Coordinate representation of the Lagrange-Poincar\'{e} equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle

Abstract

Using the dependent coordinates, the local Lagrange-Poincar\'e equations and equations for the relative equilibria are obtained for a mechanical system with a symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and the vector space) on which a free proper and isometric action of a compact semi-simple Lie group is given. As in gauge theories, dependent coordinates are implicitly determined by means of equations representing the local sections of the principal fiber bundle.