It's not clear to me that the resulting orbit is an ellipse. I would expect the radius to be a periodic function of time, indeed, but I think that in general the periapse would precess. For particular values the orbit would close after tracing a p/q rosette, but for general initial conditions the orbit would never close. On Wed, Dec 7, 2011 at 10:03 AM, Cordwell, William R <wrcordw@sandia.gov>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
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