  Hamiltonian Monte Carlo (HMC) exploits Hamiltonian dynamics to construct
efficient proposals for Markov chain Monte Carlo (MCMC). In this paper, we
present a generalization of HMC which exploits \textit{non-canonical}
Hamiltonian dynamics. We refer to this algorithm as magnetic HMC, since in 3
dimensions a subset of the dynamics map onto the mechanics of a charged
particle coupled to a magnetic field. We establish a theoretical basis for the
use of non-canonical Hamiltonian dynamics in MCMC, and construct a symplectic,
leapfrog-like integrator allowing for the implementation of magnetic HMC.
Finally, we exhibit several examples where these non-canonical dynamics can
lead to improved mixing of magnetic HMC relative to ordinary HMC.
