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We consider a reduction procedure in Wiener-type path integral for afinite-dimensional mechanical system with a symmetry representing the motion oftwo interacting scalar particles on a manifold that is the product of the totalspace of the principal bundle and a vector space. By analogy with what is donein gauge theories, the local description of the reduced motion on orbit spaceis carried out using dependent coordinates. The factorization of the measure inthe path integral, which is necessary for the reduction, is based on theapplication of the stochastic differential equation of the optimal nonlinearfiltering from the theory of stochastic processes. The non-invariance of themeasure in the path integral under the reduction is shown. The Jacobian of thereduction is generated by the projection of the mean curvature vector field ofthe orbit onto the submanifold, which is used to determine the adaptedcoordinates in the principal fiber bundle associated with the problem understudy.

Reduction of path integrals for interacting systems: The case of using dependent coordinates in the description of reduced motion on the orbit space