On the power of normal form theory:

*Simplification* of a Hamiltonian dynamical system

using normal form theory and quasiintegrals of motion

by Martin Engel

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