Question 9.9: In the case of the one-dimensional harmonic oscillator, obta......
In the case of the one-dimensional harmonic oscillator, obtain the canonical transformation from the usual canonical variables (x, p) to the action-angle variables (\phi , J).
By (9.29), W =\int{\left(2m\alpha -m^{2}\omega ^{2}x^{2}\right)^{{1}/{2}} }dx. According to Example 9.8, J =\sqrt{{m}/{k}}\alpha or E = \alpha= \left({k}/{m}\right)^{{1}/{2}}J= \omega J. Thus, W(x, J) = \int{\left(2m\omega J-m^{2}\omega ^{2}x^{2}\right)^{{1}/{2}} }dx, whence
W=\int{\sqrt{2m\alpha -m^{2}\omega ^{2}q^{2}}dq }. (9.29)
\phi =\frac{\partial W}{\partial J} =\int{\frac{mwdx}{\sqrt{2m\omega J-m^{2}\omega ^{2}x^{2}} } } =\sin ^{-1}\left(\sqrt{\frac{m\omega }{2J} }x \right), (9.136a)
p=\frac{\partial W}{\partial x} =\sqrt{2m\omega J-m^{2}\omega ^{2}x^{2}}. (9.136b)
Therefore,
x=\sqrt{\frac{2J}{m\omega} }\sin \phi , p = \sqrt{2m\omega J} \cos \phi (9.137)
is the desired canonical transformation.