Constant of motion

In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint (which would require extra constraint forces). Common examples include specific energy, specific linear momentum, specific angular momentum and the Laplace–Runge–Lenz vector (for inverse-square force laws).

Applications

Constants of motion are useful because they allow properties of the motion to be derived without solving the equations of motion. In fortunate cases, even the trajectory of the motion can be derived as the intersection of isosurfaces corresponding to the constants of motion. For example, Poinsot's construction shows that the torque-free rotation of a rigid body is the intersection of a sphere (conservation of total angular momentum) and an ellipsoid (conservation of energy), a trajectory that might be otherwise hard to derive and visualize. Therefore, the identification of constants of motion is an important objective in mechanics.

Methods for identifying constants of motion

There are several methods for identifying constants of motion.

Another useful result is Poisson's theorem, which states that if two quantities and are constants of motion, so is their Poisson bracket .

A system with n degrees of freedom, and n constants of motion, such that the Poisson bracket of any pair of constants of motion vanishes, is known as a completely integrable system. Such a collection of constants of motion are said to be in involution with each other.

In quantum mechanics

An observable quantity Q will be a constant of motion if it commutes with the hamiltonian, H, and it does not itself depend explicitly on time. This is because

where

is the commutator relation.

Derivation

Say there is some observable quantity Q which depends on position, momentum and time,

And also, that there is a wave function which obeys Schrödinger's equation

Taking the time derivative of the expectation value of Q requires use of the product rule, and results in

So finally,

Comment

For an arbitrary state of a Quantum Mechanical system, if H and Q commute, i.e. if

and Q is not explicitly dependent on time, then

But if is an eigenfunction of Hamiltonian, then even if

it is still the case that

provided Q is independent on time.

Derivation

Since

then

This is the reason why Eigenstates of the Hamiltonian are also called stationary states.

Relevance for quantum chaos

In general, an integrable system has constants of motion other than the energy. By contrast, energy is the only constant of motion in a non-integrable system; such systems are termed chaotic. In general, a classical mechanical system can be quantized only if it is integrable; as of 2006, there is no known consistent method for quantizing chaotic dynamical systems.

Integral of motion

A constant of motion may be defined in a given force field as any function of phase-space coordinates (position and velocity, or position and momentum) and time that is constant throughout a trajectory. A subset of the constants of motion are the integrals of motion, or first integrals, defined as any functions of only the phase-space coordinates that are constant along an orbit. Every integral of motion is a constant of motion, but the converse is not true because a constant of motion may depend on time.[1] Examples of integrals of motion are the angular momentum vector, , or a Hamiltonian without time dependence, such as . An example of a function that is a constant of motion but not an integral of motion would be the function for an object moving at a constant speed in one dimension.

Dirac observables

In order to extract physical information from gauge theories, one either constructs gauge invariant observables or fixes a gauge. In a canonical language, this usually means either constructing functions which Poisson-commute on the constraint surface with the gauge generating first class constraints or to fix the flow of the latter by singling out points within each gauge orbit. Such gauge invariant observables are thus the `constants of motion' of the gauge generators and referred to as Dirac observables.

References

  1. ^ "Binney, J. and Tremaine, S.: Galactic Dynamics". Princeton University Press. Retrieved 2011-05-05.
Angular momentum coupling

In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin–orbit interaction, in which case the complete physical picture must include spin-orbit coupling. Or two charged particles, each with a well-defined angular momentum, may interact by Coulomb forces, in which case coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle Schrödinger equation.

In both cases the separate angular momenta are no longer constants of motion, but the sum of the two angular momenta usually still is. Angular momentum coupling in atoms is of importance in atomic spectroscopy. Angular momentum coupling of electron spins is of importance in quantum chemistry. Also in the nuclear shell model angular momentum coupling is ubiquitous.In astronomy, spin-orbit coupling reflects the general law of conservation of angular momentum, which holds for celestial systems as well. In simple cases, the direction of the angular momentum vector is neglected, and the spin-orbit coupling is the ratio between the frequency with which a planet or other celestial body spins about its own axis to that with which it orbits another body. This is more commonly known as orbital resonance. Often, the underlying physical effects are tidal forces.

