Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report
The Birkhoff-Gustavson normal form has been generated to a high order for the Henon-Heiles model Hamiltonian. This analysis provides approximate integrable classical dynamics for the well-studied stochastic behavior of this system. Accelerated convergence of the series expansion for a second integral of the motion shows divergence of the series only in small regions of phase space, indicating approximate local integrability of the dynamics. The regions of divergence correspond well to the regions where stochastic motion first appears. If the regions of divergence are much smaller than Planck’s constant, one would expect such behavior would be irrelevant with respect to quantum mechanics. This result lends support to the semiclassical quantization methods of Swimm and Delos and of Jaffé and Reinhardt since most of the invariant manifold structure remains intact.
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