Geometry of classical mechanics with non-Abelian gauge symmetry

Geometry of the gauge orbit bundle is considered for the $SO(3)$ Yang–Mills mechanics. It is shown that it may serve as a finite-dimensional model of the geometry of nonabelian field theory with respect to such properties as the nonexistence of a global gauge, the Gribov ambiguities, convexity of the region within the corresponding horizon and stratification. The connection and the Lagrangian on the orbit space are obtained.