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In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory.

A gauge symmetry of a Lagrangian is defined as a differential operator on some vector bundle taking its values in the linear space of (variational or exact) symmetries of . Therefore, a gauge symmetry of
depends on sections of and their partial derivatives.^{[1]} For instance, this is the case of gauge symmetries in classical field theory.^{[2]} Yang–Mills gauge theory and gauge gravitation theory exemplify classical field theories with gauge symmetries.^{[3]}

Gauge symmetries possess the following two peculiarities.

- Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noether's first theorem, but the corresponding conserved current takes a particular superpotential form where the first term vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where is called a superpotential.
^{[4]} - In accordance with Noether's second theorem, there is one-to-one correspondence between the gauge symmetries of a Lagrangian and the Noether identities which the Euler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system.
^{[5]}

Note that, in quantum field theory, a generating functional fail to be invariant under gauge transformations, and gauge symmetries are replaced with the BRST symmetries, depending on ghosts and acting both on fields and ghosts.^{[6]}

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