Fluctuations around classical solutions for gauge theories in Lagrangian and Hamiltonian approach

We analyze the dynamics of gauge theories and constrained systems in general under small perturbations around a classical solution (background) in both Lagrangian and Hamiltonian formalisms. We prove that a fluctuations theory, described by a quadratic Lagrangian, has the same constraint structure and number of physical degrees of freedom as the original non-perturbed theory, assuming the non-degenerate solution has been chosen. We show that the number of Noether gauge symmetries is the same in both theories, but that the gauge algebra in the fluctuations theory becomes Abelianized. We also show that the fluctuations theory inherits all functionally independent rigid symmetries from the original theory, and that these symmetries are generated by linear or quadratic generators according to whether the original symmetry is preserved by the background, or is broken by it. We illustrate these results with the examples.