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.

## Differential equations

For a first order system of differential equations

- ${\frac {d\mathbf {r} }{dt}}=\mathbf {f} (\mathbf {r} ,t)$

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,

- ${\frac {dH}{dt}}=0$

Note that by using the multivariate chain rule,

- ${\frac {dH}{dt}}=\nabla H\cdot {\frac {d\mathbf {r} }{dt}}=\nabla H\cdot \mathbf {f} (\mathbf {r} ,t)$

so that the definition may be written as

- $\nabla H\cdot \mathbf {f} (\mathbf {r} ,t)=0$

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 mechanics

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

- ${\frac {\mathrm {d} f}{\mathrm {d} t}}=\{f,{\mathcal {H}}\}+{\frac {\partial f}{\partial t}}$

and hence is conserved if and only if $\{f,{\mathcal {H}}\}+{\frac {\partial f}{\partial t}}=0$. Here $\{f,{\mathcal {H}}\}$ denotes the Poisson Bracket.

## Lagrangian mechanics

Suppose a system is defined by the Lagrangian *L* with generalized coordinates *q*. If *L* has no explicit time dependence (so ${\frac {\partial L}{\partial t}}=0$), then the energy *E* defined by

- $E=\sum _{i}\left[{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right]-L$

is conserved.

Furthermore, if ${\frac {\partial L}{\partial q}}=0$, then *q* is said to be a cyclic coordinate and the generalized momentum *p* defined by

- $p={\frac {\partial L}{\partial {\dot {q}}}}$

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

## See also

## References

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