Brandon Carter

Brandon Carter, FRS (born 1942) is an Australian theoretical physicist, best known for his work on the properties of black holes and for being the first to name and employ the anthropic principle in its contemporary form. He is a researcher at the Meudon campus of the Laboratoire Univers et Théories, part of the CNRS.

Crystal momentum

In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors of this lattice, according to

(where is the reduced Planck's constant). Frequently[clarification needed], crystal momentum is conserved like mechanical momentum, making it useful to physicists and materials scientists as an analytical tool.

Eccentricity vector

In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vector is a constant of motion. Its main use is in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously. For the eccentricity and argument of periapsis parameters, eccentricity zero (circular orbit) corresponds to a singularity. The magnitude of the eccentricity vector represents the eccentricity of the orbit. Note that the velocity and position vectors need to be relative to the inertial frame of the central body.

Energy drift

In computer simulations of mechanical systems, energy drift is the gradual change in the total energy of a closed system over time. According to the laws of mechanics, the energy should be a constant of motion and should not change. However, in simulations the energy might fluctuate on a short time scale and increase or decrease on a very long time scale due to numerical integration artifacts that arise with the use of a finite time step Δt. This is somewhat similar to the flying ice cube problem, whereby numerical errors in handling equipartition of energy can change vibrational energy into translational energy.

More specifically, the energy tends to increase exponentially; its increase can be understood intuitively because each step introduces a small perturbation δv to the true velocity vtrue, which (if uncorrelated with v, which will be true for simple integration methods) results in a second-order increase in the energy

(The cross term in v · δv is zero because of no correlation.)

Energy drift - usually damping - is substantial for numerical integration schemes that are not symplectic, such as the Runge-Kutta family. Symplectic integrators usually used in molecular dynamics, such as the Verlet integrator family, exhibit increases in energy over very long time scales, though the error remains roughly constant. These integrators do not in fact reproduce the actual Hamiltonian mechanics of the system; instead, they reproduce a closely related "shadow" Hamiltonian whose value they conserve many orders of magnitude more closely. The accuracy of the energy conservation for the true Hamiltonian is dependent on the time step. The energy computed from the modified Hamiltonian of a symplectic integrator is from the true Hamiltonian.

Energy drift is similar to parametric resonance in that a finite, discrete timestepping scheme will result in nonphysical, limited sampling of motions with frequencies close to the frequency of velocity updates. Thus the restriction on the maximum step size that will be stable for a given system is proportional to the period of the fastest fundamental modes of the system's motion. For a motion with a natural frequency ω, artificial resonances are introduced when the frequency of velocity updates, is related to ω as

where n and m are integers describing the resonance order. For Verlet integration, resonances up to the fourth order frequently lead to numerical instability, leading to a restriction on the timestep size of

where ω is the frequency of the fastest motion in the system and p is its period. The fastest motions in most biomolecular systems involve the motions of hydrogen atoms; it is thus common to use constraint algorithms to restrict hydrogen motion and thus increase the maximum stable time step that can be used in the simulation. However, because the time scales of heavy-atom motions are not widely divergent from those of hydrogen motions, in practice this allows only about a twofold increase in time step. Common practice in all-atom biomolecular simulation is to use a time step of 1 femtosecond (fs) for unconstrained simulations and 2 fs for constrained simulations, although larger time steps may be possible for certain systems or choices of parameters.

