Probing a regular orbit with spectral dynamics

We have extended the spectral dynamics formalism introduced by Binney & Spe rgel, and have implemented a semi-analytic method to represent regular orbi ts in any potential, making full use of their regularity. We use the spectr al analysis code of Carpintero & Aguilar to determine the nature of an orbi t (irregular, regular, resonant, periodic) from a short-time numerical inte gration. If the orbit is regular, we approximate it by a truncated Fourier time series of a few tens of terms per coordinate. Switching to a descripti on in action-angle variables, this corresponds to a reconstruction of the u nderlying invariant torus. We then relate the uniform distribution of a reg ular orbit on its torus to the non-uniform distribution in the space of obs ervables by a simple Jacobian transformation between the two sets of coordi nates. This allows us to compute, in a cell-independent way, all the physic al quantities needed in the study of the orbit, including the density and t he line-of-sight velocity distribution, with much increased accuracy. The r esulting flexibility in the determination of the orbital properties, and th e drastic reduction of storage space for the orbit library, provide a signi ficant improvement in the practical application of Schwarzschild's orbit su perposition method for constructing galaxy models. We test and apply our me thod to two-dimensional orbits in elongated discs, and to the meridional mo tion in axisymmetric potentials, and show that for a given accuracy, the sp ectral dynamics formalism requires an order of magnitude fewer computations than the more traditional approaches.