Hénon-Heiles equation

While at Princeton in 1962, Michel Hénon and Carl Heiles worked on the non-linear motion of a star around a galactic center with the motion restricted to a plane. In 1964 they published an article 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 the initial conditions that do not have the third integral of motion are called chaotic orbits.

The Hénon–Heiles potential can be expressed as

The Hénon–Heiles Hamiltonian can be written as

The Hénon–Heiles system (HHS) is defined by the following four equations:

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

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