Noether's theorem

Noether's theorem links differentiable continuous symmetries of a physical action to conserved quantities. Its first theorem applies when the symmetry depends on finitely many continuous parameters.

Action (physics): A quantity defined from a system's Lagrangian over time or spacetime; classical paths make it stationary under allowed variations. The theorem starts from transformations that leave this quantity invariant.
Calculus of variations: The mathematical study of functionals and the functions or paths that make them stationary. Noether's original reasoning uses variations of the action under infinitesimal transformations.
Energy conservation: The principle that total energy remains constant in an isolated system whose dynamics do not change with time. Time-translation symmetry yields conserved energy when the system's action is time independent.
Emmy Noether: A German mathematician whose work transformed abstract algebra and established foundational results linking symmetries and conservation laws. She proved the theorem in 1918 while studying invariance problems in the calculus of variations.
Noether's second theorem: A theorem relating symmetries depending on arbitrary functions to differential identities among the Euler–Lagrange equations. Unlike the first theorem's finite-parameter symmetries, it treats local gauge symmetries and yields identities.
Lagrangian mechanics: A formulation of classical mechanics in which equations of motion follow from a system's Lagrangian. Its variational equations provide the setting for the theorem's standard derivation.
Infinitesimal transformation: A transformation expressed to first order in a small parameter, often used to represent a continuous symmetry locally. The theorem extracts a conservation law from the action's first-order change.
Momentum conservation: The principle that total linear momentum remains constant in an isolated system with translationally invariant dynamics. Spatial translation symmetry produces conserved momentum.
Invariant variation problems: Variational problems whose equations or action remain unchanged under specified groups of transformations. Noether formulated her results to address these problems, including those arising in relativity.
Gauge symmetry: A redundancy in a field description expressed through transformations that may vary independently at each spacetime point. Its arbitrary-function parameters place it in the domain of Noether's second theorem.

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