Hénon-Heiles Equation

A nonlinear nonintegrable Hamiltonian System with

$$\ddot x = -{\partial V\over\partial x}$$ (1)

$$\ddot y = -{\partial V\over\partial y},$$ (2)

where

$$V(x,y) = {\textstyle{1\over 2}}(x^2+y^2+2x^2y-{\textstyle{2\over 3}}y^3)$$ (3)

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

The energy is

$$E=V(x,y)+{\textstyle{1\over 2}}({\dot x}^2+{\dot y}^2).$$ (5)

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

$$H = {\textstyle{1\over 2}}({p_x}^2+{p_y}^2+Ax^2+By^2)+Dx^2y-{\textstyle{1\over 3}}Cy^3.$$ (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. ${D/C}=-{1/16}, {A/B}={1/6}$.

References

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.