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Picture the system below:

enter image description here

If you give the pendulum on the left a push, it will slowly transfer its energy to the one on the right, until it stops completely and the right one is in full swing, then the pendulum on the right will transfer its energy to the left one in the same manner.

Now picture this: at some time after you push the left pendulum, it will have tranferred 70% of its energy to the one on the right. Let's call this scenario 1. Then the pendulum on the left will keep transferring its energy, until it completely stops. The pendulum on the right will be at full swing.

Scenario 2 is when the pendulum on the right, some time after receiving all the energy, has given 30% of it back to the left one. The pendulum on the right will continue transferring its energy, until it stops.

Scenario 1 and 2 are identical in terms of energy, so why is it the pendulum on the left losing energy on scenario 1 and the one on the right on scenario 2? Shouldn't the same initial conditions evolve in the same way in classical mechanics? Why does one pendulum keep losing its energy until it stops, instead of reaching an equilibrium where both have half the energy and by symmetry neither should be giving energy to the other? Does it have something to do with the phase of the pendulums?

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The transfer only happens at certain phase angles. When the pendulums are in phase or exactly $\pi$ out of phase, there is no transfer.

But when they are moving $\frac \pi 2$ out of phase, the leading pendulum is losing energy and the trailing pendulum is gaining energy.

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