TheoremProved

Noether's theorem

Statement

For a physical system described by a Lagrangian $L(q, \dot q, t)$ whose action $S = \int L\, dt$ is invariant under a continuous one-parameter family of transformations of the coordinates $q$ (a continuous symmetry), there is a corresponding quantity $Q(q, \dot q, t)$ that is conserved along every solution of the Euler–Lagrange equations, i.e. $\dfrac{dQ}{dt} = 0$ .

Why is it true?

Every continuous symmetry of a physical law hides a conservation law inside it. If the laws of physics look the same after you shift every clock forward by the same amount (time-translation symmetry), energy is conserved; if they look the same after you shift every position by the same vector (space-translation symmetry), momentum is conserved; if they look the same after you rotate the whole system (rotation symmetry), angular momentum is conserved. Noether's theorem is the precise dictionary that turns 'the physics does not care about this transformation' into 'this quantity never changes.'

Proof sketch

Let $q_s(t)$ be the family of curves obtained by applying the symmetry transformation with parameter $s$ to a solution $q(t)$ , with $q_0 = q$ . Invariance of the action means $\frac{d}{ds}\Big|_{s=0} S[q_s] = 0$ for every solution. Expanding this derivative and integrating the term involving $\dot{(\delta q)}$ by parts turns it into a boundary term plus a term proportional to the Euler–Lagrange expression, which vanishes on-shell. What remains is exactly $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\, \delta q\right) = 0$ along solutions, identifying $Q = \frac{\partial L}{\partial \dot q}\, \delta q$ as the conserved quantity, where $\delta q = \frac{d}{ds}\big|_{s=0} q_s$ is the generator of the symmetry.

Stated by

Emmy Noether

Proved by

proved 1918Emmy Noether

Topics that use this theorem

Lagrangian and Hamiltonian mechanics

Related theorems

Euler–Lagrange equation Liouville's theorem (Hamiltonian mechanics)

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Emmy Noether (1918). Invariante Variationsprobleme
Herbert Goldstein, Charles P. Poole, John L. Safko (2002). Classical Mechanics