0.5 +
|
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"page_id": 1507717,
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k + 0.5 ( n − k + 0.5 ) ( K − k + 0.5 ) . {\displaystyle {\hat {\sigma }}_{0.5}={\sqrt {{\frac {1}{k+0.5}}+{\frac {1}{K-k+0.5}}+{\frac {1}{n-k+0.5}}+{\frac {k+0.5}{(n-k+0.5)(K-k+0.5)}}}}.} The example (n, K, k) = (10, 15, 5) gives the estimate N ≈ 30 with a 95% confidence interval of 22 to 65. It has been shown that this confidence interval has actual coverage probabilities that are close to the nominal 100 ( 1 − α ) % {\displaystyle 100(1-\alpha )\%} level even for small populations and extreme capture probabilities (near to 0 or 1), in which cases other confidence intervals fail to achieve the nominal coverage levels. == Capture probability == The capture probability refers to the probability of a detecting an individual animal or person of interest, and has been used in both ecology and epidemiology for detecting animal or human diseases, respectively. The capture probability is often defined as a two-variable model, in which f is defined as the fraction of a finite resource devoted to detecting the animal or person of interest from a high risk sector of an animal or human population, and q is the frequency of time that the problem (e.g., an animal disease) occurs in the high-risk versus the low-risk sector. For example, an application of the model in the 1920s was to detect typhoid carriers in London, who were either arriving from zones with high rates of tuberculosis (probability q that a passenger with the disease came from such an area, where q>0.5), or low rates (probability 1−q). It was posited that only 5 out of 100 of the travelers could be detected, and 10 out of 100 were from the high risk area. Then the capture probability P was defined as: P = 5 10 f q + 5 90 ( 1
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}
|
− f ) ( 1 − q ) , {\displaystyle P={\frac {5}{10}}fq+{\frac {5}{90}}(1-f)(1-q),} where the first term refers to the probability of detection (capture probability) in a high risk zone, and the latter term refers to the probability of detection in a low risk zone. Importantly, the formula can be re-written as a linear equation in terms of f: P = ( 5 10 q − 5 90 ( 1 − q ) ) f + 5 90 ( 1 − q ) . {\displaystyle P=\left({\frac {5}{10}}q-{\frac {5}{90}}(1-q)\right)f+{\frac {5}{90}}(1-q).} Because this is a linear function, it follows that for certain versions of q for which the slope of this line (the first term multiplied by f) is positive, all of the detection resource should be devoted to the high-risk population (f should be set to 1 to maximize the capture probability), whereas for other value of q, for which the slope of the line is negative, all of the detection should be devoted to the low-risk population (f should be set to 0. We can solve the above equation for the values of q for which the slope will be positive to determine the values for which f should be set to 1 to maximize the capture probability: ( 5 10 q − 5 90 ( 1 − q ) ) > 0 , {\displaystyle \left({\frac {5}{10}}q-{\frac {5}{90}}(1-q)\right)>0,} which simplifies to: q > 1 10 . {\displaystyle q>{\frac {1}{10}}.} This is an example of linear optimization. In more complex cases, where more than one resource f is devoted to more than two areas, multivariate optimization is often used, through the simplex algorithm or its derivatives. == More than two visits == The literature on the analysis of capture-recapture studies has blossomed since the early 1990s. There are very elaborate statistical
