# Why coupled oscillators tend to seek integer frequency ratios?

In this document, the author writes (page 225)

Coupled oscillators have a tendency to seek frequency ratios which can be expressed as rational numbers with small numerators and denominators. For example, Mercury rotates on its axis exactly three times for every two rotations around the sun, so that one Mercurial day lasts two Mercurial years. In a similar way, the orbital times of Jupiter and the minor planet Pallas around the sun are locked in a ratio of 18 to 7 (Gauss calculated in 1812 that this would be true, and observation has confirmed it). This is also why the moon rotates once around its axis for each rotation around the earth so that it always shows us the same face.

Is that true? Can we prove mathematically that Coupled oscillators love rational frequency ratios?

Oh, it appears that planetary motion is not an oscillator. But anyway, I just want some reference to verify whether this is true, preferably with mathematical derivations.

• Related: physics.stackexchange.com/q/282698/2451 and linkst herein. – Qmechanic Nov 11 '19 at 9:17
• I think the notion you are looking for is systems which have limit cycles, which is pretty much equivalent to systems which are periodic (which, for instance, coupled oscillators with rational ratios between their frequencies are). I think this then turns into a huge and complicated subject area mathematically. – tfb Nov 11 '19 at 16:48

## 2 Answers

They may in special cases, but this is by no means a rule, and most certainly doesn't hold for all cases. As I'm a firm believer in concrete examples, I'll illustrate this with the following simple case of two blocks, each of mass $$m_1=m_2=m$$, connected to walls and each other by Hookean springs (which are harmonic oscillators) of spring constant $$k$$, as shown in the following image Now, the Lagrangian for this system is, of course, $$L=\frac{1}{2}m\left(\dot{x}_1^2+\dot{x}_2^2\right)-\frac{1}{2}kx_1^2-\frac{1}{2}k\left(x_2-x_1\right)^2-\frac{1}{2}kx_2^2$$ which returns the equations of motion \begin{align} \ddot{x}_1 &=-\frac{2k}{m}x_1+\frac{k}{m}x_2 \\ \ddot{x}_2 &=\frac{k}{m}x_1-\frac{2k}{m}x_2\\ \end{align} One can solve these equations with a bit of linear algebra, and one gets, with vanishing initial velocity, \begin{align} x_1(t) &= \frac{1}{2}\left(x_1(0)+x_2(0)\right)\cos\left(\sqrt{\frac{k}{m}}\,t\right) +\frac{1}{2}\left(x_1(0)-x_2(0)\right)\cos\left(\sqrt{\frac{3k}{m}}\,t\right) \\ x_2(t) &=\frac{1}{2}\left(x_1(0)+x_2(0)\right)\cos\left(\sqrt{\frac{k}{m}}\,t\right) -\frac{1}{2}\left(x_1(0)-x_2(0)\right)\cos\left(\sqrt{\frac{3k}{m}}\,t\right) \end{align} Now, even this simple case isn't periodic except in cases where $$x_1(0)=x_2(0)$$ or $$x_1(0)=-x_2(0)$$, as the terms in the cosines differ by a factor of $$\sqrt{3}$$. This nonperiodicity as can be seen in these plots, where I have set $$k=1.2321$$, $$m=0.771203$$, $$x_1(0)=e+0.231$$, and $$x_2(0)=1$$.

There are many, many more examples I could show. In general, the motion of coupled oscillators is complex, is not necessarily periodic, and does not tend toward rational frequencies. Now, are there cases where the motion is periodic and they are in sync? Of course, but they are not the general rule. For this example, set $$x_2(0)=-x_1(0)$$, and they are in a ratio of $$1:1$$, as shown in the below plots (everything is the same, except $$x_1(0)=-1$$).

To sum it up, while there are some cases where oscillators tend towards being in a rational sync, this is not always true.

• L is not correct – Eli Nov 25 '19 at 17:31
• It has been corrected. Thanks to you. – John Dumancic Nov 25 '19 at 18:02
• Your L is the Energy it Schuld be T - V not T +V also the equations of motion wrong sign – Eli Nov 25 '19 at 18:43

The motion of two coupled harmonic oscillators is the sum of two "beat" frequency oscillations. The frequencies are functions of the masses and spring constants and can take any value, not necessarily "rational numbers with small numerators and denominators". I don't think this is a correct analogy to orbital resonances.

This page gives some mathematics of orbital resonances. It explains how resonances can be stable, but it is not clear to me how planets get into these states - it seems to me that these must be low energy states in some sense, so there must be some dissipative mechanism leading to them. Maybe that is covered in a more advanced course.