Discrete Hamiltonian Variational Integrators

We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators, which include the symplectic partitioned Runge-Kutta methods. We also characterize the group invariance properties of discrete Hamiltonians which lead to a discrete Noether's theorem.Comment: 23 page

Dirac Structures and Hamilton-Jacobi Theory for Lagrangian Mechanics on Lie Algebroids

This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation \T^EE^*. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.Comment: 22 pages. arXiv admin note: text overlap with arXiv:0706.278