Properties of hamiltonian variational integrators

Abstract

Discrete Hamiltonian variational integrators are derived from type II and type III generating functions for symplectic maps, and in this article, we establish a variational error analysis result that relates the order of accuracy of the associated numerical methods with the extent to which these generating functions approximate the exact discrete Hamiltonians. We also introduce the notion of an adjoint discrete Hamiltonian and relate it to the adjoint of the associated symplectic integrator. We show that when constructing discrete Lagrangians and discrete Hamiltonians using the same approximation method, this does not necessarily lead to the same symplectic integrator. Numerical experiments also indicate that the resonance behavior of variational integrators based on averaging methods depends on the type of generating functions used, and we relate this resonance behavior to the ill-posedness of the boundary-value problems used to define the exact discrete Lagrangian and exact discrete Hamiltonian.