Note on the Three-Particle Lattice and the Henon-Heiles Problems

摘要

The Lagrangian of a three-particle lattice may be reduced to the form L = 1/2((x-dot)sq + (y-dot)sq) - V(x,y), where x and y are normal coordinates (the third coordinate being ignorable). If only the quadratic and cubic terms are retained in the expansion of V about x = y = 0, the problem reduces to that of Henon and Heiles and is characterized by a stable center at the origin and three saddle points at the vertices of an equilateral triangle within (outside of) which V > (<) 0. The inclusion of the quartic term in V alters the Henon-Heiles problem by rendering V > 0 for all x,y for an appropriate range of parameters (including that of Toda's exponential potential). The problem then is integrable if the cubic component of V vanishes identically; otherwise, the motion may be chaotic, but only for much larger energies than those required for comparable chaos in the Henon-Heiles problem. Representative numerical results are presented. Keywords: Three particle lattice; Henon Heiles problem; Strange attractors; Chaotic motion; Reprints. (Author)