Hamilton-Dirac systems for charged particles in gauge fields

TL;DR

This paper develops a geometric framework using Hamilton-Dirac systems and the Sternberg phase space to analyze the dynamics of charged particles in gauge fields, providing new variational principles and structures.

Contribution

It introduces a novel Hamilton-Dirac structure on Sternberg phase space and links it to the magnetized Tulczyjew triple, advancing the geometric understanding of particles in gauge fields.

Findings

Established a Dirac structure on Sternberg-Pontryagin bundle.

Linked the new bundle to the magnetized Tulczyjew triple.

Applied the framework to a charged particle in electromagnetic field.

Abstract

In this work, we use the Sternberg phase space (which may be considered as the classical phase space of particles in gauge fields) in order to explore the dynamics of such particles in the context of Hamilton-Dirac systems and their associated Hamilton-Pontryagin variational principles. For this, we develop an analogue of the Pontryagin bundle in the case of the Sternberg phase space. Moreover, we show the link of this new bundle to the so-called magnetized Tulczyjew triple, which is an analogue of the link between the Pontryagin bundle and the usual Tulczyjew triple. Taking advantage of the symplectic nature of the Sternberg space, we induce a Dirac structure on the Sternberg-Pontryagin bundle which leads to the Hamilton-Dirac structure that we are looking for. We also analyze the intrinsic and variational nature of the equations of motion of particles in gauge fields in regards of the defined new geometry. Lastly, we illustrate our theory through the case of a $U(1)$ gauge group, leading to the paradigmatic example of an electrically charged particle in an electromagnetic field.