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.
For a first order system of differential equations
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.
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.
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
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.
- Blanchard, Devaney, Hall (2005). Differential Equations. Brooks/Cole Publishing Co. p. 486. ISBN 0-495-01265-3.CS1 maint: multiple names: authors list (link)