  We develop a new, coordinate-free formulation of Hamiltonian mechanics on the
dual of a Lie algebroid. Our approach uses a connection, rather than
coordinates in a local trivialization, to obtain global expressions for the
horizontal and vertical dynamics. We show that these dynamics can be obtained
in two equivalent ways: (1) using the canonical Lie-Poisson structure,
expressed in terms of the connection; or (2) using a novel variational
principle that generalizes Hamilton's phase space principle.
