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Given the following ODE: $$\ddot{x}(t)+\sin(\omega t)x(t)=0$$ its solution can be expressed in terms of the Mathieu functions. Plotting this solutions and assuming known the initial conditions it can be found a main oscillatory behaviour with another superimposed oscillation close to the maxima and minima of the solution. Now my question is: how can I find the frequencies of these oscillations knowing $\omega$? Or in other words, is there a formula linking the frequencies to $\omega$?

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Isn't w = 2 * Pi * f? – Amzoti Sep 7 '12 at 17:43
@amzoti: yes it is. – Riccardo.Alestra Sep 8 '12 at 18:30
When $\omega\rightarrow 0$, there is an adiabatic invariant. – Jon Sep 10 '12 at 8:48
@Jon: It's true, but I'm interested also in the case $\omega \to \infty$ – Riccardo.Alestra Sep 10 '12 at 11:29
up vote 2 down vote accepted

It is important to note that, if $\omega$ has physical dimensions, talking about it being large or small is meaningless. In order to treat this equation on a mathematical standpoint we need to introduce an "observation time" $\tau$ that gives the proper scale to time variable. In this case we have the equation $$ \ddot x+\frac{1}{\tau^2}\sin\left(\omega\tau\frac{t}{\tau}\right)x=0. $$ Then, changing the time variable to $\theta=\frac{t}{\tau}$ and putting $\omega\tau=\lambda$, we are left with the equation $$ \ddot x+\sin(\lambda\theta)x=0. $$ For this equation, Floquet theorem applies and a general form to the solution is given that is the product of two periodic functions.

It is worthwhile to note the two cases $\lambda\ll 1$ and $\lambda\gg 1$. The former produces an adiabatic invariant as is well known. This can be written in the form $(\dot x)^2+x^2$ (see here for the theory). The other limit is intriguing as perturbation theory applies. This can be seen by the rescaling $w=\lambda\theta$. The equation takes the form $$ \lambda^2\ddot x+\sin(w)x=0 $$ and diving by $\lambda$ and setting $\epsilon=\frac{1}{\lambda^2}\ll 1$ one can do perturbation theory. The solution, assuming for the sake of simplicity $x(0)=a$ and $\dot x(0)=0$, has the form $$ x(w)=a+\epsilon a\sin(w)-\epsilon^2\left(\frac{a}{8}\cos(2w)+\frac{a}{8}+\frac{a}{4}w^2\right)+O(\epsilon^3) $$ where another harmonic is seen to appear as a second order effect.

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