# Lagrangian formulation for electric charge in a magnetic monopole distribution

## TL;DR

This paper develops a Lagrangian framework for describing an electric charge in a magnetic monopole field, including relativistic and string generalizations, revealing new gauge symmetries and connections to Hamiltonian formulations.

## Contribution

It introduces a novel Lagrangian formulation for electric charges in magnetic monopole distributions, incorporating a doubled configuration space and extending to relativistic, string, and non-Abelian cases.

## Key findings

- Legendre transform matches recent Hamiltonian descriptions.
- Relativistic Lagrangian exhibits a new gauge symmetry.
- Generalizations include open strings and non-Abelian gauge fields.

## Abstract

We give a Lagrangian description of an electric charge in a field sourced by a continuous magnetic monopole distribution. The description is made possible thanks to a doubling of the configuration space. The Legendre transform of the nonrelativistic Lagrangian agrees with the Hamiltonian description given recently by Kupriyanov and Szabo. The covariant relativistic version of the Lagrangian is shown to introduce a new gauge symmetry, in addition to standard reparametrizations. The generalization of the system to open strings coupled to a magnetic monopole distribution is also given, as well as the generalization to particles in a non-Abelian gauge field which does not satisfy Bianchi identities in some region of the space-time.