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

The Lagrange-Poincar\'e equations for a mechanical system with symmetry  on the principal fiber bundle over the base represented by the bundle space  of the associated bundle