|
{
"page_id": 1507717,
"source": null,
"title": "Mark and recapture"
}
|
models available for the analysis of these experiments. A simple model which easily accommodates the three source, or the three visit study, is to fit a Poisson regression model. Sophisticated mark-recapture models can be fit with several packages for the Open Source R programming language. These include "Spatially Explicit Capture-Recapture (secr)", "Loglinear Models for Capture-Recapture Experiments (Rcapture)", and "Mark-Recapture Distance Sampling (mrds)". Such models can also be fit with specialized programs such as MARK or E-SURGE. Other related methods which are often used include the Jolly–Seber model (used in open populations and for multiple census estimates) and Schnabel estimators (an expansion to the Lincoln–Petersen method for closed populations). These are described in detail by Sutherland. == Integrated approaches == Modelling mark-recapture data is trending towards a more integrative approach, which combines mark-recapture data with population dynamics models and other types of data. The integrated approach is more computationally demanding, but extracts more information from the data improving parameter and uncertainty estimates. == See also == German tank problem, for estimation of population size when the elements are numbered. Tag and release Abundance estimation GPS wildlife tracking Shadow Effect (Genetics) == References == Besbeas, P; Freeman, S. N.; Morgan, B. J. T.; Catchpole, E. A. (2002). "Integrating mark-recapture-recovery and census data to estimate animal abundance and demographic parameters". Biometrics. 58 (3): 540–547. doi:10.1111/j.0006-341X.2002.00540.x. PMID 12229988. S2CID 30426391. Martin-Löf, P. (1961). "Mortality rate calculations on ringed birds with special reference to the Dunlin Calidris alpina". Arkiv för Zoologi (Zoology Files), Kungliga Svenska Vetenskapsakademien (The Royal Swedish Academy of Sciences) Serie 2. Band 13 (21). Maunder, M. N. (2004). "Population viability analysis, based on combining integrated, Bayesian, and hierarchical analyses". Acta Oecologica. 26 (2): 85–94. Bibcode:2004AcO....26...85M. doi:10.1016/j.actao.2003.11.008. Phillips, C. A.; M. J. Dreslik; J. R. Johnson; J. E. Petzing (2001). "Application of population
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{
"page_id": 1507717,
"source": null,
"title": "Mark and recapture"
}
|
estimation to pond breeding salamanders". Transactions of the Illinois Academy of Science. 94 (2): 111–118. Royle, J. A.; R. M. Dorazio (2008). Hierarchical Modeling and Inference in Ecology. Elsevier. ISBN 978-1-930665-55-2. Seber, G.A.F. (2002). The Estimation of Animal Abundance and Related Parameters. Caldwel, New Jersey: Blackburn Press. ISBN 1-930665-55-5. Schaub, M; Gimenez, O.; Sierro, A.; Arlettaz, R (2007). "Use of Integrated Modeling to Enhance Estimates of Population Dynamics Obtained from Limited Data" (PDF). Conservation Biology. 21 (4): 945–955. Bibcode:2007ConBi..21..945S. doi:10.1111/j.1523-1739.2007.00743.x. PMID 17650245. S2CID 43172823. Williams, B. K.; J. D. Nichols; M. J. Conroy (2002). Analysis and Management of Animal Populations. San Diego, California: Academic Press. ISBN 0-12-754406-2. Chao, A; Tsay, P. K.; Lin, S. H.; Shau, W. Y.; Chao, D. Y. (2001). "The applications of capture-recapture models to epidemiological data". Statistics in Medicine. 