0
$\begingroup$

Suppose I have a completely integrable $1$ degree of freedom Hamiltonian $H(I, \varphi) = \frac{I^{2}}{2}$ written in action-angle variables $(I, \varphi) \in \mathbb{R} \times \mathbb{S}$.

What would the corresponding Hamiltonian be in canonically conjugated variables $(p,q)$, and the corresponding example of the physical system it describes?

Naturally we are looking for some kind of an oscillator (since all orbits are periodic) but with frequency depending on $I$ (the amplitude).

Improve this question
$\endgroup$
8
  • 1
    $\begingroup$ There is no unique answer to this question as posed , in the sense that the Hamiltonian for many systems can be reduced to this form using suitable canonical transformations. $\endgroup$
    – ZeroTheHero
    Commented Aug 14, 2017 at 15:54
  • $\begingroup$ What would be a good example of such a system? Some kind of anharmonic oscillator? @ZeroTheHero $\endgroup$
    – Alex
    Commented Aug 14, 2017 at 15:57
  • $\begingroup$ the usual harmonic oscillator. $\endgroup$
    – ZeroTheHero
    Commented Aug 14, 2017 at 15:58
  • $\begingroup$ But for the usual harmonic oscillator, the typical Hamiltonian is $H = I$ and frequency $\omega(I) = 1$ (say after normalisation of constants), i.e. does not vary with action. What I specify is that in my case, frequency varies with action $I$ (I need this for application of KAM theory etc) @ZeroTheHero $\endgroup$
    – Alex
    Commented Aug 14, 2017 at 16:02
  • $\begingroup$ P.S. in general, for KAM theory one needs nondegenerate frequencies; the Hamiltonians of the form $H = \frac{I^{2}}{2}$ also arise in so-called apriori unstable systems as models for Arnol'd diffusion @ZeroTheHero $\endgroup$
    – Alex
    Commented Aug 14, 2017 at 16:07

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.