  The Lagrange--Poincar\'e equations for the mechanical system describing the
motion of a scalar particle on a Riemannian manifold with a given free and
isometric action of a compact Lie group is obtained. In an arising principle
fibre bundle, the total space of which serves as a configuration space of the
considered mechanical system, the local description of the reduced motion is
done in terms of dependent coordinates. In obtaining of the equations we use
the variational principle developed by Poincar\'e for the mechanical systems
with a symmetry.
