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mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, 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 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.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
, rather than a ''physical'' constraint (which would require extra
constraint force Constraint may refer to: * Constraint (computer-aided design), a demarcation of geometrical characteristics between two or more entities or solid modeling bodies * Constraint (mathematics), a condition of an optimization problem that the solution m ...
s). Common examples include
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
,
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
,
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and the
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 binary star or a planet revolving around a star. For t ...
(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 physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
. In fortunate cases, even the
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
of the motion can be derived as the intersection of
isosurface An isosurface is a three-dimensional analog of an isoline. It is a surface that represents points of a constant value (e.g. pressure, temperature, velocity, density) within a volume of space; in other words, it is a level set of a continuous f ...
s corresponding to the constants of motion. For example,
Poinsot's construction In classical mechanics, Poinsot's construction (after Louis Poinsot) is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion ha ...
shows that the torque-free
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
of a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
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 Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
.


Methods for identifying constants of motion

There are several methods for identifying constants of motion. * The simplest but least systematic approach is the intuitive ("psychic") derivation, in which a quantity is hypothesized to be constant (perhaps because of
experimental data Experimental data in science and engineering is data produced by a measurement, test method, experimental design or quasi-experimental design. In clinical research any data produced are the result of a clinical trial. Experimental data may be qua ...
) and later shown mathematically to be conserved throughout the motion. * The
Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
s provide a commonly used and straightforward method for identifying constants of motion, particularly when the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
adopts recognizable functional forms in
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
. * Another approach is to recognize that a
conserved quantity In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are ...
corresponds to a
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
.
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
provides a systematic way of deriving such quantities from the symmetry. For example,
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
results from the invariance of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
under shifts in the origin of
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
,
conservation of linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
results from the invariance of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
under shifts in the origin of
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
(''translational symmetry'') and
conservation of angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system ...
results from the invariance of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
under
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s. The converse is also true; every symmetry of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
corresponds to a constant of motion, often called a ''conserved charge'' or ''current''. * A quantity A is a constant of the motion if its total time derivative is zero : 0 = \frac = \frac + \, :which occurs when A's
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. Th ...
with the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
equals minus its partial derivative with respect to time : \frac = -\. Another useful result is Poisson's theorem, which states that if two quantities A and B 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 In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. Such a collection of constants of motion are said to be in
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
with each other. For a
closed system A closed system is a natural physical system that does not allow transfer of matter in or out of the system, although — in contexts such as physics, chemistry or engineering — the transfer of energy (''e.g.'' as work or heat) is allowed. In ...
(
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
not explicitly dependent on time), the energy of the system is a constant of motion (a
conserved quantity In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are ...
).


In quantum mechanics

An observable quantity ''Q'' will be a constant of motion if it commutes with the
hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, ''H'', and it does not itself depend explicitly on time. This is because ::\frac \langle \psi , Q , \psi \rangle = \frac \langle \psi, \left H,Q \right\psi \rangle + \langle \psi , \frac , \psi \rangle \, where : ,Q= HQ - QH \, is the commutator relation.


Derivation

Say there is some observable quantity ''Q'' which depends on position, momentum and time, ::Q = Q(x,p,t) \, And also, that there is a
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
which obeys Schrödinger's equation ::i\hbar \frac = H \psi .\, Taking the time derivative of the expectation value of ''Q'' requires use of the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
, and results in :: So finally, ::


Comment

For an arbitrary state of a Quantum Mechanical system, if H and Q commute, i.e. if ::\left H,Q \right= 0 and Q is not explicitly dependent on time, then ::\frac \langle Q \rangle = 0 But if \psi is an eigenfunction of Hamiltonian, then even if ::\left ,Q\right \neq 0 it is still the case that ::\frac\langle Q \rangle = 0 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 In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
has constants of motion other than the energy. By contrast,
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
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. Examples of integrals of motion are the angular momentum vector, \mathbf = \mathbf \times \mathbf, or a Hamiltonian without time dependence, such as H(\mathbf,\mathbf) = \frac v^2 + \Phi(\mathbf). An example of a function that is a constant of motion but not an integral of motion would be the function C(x,v,t) = x - vt for an object moving at a constant speed in one dimension.


Dirac observables

In order to extract physical information from
gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
, one either constructs
gauge invariant In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
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 A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanis ...
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

* {{DEFAULTSORT:Constant Of Motion Classical mechanics