system
Given the following post, respond as if you're commenting on Reddit:
human
When on a playground swing, why does the act of "pumping" your legs (and torso) accelerate the swing?

My daughter asked me this the other day, and my first thought was that when you shift position on the swing, it changes where the weight on the pendulum is - kind of the same way a metronome works. But I don't know the actual reason, and if that's correct, I don't know why shifting the weight up and down would work: wouldn't one cancel out the other?
gpt
Pumping is a dynamic phenomenon.  It's pretty obvious that you can't get yourself going from a dead stop with static processes alone.  It turns out you *can* pump yourself from a dead stop, but it's hard, which is why most people give a little kick-start.

The simplest-to-describe pump is not what kids do at all, it's a type of standing pump you can do on a board swing.  That involves crouching on the board at the top of the swing, and standing up at the bottom of the swing.  That type of pumping works by conservation of angular momentum around the fulcrum at the top of the swing:  you lower yourself toward the board swing when you are at the top of the arc and not moving, and raise yourself when you are at the bottom and moving fast -- which increases your speed (just like a ballerina pulling her hands in while spinning).  The energy comes from the fact that you're *lowering* yourself in a low-G environment (at the top of the arc) and *raising* yourself in a high-G environment (at the bottom of the arc).  This technique is so effective that you can easily top out a typical swingset in ~10 arcs.

The back-and-forth seated pumping that kids do works sort of the same way.  When you lean you are kinking the chain/rope at your hand, which raises your center-of-gravity slightly.  The kink generally increases as you pass through the bottom of the arc, giving you the same effect as the standing pump in the last paragraph (but with less magnitude).  The kink increases because the period of the pendulum formed by the child under the kink is shorter than the period of the pendulum formed by the child and swing under the swingset, so the chain-kink-child system runs ahead of the swingset-chain-child system.  There's also an asymmetry in the weight distribution in the two different directions of leaning, that helps force the smaller pendulum into motion.

Anyway, swinging is all about breaking the symmetry of the swing system and exploiting that broken symmetry to excite the system.  The oscillating motion itself breaks the obvious symmetry of the static pendulum.

How to start from a dead stop?  One way is to whirl your arms around a horizontal axis.  As you *start*  or *stop* your arms turning, you apply a torque that can start the swing oscillating slightly.  The trick is to phase the torques with the oscillations of the pendulum formed by you on the swing.   If you time starting and stopping (and reversing) the whirling motion appropriately, you can actually get a reasonable amount of height:  say, a few inches above the bottom of the arc.  Then you can start conventional pumping.

**tl;dr**: even though the swing/pendulum itself is symmetric, when it's moving the symmetry is broken.  Pumping exploits that.  Most pumping works by moving your center-of-gravity up and down as you swing.

Edits: some typos and a tl;dr.