Iterative determination of invariant tori for a time-periodic Hamiltonian with two degrees of freedom

Journal Article · Tue Sep 15 04:00:00 EDT 1992 · Physical Review A. General Physics; (United States)

DOI:https://doi.org/10.1103/PhysRevA.46.3493· OSTI ID:7158579

Gabella, W E ^[1]; Ruth, R D; Warnock, R L ^[2]

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1. Department of Physics, University of Colorado, Boulder, Colorado 84309 (United States) Stanford Linear Accelerator Center, Stanford University, Stanford, Californi a 94309 (United States)
2. Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309 (United States)

We describe a nonperturbative numerical technique for solving the Hamilton-Jacobi equation of a nonlinear Hamiltonian system. We find the time-periodic solutions that yield accurate approximations to invariant tori. The method is suited to the case in which the perturbation to the underlying integrable system has a periodic and not necessarily smooth dependence on the time. This case is important in accelerator theory, where the perturbation is a periodic step function in time. The Hamilton-Jacobi equation is approximated by its finite-dimensional Fourier projection with respect to angle variables, then solved by Newton's method. To avoid Fourier analysis in time, which is not appropriate in the presence of step functions, we enforce time periodicity of solutions by a shooting algorithm. The method is tested in soluble models, and finally applied to a nonintegrable example, the transverse oscillations of a particle beam in a storage ring, in two degrees of freedom. In view of the time dependence of the Hamiltonian, this is a case with 21/2 degrees of freedom,'' in which phenomena like Arnol'd diffusion can occur.

• DOE Contract Number: FG02-86ER40302; AC03-76SF00515
• OSTI ID: 7158579
• Journal Information: Physical Review A. General Physics; (United States), Journal Name: Physical Review A. General Physics; (United States) Vol. 46:6; ISSN 1050-2947; ISSN PLRAA
• Country of Publication: United States
• Language: English

Related Subjects

661100* -- Classical & Quantum Mechanics-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ACCELERATORS
DIFFERENTIAL EQUATIONS
DYNAMICS
EQUATIONS
FUNCTIONS
HAMILTON-JACOBI EQUATIONS
HAMILTONIAN FUNCTION
MATHEMATICS
MECHANICS
NONLINEAR PROBLEMS
NUMERICAL ANALYSIS
PARTIAL DIFFERENTIAL EQUATIONS
STABILITY
STORAGE RINGS