Analytic simulation of the Poincare surface of sections for the diamagnetic Kepler problem

The Poincare surface-of-section analysis which the authors previously reported (ibid., vol.16, p.L503, 1983) on the diamagnetic Kepler problem (classical hydrogen atom in a uniform magnetic field) in a transition region from regular to chaotic motions is simulated by an analytic means, by taking intersections of the energy integral and the approximate integral Lambda of Solovev to obtain sections of the two separate regions of the motion that exist in the limit of a weak magnetic field (B to 0). The origin of the unique hyperbolic point and the separatrix around which the onset of chaos takes place are thus identified. The invariant tori arising near the full chaos are shown to be simulated by this method but with modified parameter values in the expression Lambda .