Energy drift can also result from imperfections in evaluating the energy function, usually due to simulation parameters that sacrifice accuracy for computational speed. For example, cutoff schemes for evaluating the electrostatic forces introduce systematic errors in the energy with each time step as particles move back and forth across the cutoff radius if sufficient smoothing is not used. Particle mesh Ewald summation is one solution for this effect, but introduces artifacts of its own. Errors in the system being simulated can also induce energy drifts characterized as "explosive" that are not artifacts, but are reflective of the instability of the initial conditions; this may occur when the system has not been subjected to sufficient structural minimization before beginning production dynamics. In practice, energy drift may be measured as a percent increase over time, or as a time needed to add a given amount of energy to the system. The practical effects of energy drift depend on the simulation conditions, the thermodynamic ensemble being simulated, and the intended use of the simulation under study; for example, energy drift has much more severe consequences for simulations of the microcanonical ensemble than the canonical ensemble where the temperature is held constant. Energy drift is often used as a measure of the quality of the simulation, and has been proposed as one quality metric to be routinely reported in a mass repository of molecular dynamics trajectory data analogous to the Protein Data Bank.

Euler's three-body problem

In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760. Important extensions and analyses were contributed subsequently by Lagrange, Liouville, Laplace, Jacobi, Darboux, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff and E. T. Whittaker, among others.Euler's problem also covers the case when the particle is acted upon by other inverse-square central forces, such as the electrostatic interaction described by Coulomb's law. The classical solutions of the Euler problem have been used to study chemical bonding, using a semiclassical approximation of the energy levels of a single electron moving in the field of two atomic nuclei, such as the diatomic ion HeH2+. This was first done by Wolfgang Pauli in his doctoral dissertation under Arnold Sommerfeld, a study of the first ion of molecular hydrogen, namely the Hydrogen molecule-ion H2+. These energy levels can be calculated with reasonable accuracy using the Einstein–Brillouin–Keller method, which is also the basis of the Bohr model of atomic hydrogen. More recently, as explained further in the quantum-mechanical version, analytical solutions to the eigenenergies have been obtained: these are a generalization of the Lambert W function.

The exact solution, in the full three dimensional case, is explicitated in terms of Weierstrass's elliptic functions For convenience, the problem may also be solved by numerical methods, such as Runge–Kutta integration of the equations of motion. The total energy of the moving particle is conserved, but its linear and angular momentum are not, since the two fixed centers can apply a net force and torque. Nevertheless, the particle has a second conserved quantity that corresponds to the angular momentum or to the Laplace–Runge–Lenz vector as limiting cases.

The Euler three-body problem is known by a variety of names, such as the problem of two fixed centers, the Euler–Jacobi problem, and the two-center Kepler problem. Various generalizations of Euler's problem are known; these generalizations add linear and inverse cubic forces and up to five centers of force. Special cases of these generalized problems include Darboux's problem and Velde's problem.

Good quantum number

In quantum mechanics, given a particular Hamiltonian and an operator with corresponding eigenvalues and eigenvectors given by , then the numbers (or the eigenvalues) are said to be "good quantum numbers" if every eigenvector remains an eigenvector of with the same eigenvalue as time evolves.

Hence, if:

then we require

for all eigenvectors in order to call a good quantum number (where s represent the eigenvectors of the Hamiltonian).

In other words, the eigenvalues are good quantum numbers if the corresponding operator is a constant of motion (commutes with the time evolution). Good quantum numbers are often used to label initial and final states in experiments. For example, in particle colliders particles are initially prepared in approximate momentum eigenstates (the particle momentum being a good quantum number for non-interacting particles), then the particles are made collide (particle momentum is not a good quantum number for the interacting particles and thus changes), and after the collision particles are measured in momentum eigenstates (a long time after the collision, particle momentum again is a good quantum number).

Theorem: A necessary and sufficient condition for q (which is an eigenvalue of an operator O) to be good is that commutes with the Hamiltonian .

Proof: Assume .

If is an eigenvector of , then we have (by definition) that , and so :
Hamiltonian system

A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.

Hypothesis

A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories. Even though the words "hypothesis" and "theory" are often used synonymously, a scientific hypothesis is not the same as a scientific theory. A working hypothesis is a provisionally accepted hypothesis proposed for further research, in a process beginning with an educated guess or thought.A different meaning of the term hypothesis is used in formal logic, to denote the antecedent of a proposition; thus in the proposition "If P, then Q", P denotes the hypothesis (or antecedent); Q can be called a consequent. P is the assumption in a (possibly counterfactual) What If question.

