Lagrangian formulation for electric charge in a magnetic monopole distribution

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 [Phys. Rev. D 98, 045005 (2018)]. 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 is the generalization to particles in a non-Abelian gauge field which does not satisfy Bianchi identities in some region of the space-time.