Covariant Hamiltonian approach for time-dependent potentials applied to a pill-box cavity

The common treatment of time-dependent potentials, such as those used for radio frequency cavities, is to average a potential's time component through the interval that the reference particle spends in the cavity. Such an approach, using the so-called transit-time factor, uses time as the independent variable in the Hamiltonian. In this paper, we instead propose a fully covariant Hamiltonian to treat the time component of the potential like any other space component. We show how to calculate the dynamics of the particles in a pill-box cavity using an explicit sympletic integrator. Finally, we compare the results with the simulator TraceWin.