Variational order for forced Lagrangian systems II: Euler-Poincar\'e equations with forcing

TL;DR

This paper develops a variational framework for forced Euler-Poincaré equations, exploring their geometric structure and applying the theory to derive and analyze geometric numerical integrators for forced systems.

Contribution

It introduces a variational derivation for forced Euler-Poincaré equations and connects the geometry to Poisson groupoids, advancing the understanding of forced geometric mechanics.

Findings

Derived variational principles for forced Euler-Poincaré equations

Analyzed geometric structure related to Poisson groupoids

Applied theory to develop geometric integrators for forced systems

Abstract

In this paper we provide a variational derivation of the Euler-Poincar\'e equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the previous construction to the formal derivation of the variational error for numerical integrators of forced Euler-Poincar\'e equations and the application of this theory to the derivation of geometric integrators for forced systems.