Hénon-Heiles System

Contour plot of the Hénon-Heiles potential

While at Princeton in 1962, Michel Hénon and Carl Heiles worked on the non-linear motion of a star around a galactic center where the motion is restricted to a plane. In 1964 they have published an article [1] titled 'The applicability of the third integral of motion: Some numerical experiments'. Their original idea was to find a third integral of motion in a galactic dynamics. For that purpose they have taken a simplified two-dimensional nonlinear axi-symmetric potential and found that the third integral exist only for a limited number of initial conditions. In the modern perspective these initial conditions which doesn't have the third integral motion are called chaotic orbits.

Introduction

The Hénon-Heiles Potential can be expressed as[2]

V(x,y) = \frac{1}{2}(x^2+y^2)+\lambda (x^2y - \frac{y^3}{3})

The Hénon-Heiles Hamiltonian can be written as

H=\frac1 2 (p_{x}^2+p_{y}^2)+\frac{1}{2}(x^2+y^2)+\lambda (x^2y - \frac{y^3}{3})

The Hénon-Heiles System (HHS) is defined by the following four equations:

\dot{x}=p_{x}
\dot{p_{x}}=-x-2\lambda xy
\dot{y}=p_{y}
\dot{p_{y}}=-y-\lambda(x^2-y^2)

In the classical chaos community, the value of the parameter \lambda is usually taken as unity. Since HHS is specified in \R^2, we need a Hamiltonian of degrees of freedom two to model it. It can be solved for some cases using Painlevé Analysis.

Quantum Henon-Heiles Hamiltonian

In the quantum case the Henon-Heiles Hamiltonian can be written as a two-dimensional Schrödinger equation.

The corresponding two-dimensional Schrödinger equation is given by

i\hbar\frac{\partial}{\partial t} \Psi(x,y) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + \frac1 2 (x^2+y^2 )+\lambda (x^2 y-\frac 1 3 y^3) \right ] \Psi(x,y).

Wada Property of the Exit Basins

Hénon-Heiles system shows rich dynamical behavior. Usually the wada property cannot be seen in the Hamiltonian system but Hénon-Heiles exit basin shows an interesting wada property. It can be seen that when the energy is greater than the critical energy, the Hénon-Heiles system has three exit basins. In 2001 M. A. F. Sanjuán et al. [3] had shown that in the Henon-Heiles system the exit basins have the wada property.

References

  1. ↑ Hénon, M.; Heiles, C. (1964). "The applicability of the third integral of motion: Some numerical experiments". The Astrophysical Journal 69: 73–79. Bibcode:1964AJ.....69...73H. doi:10.1086/109234.
  2. ↑ Hénon, Michel (1983), "Numerical exploration of Hamiltonian Systems", in Iooss, G., Chaotic Behaviour of Deterministic Systems, Elsevier Science Ltd, pp. 53–170, ISBN 044486542X
  3. ↑ Wada basins and chaotic invariant sets in the Henon-Heiles system, Phys. Rev. E 64, 066208 (2001)

External links

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