Question 9.7: Find the projections on the phase planes of the phase space ......

In Cartesian coordinates,

H =\frac{1}{2m}\left(p^{2}_{x}+p^{2}_{y}\right) +\frac{k_{x}}{2}x^{2}+ \frac{k_{y}}{2}y^{2}   (9.114)

and the Hamilton-Jacobi equation (9.54) is separable in the form W(x, y) = W_{1}(x) +W_{2}(y) where

H=\left(q_{1},. . . ,q_{n},\frac{\partial W}{\partial q_{1}}, . . . ,\frac{\partial W}{\partial q_{n}} \right) =\alpha_{1}.  (9.54)

\frac{1}{2m}\left(\frac{dW_{1}}{dx}\right)^{2}+\frac{k_{x}}{2}x^{2}=\alpha _{x},  (9.115a)

\frac{1}{2m}\left(\frac{dW_{2}}{dy}\right)^{2}+\frac{k_{y}}{2}y^{2}=\alpha _{y},  (9.115b)

with

H = α_{1} = α_{x} + α_{y} . (9.116)

Since p_{x} ={dW_{1}}/{dx} and p_{y} ={dW_{2}}/{dy}, Eqs. (9.115) represent ellipses on the phase planes (x, p_{x} ) and (y, p_{y} ), respectively. Although both projections are periodic with periods \tau _{x}= {2\pi }/{\omega_{x} }=2\pi \left({m}/{k_{x}}\right)^{{1}/{2}} and \tau _{y}= {2\pi }/{\omega_{y} }=2\pi \left({m}/{k_{y}}\right)^{{1}/{2}}, the motion in phase space will only be periodic if the frequencies \omega_{x} and \omega_{y} are commensurate – that is, if {\omega_{x}} /{\omega_{y}}  is a rational number. Indeed, if τ is the period of the motion, the phase space path closes in the course of the time interval τ .The projections also close, so τ has to be an integer number of periods τ_{x} and τ_{y} – that is, there exist integers m and n such that \tau= {m2\pi}/{\omega_{x}}={n2\pi}/{\omega_{y}}, whence {\omega_{x}} /{\omega_{y}}=m/n is a rational number.