22.6: Frequency Multiples

The above analysis is for frequencies not very far from $\omega_{0}$. But nonlinear terms can cause resonance to occur at frequencies which are rational multiples of $\omega_{0}$. Landau shows that a small $\dfrac{1}{3} \alpha x^{3}$ in the potential (so an additional force $\alpha x^{2}$ in the equation of motion) can generate a resonance near $\gamma=\dfrac{1}{2} \omega_{0}$. We’ve only considered a quartic addition to the potential, $\dfrac{1}{4} \beta x^{4}, \text { a force } \beta x^{3}$, we can show that gives a resonance near $\gamma=\dfrac{1}{3} \omega_{0}$, and presumably this is the small bump near the beginning of the curves above for large driving strength.

$\text { We have } \ddot{x}+2 \lambda \dot{x}+\omega_{0}^{2} x=(f / m) \cos \gamma t-\beta x^{3}$

$\text { We'll write } x=x^{(0)}+x^{(1)}+\ldots$

$\text { Let's define } x^{(0)} \text { by }$

$$\ddot{x}^{(0)}+2 \lambda \dot{x}^{(0)}+\omega_{0}^{2} x^{(0)}=(f / m) \cos \gamma t$$

$\text { So } x^{(0)}=b \cos (\gamma t+\delta) . \text { Then }$

$$\begin{aligned} \ddot{x}^{(1)}+2 \lambda \dot{x}^{(1)}+\omega_{0}^{2} x^{(1)} &=-\beta\left(x^{(0)}\right)^{3} \\ &=-\beta b^{3} \cos ^{3}(\gamma t+\delta) \\ &=-\beta b^{3}\left[\dfrac{3}{4} \cos (\gamma t+\delta)+\dfrac{1}{4} \cos (3 \gamma t+\delta)\right] \end{aligned}$$

Then, for $\gamma=\dfrac{1}{3} \omega_{0}$, the second term, $-\beta b^{3} \dfrac{1}{4} \cos (3 \gamma t+\delta)=-\beta b^{3} \dfrac{1}{4} \cos \left(\omega_{0} t+\delta\right)$, will have a resonant response, although it is proportional to the (small) amplitude cubed. Similar arguments work for other fractional frequencies.