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 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 took a simplified two-dimensional nonlinear axi-symmetric potential and found that the third integral existed only for a limited number of initial conditions. In the modern perspective these initial conditions which do not 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^{2}y-{\frac {y^{3}}{3}})

The Hénon-Heiles Hamiltonian can be written as

H={\frac 12}(p_{{x}}^{2}+p_{{y}}^{2})+{\frac {1}{2}}(x^{2}+y^{2})+\lambda (x^{2}y-{\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}+{\frac 12}(x^{2}+y^{2})+\lambda (x^{2}y-{\frac 13}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

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

External links

http://mathworld.wolfram.com/Henon-HeilesEquation.html

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