Implementation of a Thick Quadrupole in the MAD-X tracking module

MAD-X performs, element by element, the tracking of one or many particle trajectories in the last used sequence. Until pro release 5.02.00 only thin elements were allowed in tracking, with the exception of the element drift, to guarantee the simplecticity of the coordinate transformation. For this reason, any lattice needed to be converted into a ``thin-element'' lattice by invoking the command MAKETHIN. This behaviour has recently been changed in order to allow the tracking through thick quadrupoles. In this note the equations of motion in the thick quadrupole are worked out from the expanded Hamiltonian, and the thick map is derived. This map is symplectic and takes into account the path length difference due to second order terms in the canonical variables.