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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 that M is Minkowski space and G = 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.