Noether's second theorem

A theorem stating that continuous symmetries depending on arbitrary functions imply differential identities among the Euler–Lagrange equations of a variational problem.

Euler–Lagrange equation: A differential equation obtained by requiring a functional to be stationary under variations of its fields. The theorem’s identities constrain these equations when the action has arbitrary-function symmetries.
Calculus of variations: The mathematical study of extrema of functionals, including the equations satisfied by stationary fields. Noether’s theorem is derived by varying an action functional.
Electromagnetism: The classical field theory of electric and magnetic interactions, described by the electromagnetic field tensor. Gauge invariance of its action entails a differential identity among its field equations.
Emmy Noether: A German mathematician whose work transformed abstract algebra and the mathematical foundations of physics. She proved the theorems connecting continuous variational symmetries with conservation laws and identities.
Variational symmetry: A transformation of fields and coordinates that leaves an action invariant, possibly up to a boundary term. Noether’s second theorem specializes symmetry analysis to transformations with arbitrary-function parameters.
Action (physics): A functional of fields whose stationary points describe the classical equations of motion. The symmetry condition applies to the action whose variation produces the field equations.
Yang–Mills theory: A gauge field theory based on a non-Abelian symmetry group, used in the Standard Model of particle physics. Its local gauge symmetry produces identities that relate the field equations.
Invariant variation problems: Variational problems whose action remains invariant under a family of transformations. Noether developed her results while studying invariance in variational problems.
Gauge symmetry: A redundancy in a mathematical description generated by transformations that leave physical predictions unchanged. Gauge transformations are the standard source of arbitrary-function symmetries covered by the theorem.
Infinitesimal transformation: A transformation specified to first order in a small parameter, often used to express continuous symmetries. Arbitrary-function symmetry parameters enter the theorem through infinitesimal field variations.

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Gauge theoryRelated: It explains the constraints associated with local gauge redundancy.