Chronology | Current Month | Current Thread | Current Date |
[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
Despite our propensity for giving hints, I'll just post a relatively nice treatment of the problem:
http://www.scribd.com/doc/50526620/26/Adiabatic-invariant-of-a-pendulum
V.I. Arnold, in his "Mathematical Methods of Classical Mechanics" (Section 52 in the "2nd, corrected printing" I have access to right now), makes it very clear as to which parameters we may change and have Liouville's Theorem still apply. The fact that we can parametrically resonate the pendulum (your swingset example) is well-known, and requires knowledge of the phase of the pendulum. It turns out that ignorance of this phase is equivalent to a requirement that the parameter we change is changed in a twice-continuously-differentiable way.
It's a very interesting problem!