Partial Hamiltonian formalism, multi-time dynamics and singular theories

Abstract

In this paper we formulate singular theories (determined by degenerate Lagrangians) without involving constraints. We construct a partial Hamiltonian formalism in reduced phase space (with arbitrary number of momenta). The equations of motion are first-order differential equations, and they coincide with ones of the multi-time dynamics under a certain condition, which in a singular theory is coincidence of number of generalized momenta to the rank of the Hessian matrix. Non-canonical generalized velocities satisfy the system of linear algebraic equations, which sets the appropriate classification of singular theories (gauge and nongauge). To describe the time evolution of physical quantities we introduce a new anti-symmetric bracket (similar to the Poisson bracket). It is shown how the extension of the phase space leads to constraints, and the new bracket goes into the Dirac bracket. Quantization is briefly discussed.