Affine connection structure of the charged symplectic 2-form

Abstract

It is shown that the charged symplectic form in Hamiltonian dynamics of classical charged particles in electromagnetic fields defines a generalized affine connection on an affine frame bundle associated with spacetime. Conversely, a generalized affine connection can be used to construct a symplectic 2-form if the associated linear connection is torsion-free and the antisymmetric part of theR 4* translational connection is locally derivable from a potential. Hamiltonian dynamics for classical charged particles in combined gravitational and electromagnetic fields can therefore be reformulated as aP(4)=O(1, 3)⊗R 4* geometric theory with phase space the affine cotangent bundleAT * M of spacetime. The sourcefree Maxwell equations are reformulated as a pair of geometrical conditions on the ℝ4* curvature that are exactly analogous to the source-free Einstein equations.