I still don't think I fully agree with that explanation. But I do see where my earlier explanation was incomplete. So I'll try again.
Note that I uninstalled quicktime a few years ago, and I don't feel like reinstalling. So I was unable to watch those movie clips. I did find
another site with diagrams, which I think covers the same thing.
The explanation given seems to be that you rotate, and transfer angular momentum to the system. Then when the swing is rotating the other way (on the return) you rotate the other way. So that you are adding each time with no net change in your own angular momentum. And I guess that works to an extent, but I think the effect would be small.
However, what I do see happening, is that this strategic motion (or rotation) of the person's body, if done with the right timing, has the effect of lengthening the down motion period and shortening the up motion period. And then reversing that motion for the swing return again lengthens the down swing in the other direction and shortens the upswing in the other direction.
Since the amount of angular momentum change caused by gravity depends on how long the gravitational force is applied, lengthening the down swing and shortening the upswing will increase the amount of angular momentum added in the downswing and reduce the amount of angular momentum removed in the upswing.
I'll still think about that some more.
Thanks for persisting with this. I did learn some more about it as a result.
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As indicated, I was not able to watch those clips. But let's talk about the theory they give of transferring angular momentum by rotating that wheel.
It seems to me that if that wheel rotation were done only when the swing is near the bottom, you would get the effect of transferring angular momentum. If the rotation is done near the top, you also get the effect of lengthening the down swing and shortening the upswing, which I think is the more important part. If I had the equipment, I would want to experiment with that to see if my analysis is correct.