  We consider a reduction procedure in Wiener-type path integral for a
finite-dimensional mechanical system with a symmetry representing the motion of
two interacting scalar particles on a manifold that is the product of the total
space of the principal bundle and a vector space. By analogy with what is done
in gauge theories, the local description of the reduced motion on orbit space
is carried out using dependent coordinates. The factorization of the measure in
the path integral, which is necessary for the reduction, is based on the
application of the stochastic differential equation of the optimal nonlinear
filtering from the theory of stochastic processes. The non-invariance of the
measure in the path integral under the reduction is shown. The Jacobian of the
reduction is generated by the projection of the mean curvature vector field of
the orbit onto the submanifold, which is used to determine the adapted
coordinates in the principal fiber bundle associated with the problem under
study.
