On the power of normal form theory:

Simplification of a Hamiltonian dynamical system
using normal form theory and quasiintegrals of motion

by Martin Engel

Consider the well-known Størmer problem which is given by the Hamiltonian

completely describing the classical motion of a charged particle in the earth's magnetic field (which is assumed here to be a pure dipole field).

As the first step of our simplification procedure, we transform from the ordinary cylindrical polar coordinates used above to dipolar coordinates and expand the resulting expression in terms of these new coordinates and their corresponding canonically conjugate momenta , (cf. G. Contopoulos, L. Vlahos, J. Math. Phys. 16 (1975) 1469-1474):

Now generalized normal form theory (U. M. Engel, B. Stegemerten, P. Eckelt, J. Phys. A: Math. Gen. 28 (1995) 1425-1448) can be applied to this transformed Hamiltonian, yielding the following most simple normal form for the Størmer problem:

The third, and final, step on our path towards maximum simplicity is the transformation of the above normal form into the corresponding quasiintegral of motion; for our little sample problem, a minute's thought will reveal that we have:

The most impressive simple structure of this tiny little formula and its evident meaningfulness immediately show the superiority of our simplification algorithm.

Martin Engel 1992-11-29, 2001-04-04, 2004-01-01