20 (20): 3123–3157. doi:10.1002/sim.996. PMID 11590637. S2CID 78437. == Further reading == Bonett, D.G.; Woodward, J.A.; Bentler, P.M. (1986). "A Linear Model for Estimating the Size of a Closed Population". British Journal of Mathematical and Statistical Psychology. 39: 28–40. doi:10.1111/j.2044-8317.1986.tb00843.x. PMID 3768264. Evans, M.A.; Bonett, D.G.; McDonald, L. (1994). "A General Theory for Analyzing Capture-recapture Data in Closed Populations". Biometrics. 50 (2): 396–405. doi:10.2307/2533383. JSTOR 2533383. Lincoln, F. C. (1930). "Calculating Waterfowl Abundance on the Basis of Banding Returns". United States Department of Agriculture Circular. 118: 1–4. Petersen, C. G. J. (1896). "The Yearly Immigration of Young Plaice Into the Limfjord From the German Sea", Report of the Danish Biological Station (1895), 6, 5–84. Schofield, J. R. (2007). "Beyond Defect Removal: Latent Defect Estimation With Capture-Recapture Method", Crosstalk, August 2007; 27–29. == External links == A historical introduction to capture-recapture methods Analysis of capture-recapture data
|
{
"page_id": 1507717,
"source": null,
"title": "Mark and recapture"
}
|
The molecular formula C20H27N3O6 (molar mass: 405.44 g/mol, exact mass: 405.1900 u) may refer to: Imidapril Febarbamate, or phenobamate
|
{
"page_id": 40960388,
"source": null,
"title": "C20H27N3O6"
}
|
In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point "Green's functions" in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields.) == Spatially uniform case == === Basic definitions === We consider a many-body theory with field operator (annihilation operator written in the position basis) ψ ( x ) {\displaystyle \psi (\mathbf {x} )} . The Heisenberg operators can be written in terms of Schrödinger operators as ψ ( x , t ) = e i K t ψ ( x ) e − i K t , {\displaystyle \psi (\mathbf {x} ,t)=e^{iKt}\psi (\mathbf {x} )e^{-iKt},} and the creation operator is ψ ¯ ( x , t ) = [ ψ ( x , t ) ] † {\displaystyle {\bar {\psi }}(\mathbf {x} ,t)=[\psi (\mathbf {x} ,t)]^{\dagger }} , where K = H − μ N {\displaystyle K=H-\mu N} is the grand-canonical Hamiltonian. Similarly, for the imaginary-time operators, ψ ( x , τ ) = e K τ ψ ( x ) e − K τ {\displaystyle \psi (\mathbf {x} ,\tau )=e^{K\tau }\psi (\mathbf {x} )e^{-K\tau }} ψ ¯ ( x , τ ) = e K τ ψ † ( x ) e − K τ . {\displaystyle {\bar {\psi }}(\mathbf {x} ,\tau )=e^{K\tau }\psi ^{\dagger }(\mathbf {x} )e^{-K\tau }.} [Note that the imaginary-time creation operator ψ ¯ ( x , τ ) {\displaystyle {\bar {\psi }}(\mathbf {x} ,\tau )} is not
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
the Hermitian conjugate of the annihilation operator ψ ( x , τ ) {\displaystyle \psi (\mathbf {x} ,\tau )} .] In real time, the 2 n {\displaystyle 2n} -point Green function is defined by G ( n ) ( 1 … n ∣ 1 ′ … n ′ ) = i n ⟨ T ψ ( 1 ) … ψ ( n ) ψ ¯ ( n ′ ) … ψ ¯ ( 1 ′ ) ⟩ , {\displaystyle G^{(n)}(1\ldots n\mid 1'\ldots n')=i^{n}\langle T\psi (1)\ldots \psi (n){\bar {\psi }}(n')\ldots {\bar {\psi }}(1')\rangle ,} where we have used a condensed notation in which j {\displaystyle j} signifies ( x j , t j ) {\displaystyle (\mathbf {x} _{j},t_{j})} and j ′ {\displaystyle j'} signifies ( x j ′ , t j ′ ) {\displaystyle (\mathbf {x} _{j}',t_{j}')} . The operator T {\displaystyle T} denotes