Noether's theorem

One of the deepest results in all of physics: every continuous symmetry of a system corresponds to a conserved quantity. Why is momentum conserved? Because the laws do not change if you shift everything sideways. Why energy? Because they do not change as time passes. Angular momentum? Rotational symmetry. Symmetry and conservation are two faces of a single coin.

Precisely: if the action S = integral of L dt is invariant under a continuous one-parameter transformation of the coordinates (and possibly of time), then there is a quantity that stays constant along the equations of motion. Invariance under translating a coordinate gives conservation of its conjugate momentum; invariance under time translation gives conservation of energy (the Hamiltonian); invariance under rotation gives conservation of angular momentum. A cyclic coordinate is the simplest special case.

Two honesties matter. First, the symmetry must be continuous, a smooth family labeled by a parameter; a discrete symmetry such as parity or time reversal does not yield a conserved quantity by this route, though it does give selection rules. Second, strictly the action need only be invariant up to a boundary or total-derivative term. The theorem is the backbone of modern field theory, where it produces conserved currents, for instance electric charge from a phase symmetry of the fields.