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Hénon-Heiles Equation


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

x^..=-(partialV)/(partialx)
(1)
y^..=-(partialV)/(partialy),
(2)

where the potential energy function is defined by the polar equation

 V(r,theta)=1/2r^2+1/3r^3sin(3theta),
(3)

giving Cartesian potential

 V(x,y)=1/2(x^2+y^2+2x^2y-2/3y^3).
(4)

The total energy of the system is then given by

 E=V(x,y)+1/2(x^.^2+y^.^2),
(5)

which is conserved during motion.

HenonHeilesODE

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.

HenonHeiles

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

 H=1/2(p_x^2+p_y^2+Ax^2+By^2)+Dx^2y-1/3Cy^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.

HenonHeilesModes

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

 V(r,theta)=r^4+ar^2+br^3cos(3theta)
(7)

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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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.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." http://www.mathematicaguidebooks.org/additions.shtml#S_2_01.

Referenced on Wolfram|Alpha

Hénon-Heiles Equation

Cite this as:

Weisstein, Eric W. "Hénon-Heiles Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Henon-HeilesEquation.html

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