Detecting Regularity in Hamiltonian Systems by Tracing Critical Parameters of Potential Functions

概要

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Applying the Painleve Test (P. T.) for Partial Differential Equations to the Hamilton-Jacobiequation of two-degree-of-freedom Hamiltonian Systerns, a hierarchy of highly nonlinear equa-tions satisfied by the potential function of these systeuns is derived. Using these equations asconditions for integrability we identify for a nttraaber of potentials of polynomial type (Henon-Heiles, Quartic and Sextic) critical parameter values for which the corresponding HamiltonianSystems are Painleve Integrable (P. I.), (in the sense that they pass the P. T.). For a polynomialpotential ftnnction which is not P. I. the conditions give precisely the parameter value for whichthe system has regular behavior (in terms of regular Poincare plots). Applying the first andsimplest nonlilaear equation to the Restricted Circular Three Body Problem (a non-polynomialpotential ftrnction) we have an indication of the known stability region determined by the pa-rameter p where the critical value 0.0385 is impressively revealed. A direct attempt to solvethe first equation of the hierarchy gave the obvious solution of a rotational invariant potentialfunction and a new class of potentials which give regttlar scattering behavior. It is conjecturedthat other nontrivial solutions of this sinaple equation would give nontrivial classes of potentialfunctions with regular behavior.
社団法人日本物理学会の論文
1996-08-15