Lyapunov exponents for classical orbits of the hydrogen atom in a magnetic field

Lyapunov characteristic exponents are calculated for classical trajectories of the Hamiltonian describing a hydrogen atom in a uniform magnetic field, and particular attention is given to periodic orbits. As the magnetic field is turned on, instability grows around the almost circular orbit which is a precise circle in the integrable limit ɛ=-∞, ɛ being the scaled energy of the system. The Lyapunov exponent of the almost circular orbit is proportional to ‖ɛ${\Vert }^{\mathrm{-}3/2}$ near the integrable limit, and this is consistent with a square-root law found by G. Benettin [Physica D 13, 211 (1984)] for the onset of instability in certain billiards.