explanation of random motion of multiple pendulums

Question

is there an explanation of random motion of multiple pendulums , it seems to be 'orderly chaos'. example : https://www.youtube.com/watch?v=1M8ciWSgc_k

Answer 1

For simplicity I suggest to look at a simpler situation where we consider a set of harmonic osillators lying on a plane (like the shadows of the pendulums int he video on the floor).

Let's assume that pendulums are a set of $N$ decoupled harmonic oscillators moving in the y direction with equilibrium positions at $x=n a$, where $n=1,...,N$. Let $\omega_m$ be the proper oscillation frequency of the $n$-th oscillator. Moreover, let the oscillation amplitudes $A_n = A$ and the phases $\phi_n = \phi = 0$ $\forall n$ (in the video you show all the oscillators starting from the same initial y coordinate with zero velocity). Since the oscillators are decoupled, their cynematic equations are $$y_n(t) = A \cos{(\omega_n t)}, \;\;\;\; x_n(t) = n a $$

Now let's take an example and assume that the proper frequency is linear with $n$: $\omega_n=\omega_0 n$. From the equation of motion you immediately get $\omega_n = \frac{\omega_0}{a}x_n$. Now let's consider any finite time $t_0$ of the process and see what happens at $y_n(t)$: $$y_n(t_0) = A \cos{\left(\frac{\omega_0t_0}{a} x_n \right)} \;\;\; \rightarrow \;\;\; y_{t_0}(x) = A \cos{\left(\frac{\omega_0t_0}{a} x \right)}, $$ where after the arrow I have changed the perspective, regarding the spatial profile of $y$ at fixed time (rather than the time evolution of a single $y_n$ coordinate). From this you immediately see that the spatial pattern at any given time is harmonic with a period that changes in time like $\lambda(t) = 2\pi a/\omega_0t$. How can you realize this with a set of pendulums? Well, you know that in a pendulum $\omega_n$ is related to the length of the rope as $\omega_n \propto \sqrt{1/l_n}$, so you simply need to take $l_n \propto 1/n^2$.

In a more general situation, where $\omega_n$ is not linear with $n$, the description doesn't change much: you don't get an harmonic spatial pattern of $y(x)$, but still you can write $$ y_t(x) = A \cos{(tf(x))} $$ for a suitable $f(x)$ which depends on the situation. Still this is somehow regular at any given time.

Hope this helps! Let me know :)