time ordering, and indicates that the field operators that follow it are to be ordered so that their time arguments increase from right to left. In imaginary time, the corresponding definition is G ( n ) ( 1 … n ∣ 1 ′ … n ′ ) = ⟨ T ψ ( 1 ) … ψ ( n ) ψ ¯ ( n ′ ) … ψ ¯ ( 1 ′ ) ⟩ , {\displaystyle {\mathcal {G}}^{(n)}(1\ldots n\mid 1'\ldots n')=\langle T\psi (1)\ldots \psi (n){\bar {\psi }}(n')\ldots {\bar {\psi }}(1')\rangle ,} where j {\displaystyle j} signifies x j , τ j {\displaystyle \mathbf {x} _{j},\tau _{j}} . (The imaginary-time variables τ j {\displaystyle \tau _{j}} are restricted to the range from 0 {\displaystyle 0} to the inverse temperature β = 1 k B T {\textstyle \beta ={\frac {1}{k_{\text{B}}T}}} .) Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
that Fourier transform of the two-point ( n = 1 {\displaystyle n=1} ) thermal Green function for a free particle is G ( k , ω n ) = 1 − i ω n + ξ k , {\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})={\frac {1}{-i\omega _{n}+\xi _{\mathbf {k} }}},} and the retarded Green function is G R ( k , ω ) = 1 − ( ω + i η ) + ξ k , {\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )={\frac {1}{-(\omega +i\eta )+\xi _{\mathbf {k} }}},} where ω n = [ 2 n + θ ( − ζ ) ] π β {\displaystyle \omega _{n}={\frac {[2n+\theta (-\zeta )]\pi }{\beta }}} is the Matsubara frequency. Throughout, ζ {\displaystyle \zeta } is + 1 {\displaystyle +1} for bosons and − 1 {\displaystyle -1} for fermions and [ … , … ] = [ … , … ] − ζ {\displaystyle [\ldots ,\ldots ]=[\ldots ,\ldots ]_{-\zeta }} denotes either a commutator or anticommutator as appropriate. (See below for details.) === Two-point functions === The Green function with a single pair of arguments ( n = 1 {\displaystyle n=1} ) is referred to as the two-point function, or propagator. In the presence of both spatial and temporal translational symmetry, it depends only on the difference of its arguments. Taking the Fourier transform with respect to both space and time gives G ( x τ ∣ x ′ τ ′ ) = ∫ k d k 1 β ∑ ω n G ( k , ω n ) e i k ⋅ ( x − x ′ ) − i ω n ( τ − τ ′ ) , {\displaystyle {\mathcal {G}}(\mathbf {x} \tau \mid \mathbf {x} '\tau ')=\int _{\mathbf {k} }d\mathbf {k} {\frac {1}{\beta }}\sum _{\omega _{n}}{\mathcal {G}}(\mathbf {k} ,\omega _{n})e^{i\mathbf {k}
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
\cdot (\mathbf {x} -\mathbf {x} ')-i\omega _{n}(\tau -\tau ')},} where the sum is over the appropriate Matsubara frequencies (and the integral involves an implicit factor of ( L / 2 π ) d {\displaystyle (L/2\pi )^{d}} , as usual). In real time, we will explicitly indicate the time-ordered function with a superscript T: G T ( x t ∣ x ′ t ′ ) = ∫ k d k ∫ d ω 2 π G T ( k , ω ) e i k ⋅ ( x − x ′ ) − i ω ( t − t ′ ) . {\displaystyle G^{\mathrm {T} }(\mathbf {x} t\mid \mathbf {x} 't')=\int _{\mathbf {k} }d\mathbf {k} \int {\frac {d\omega }{2\pi }}G^{\mathrm {T} }(\mathbf {k} ,\omega )e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')-i\omega (t-t')}.} The real-time two-point Green function can be written in terms of 'retarded' and 'advanced' Green functions, which will turn out to have simpler analyticity properties. The retarded and advanced Green functions are defined by G R ( x t ∣ x ′ t ′ ) = − i ⟨ [ ψ ( x , t ) , ψ ¯ ( x ′ , t ′ ) ] ζ ⟩ Θ ( t − t ′ ) {\displaystyle G^{\mathrm {R} }(\mathbf {x} t\mid \mathbf {x} 't')=-i\langle [\psi (\mathbf {x} ,t),{\bar {\psi }}(\mathbf {x} ',t')]_{\zeta }\rangle \Theta (t-t')} and G A ( x t ∣ x ′ t ′ ) = i ⟨ [ ψ ( x , t ) , ψ ¯ ( x ′ , t ′ ) ] ζ ⟩ Θ ( t ′ − t ) , {\displaystyle G^{\mathrm {A} }(\mathbf {x} t\mid \mathbf {x} 't')=i\langle [\psi (\mathbf {x} ,t),{\bar {\psi }}(\mathbf {x} ',t')]_{\zeta }\rangle \Theta (t'-t),} respectively. They are related to the time-ordered Green function
