Reduction in systems with local symmetry

This review is devoted to problems associated with the study of dynamical systems with a finite number of degrees of freedom possessing local symmetry. The procedure of reduction of the system of dynamical equations to the normal form, where the Cauchy problem has a unique solution, is discussed within the framework of the classical Lagrangian and Hamiltonian theory. Special attention is given to the geometrical reduction scheme, which allows the physical subspace in the phase space of a degenerate dynamical system to be distinguished, and makes it possible to find the explicit form of the corresponding canonical variables without introducing additional gauge-fixing conditions (gauges) into the theory. The two reduction procedures, the geometrical method and the gauge-fixing method, are compared in order to understand what conditions on the gauges guarantee the correctness of the reduction procedure.