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| In [[mechanics]], a '''constant of motion''' is a [[conservation law|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 [[Equation of motion|equations of motion]], rather than a ''physical'' [[Constraint (mathematics)|constraint]] (which would require extra [[constraint force]]s). Common examples include [[conservation of energy|specific energy]], [[momentum#Conservation_of_linear_momentum|specific linear momentum]], [[angular_momentum#Conservation_of_angular_momentum|specific angular momentum]] and the [[Laplace–Runge–Lenz vector]] (for [[inverse-square law|inverse-square force laws]]).
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| ==Applications==
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| Constants of motion are useful because they allow properties of the motion to be derived without solving the [[Equation of motion|equations of motion]]. In fortunate cases, even the [[trajectory]] of the motion can be derived as the [[Intersection (set theory)|intersection]] of [[isosurface]]s corresponding to the constants of motion. For example, [[Poinsot's ellipsoid|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]].
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| ==Methods for identifying constants of motion==
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| There are several methods for identifying constants of motion.
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| * 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]]) and later shown mathematically to be conserved throughout the motion.
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| * The [[Hamilton–Jacobi equation]]s provide a commonly used and straightforward method for identifying constants of motion, particularly when the [[Hamiltonian mechanics|Hamiltonian]] adopts recognizable functional forms in [[orthogonal coordinates]].
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| * Another approach is to recognize that a [[conservation law|conserved quantity]] corresponds to a [[symmetry]] of the [[Lagrangian]]. [[Noether's theorem]] provides a systematic way of deriving such quantities from the symmetry. For example, [[conservation of energy]] results from the invariance of the [[Lagrangian]] under shifts in the origin of [[time]], [[momentum#Conservation_of_linear_momentum|conservation of linear momentum]] results from the invariance of the [[Lagrangian]] under shifts in the origin of [[space]] (''translational symmetry'') and [[angular_momentum#Conservation_of_angular_momentum|conservation of angular momentum]] results from the invariance of the [[Lagrangian]] under [[rotation]]s. The converse is also true; every symmetry of the [[Lagrangian]] corresponds to a constant of motion, often called a ''conserved charge'' or ''current''.
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| * A quantity <math>A</math> is conserved if it is not explicitly time-dependent and if its [[Poisson bracket]] with the [[Hamiltonian mechanics|Hamiltonian]] is zero
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| :<math>
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| \frac{dA}{dt} = \frac{\partial A}{\partial t} + \{A, H\}
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| </math>
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| Another useful result is '''Poisson's theorem''', which states that if two quantities <math>A</math> and <math>B</math> are constants of motion, so is their Poisson bracket <math>\{A, B\}</math>.
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| 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 (mathematics)|involution]] with each other.
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| ==In quantum mechanics==
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| An observable quantity ''Q'' will be a constant of motion if it [[Commutator|commutes]] with the [[Hamiltonian mechanics|hamiltonian]], ''H'', and it does not itself depend explicitly on time. This is because
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| ::<math>\frac{d}{dt} \langle \psi | Q | \psi \rangle = \frac{-1}{i \hbar} \langle \psi|\left[ H,Q \right]|\psi \rangle + \langle \psi | \frac{dQ}{dt} | \psi \rangle \,</math>
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| where
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| :<math>[H,Q] = HQ - QH \,</math>
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| is the commutator relation.
