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We analyze the dynamics of gauge theories and constrained systems in generalunder small perturbations around a classical solution (background) in bothLagrangian and Hamiltonian formalisms. We prove that a fluctuations theory,described by a quadratic Lagrangian, has the same constraint structure andnumber of physical degrees of freedom as the original non-perturbed theory,assuming the non-degenerate solution has been chosen. We show that the numberof Noether gauge symmetries is the same in both theories, but that the gaugealgebra in the fluctuations theory becomes Abelianized. We also show that thefluctuations theory inherits all functionally independent rigid symmetries fromthe original theory, and that these symmetries are generated by linear orquadratic generators according to whether the original symmetry is preserved bythe background, or is broken by it. We illustrate these results with theexamples.Comment: 27 pages; non-essential but clarifying changes in Introduction, Sec.  3 and Conclusions; the version to appear in J.Phys.

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