Tulczyjew's Approach for Particles in Gauge Fields

Abstract

Around mid-1970s W. M. Tulczyjew discovered an approach which brings the two formalisms under a common geometric roof: the dynamics of a particle with configuration space $X$ is determined by a Lagrangian submanifold $D$ of $TT^*X$ (the total tangent space of $T^*X$), and the description of $D$ by its Hamiltonian $H$: $T^*X\to \mathbb R$ (resp. its Lagrangian $L$: $TX\to\mathbb R$) 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 $T^*X$ 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 $L(q, \dot q)$, 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 $\mathrm{U}(1)$ gauge group to any compact connected gauge group.