Contents

• Contents
• Introduction
• Normal forms
  • Lie transformations and the Birkhoff-Gustavson normal form
  • The generalized normal form
  • Quasi-integrals of motion
• Normalizing a magnetic bottle
  • The model system
  • Local and global analysis of the quasi-integral
• Concluding remarks
• Appendix: Details of the normalization pro-
  cess for a magnetic bottle
• Bibliography
• Figure captions
• About this document ...

Normal forms and quasi-integrals for the Hamiltonians of magnetic bottles

U. M. Engel, B. Stegemerten and P. Eckelt

Institut für Theoretische Physik I, Universität Münster,
Wilhelm-Klemm-Straße 9, 48149 Münster, Germany

Short title: Normal forms and quasi-integrals of magnetic bottles

PACS numbers: 05.45, 02.90

Submitted to: J. Phys. A: Math. Gen.

Date: 16 May 1994,

Abstract:

The well-known Birkhoff-Gustavson normal form theory suffers from the restraint that the quadratic part of the Hamiltonian must be of the harmonic oscillator type. In this paper we describe a generalized normal form concept which can be applied to any polynomial Hamiltonian, thus rendering the above restriction to harmonic oscillators unnecessary. As in the classical theory we can derive an asymptotic expression for a second integral of motion. The truncated formal integral, the quasi-integral, exhibits good convergence properties in regions of phase space where the dynamics is regular, whereas in chaotic regions the convergence deteriorates. In order to exemplify these findings we apply the theory to a Hamiltonian describing a particular type of magnetic bottle which cannot be analyzed using the Birkhoff-Gustavson normal form. We calculate the quasi-integrals up to and including the 14th order and analyze their convergence properties.