|Nov13-05, 01:51 PM||#1|
2-Body inverse cube problem
Hello, everyone. Here is my question and my thoughts on it:
Suppose we lived in a bizarre world in which gravitation, instead of being an inverse square law, were an inverse cube law.
(a) In this world, show that the 2-body problem can be brought to a central force problem in a plane P by an appropiate choice of coordinates.
(b) Write this central force problem in polar coordinates in P.
(c) Describe the motions. Are there any where r(t) is bounded away from 0 and infinity?
(a) This has nothing to do with the fact that gravity is an inverse cube law. Just use radial coordinates and the reduced mass to write it in terms of radial vectors. My only qualm is that reduced mass comes from Newton's 3rd Law and maybe Newton's 3rd Law doesn't hold in inverse-cube gravity?
(b) Normally, the central force problem is written as: http://www.answers.com/main/ntquery;...body%20problem
but I am wondering whether the tangential component of acceleration is not equal to zero as it is here because the radius will not be constant...
(c) I'm pretty sure that the motions will be spirals and are thus unstable so there is no r(t) where the motions are stable (or not bounded away from 0 and infinity). I'm pretty sure this is the right answer and is partially what is tripping me up on (b) as now the radius won't be constant therefore there will be a tangential component to acceleration in polar coordinates.
Any thoughts or suggestions are appreciated.
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