  We construct a symplectic realisation of the twisted Poisson structure on the
phase space of an electric charge in the background of an arbitrary smooth
magnetic monopole density in three dimensions. We use the extended phase space
variables to study the classical and quantum dynamics of charged particles in
arbitrary magnetic fields by constructing a suitable Hamiltonian that
reproduces the Lorentz force law for the physical degrees of freedom. In the
source-free case the auxiliary variables can be eliminated via Hamiltonian
reduction, while for non-zero monopole densities they are necessary for a
consistent formulation and are related to the extra degrees of freedom usually
required in the Hamiltonian description of dissipative systems. We obtain new
perspectives on the dynamics of dyons and motion in the field of a Dirac
monopole, which can be formulated without Dirac strings. We compare our
associative phase space formalism with the approach based on nonassociative
quantum mechanics, reproducing extended versions of the characteristic
translation group three-cocycles and minimal momentum space volumes, and prove
that the two approaches are formally equivalent. We also comment on the
implications of our symplectic realisation in the dual framework of
non-geometric string theory and double field theory.
