Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field

This note is to show how to use symplectic geometry to write equations of motion of a “classical particle” in the presence of a Yang-Mills field, for any gauge group,G , and any differentiable manifold,M . In the case thatM is Minkowski space andG =U (1), the equations reduce to the Lorentz equations for a charged particle in an electromagnetic field. Our procedure in the general case uses the connection form as defined on the principle bundle to introduce a symplectic structure on certain associated bundles and is automatically gauge invariant.