Given a 1 d.o.f Hamiltonian H(q,p) what is the general procedure for finding action angle variables (I,θ)?
I have read the Wikipedia page on action angle variables and canonical transforms but have difficulty applying the general methods to specific problems. Can someone explain the method to me using a simple general example?
Answer
In local coordinates the canonical transformation to action angle coordinates (q,p)→(Q,P) can be related by, Pi=12π∮pidqi and Qi=∂∂Pi∫pidqi
Consider the one dimensional harmonic oscillator with the following Hamiltonian H=12m[p2+m2ω2q2]. Rearrange this for p and take the hypersurface H=E. p=±√2mE−m2ω2q2
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