  In this paper we show that the Hamiltonian Monte Carlo method for compact Lie
groups constructed in \cite{kennedy88b} using a symplectic structure can be
recovered from canonical geometric mechanics with a bi-invariant metric. Hence
we obtain the correspondence between the various formulations of Hamiltonian
mechanics on Lie groups, and their induced HMC algorithms. Working on $\G\times
\g$ we recover the Euler-Arnold formulation of geodesic motion, and construct
explicit HMC schemes that extend \cite{kennedy88b,Kennedy:2012} to non-compact
Lie groups by choosing metrics with appropriate invariances. Finally we explain
how mechanics on homogeneous spaces can be formulated as a constrained system
over their associated Lie groups, and how in some important cases the
constraints can be naturally handled by the symmetries of the Hamiltonian.
