Some remarks about pseudo-Hamiltonian

Technical Report · Mon Nov 01 04:00:00 EST 1993

DOI:https://doi.org/10.2172/10194905· OSTI ID:10194905

Malitsky, N; Bourianoff, G ^[1]; Severgin, Yu ^[2]

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1. Superconducting Super Collider Lab., Dallas, TX (United States)
2. D. V. Efremov Scientific Research Inst., St. Petersburg (Russian Federation)

For the many applied tasks of accelerator physics, the 4D single particle pseudo-Hamiltonian may be presented as the Hamiltonian of the near-integrable system consisting of integrable and perturbed terms. The KAM theorem states that for sufficiently small perturbation the invariant surfaces continue to exist and, for the systems with two degrees of freedom, completely isolate the thin stochastic layers. As the perturbation strength increases, a transition can occur in which these surfaces disappear and the stochastic layers connect, resulting in globally stochastic motion. One of the important problems is to determine this {open_quotes}boundary{close_quotes} invariant surface. There are several approaches that may be used to describe the regular trajectories in the small limited region. The most powerful method is the perturbation theory which allows us to study the combined influence of the different multipoles. The inclusion of Lie operators improved this method and developed it up to high order perturbation. But the perturbation theory failed to describe the change in topology and since the regular trajectories depend discontinuously on choice of initial coordinates, it cannot be used in the whole region of the stable motion. The authors suggest to limit the attention to the study of the {open_quotes}boundary{close_quotes} invariant and implement the additional {open_quotes}local{close_quotes} transformation. The authors briefly review the well known theories, their advantages and imperfections, and the necessity of the {open_quotes}local{close_quotes} transformation. They present the comparison of the map tracking with the invariants determined by the perturbation methods.

• Research Organization: Superconducting Super Collider Lab., Dallas, TX (United States)
• Sponsoring Organization: USDOE, Washington, DC (United States)
• DOE Contract Number: AC35-89ER40486
• OSTI ID: 10194905
• Report Number(s): SSCL--648; ON: DE94003273
• Country of Publication: United States
• Language: English

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