Lagrange-d'Alembert-Poincar\`e equations by several stages

TL;DR

This paper derives explicit multi-stage Lagrange-d'Alembert-Poincaré equations using principal connections, extending previous two-stage methods, with an application to Euler's disk.

Contribution

It provides a new explicit formulation for multi-stage equations in geometric mechanics, generalizing earlier two-stage approaches.

Findings

Explicit multi-stage equations derived

Extension of two-stage methods to multiple stages

Application demonstrated on Euler's disk

Abstract

The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincar\`{e} equations in several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle in several stages, extending methods introduced for the case of two stages by one of the authors and collaborators. The case of the Euler's disk is described as an illustrative example.