nLab
conservation law

Contents

Idea

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 NoetherNoether'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.

References

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

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)