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Hénon-Heiles Equation

A nonlinear nonintegrable Hamiltonian System with

$\displaystyle \ddot x$ $\textstyle =$ $\displaystyle -{\partial V\over\partial x}$ (1)
$\displaystyle \ddot y$ $\textstyle =$ $\displaystyle -{\partial V\over\partial y},$ (2)

$\displaystyle V(x,y)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(x^2+y^2+2x^2y-{\textstyle{2\over 3}}y^3)$ (3)
$\displaystyle V(r,\theta)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}r^2+{1\over 3} r^3\sin(3\theta).$ (4)

The energy is
E=V(x,y)+{\textstyle{1\over 2}}({\dot x}^2+{\dot y}^2).
\end{displaymath} (5)

\begin{figure}\begin{center}\BoxedEPSF{henon_heiles083.epsf scaled 320}\quad\BoxedEPSF{henon_heiles125.epsf scaled 320}\end{center}\end{figure}

The above plots are Surfaces of Section for $E=1/12$ and $E=1/8$. The Hamiltonian for a generalized Hénon-Heiles potential is

H = {\textstyle{1\over 2}}({p_x}^2+{p_y}^2+Ax^2+By^2)+Dx^2y-{\textstyle{1\over 3}}Cy^3.
\end{displaymath} (6)

The equations of motion are integrable only for
1. ${D/ C}=0$,

2. ${D/C}=-1, {A/B}=1$,

3. ${D/C}=-{1/6}$, and

4. ${D/C}=-{1/16}, {A/B}={1/6}$.


Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 144-153, 1988.

Hénon, M. and Heiles, C. ``The Applicability of the Third Integral of Motion: Some Numerical Experiments.'' Astron. J. 69, 73-79, 1964.

© 1996-9 Eric W. Weisstein