Re: [Phys-l] Rocket Science

[John Denker <jsd@av8n.com>]

Carl Mungan wrote:

> How can I prove mathematically 
> that the orbits aren't closed for any positive values of the two 
> initial parameters?

One conceptually-simple approach is to hypothesize that the orbit
is periodic and then express r(theta) as a Fourier series.  That
replaces the differential equation (DE) by a system of algebraic
equations (AEs).
 -- If the AEs are insoluble, i.e. self-inconsistent, this proves
  that the orbits are not periodic.  Proof by contradiction.
 -- If the AEs are soluble under certain conditions, that tells
  you the conditions under which periodic orbits exist.

If you started with a not-too-complex differential equation, you
can hope to find a not-too-complex _recurrence relation_ for the
Fourier coefficients.

It is fairly common in physics for such recurrence relations to
have acceptable solutions for some but not all values of the
boundary conditions.  A loosely-analogous but interesting example
can be seen in Feynman volume III equations 19.22 and/or 19.50.
This was foreshadowed in the discussion (including pictures) at
the end of chapter 16.

Another loosely-analogous but interesting example can be found
in chapter 13, especially section 13-2.