Hénon-Heiles Equation

The Hénon-Heiles equation is a nonlinear nonintegrable Hamiltonian system with


where the potential energy function is defined by the polar equation


giving Cartesian potential


The total energy of the system is then given by


which is conserved during motion.


Integrating the above coupled ordinary differential equations from an arbitrary starting point with x(t=0)=0 and E=1/8 gives the motion illustrated above.


Surfaces of section are illustrated above for various initial energies E, y(t) is plotted vs. y^.(t) at values where x(t)=0.

The Hamiltonian for a generalized Hénon-Heiles potential is


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.


The plots above show a number of eigenfunctions of the Schrödinger equation with a generalized Hénon-Heiles potential


for certain specific values of (a,b) (M. Trott, pers. comm., Jan. 6, 2004).

See also

Standard Map, Surface of Section

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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.Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 171-172, 1990.Tabor, M. "The Hénon-Heiles Hamiltonian." §4.1.b in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 121-122, 1989.Trott, M. "The Mathematica Guidebooks Additional Material: Hénon-Heiles Eigenfunctions."

Referenced on Wolfram|Alpha

Hénon-Heiles Equation

Cite this as:

Weisstein, Eric W. "Hénon-Heiles Equation." From MathWorld--A Wolfram Web Resource.

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