The inverse problem of the calculus of variations for discrete systems

TL;DR

This paper develops a geometric framework for the inverse calculus of variations in discrete mechanics, linking discrete and continuous problems and emphasizing variationality in numerical integrator design.

Contribution

It introduces a geometric approach using Lagrangian and isotropic submanifolds for the inverse problem in discrete systems, bridging discrete and continuous variational problems.

Findings

Geometric formulation of the inverse problem for discrete mechanics.

Transition method between discrete and continuous variational problems.

Variationality as a key property in numerical integrator design.

Abstract

We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design and computer simulation of numerical integrators.