  Hamiltonian Monte Carlo is typically based on the assumption of an underlying
canonical symplectic structure. Numerical integrators designed for the
canonical structure are incompatible with motion generated by non-canonical
dynamics. These non-canonical dynamics, motivated by examples in physics and
symplectic geometry, correspond to techniques such as preconditioning which are
routinely used to improve algorithmic performance. Indeed, recently, a special
case of non-canonical structure, magnetic Hamiltonian Monte Carlo, was
demonstrated to provide advantageous sampling properties. We present a
framework for Hamiltonian Monte Carlo using non-canonical symplectic
structures. Our experimental results demonstrate sampling advantages associated
to Hamiltonian Monte Carlo with non-canonical structure. To summarize our
contributions: (i) we develop non-canonical HMC from foundations in symplectic
geomtry; (ii) we construct an HMC procedure using implicit integration that
satisfies the detailed balance; (iii) we propose to accelerate the sampling
using an {\em approximate} explicit methodology; (iv) we study two novel,
randomly-generated non-canonical structures: magnetic momentum and the coupled
magnet structure, with implicit and explicit integration.
