Computer Algebra Recipes

8.1. THE POINCAR ESECTION 325
To aid in interpreting the contour plot, let's ¯nd the stationary points where the
force vanishes. Di®erentiating V with respect to each coordinate and solving
for the coordinate values that make the force components equal to zero,
> sol:=solve(fdiff(V,q1),diff(V,q2)g,fq1,q2g);
sol := fq2 =0; q1 =0g; fq2 =1; q1 =0g;
fq2 = ¡1 1
; q1 =
2 2 RootOf(¡3+ Z 2 ; label = L2 )g
yields a stationary point at q1 = q2 =0andq1 =0; q2 = 1, as well as others
at q2 = ¡ 1 and q1 given by the RootOf placeholder. These latter values may
2
be found by applying the allvalues command to the third entry in sol.
> sol3:=allvalues(sol[3]);
p p
3 ¡1 ¡1 3
sol3 := fq1 = ; q2 = g; fq2 = ; q1 = ¡
2 2 2 2 g
1 ; q2 = ¡ 2 2 and at q1 = ¡ p 3;
q2 = 2
. The four stationary points are now extracted separately and labeled,
> s1:=sol[1]; s2:=sol[2]; s3:=sol3[1]; s4:=sol3[2];
There are two more ¯xed points at q1 = p 3
¡ 1
2
s1 := fq2 =0; q1 =0g s2 := fq2 =1; q1 =0g
p p
3 ¡1
¡1 3
s3 := fq1 = ; q2 = g s4 := fq2 = ; q1 = ¡
2 2 2 2 g
and the potential energy at each stationary point determined.
> U:=v->eval(V,v): V1:=U(s1); V2:=U(s2); V3:=U(s3); V4:=U(s4);
V1 := 0 V2 := 1 1 1
V3 := V4 :=
6 6 6
The stationary point s1 at the origin is the minimum of what would be a
parabolic potential well if the cubic terms were not present in V . Referring
to the contour plot, the shape of the contour lines changes as one moves away
from the origin, the contour lines near the other three stationary points being
characteristic of saddle points. 1 If the particle has a total energy below the
potential energy 1
6 at the saddle point and starts inside the region bounded
by the three saddle points, it will have a bounded orbit inside this region. If
E> 1
6 , the particle could escape through one of the saddle points to in¯nity.
The Hamiltonian is entered, and the command hamilton eqs used to generate
Hamilton's equations and a list of the four dependent variables.
> H:=p1^2/2+p2^2/2+V;
> hamilton_eqs(H);
[ d
d
p1(t) =¡q1 (t) ¡ 2 q1(t) q2 (t);
dt dt p2 (t) =¡q2 (t) ¡ q1 (t)2 + q2 (t) 2 ;
d
d
q1 (t) =p1(t); q2 (t) =p2 (t)]; [p1(t); p2 (t); q1 (t); q2 (t)]
dt dt
1 You can con¯rm this by making a 3-dimensional contour plot using the plot3d command.