Non-linear connections on phase space and the Lorentz force law
The equations of parallel transport for a non-linear connection on phase space are examined. It is shown that, for a free-particle Lagrangian, the connection term first-order in momentum reproduces the geodesic equation of General Relativity and the term zeroth-order in the momentum reproduces the Lorentz electromagnetic force law. Hence from one mathematical expression, a non-linear parallel transport equation, one can derive the interaction laws for both the gravitational and electromagnetic forces. These equations are free of the difficulties associated with formalism of Weyl, which forms the basis for the theory of Yang and Mills.