Betatron frequency and the Poincaré rotation number

Abstract

Symplectic maps are routinely used to describe single-particle dynamics in circular accelerators. In the case of a linear accelerator map, the rotation number (the betatron frequency) can be easily calculated from the map itself. In the case of a symplectic nonlinear map, the rotation number is normally obtained numerically, by iterating the map for given initial conditions, or through a perturbation approach. Integrable maps, a subclass of symplectic maps, allow for an analytic evaluation of their rotation numbers. In this paper we propose an analytic expression to determine the rotation number for integrable symplectic maps of the plane, if an integral is explicitly known, and present several examples, relevant to accelerators. If the integral of motion is not explicitly known, one can obtain the rotation number numerically as outlined in Appendix B. These new results can be used to analyze the topology of the accelerator Hamiltonians as well as to serve as the starting point for a perturbation theory for maps.