Reference: A.J. Lichtenberg and M.A. Lieberman, "Regular and Stochastic Motion", Applied Mathematical Sciences 38 (New York: Springer Verlag, 1994).
Figure 1.a. shows a 2-D surface-of-section (2PS) over the q2=0 plane, with 127 intersection points lying on smooth curves.
Figure 1.b is a 2PS over the q1=0 plane with 146 intersection points. The smoothness of the curves in both (p,q) planes is related to the integrability of the system.
A Poincare space-of-section corresponding to Figure1.a can be manipulated with the mouse to obtain the following illustrative perspectives:
Figure 2.a - 3-D projection of a surface of section (3PS) showing a KAM surface of regular trajectories. The plot has been manipulated with the mouse to produce a view at , .
Figure 2.b The same figure was manipulated with the mouse to display a plane projection of the 3PS (at , ) showing how the intersection points are joined outside the 2PS.
Another indication of the integrability of the system is that regular curves exist whatever the value of H. As an example of this, a surface-of-section (one solution curve), and a related 3-D projection, at H=256, can be built as follows:
A set with one list of initial conditions satisfying the Hamiltonian constraint (H0=256):
Figure 3.a shows smooth curves on the 2PS, q1=0 plane.
Figure 3.b is a 3PS corresponding to Figure 3.a, displaying a KAM surface constituted by just one regular curve. The plot has been manipulated with the mouse to produce a view at , , and the Projection was set to Far (in menu bar when the plot is selected.)
