http://www.mathhelpforum.com/math-he...1&d=1210543347

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- May 11th 2008, 02:02 PMszpengchaooscillation of a ring on parabola
- May 11th 2008, 02:03 PMszpengchaono idea with (ii)
no idea with ii

- May 11th 2008, 06:59 PMmr fantastic
Start by getting the equations of motion using Lagranges equation. Here is a reference that saves me the trouble of doing it: (Example 3.2)

Schaum's Outline of Theory and ... - Google Book Search - May 11th 2008, 07:27 PMtopsquark
Or you could attack the thing head-on.

Newton's Law in the two coordinate directions says

That system looks horrible to solve, but the thing is that I can't come up with a good generalized coordinate for this problem. The best I can offer is to put an angular variable at the focus of the parabola.

However, things aren't all bleak. All we need is the coefficients of the expansion:

Presumably if h is small then A = 0, the coefficients of the powers of x higher than 1 are small, and B is negative. In that case

which is the harmonic oscillator equation. I leave it to you to check that these conditions come out. (I haven't checked.)

-Dan

Edit: After seeing the original posting of this problem it is clear that the problem is requiring an exact, not approximate solution. The only way I can think of to get this is to use the Lagrangian technique. But again, I can't think of a good generalized coordinate here.