The Poincaré variational principle in the Lagrange–Poincaré reduction of mechanical systems with symmetry

Abstract

The local Lagrange–Poincaré equations (the reduced Euler–Lagrange equations) for the mechanical system describing the motion of a scalar particle on a finite-dimensional Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are obtained. The equations are written in terms of dependent coordinates which are used to represent the local dynamic given on the orbit space of the principal fiber bundle. The derivation of the equations is performed with the help of the variational principle developed by Poincaré for mechanical systems with symmetry.