Variational Principles for Hamiltonian Systems

TL;DR

This paper introduces a new variational principle for Hamiltonian systems that is applicable on manifolds and infinite-dimensional spaces, enhancing the theoretical framework and potential applications in optimization and control.

Contribution

It develops a novel Type II variational approach based on a virtual work principle, extending Hamiltonian variational principles to manifolds and infinite-dimensional systems.

Findings

Established a variational principle on vector spaces and manifolds.

Extended the approach to infinite-dimensional Hamiltonian systems.

Reviewed existing variational principles and applications.

Abstract

Motivated by recent developments in Hamiltonian variational principles, Hamiltonian variational integrators, and their applications such as to optimization and control, we present a new Type II variational approach for Hamiltonian systems, based on a virtual work principle that enforces the Type II boundary conditions through a combination of essential and natural boundary conditions; particularly, this approach allows us to define this variational principle intrinsically on manifolds. We first develop this variational principle on vector spaces and subsequently extend it to parallelizable manifolds, general manifolds, as well as to the infinite-dimensional setting. Furthermore, we provide a review of variational principles for Hamiltonian systems in various settings as well as their applications.