Tulczyjew’s approach for particles in gauge fields

Abstract

In mid-1970s Tulczyjew discovered an approach to classical mechanics which brings the Hamiltonian formalism and the Lagrangian formalism under a common geometric roof: the dynamics of a particle with configuration space X is determined by a Lagrangian submanifold D of (the total tangent space of ), and the description of D by its Hamiltonian H: (resp. its Lagrangian L: ) yields the Hamilton (resp. Euler–Lagrange) equation. It is reported here that Tulczyjew’s approach also works for the dynamics of (charged) particles in gauge fields, in which the role of the total cotangent space is played by Sternberg phase spaces. In particular, it is shown that, for a particle in a gauge field, the equation of motion can be locally presented as the Euler–Lagrange equation for a Lagrangian which is the sum of the ordinary Lagrangian , the Lorentz term, and an extra new term which vanishes whenever the gauge group is abelian. A charge quantization condition is also derived, generalizing Dirac’s charge quantization condition from gauge group to any compact connected gauge group.