Normal forms and quasi-integrals for the Hamiltonians of
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,
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
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
, 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.