  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.
