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.