  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.