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| ===Derivation===
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| Say there is some observable quantity ''Q'' which depends on position, momentum and time,
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| ::<math>Q = Q(x,p,t) \,</math>
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| And also, that there is a [[wave function]] which obeys [[Schrödinger equation|Schrödinger's equation]]
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| ::<math>i\hbar \frac{\partial\psi}{\partial t} = H \psi .\,</math>
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| Taking the time derivative of the expectation value of ''Q'' requires use of the [[product rule]], and results in
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| ::{|
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| |<math>\frac{d}{dt} \langle Q \rangle \,</math>
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| |<math> = \frac{d}{dt} \langle \psi | Q | \psi \rangle \,</math>
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| |-
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| |<math> = \langle \frac{d\psi}{dt} | Q | \psi \rangle + \langle \psi | \frac{dQ}{dt} | \psi \rangle + \langle \psi | Q | \frac{d\psi}{dt} \rangle\,</math>
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| |-
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| |<math> = \frac{-1}{i\hbar} \langle H \psi | Q | \psi \rangle + \langle \psi | \frac{dQ}{dt} | \psi \rangle + \frac{1}{i\hbar}\langle \psi | Q | H \psi \rangle \,</math>
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| |-
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| |<math> = \frac{-1}{i\hbar} \langle \psi | HQ | \psi \rangle + \langle \psi | \frac{dQ}{dt} | \psi \rangle + \frac{1}{i\hbar}\langle \psi | QH | \psi \rangle \,</math>
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| |-
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| |<math>= \frac{-1}{i \hbar} \langle \psi|\left[H,Q\right]|\psi \rangle + \langle \psi | \frac{dQ}{dt} | \psi \rangle \,</math>
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| |}
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| So finally,
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| ::{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>\frac{d}{dt} \langle \psi | Q | \psi \rangle = \frac{-1}{i \hbar} \langle \psi| \left[ H,Q \right]|\psi \rangle + \langle \psi | \frac{dQ}{dt} | \psi \rangle \,</math>
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| |}
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| ===Comment===
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| For an arbitrary state of a Quantum Mechanical system, if H and Q commute, i.e. if
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| ::<math>\left[ H,Q \right] = 0 </math>
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| and Q is not explicitly dependent on time, then
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| ::<math>\frac{d}{dt} \langle Q \rangle = 0 </math>
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| But if <math>\psi</math> is an eigenfunction of Hamiltonian, then even if
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| ::<math>\left[H,Q\right] \neq 0 </math>
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| ::<math>\frac{d}{dt}\langle Q \rangle = 0 </math>
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| provided Q is not explicitly dependent on time.
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| ===Derivation===
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| ::{|
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| |<math> \frac{d}{dt} \langle Q \rangle = \frac{-1}{i\hbar} \langle \psi | \left[ H,Q \right] | \psi\rangle \,</math>
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| |-
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| |<math>= \frac{-1}{i\hbar} \langle \psi | HQ - QH | \psi \rangle \,</math>
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| |}
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| Since
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| :{|
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| |<math> H|\psi\rangle = E |\psi \rangle \,</math>
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| |-
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| |<math> \frac{d}{dt} \langle Q \rangle = \frac{-1}{i\hbar} \left( E \langle \psi | Q | \psi \rangle - E \langle \psi | Q | \psi \rangle \right) \,</math>
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| |-
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| |<math> = 0 </math>
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| |}
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| This is the reason why Eigen states of Hamiltonian are also called as stationary states.
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| ==Relevance for quantum chaos==
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| In general, an [[integrable system]] has constants of motion other than the energy. By contrast, [[energy]] is the only constant of motion in a [[Dynamical system|non-integrable system]]; such systems are termed chaotic. In general, a classical mechanical system can be [[quantum mechanics|quantized]] only if it is integrable; as of 2006, there is no known consistent method for quantizing chaotic dynamical systems.
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| ==Integral of motion==
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| A constant of motion may be defined in a given force field as any function of [[phase space|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.<ref>{{cite web|url = http://press.princeton.edu/titles/8697.html|title = Binney, J. and Tremaine, S.: Galactic Dynamics.|publisher = Princeton University Press|accessdate = 2011-05-05}}</ref> Examples of integrals of motion are the angular momentum vector, <math>\mathbf{L} = \mathbf{x} \times \mathbf{v}</math>, or a Hamiltonian without time dependence, such as <math>H(\mathbf{x},\mathbf{v}) = \frac{1}{2} v^2 + \Phi</math>. An example of a function that is a constant of motion but not an integral of motion would be the function <math>C(x,v,t) = x - vt</math> for an object moving at a constant speed in one dimension.
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| ==References==
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| {{reflist|1}}
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| *{{cite book | author=Griffiths, David J. | authorlink = David J. Griffiths | title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall | year=2004 | isbn=0-13-805326-X}}
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| {{DEFAULTSORT:Constant Of Motion}}
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| [[Category:Classical mechanics]]
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The author's name is Christy. For a while I've been in Mississippi but now I'm contemplating other options. Distributing manufacturing is exactly where my primary income comes from and it's something I really enjoy. As a lady what she really likes is style and she's been doing it for fairly a whilst.
Here is my web blog - online psychic reading (her comment is here)