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

Abstract

Applying the Painleve Test (P. T.) for Partial Differential Equations to the Hamilton-Jacobi equation of two-degree-of-freedom Hamiltonian Systems, a hierarchy of highly nonlinear equations satisfied by the potential function of these systems is derived. Using these equations as conditions for integrability we identify for a number of potentials of polynomial type (Henon-Heiles, Quartic and Sextic) critical parameter values for which the corresponding Hamiltonian Systems are Painleve Integrable (P. I.), (in the sense that they pass the P. T.). For a polynomial potential function which is not P. I. the conditions give precisely the parameter value for which the system has regular behavior (in terms of regular Poincare plots). Applying the first and simplest nonlihear equation to the Restricted Circular Three Body Problem (a non-polynomial potential function) we have an indication of the known stability region determined by the parameter μ where the critical value 0.0385 is impressively revealed. A direct attempt to solve the first equation of the hierarchy gave the obvious solution of a rotational invariant potential function and a new class of potentials which give regular scattering behavior. It is conjectured that other nontrivial solutions of this simple equation would give nontrivial classes of potential functions with regular behavior.