Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings

TL;DR

This paper explores the connection between Hamiltonian dynamics and constrained variational calculus using symplectic geometry, extending to discrete systems and nonholonomic mechanics, with applications in geometric integration.

Contribution

It introduces a unified geometric framework for continuous and discrete Hamiltonian and variational systems, including nonholonomic constraints, using Tulczyjew's triples.

Findings

Unified geometric description of Hamiltonian and variational systems

Extension to discrete dynamics and nonholonomic mechanics

Applications to geometric integration methods

Abstract

The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew's triples. The results are also extended to the case of discrete dynamics and nonholonomic mechanics. Interesting applications to geometrical integration of Hamiltonian systems are obtained.