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

Dependent coordinates in the Lagrange-Poincar\'e equations for mechanical systems with symmetry