I Goldstein Action-angle Variables

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1. Mar 30, 2017

crossword.bob

I'm currently working (slowly) through Goldstein (et al), 3rd Edition, and a remark in the section on Action-angle Varibles for Completely Separable Systems (10.7) is giving me pause. We're told that the orbit equations for all $(q_i, p_i)$ pairs in phase space describe libration or periodic forms.

However, in the following paragraph, we are warned that this does not mean that each $q_i$ and $p_i$ will necessarily be periodic functions of time, and I am having some trouble imagining a situation where that might not be the case.

First of all, it is clear to me that even if they are each individually periodic, the overall particle motion need not be (if, say, periods are not co-rational), as he goes on to say. That bit's fine. It's the case where they would not be individually periodic that I'm unclear on.

Would I be correct in thinking that the trick here is that the orbit equation $$p_i = p_i(q_i; \alpha_1,\ldots, \alpha_n),$$ simply relates $q_i$ and $p_i$, and that the actual particle motion (projected onto this coordinate plane), parameterised by $t$, doesn't necessarily trace this orbit in a simple motion?

If this is the case, is there a (relatively) straightforward example where this happens? As far as I can tell, the usual suspects, (an)isotropic harmonic oscillators, Kepler, do have orbits that lead to individually periodic $p_i$ and $q_i$; this seems to be built into the simple connections between $\dot{q}_i$ and $p_i$. Does this break down for more generalised coordinates?

2. Mar 30, 2017