Dependent coordinates in the Lagrange-Poincar\'e equations for mechanical systems with symmetry

TL;DR

This paper derives the Lagrange-Poincaré equations for a scalar particle moving on a Riemannian manifold with symmetry, using dependent coordinates and Poincaré's variational principle, to describe reduced motion.

Contribution

It introduces a local description of reduced motion in terms of dependent coordinates within the Lagrange-Poincaré framework for symmetric mechanical systems.

Findings

Derived explicit Lagrange-Poincaré equations using dependent coordinates.

Applied variational principle to systems with symmetry on Riemannian manifolds.

Provided a geometric framework for analyzing symmetric mechanical systems.

Abstract

The Lagrange--Poincar\'e equations for the mechanical system describing the motion of a scalar particle on a Riemannian manifold with a given free and isometric action of a compact Lie group is obtained. In an arising principle fibre bundle, the total space of which serves as a configuration space of the considered mechanical system, the local description of the reduced motion is done in terms of dependent coordinates. In obtaining of the equations we use the variational principle developed by Poincar\'e for the mechanical systems with a symmetry.