Generalized Phase Space Formulation of the Hamiltonian Dynamics

In an n dimensional phase space, the generator of the time translation can be written in terms of a Hamiltonian and a set of Poisson brackets for the phase space variables. When the velocity vector in this phase space is divergenceless, then the equations of motion reduce to those obtained by Nambu. The extension of the Hamiltonian dynamics to the phase space of arbitrary dimensions enables one to find a generalized Hamiltonian function for equations of motion involving time derivatives of any order (even or odd) of the coordinates. The problem of quantization of Nambu's generalized dynamics is studied, and it is shown that in certain cases, for a system moving under a set of constraints, it is possible to replace the Hamiltonian operator by an infinite number of generators of time translation functions. Some examples from classical dynamics and quantum mechanics are given to show the range of applicability of the generalized phase space formulation.