Several years ago, I was considering small perturbations of geostationary orbits. Assuming that the satellites are small enough not to have any attraction for one another, the orbits are isochronous only when their semimajor axes are the same length. So if one is in a circular orbit, and the other is in a very slightly elliptical orbit, then one will act as an approximately day-long yoyo, going down & then coming back up, but never going past on the up portion. Of course, the perturbations from variations in the Earth's gravity are more significant; apparently, Mt Everest pulls enough harder to cause all the geostationary objects to eventually park themselves above the Indian Ocean. Q: suppose 2 approx geostationary objects are in the same plane with the same (extremely small) eccentricity, but their semimajor axes are at right angles to one another. Furthermore, the objects are never very far apart -- e.g., max 1 mile. In the frame of reference of one of the objects, what path does the other object take? At 07:03 AM 12/7/2011, Cordwell, William R wrote:

For the cone, if the particle is in a stable circle, I would expect that a small impulse (down or up) would give it an oscillatory behavior, similar to being in an orbit with constant angular momentum, with the resulting ellipse being viewed as an oscillation about the circular orbit. Harmonic is not clear.

-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker Sent: Monday, December 05, 2011 8:28 AM To: Bill Gosper Cc: math-fun@mailman.xmission.com Subject: [EXTERNAL] Re: [math-fun] spherical pendulums?

Here's a related problem:

A point particle of mass M (i.e., zero moment of inertia) is sliding frictionlessly around inside a vertical cylinder; gravity is downwards of strength G. What are the paths?

Ditto with a vertical cone instead of a cylinder.

At 12:44 AM 12/5/2011, Bill Gosper wrote:

...

Are they chaotic or just hairy? The Wolfram demonstration seems quasiperiodic.? Gene once disabused me of the folly of trying to resolve the motion into x and y. But suppose we hung the string from the inside of an upward cusp of a cycloid of revolution. Would the pendulum simply describe an ellipse? --rwg