Discrete Hamiltonian variational integrators

Abstract

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.

Variational integrators – A continuous time approach

“…The recent literature on variational integrators, for example , relies typically on Hamilton's principle, which relates the dynamic equilibrium to stationary trajectories of a certain action functional. In the discrete mechanics framework, the action functional is discretized in the first place.…”

“…The Hamiltonian formalism is introduced in by performing so‐called left or right discrete Legendre transforms. A more direct approach is proposed in , where the action, expressed using Hamiltonian formalism, is discretized without recourse to Lagrangian formalism.…”

“…The parametrization using basis functions allows to evaluate time derivatives in a straightforward and unambiguous manner, without referring to discrete Legendre transformations, as carried out in . The proposed approach is also flexible enough to encompass, for instance, the Galerkin variational integration approach from (in case of a constant mass matrix) or the approach from as special cases.…”

Phase fitted variational integrators using interpolation techniques on non regular grids

“…The number of intermediate points in a time interval increases the order of the method, see for example Refs. [1,2,5] where the intermediate points have been chosen to subdivide the time interval into equal subintervals. In the present work, high order variational integrators are derived that use different choices of intermediate points for the action integral approximation by employing general interpolation techniques.…”

Prolongation-collocation variational integrators

“…Furthermore, it would be interesting to extend the prolongation-collocation techniques to Hamiltonian variational integrators (Leok & Zhang, 2011) and Hamilton-Pontryagin variational integrators (Leok & Ohsawa, 2011) by considering prolongations of Hamilton's equations, and the implicit Euler-Lagrange equations.…”