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This note is a continuation of our earlier articles arXiv:1612.08897 andarXiv:1709.09030, where using the dependent coordinates the localLagrange-Poincar\'e equations were obtained for a mechanical system withsymmetry describing the motion of two interacting scalar particles on a specialRiemannian manifold (the product of the total space of the principal fiberbundle and vector space), on which a free proper and isometric action of acompact semisimple Lie group is given. Assuming the existence of the parametricrepresentations for local sections in the principal bundle, we make thetransition to independent coordinates in the obtained Lagrange-Poincar\'eequations.

Transition to the case of "resolved gauge" in the Lagrange-Poincar\'e  equations for a mechanical system with symmetry on the total space of a  principal fiber bundle whose base is the bundle space of the associated  bundle