Of Extended Generating Functions

The time t maps of Hamiltonian flows are symplectic. The order n Taylor series approximation with respect to initial conditions of such a map is symplectic through terms of order n. Given an order n Hamiltonian symplectic map, there are a variety of procedures, called symplectification methods, which produce exactly symplectic maps with Taylor series that agree with the initial Taylor map through terms of order n. Here we extend the generating function method of symplectification. To this end, we develop a general theory of generating functions of canonical transformations. It is shown that locally any symplectic map has uncountably many generating functions, each of which is associated with a conformal symplectic map. Within the subgroup of linear conformal symplectic maps, the available types can be organized into equivalence classes represented by symmetric matrices. Furthermore, equivalence of symplectification with and without factorization of the symplectic maps into linear and nonlinear parts is proved. The method is illustrated with two examples; an anharmonic oscillator, and the dynamics in a proposed new particle accelerator, the so-called Neutrino Factory, which is known to exhibit a wide spectrum of nonlinear effects. Received: October 22, 2003 c © 2004 Academic Publications Correspondence author 242 B. Erdélyi, M. Berz AMS Subject Classification: 81S10, 58F05