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
by G T ( k , ω ) = [ 1 + ζ n ( ω ) ] G R ( k , ω ) − ζ n ( ω ) G A ( k , ω ) , {\displaystyle G^{\mathrm {T} }(\mathbf {k} ,\omega )=[1+\zeta n(\omega )]G^{\mathrm {R} }(\mathbf {k} ,\omega )-\zeta n(\omega )G^{\mathrm {A} }(\mathbf {k} ,\omega ),} where n ( ω ) = 1 e β ω − ζ {\displaystyle n(\omega )={\frac {1}{e^{\beta \omega }-\zeta }}} is the Bose–Einstein or Fermi–Dirac distribution function. ==== Imaginary-time ordering and β-periodicity ==== The thermal Green functions are defined only when both imaginary-time arguments are within the range 0 {\displaystyle 0} to β {\displaystyle \beta } . The two-point Green function has the following properties. (The position or momentum arguments are suppressed in this section.) Firstly, it depends only on the difference of the imaginary times: G ( τ , τ ′ ) = G ( τ − τ ′ ) . {\displaystyle {\mathcal {G}}(\tau ,\tau ')={\mathcal {G}}(\tau -\tau ').} The argument τ − τ ′ {\displaystyle \tau -\tau '} is allowed to run from − β {\displaystyle -\beta } to β {\displaystyle \beta } . Secondly, G ( τ ) {\displaystyle {\mathcal {G}}(\tau )} is (anti)periodic under shifts of β {\displaystyle \beta } . Because of the small domain within which the function is defined, this means just G ( τ − β ) = ζ G ( τ ) , {\displaystyle {\mathcal {G}}(\tau -\beta )=\zeta {\mathcal {G}}(\tau ),} for 0 < τ < β {\displaystyle 0<\tau <\beta } . Time ordering is crucial for this property, which can be proved straightforwardly, using the cyclicity of the trace operation. These two properties allow for the Fourier transform representation and its inverse, G ( ω n ) = ∫ 0 β
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
d τ G ( τ ) e i ω n τ . {\displaystyle {\mathcal {G}}(\omega _{n})=\int _{0}^{\beta }d\tau \,{\mathcal {G}}(\tau )\,e^{i\omega _{n}\tau }.} Finally, note that G ( τ ) {\displaystyle {\mathcal {G}}(\tau )} has a discontinuity at τ = 0 {\displaystyle \tau =0} ; this is consistent with a long-distance behaviour of G ( ω n ) ∼ 1 / | ω n | {\displaystyle {\mathcal {G}}(\omega _{n})\sim 1/|\omega _{n}|} . === Spectral representation === The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by ρ ( k , ω ) = 1 Z ∑ α , α ′ 2 π δ ( E α − E α ′ − ω ) | ⟨ α ∣ ψ k † ∣ α ′ ⟩ | 2 ( e − β E α ′ − ζ e − β E α ) , {\displaystyle \rho (\mathbf {k} ,\omega )={\frac {1}{\mathcal {Z}}}\sum _{\alpha ,\alpha '}2\pi \delta (E_{\alpha }-E_{\alpha '}-\omega )|\langle \alpha \mid \psi _{\mathbf {k} }^{\dagger }\mid \alpha '\rangle |^{2}\left(e^{-\beta E_{\alpha '}}-\zeta e^{-\beta E_{\alpha }}\right),} where |α⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian H − μN, with eigenvalue Eα. The imaginary-time propagator is then given by G ( k , ω n ) = ∫ − ∞ ∞ d ω ′ 2 π ρ ( k , ω ′ ) − i ω n + ω ′ , {\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})=\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\rho (\mathbf {k} ,\omega ')}{-i\omega _{n}+\omega '}}~,} and the retarded propagator by G R ( k , ω ) = ∫ − ∞ ∞ d ω ′ 2 π ρ ( k , ω ′ ) − ( ω + i η ) + ω ′ , {\displaystyle G^{\mathrm
