Integrals of motion and resonances in a dipole magnetic field

A method is developed for deriving a third integral of motion, besides the Hamiltonian and the angular momentum, of a charged particle, in a dipole magnetic field. This method is particularly useful in resonance cases, where the usual adiabatic invariants are not applicable. First the Hamiltonian is reduced to a ’’regular’’ form, i.e., its lowest order terms are written as H2=Φ12 + Φ22, where Φ12= (1/2)(a2+ p2a), Φ22= (1/2)(ω2b2+ p2b). Then the third integral can be constructed step by step as a series; in every resonance case a different form of the integral is derived. In the nonresonant cases the Hamiltonian is written in a normal form H=H* (Φ1, Φ2), where Φ1, Φ2 are canonical variables introduced by using von Zeipel’s method. The nonresonant orbits are quasiperiodic with frequencies ω1=∂H*/∂Φ1, ω2=ω∂H*/∂Φ2 and rotation number Rot=ω1/ω2. As an example the location of a particular resonance is found. The comparison with the numerical integrations is satisfactory.