Continuous versus discrete structures II -- Discrete Hamiltonian systems and Helmholtz conditions

TL;DR

This paper introduces discrete Hamiltonian systems within the discrete embedding framework, compares with prior approaches, and solves the discrete Helmholtz inverse problem, demonstrating various applications.

Contribution

It defines discrete Hamiltonian systems and addresses the discrete Helmholtz inverse problem, providing new insights and solutions in the discrete calculus of variation.

Findings

Explicit formulation of discrete Hamiltonian systems

Solution to the discrete Helmholtz inverse problem

Discussion of multiple applications

Abstract

We define discrete Hamiltonian systems in the framework of discrete embeddings. An explicit comparison with previous attempts is given. We then solve the discrete Helmholtz's inverse problem for the discrete calculus of variation in the Hamiltonian setting. Several applications are discussed.