The Hénon-Heiles equation is a nonlinear nonintegrable Hamiltonian system with
(1)
| |||
(2)
|
where the potential energy function is defined by the polar equation
(3)
|
giving Cartesian potential
(4)
|
The total energy of the system is then given by
(5)
|
which is conserved during motion.
Integrating the above coupled ordinary differential equations from an arbitrary starting point with
and
gives the motion illustrated above.
![HenonHeiles](images/eps-svg/HenonHeiles_850.png)
Surfaces of section are illustrated above for various initial energies ,
is plotted vs.
at values where
.
The Hamiltonian for a generalized Hénon-Heiles potential is
(6)
|
The equations of motion are integrable only for
1. ,
2. ,
3. ,
and
4. .
![HenonHeilesModes](images/eps-svg/HenonHeilesModes_700.png)
The plots above show a number of eigenfunctions of the Schrödinger equation with a generalized Hénon-Heiles potential
(7)
|
for certain specific values of (M. Trott, pers. comm., Jan. 6, 2004).