Computer Algebra Recipes

324 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS
For N = 1, the motion can be described by a trajectory in the q1 vs. p1 phase
plane. For N>1, the trajectory is in a 2N-dimensional phase space.
Originally motivated to study the motion of a star inside a galaxy, Henon
and Heiles [HH64] introduced a conservative Hamiltonian describing the motion
of a unit mass in the two-dimensional potential
V = 1
2 q2 1 + 1
2 q2 2 + q 2 1 q2 ¡ 1
3 q3 2: (8.4)
The ¯rst two terms in V would generate a paraboloid of revolution characteristic
of a two-dimensional harmonic oscillator. The force ~ F = ¡rV in this case is
just the two-dimensional form of Hooke's law. The inclusion of the two cubic
terms in V distort the shape of the potential away from a paraboloid, add
nonlinear terms to Hooke's law, and Hamilton's equations generate nonlinear
ODEs in the 4-dimensional phase space.
In this recipe, we shall use specialized commands found in the DEtools
library package to generate these equations, numerically solve them to produce
the trajectory for a speci¯ed energy and initial conditions, and produce a
Poincare section.
> restart: with(plots): with(DEtools):
Entering the Henon{Heiles potential (8.4), a two-dimensional contour plot is
generated, the contour lines given by V =0:04 i with i = 0 to 9. To obtain
smooth curves, the number of plotting points is taken to be 5000.
> V:=q1^2/2+q2^2/2+q1^2*q2-q2^3/3:
> contourplot(V,q1=-2..2,q2=-2..2,contours=[seq(0.04*i,
i=0..9)],numpoints=5000,color=black);
q2
2
1
–2 –1 0
1 q1 2
–1
–2
Figure 8.4: Contour plot of the Henon{Heiles potential.