Poincaré section of the Hénon-Heiles Hamiltonian

This simulation explores the solution of the Hamiltonian used by Hénon and Heiles to study the motion of a star in a galaxy:
H = ½ (p_x² + p_y² + x² + y²) + x² y - y³/3
Since it has two degrees of freedom, the phase-space has dimension 4. To get a three-dimensional dynamical system we choose for each value of the conserved energy the corresponding energy surface.

The initial value for x is always 0. You can select below (in the corresponding edit boxes or by using the mouse in the plane x = 0) the initial values y and p_y. The initial value for p_x is automatically computed by the program to make sure H = E for the energy value E selected in the corresponding edit box.
The simulation will display on the left a projection of the three-dimensional orbit (x,y,p_y) on the energy surface and on the right the (y,p_y) coordinates of the solution points at which x = 0. The green points are the border of the intersection of the energy surface and the (y,p_y) plane. Attempting to choose initial values outside the energy surface will issue an error message.
The point of view of the three-dimensional projection can be changed with the mouse.
The whole image can be moved with the mouse while pressing Ctrl.
To change the zoom in the projections, press Shift when moving up or down the mouse pointer.
Put the mouse pointer over an element to get information about

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

Activities

Press Start to start the simulation. From time to time you may use Clear orbit, to see better only the orbit.
To fully appreciate the form of the invariant set containin the orbit, uncheck Plane and use the mouse to change the projection view. (You may also want to set Ω = 0 to avoid the constant change of viewpoint.)
Check that for E = 1/12= 0.083... the system is predominantly integrable, since most orbits (but not all) lie in an invariant torus.
Most orbits are biperiodic: find an (approximately) periodic solution (try around y = 0.102, p_y = 0).
Select E = 1/8 = 0.125 and you should be able to see that many toruses have been destroyed but there are still a chain of islands of integrability.
For E = 1/6 = 0.166... the system is predominantly chaotic. Check this and find some of the remaining small toruses.

This is an English translation of the Basque original for a course on theoretical mechanics.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.