Noether's symmetry theorem associates a conservation law with each continuous one-parameter variational symmetry of an action integral, under suitable regularity hypotheses (Noether 1918, Olver 1986). The invariance may hold up to a total derivative. In mechanics the conserved quantity is a first integral; for field equations the theorem gives a local conservation law.
For a simple example, let 
be a Lagrangian
function satisfying the Euler-Lagrange
differential equation. If 
is independent of a coordinate 
, translation in that coordinate is a symmetry,
and the conjugate momentum obeys
 |
If 
is independent of time, the conserved energy function is
 |
More generally, spatial translation symmetry corresponds to conservation of momentum, rotation symmetry to conservation of angular momentum, and time-translation symmetry to conservation of energy. The theorem does not assert that every discrete symmetry produces such a conserved quantity.
