conservation law

Given a dynamical system, a **conservation law** is the statement that some observable is time independent *in every solution*. Thus it is the same as an observable whose value is time-independent in each solution. Of course, in some particular solution there can be a conserved quantity which is not coming from a conservation law, but is an incidental feature of just that particular solution of equations of motion. Thus a conservation law is the same as a “universally” **conserved observable**.

In quantum mechanics an observable is conserved in time if it commutes with the Hamiltonian operator. In classical mechanics, there is a famous theorem of Emmy Noether – Noether's theorem – which assigns a conservation law to any smooth symmetry of system. For example the isotropy of a space is related to the conservation of angular momentum, and the homogeneity of the space to the conservation of usual momentum.

For 1-parameter groups of symmetries in classical mechanics, the formulation and the proof of Noether's theorem can be found in the monograph

- Vladimir Arnold,
*Mathematical methods of classical mechanics*

For more general case see the books by Peter Olver.

Revised on August 27, 2012 21:42:41
by Urs Schreiber
(89.204.130.6)