Hamiltonian mechanics and Lie algebroid connections

Abstract

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.

17 pages. v2: edits and revisions throughout paper; added Section 3.5 on systems arising from Lie algebroid metrics