Hénon-Heiles equation

The Henon-Heiles equation is used to model stars. It is expressed as

$V(x,y) = \frac{1}{2}(x^2%2By^2%2B\frac{1}{2}x^2y %2B \frac{2}{3}y^3)$

While at Princeton in 1964, Michel Hénon and Carl Heiles published a paper that describes the non-linear motion of a star around a galactic center where the motion is restricted to a plane.

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

$dx/dt=u$

$dy/dt=v$

$dv/dt=-Ax%2B2xy$

$dv/dt=-By%2B\epsilon y^2%2Bx^2$

where $A,B, \epsilon \in \R, A > 0$ and $B > 0$ . 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 Painleve Analysis. The Hamiltonian for the HHS is

$H=\frac1 2 (u^2%2Bv^2%2BAx^2%2BBy^2 )-x^2 y-\frac 1 3 \epsilon y^3$

External links

http://bbs.sachina.pku.edu.cn/Stat/Math_World/math/h/h183.htm