A Hamiltonian-free description of single particle dynamics for hopelessly complex periodic systems

Abstract

A picture of periodic systems that does not rely on the Hamiltonian of the system, but on maps between a finite number of time locations, is developed. Moser or Deprit-like normalizations are done directly on the maps, thereby avoiding the complex time-dependent theory. Linear and nonlinear Floquet variables are redefined entirely in terms of maps. This approach relies heavily on the Lie representation of maps introduced by Dragt and Finn [J. Math. Phys. 20, 2649 (1979); J. Geophys. Res. 81, 13 (1976)]. One might say that although the Hamiltonian is not used in the normalization transformation, Lie operators are used, which are themselves, in some sense, pseudo-Hamiltonians for the maps they represent. The techniques find application in accelerator dynamics or in any field where the Hamiltonian is periodic, but hopelessly complex, such as magnetic field design in stellarators.