Carlo Rovelli

I have received a Newton's pendulum as a gift. A Newton's pendulum is formed by 5 metal balls forming a row, each hanging on two cords that allow it to oscillate -like a single pendulum- in the direction of the lines of the row itself. Quite pleasantly, if the leftmost ball is raised and let fall down against the other four, three of these almost not move at all, while the fourth springs out alone, rises maximal height and then falls down. This gives a periodic motion in which only the two outer balls swing. But there is more. If the two leftmost, rather than one, balls are raised and let swing down, then only the central ball stays put, and the two rightmost swing out. Furthermore, quite interesting, if the three leftmost are raised and let swing down, then three continue the swing on the opposite side, of which two were put and one was one of the three originally moved. And so for the four: three continue and one remains put waiting for the four to come back. All this is quite cute to watch.
Can we compute this motion from first principles? Obviously energy and momentum are appropriately conserved and exchanged in each of the collisions, but this is not sufficient to fix the motion, since the conservation laws are two are the balls are five. More is needed. I searched online for a simple explanation and could not find one. So, thought of providing one. Here it is, using only symmetry.
Let me start from the head-on collision of two equal-mass balls, one of which is initially not moving. What happens after the collision? The easiest way to answer (the idea is by Huygens) is to look what happens if we move at half the velocity of the incoming ball. In this reference frame, the two balls move against each other at equal opposite speed, therefore by symmetry and energy conservation the only thing they can do is to both bounce back reversing exactly their velocity. We now see clearly that each ball acquires, after the collision, the velocity that the other had before the collision. This conclusion is clearly independent form the reference system, and therefore a ball hitting head on a still ball stops while the other ball starts with the velocity of the oncoming ball.
Now, what happens if a ball hits a row of 4 still balls, touching each other, all balls having the same mass? To solve this, let us first solve a slightly different problem. Instead of four balls touching each other, imagine that the four balls are at a small distance D from one another. Then the solution is easy: the incoming ball collides with the first of the still ones and stops, while the hit one starts with the same velocity. But very soon it hits the next, it stops and the next starts, and so on, until the last, which has nothing to collide upon and hence can swing out. That is: all balls move only by a small amount D, up the last that swings out. Now let's consider what happens if we take D to be smaller. Clearly the only difference is a smaller displacement of the balls that do not move much. And if we take D to zero, we obtain the result we observe: three balls do not move and the last swings out, as we observe.
Now, what happens if two ball hit a row of three still balls? Again, let us imagine of separating all balls (the two incoming as well as the three still) by a small D amount. Then is nice: the first of the two incoming balls hits the first still. It stops and pushes the other ball, that starts the same sequence as the previous case, but immediately later the second incoming balls hits its companion that just stopped, starting a second equal sequence of pushes that pushes out a second ball, immediately following the first swinging out.
It is clear that all the observed movements follow very easily, simply from energy conservation, symmetry, and invariance under change of reference frame.