Re: [Phys-l] Rocket Science

[John Denker <jsd@av8n.com>]

Carl Mungan wrote:

> I seem to recall reading in
> mechanics books that only the inverse square and harmonic oscillator force laws give rise to
> closed orbits.
>
> But how does one prove in general that orbits are closed in a given problem? 

Wow, what an interesting question.

John M. has outlined a general, elegant answer.

Let me give a much less general answer, addressing only the two cases
mentioned.

 -- For the -1/r potential, you can exhibit the exact solution and
  observe that it is a closed ellipse.  You can also show that closure
  is unstable w.r.t a wide class of perturbations;  for instance, a
  -r^(-1.0001) potential can be described as an ellipse that precesses.

 -- For the r^2 potential, separation of variables gets the job done.
  Write r^2 = x^2 + y^2 and solve the x-part and the y-part separately.


> Suppose either you
> are given differential equations for r(t) and phi(t), which one should probably be able to
> translate into a second-order equation for r(phi), or equivalently say you are given the force
> expression and relevant conservation laws? 

I'm not sure about the "equivalently" there.  Getting from differential
equations to conservation laws is nontrivial.  There are important cases where
it *can* be done, notably if you start with a Hamiltonian, so that Noether's
theorem applies ... but if you start with some more-general (non-Hamiltonian)
differential equations for r(t) and phi(t), I don't know of any systematic
way of discovering what (if anything) is conserved.

Noether's theorem is a thing of beauty.  It's worth learning classical
physics just to get to this result.
  Hamiltonian independent of time     <==> conservation of energy
  Hamiltonian independent of position <==> conservation of momentum
  Hamiltonian independent of rotation <==> conservation of angular momentum
  Gauge independence of E field       <==> conservation of charge
  etc.

http://en.wikipedia.org/wiki/Noether's_theorem

==================

Also: Returning to a previous part of this thread:  The virial theorem is
more robust and less of a kludge than you might have gathered from intro-level
textbooks.  Wikipedia has a relatively nice, clear, general derivation:
  http://en.wikipedia.org/wiki/Virial_Theorem