The “geometric” and the “dynamic” variational principles are extended to the case of a classical test particle coupled with an external field through a scalar, velocity-independent term $\mathcal{U}.$ The relative Lagrangians are determined, and a super-Hamiltonian derivation of Hamilton-Jacobi equation for this case is also given. Consistency of results is shown throughout. Results thus found are then specialized to the case of the curved-spacetime motion of a classical polarized test particle, subject to the influence of an external electromagnetic field. We also consider the behavior of a spinning classical polarized test particle in a curved spacetime: the equations of motion are given, together with the equations describing the evolution of the spin.

• Received 14 October 2003

DOI:https://doi.org/10.1103/PhysRevD.70.024012