Beam dynamics with the Hamilton-Jacobi equation

A nonperturbational method of solving the Hamilton-Jacobi equation for invariant surfaces in phase space is described. The problem is formulated in action-angle variables with a general nonlinear perturbation. The solution of the Hamilton-Jacobi equation is regarded as the fixed point of a map on the Fourier coefficients of the generating function. Periodicity of the generator in the independent variable is enforced with a shooting method. Two methods for finding the fixed point and hence the invariant surface are presented. A solution by plain iteration is found to be economical but has a restricted domain of convergence. The Newton iteration is costly but yields solutions up to the dynamic aperture. Examples of lattices with sextupoles for chromatic correction are discussed.<>