  The Lagrange--Poincar\'{e} equations for a mechanical system which describes
the interaction of two scalar particles that move on a special Riemannian
manifold, consisting of the product of two manifolds, the total space of a
principal fiber bundle and the vector space, are obtained. The derivation of
equations is performed by using the variational principle developed by
Poincar\'e for the mechanical systems with a symmetry. The obtained equations
are written in terms of the dependent variables which, as in gauge theories,
are implicitly determined by means of equations representing the local sections
of the principal fiber bundle.
