- #1
giraffe714
- 21
- 2
- TL;DR Summary
- Because the action variable is not necessarily the same between oscillatory and rotational motion, how can it be ensured that the Hamiltonian is differentiable, or at least continuous at the separatrix?
As said in the tl;dr: is the Hamiltonian necessarily differentiable (hence continuous) at the separatrix in the action-angle formalism? After all, the action variables are different depending on the type of motion. As far as I know the Hamiltonian H = H(J) can be found by inverting J for E, and the most obvious thing I could come up with (which may just be where I'm going wrong) is to define H as a piecewise function, with:
$$ H = \begin{cases} H(J_{osc}) & E < E_{osc} \\ H(J_{rot}) & E > E_{rot} \end{cases} $$
But (a), it isn't defined at the separatrix, mainly because I'm not entirely sure how I would go about calculating J at the separatrix as, from what I can understand, at that energy there's basically a 50/50 chance the next "cycle" will be rotation or oscillation (hence, what bounds should be used?) And (b), even if for example ## J_{sep} = J_{rot} ##, would the Hamiltonian then necessarily be continuous? Again, perhaps it's the way I'm thinking about the Hamiltonian as a piecewise function, perhaps there's a better way to combine the two. While my intuition for the fact that, yes, it should be continuous is there: for example, for p = p(x, E) the energy slowly climbs up in oscillations until in reaches the separatrix, at which there is a point where p = 0, and then that bottom-most part of the separatrix where p = 0 smoothes itself out and rises to become rotational motion. And of course this shouldn't change in the action-angle formalism because after all it's a canonical transformation. But despite having this intuition, I think it's important to have a proof for a more concrete understanding. Might be just missing something though, in which case do point that out. But if I'm not just making a mistake, please point me to a resource which explains this or maybe gives a proof sketch. Thanks!
$$ H = \begin{cases} H(J_{osc}) & E < E_{osc} \\ H(J_{rot}) & E > E_{rot} \end{cases} $$
But (a), it isn't defined at the separatrix, mainly because I'm not entirely sure how I would go about calculating J at the separatrix as, from what I can understand, at that energy there's basically a 50/50 chance the next "cycle" will be rotation or oscillation (hence, what bounds should be used?) And (b), even if for example ## J_{sep} = J_{rot} ##, would the Hamiltonian then necessarily be continuous? Again, perhaps it's the way I'm thinking about the Hamiltonian as a piecewise function, perhaps there's a better way to combine the two. While my intuition for the fact that, yes, it should be continuous is there: for example, for p = p(x, E) the energy slowly climbs up in oscillations until in reaches the separatrix, at which there is a point where p = 0, and then that bottom-most part of the separatrix where p = 0 smoothes itself out and rises to become rotational motion. And of course this shouldn't change in the action-angle formalism because after all it's a canonical transformation. But despite having this intuition, I think it's important to have a proof for a more concrete understanding. Might be just missing something though, in which case do point that out. But if I'm not just making a mistake, please point me to a resource which explains this or maybe gives a proof sketch. Thanks!
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