Multisymplectic Hamiltonian Variational Integrators

TL;DR

This paper extends Hamiltonian variational integrators to multisymplectic field theories, introducing systematic construction methods that preserve multisymplectic structure and Noether's theorem in a discrete setting.

Contribution

It develops a framework for multisymplectic Hamiltonian variational integrators using Type II generating functionals, ensuring conservation laws and connecting to existing integrators.

Findings

Discrete multisymplectic conservation law established

Discrete Noether's theorem proven for Lie group invariance

Spacetime tensor product Runge--Kutta methods are symplectic in space and time

Abstract

Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge--Kutta discretizations is well-defined if and only if the partitioned Runge--Kutta methods are symplectic in space and time.