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Conserved quantity

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In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables whose value remains constant along each trajectory of the system.[1]

Not all systems have conserved quantities, and conserved quantities are not unique, since one can always apply a function to a conserved quantity, such as adding a number.

Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have energy as a conserved quantity so long as the forces involved are conservative.

Contents

Differential equationsEdit

For a first order system of differential equations

 

where bold indicates vector quantities, a scalar-valued function H(r) is a conserved quantity of the system if, for all time and initial conditions in some specific domain,

 

Note that by using the multivariate chain rule,

 

so that the definition may be written as

 

which contains information specific to the system and can be helpful in finding conserved quantities, or establishing whether or not a conserved quantity exists.

Hamiltonian mechanicsEdit

For a system defined by the Hamiltonian H, a function f of the generalized coordinates q and generalized momenta p has time evolution

 

and hence is conserved if and only if  . Here   denotes the Poisson Bracket.

Lagrangian mechanicsEdit

Suppose a system is defined by the Lagrangian L with generalized coordinates q. If L has no explicit time dependence (so  ), then the energy E defined by

 

is conserved.

Furthermore, if  , then q is said to be a cyclic coordinate and the generalized momentum p defined by

 

is conserved. This may be derived by using the Euler–Lagrange equations.

See alsoEdit

ReferencesEdit

  1. ^ Blanchard, Devaney, Hall (2005). Differential Equations. Brooks/Cole Publishing Co. p. 486. ISBN 0-495-01265-3.