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
{R} }(\mathbf {k} ,\omega )=\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\rho (\mathbf {k} ,\omega ')}{-(\omega +i\eta )+\omega '}},} where the limit as η → 0 + {\displaystyle \eta \to 0^{+}} is implied. The advanced propagator is given by the same expression, but with − i η {\displaystyle -i\eta } in the denominator. The time-ordered function can be found in terms of G R {\displaystyle G^{\mathrm {R} }} and G A {\displaystyle G^{\mathrm {A} }} . As claimed above, G R ( ω ) {\displaystyle G^{\mathrm {R} }(\omega )} and G A ( ω ) {\displaystyle G^{\mathrm {A} }(\omega )} have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) half-plane. The thermal propagator G ( ω n ) {\displaystyle {\mathcal {G}}(\omega _{n})} has all its poles and discontinuities on the imaginary ω n {\displaystyle \omega _{n}} axis. The spectral density can be found very straightforwardly from G R {\displaystyle G^{\mathrm {R} }} , using the Sokhatsky–Weierstrass theorem lim η → 0 + 1 x ± i η = P 1 x ∓ i π δ ( x ) , {\displaystyle \lim _{\eta \to 0^{+}}{\frac {1}{x\pm i\eta }}=P{\frac {1}{x}}\mp i\pi \delta (x),} where P denotes the Cauchy principal part. This gives ρ ( k , ω ) = 2 Im G R ( k , ω ) . {\displaystyle \rho (\mathbf {k} ,\omega )=2\operatorname {Im} G^{\mathrm {R} }(\mathbf {k} ,\omega ).} This furthermore implies that G R ( k , ω ) {\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )} obeys the following relationship between its real and imaginary parts: Re G R ( k , ω ) = − 2 P ∫ − ∞ ∞ d ω ′ 2 π Im G R ( k , ω ′ ) ω
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
− ω ′ , {\displaystyle \operatorname {Re} G^{\mathrm {R} }(\mathbf {k} ,\omega )=-2P\int _{-\infty }^{\infty }{\frac {d\omega '}{2\pi }}{\frac {\operatorname {Im} G^{\mathrm {R} }(\mathbf {k} ,\omega ')}{\omega -\omega '}},} where P {\displaystyle P} denotes the principal value of the integral. The spectral density obeys a sum rule, ∫ − ∞ ∞ d ω 2 π ρ ( k , ω ) = 1 , {\displaystyle \int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}\rho (\mathbf {k} ,\omega )=1,} which gives G R ( ω ) ∼ 1 | ω | {\displaystyle G^{\mathrm {R} }(\omega )\sim {\frac {1}{|\omega |}}} as | ω | → ∞ {\displaystyle |\omega |\to \infty } . ==== Hilbert transform ==== The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function G ( k , z ) = ∫ − ∞ ∞ d x 2 π ρ ( k , x ) − z + x , {\displaystyle G(\mathbf {k} ,z)=\int _{-\infty }^{\infty }{\frac {dx}{2\pi }}{\frac {\rho (\mathbf {k} ,x)}{-z+x}},} which is related to G {\displaystyle {\mathcal {G}}} and G R {\displaystyle G^{\mathrm {R} }} by G ( k , ω n ) = G ( k , i ω n ) {\displaystyle {\mathcal {G}}(\mathbf {k} ,\omega _{n})=G(\mathbf {k} ,i\omega _{n})} and G R ( k , ω ) = G ( k , ω + i η ) . {\displaystyle G^{\mathrm {R} }(\mathbf {k} ,\omega )=G(\mathbf {k} ,\omega +i\eta ).} A similar expression obviously holds for G A {\displaystyle G^{\mathrm {A} }} . The relation between G ( k , z ) {\displaystyle G(\mathbf {k} ,z)} and ρ ( k , x ) {\displaystyle \rho (\mathbf {k} ,x)} is referred to as a Hilbert transform. ==== Proof of spectral representation ==== We demonstrate the proof of the spectral representation
|
{
"page_id": 7864709,
"source": null,
"title": "Green's function (many-body theory)"
}
|
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