A Geometrical Formulation for Classical Mechanics in Gauge Fields

We give a geometrical formulation for the classical mechanics in a non-abelian gauge field on a Riemannian manifold. The formulation is based on the reduction procedure associated to the non-abelian symmetry in the principal bundle which describes the gauge field. In the formulation we present explicitly the equation of the motion (called Wong's equation) of a charged particle by using a local coordinate system.