The adjective hypothetical, meaning "having the nature of a hypothesis", or "being assumed to exist as an immediate consequence of a hypothesis", can refer to any of these meanings of the term "hypothesis".

Jacobi integral

In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem. Unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible. The integral has been used to derive numerous solutions in special cases.

It was named after German mathematician Carl Gustav Jacob Jacobi.

Jakob Hermann

Jakob Hermann (16 July 1678 – 11 July 1733) was a mathematician who worked on problems in classical mechanics. He is the author of Phoronomia, an early treatise on Mechanics in Latin, which has been translated by Ian Bruce in 2015-16. In 1729, he proclaimed that it was as easy to graph a locus on the polar coordinate system as it was to graph it on the Cartesian coordinate system.

He appears to have been the first to show that the Laplace–Runge–Lenz vector is a constant of motion for particles acted upon by an inverse-square central force.Hermann was born and died in Basel. He received his initial training from Jacob Bernoulli and graduated with a degree in 1695. He became a member of the Berlin Academy in 1701. He was appointed to a chair in mathematics in Padua in 1707, but moved to Frankfurt an der Oder in 1713, and thence to St. Petersburg in 1724. Finally, he returned to Basel in 1731 to take a chair in ethics and natural law.Hermann was elected to the Académie Royale des Sciences (Paris) in 1733, the year of his death.

Hermann was a distant relative of Leonhard Euler.

Kerr metric

The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially-symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.

Laplace–Runge–Lenz vector

In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today.

In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system. The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carl Runge and Wilhelm Lenz. It is also known as the Laplace vector, the Runge–Lenz vector and the Lenz vector. Ironically, none of those scientists discovered it. The LRL vector has been re-discovered several times and is also equivalent to the dimensionless eccentricity vector of celestial mechanics. Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.

Liouville dynamical system

In classical mechanics, a Liouville dynamical system is an exactly soluble dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows:

The solution of this system consists of a set of separably integrable equations

where E = T + V is the conserved energy and the are constants. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs. This solution has numerous applications, such as the orbit of a small planet about two fixed stars under the influence of Newtonian gravity. The Liouville dynamical system is one of several things named after Joseph Liouville, an eminent French mathematician.

Lotka–Volterra equations

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

where

x is the number of prey (for example, rabbits);
y is the number of some predator (for example, foxes);
and represent the instantaneous growth rates of the two populations;
t represents time;
α, β, γ, δ are positive real parameters describing the interaction of the two species.

The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.

Nutation

Nutation (from Latin nūtātiō, "nodding, swaying") is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. If it is not caused by forces external to the body, it is called free nutation or Euler nutation. A pure nutation is a movement of a rotational axis such that the first Euler angle is constant. In spacecraft dynamics, precession (a change in the first Euler angle) is sometimes referred to as nutation.

Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by and , respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.

In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups.

All of these objects are named in honour of Siméon Denis Poisson.

Tisserand's criterion

Tisserand's criterion is used to determine whether or not an observed orbiting body, such as a comet or an asteroid, is the same as a previously observed orbiting body.While all the orbital parameters of an object orbiting the Sun during the close encounter with another massive body (e.g. Jupiter) can be changed dramatically, the value of a function of these parameters, called Tisserand's relation (due to Félix Tisserand) is approximately conserved, making it possible to recognize the orbit after the encounter.

Von Neumann entropy

In quantum statistical mechanics, the von Neumann entropy, named after John von Neumann, is the extension of classical Gibbs entropy concepts to the field of quantum mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is

where denotes the trace and ln denotes the (natural) matrix logarithm. If ρ is written in terms of its eigenvectors |1〉, |2〉, |3〉, ... as

then the von Neumann entropy is merely

In this form, S can be seen to amount to the information theoretic Shannon entropy.[clarification needed]

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