Explicit integration of the Hénon-Heiles Hamiltonians 1

2 00 6 Painlevé Property of the Hénon - Heiles Hamiltonians

— Time independent Hamiltonians of the physical type H = (P 2 1 + P 2 2 )/2 + V (Q1, Q2) pass the Painlevé test for only seven potentials V , known as the Hénon-Heiles Hamiltonians, each depending on a finite number of free constants. Proving the Painlevé property was not yet achieved for generic values of the free constants. We integrate each missing case by building a birational transformatio...

Statistical mechanics of Hénon-Heiles oscillators.

Statistical mechanics is an asymptotic theory valid in the limit of an infinite number of degrees of freedom. The study of small-dimensional chaos, starting from the papers by Lorenz [1]and Henon and Heiles [2], raises the question: What is essential for the laws of statistical mechanics to be true —a large number of degrees of freedom or chaos? For ergodic Hamiltonian systems the answer was gi...

Possible conserved quantity for the Hénon-Heiles problem.

Level Density of the Hénon - Heiles System Above the Critical Barrier Energy

We discuss the coarse-grained level density of the Hénon-Heiles system above the barrier energy, where the system is nearly chaotic. We use periodic orbit theory to approximate its oscillating part semiclassically via Gutzwiller’s semiclassical trace formula (extended by uniform approximations for the contributions of bifurcating orbits). Including only a few stable and unstable orbits, we repr...

Construction of a quasiconserved quantity in the Hénon-Heiles problem using a single set of variables.