Conserved quantity

In mathematics, a conserved quantity of a dynamical system is a function H of the dependent variables that is a constant (in other words, conserved)[1] along each trajectory of the system. A conserved quantity can be a useful tool for qualitative analysis. Not all systems have conserved quantities, however the existence has nothing to do with linearity (a simplifying trait in a system) which means that finding and examining conserved quantities can be useful in understanding nonlinear systems.

Conserved quantities are not unique, since one can always add a constant to a conserved quantity.

Since most laws of physics express some kind of conservation, conserved quantities commonly exist in mathematic models of real 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}{d t} = \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{d H}{d t} = 0

Note that by using the multivariate chain rule,

\frac{d H}{d t} = \nabla H \cdot \frac{d \mathbf r}{d t} = \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

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