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.Comment: 27 pages; non-essential but clarifying changes in Introduction, Sec.
3 and Conclusions; the version to appear in J.Phys.