A few people have commented on my problem, but so far not along the
lines I was thinking of. My fault for not clearly spelling out the
problem. So let me now be explicit. The second-order differential
equation for r(phi) is:
r(1+r*r)(ddr) = (2+r*r)(dr)(dr) + r*r[1-(r/r0)^4]
where dr means the first derivative of r(phi) w.r.t phi, ddr means
the second derivative, and r0 is the initial value of r (ie. at
phi=0). Don't worry about the fact that this equation doesn't look
dimensionally correct - I have normalized r into dimensionless form
by absorbing constants involving m, g, etc.
So we don't have a power law and trying to find whether r is periodic
in phi is not obvious to me. You can solve the above equation
numerically and plot it for the 2-dimensional motion of r(phi),
assuming initial values r0 and dr(0). You should find bounded orbits
for any positive values of these two initial parameters.
The question again is: The resulting numerical plots don't look
closed when I randomly try various positive initial values. But maybe
I simply didn't try the right values. How can I prove mathematically
that the orbits aren't closed for any positive values of the two
initial parameters?
--
Carl E. Mungan, Asst Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002 mailto:mungan@usna.eduhttp://usna.edu/Users/physics/mungan/