Question 9.8: Find the fundamental frequencies of a two-dimensional harmon......

Taking into account that {dW_{1}}/{dx}=p_{x}, Eq. (9.115a) admits two solutions for p_{x}, namely

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

p^{\pm }_{x}=\pm \sqrt{2m\alpha_{x}-m k_{x}x^{2} },  (9.129)

in which x takes values only between x_{−} =-\sqrt{{2 \alpha_{x}}/{k_{x}}} and x_{+} =\sqrt{{2 \alpha_{x}}/{k_{x}}}. A complete cycle of the coordinate x consists in a variation from x_{−} to x_{+} and back to x_{−}. In going from x_{−} to x_{+} the variable x is increasing and p_{x} > 0, whereas p_{x} < 0 during the return from x_{+} to x_{−}. Therefore,

2\pi J_{x}=\oint{p_{x}dx}=\int_{x_{-}}^{x_{+}}{p^{+}_{x}dx} + \int_{x_{+}}^{x_{-}}{p^{-}_{x}dx}=2\int_{x_{-}}^{x_{+}}{ \sqrt{2m\alpha _{x}-mk_{x}x^{2}\alpha }dx}.  (9.130)

This integral is easily calculated by the substitution x = \sqrt{{2 \alpha _{x}}/{k_{x}}} \sin \theta , giving

J_{x}=\frac{1}{2\pi }2\int_{-{\pi }/{2}}^{{\pi }/{2}}{\sqrt{mk_{x}}\frac{2\alpha _{x}}{k_{x}} \cos ^{2} \theta d\theta }=\frac{1}{\pi } \sqrt{mk_{x}}\frac{2\alpha _{x}}{k_{x}} \frac{\pi }{2} =\sqrt{\frac{m}{k_{x}} } \alpha _{x}.  (9.131)

The generic procedure outlined above is not necessary in the case of the harmonic oscillator. The previous result for J_{x} can be obtained virtually without calculations by noting that Eq. (9.115a) can be written in the form

\frac{x^{2}}{a^{2}} +\frac{p^{2}_{x}}{b^{2}} =1, a=\sqrt{{2\alpha _{x}}/{k_{x}}}, b=\sqrt{2m\alpha _{x}} ,  (9.132)

which is the standard equation of an ellipse in the (x, p_{x}) phase plane. The action variable J_{x} multiplied by 2π is the area of this ellipse:

J_{x}=\frac{1}{2\pi } \pi ab=\sqrt{\frac{m}{k_{x}} }\alpha _{x}.   (9.133)

Similarly,J_{y}= \sqrt{{m}/{k_{y}}}\alpha _{y} and, as a consequence,

H = α_{1} = α_{x} + α_{y} =\sqrt{\frac{k_{x}}{m} }J_{x}+ \sqrt{\frac{k_{y}}{m} }J_{y},  (9.134)

from which

\omega_{x}=\frac{\partial H}{\partial J_{x}}= \sqrt{\frac{k_{x}}{m}} , \omega_{y}=\frac{\partial H}{\partial J_{y}}= \sqrt{\frac{k_{y}}{m}},   (9.135)

which is the correct result for the frequencies of the oscillations in the x, y-directions.