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
(i) Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy first Noether's theorem, but the corresponding conserved current
(ii) In accordance with second Noether's 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.
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.