Every continuous symmetry of a physical system's action gives rise to a corresponding conserved quantity.
Emmy Noether proved in 1918 that for every continuous symmetry of a system's action $S = \int L\,dt$, there exists an associated quantity that does not change with time. The symmetry condition is that the Lagrangian changes by at most a total time derivative under an infinitesimal transformation $q_i \to q_i + \epsilon\,\delta q_i$:
The conserved quantity $Q$ is called the Noether charge. The three great examples: time-translation invariance gives energy; space-translation invariance gives linear momentum; rotational invariance gives angular momentum.
Imagine a game whose rules are identical no matter when you play (time symmetry) — your score total is always preserved. If the rules are identical wherever you play (space symmetry) — momentum is preserved. If identical in every direction you face (rotational symmetry) — angular momentum is preserved. Noether showed physics is the ultimate fair game.
The canonical momentum conjugate to coordinate $q_i$ is $p_i = \partial L / \partial \dot{q}_i$. Under an infinitesimal symmetry transformation $\delta q_i$, the Noether charge is:
For time translation ($t \to t + \epsilon$): $\delta q_i = \dot{q}_i$, $F = L$, giving $Q = \sum_i p_i \dot{q}_i - L = H$ — the Hamiltonian (total energy). For spatial translation: $\delta x = 1$, $F = 0$, giving $Q = p_x$ — linear momentum. For rotation about z: $\delta x = -y$, $\delta y = x$, giving $Q = xp_y - yp_x = L_z$ — angular momentum.
In quantum field theory, every internal symmetry (e.g., U(1) gauge invariance of electromagnetism) produces a conserved Noether current. Electric charge conservation follows from U(1) symmetry; colour charge from SU(3) symmetry of QCD. Noether's theorem is the invisible scaffolding of the entire Standard Model.
Noether's theorem is a two-way street: if a symmetry is broken, the corresponding quantity is no longer conserved. In an expanding universe, time-translation symmetry is broken — energy is not globally conserved in cosmology. A crystal breaks continuous spatial symmetry, and momentum is only conserved modulo the reciprocal lattice vector (crystal momentum).
The right-hand side is zero if and only if the action is invariant. When an external field breaks the symmetry (e.g., gravity breaking spatial translation vertically), the rate of change of $Q$ equals the applied "force" — recovering familiar equations like $dp/dt = F$.
Noether developed this theorem at Göttingen in 1918 at the invitation of David Hilbert and Felix Klein, who needed a resolution to the puzzle of energy conservation in Einstein's General Relativity. Einstein called her work "the most significant creative mathematical genius thus far produced." She proved two theorems — the first (Noether's First Theorem) is the one we explore here.