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

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 analysed using the Birkhoff-Gustavson normal form. We calculate the quasi-integrals up to and including the 14th order and analyse their convergence properties.