A Generalized Formulation of Dynamics and Constraints in the Phase Space-Time

Abstract

A reformulation is introduced for the dynamics on the phase space-time (the phase space extended to contain the time and its conjugate momentum), in which the dynamics is characterized by a constraint hyper-surface in the phase space-time, rather than a Hamiltonian on the phase space (or extended phase space, the phase space and the time). In the case that the constraint hyper-surface is constructed by a Hamiltonian on the extended phase space, the reformulation produces the same result of the conventional Hamiltonian dynamics, as expected. But the reformation allows for more general cases, where the constraint is a relation between the phase space-time coordinates: an implicit rather that an explicit Hamiltonian. In this reformulation, there is no fundamental difference between constraints and dynamics: additional constraints just restrict the phase space-time further, decreasing the dimension of the restricted manifold. The setup for dynamics and constraints is introduced. Different cases are discussed, regarding the existence and uniqueness of the solution to the initial value problem: no solution, a unique solution, more than one solutions. Some examples are studied in detail. A comparison (correspondence) with the conventional setup is also presented.