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Re: [Phys-l] Rocket Science



On Jul 28, 2006, at 6:18 AM, 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.

At least among power law central forces this is correct. Other powers of the form n^2 - 3 (e.g., 6, 13, 22, etc.) give rise to approximately closed orbits in the "nearly circular" limit. See, for instance, <http://farside.ph.utexas.edu/teaching/336k/lectures/ node51.html>. Also ANY power law central force greater than -3 will give rise to lots of different closed orbits if the initial conditions are picked carefully. For instance its fun to play with high power attractive central forces like r^20 with which one can form pretty fair replicas of equilateral triangles, pentagons, octagons, etc, and even n pointed stars.

But how does one prove in general that orbits are closed in a given problem? 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? -Carl

I may be misinterpreting your question, but ...

The usual method is to compare the period for a radial oscillation to the azimuthal period. In the case of the r^-2 force law the periods are the same. In the case of the r^+1 force there are two complete radial oscillations in one azimuthal period.

John "Slo" Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://outlawsofphysics.com>