Conservation law

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Global conservation laws

A global or integral conservation law for an evolution equation is any quantity Q(t) depending on the value of all the fields at time t which is (formally) constant in time:

The conserved quantity Q(t) is typically an integral over space. For instance, in NLS, examples of conserved quantities include the total mass

the total momentum

and the total energy

Noether's theorem relates conserved quantities to symmetries of the underlying equation, in the case that the equation is Hamiltonian or Lagrangian.

Local conservation laws

A local or pointwise conservation law for any equation is any local function of the fields at or near which obeys the continuity equation

for some other local functions of the fields near . Note from Stokes' theorem that this implies (in flat space at least) that the integral is a global conserved quantity. By modifying using spatial or frequency cutoffs one can also create almost conserved quantities, virial identities and monotonicity formulae.