RF multipole implementation

The electromagnetic radio-frequency (RF) field of accelerating structures and crab-cavities can exhibit transverse field components due to asymmetries in the azimuthal direction of the element geometry. Tracking simulations must be performed to evaluate the impact of such transverse RF deflections on the beam dynamics. In an ultra-relativistic regime where the Panofsky-Wenzel theorem is applicable, these RF deflections can be modeled via a multipolar expansion of the generating RF field similarly to what is done with static magnetic elements. The element implementing such RF multipolar fields has been called RF multipole. In this note we present an analytical formulation of a thin RF multipole Hamiltonian, and we explicitly calculate the RF kick and the elements of its first- and second- order transfer matrices. Also, we present the implementation of the corresponding code in MAD-X, plus some tests of tracking, simplecticity, consistency, and reflected maps that we successfully applied to verify the correctness of our computations.