Multisymplectic Hamiltonian variational integrators

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.

Keywords:

• Multisymplectic integrators
• variational integrators
• Hamiltonian field theory
• geometric numerical integration
• symplectic integrators

2020 Mathematics Subject Classifications:

• 65P10
• 37K05
• 53D42
• 70H25

Acknowledgments

The authors would like to thank the referees for their careful review of this paper and their helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

BT was supported by the NSF Graduate Research Fellowship DGE-2038238, and by NSF under grants DMS-1411792, DMS-1813635. ML was supported by NSF under grants DMS-1411792, DMS-1345013, DMS-1813635, by Air Force Office of Scientific Research (AFOSR) under grant FA9550-18-1-0288, and by the DoD under grant HQ00342010023 (Newton Award for Transformative Ideas during the COVID-19 Pandemic).