Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism

TL;DR

This paper extends Feynman-Dyson's proof of Maxwell's equations by incorporating noncommutative geometry into velocity phase space, revealing new volume-preserving Hamiltonian systems with generalized dynamics.

Contribution

It introduces a noncommutative framework in velocity phase space, generalizing Feynman-Dyson's scheme and analyzing associated dynamical flows using Souriau formalism.

Findings

Generalized Feynman-Dyson scheme with noncommutative coordinates

Identification of volume-preserving dynamical flows

New classes of Hamiltonian systems analyzed

Abstract

We consider Feynman-Dyson's proof of Maxwell's equations using the Jacobi identities on the velocity phase space. In this paper we generalize the Feynman-Dyson's scheme by incorporating the non-commutativity between various spatial coordinates along with the velocity coordinates. This allows us to study a generalized class of Hamiltonian systems. We explore various dynamical flows associated to the Souriau form associated to this generalized Feynman-Dyson's scheme. Moreover, using the Souriau form we show that these new classes of generalized systems are volume preserving mechanical systems.