We derive a variational characterization of the exact discrete Hamiltonian, which is a Type II generating function for the exact flow of a Hamiltonian system, by considering a Legendre transformation of Jacobi's solution of the Hamilton-Jacobi equation. This provides an exact correspondence between continuous and discrete Hamiltonian mechanics, which arise from the continuous-and discrete-time Hamilton's variational principle on phase space, respectively. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators that includes the symplectic partitioned Runge-Kutta methods. This extends the framework of variational integrators to Hamiltonian systems with degenerate Hamiltonians, for which the standard theory of Lagrangian variational integrators cannot be applied. We also characterize the group invariance properties of discrete Hamiltonians that lead to a discrete Noether's theorem.

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In this article, we develop high-order symplectic integrators for solving second order differential equations which can be transformed into separable Hamiltonian systems. The construction of such high-order integrators is based on the notion of continuous-stage Runge-Kutta-Nyström methods in conjunction with the Legendre polynomial expansion techniques and simplifying assumptions of order conditions. As examples, three new one-parameter families of symplectic methods which are of order 4, 6 and 8 respectively are derived in use of Gaussian-type quadrature. Some numerical tests are well performed to verify our theoretical results.