Can you get a playground to swing from stationary? I feel this might be a FAQ but I would love a definitive answer.
Imagine a frictionless stationary idealised child's playground swing.  If you are sitting on the seat of the swing, is it possible in principle to set it into motion by simply moving your body and pulling on the chains of the swing? If so, is there a simple explanation for how you can change your center of mass without an external force acting on you?
 A: Yes. A swing does not depend on friction.  The way you start up is by leaning back and pulling hard with your hands placed on the ropes.  Your hand-hold positions effectively create a compound pendulum, so your mass is offset from null. By timing your shift in weight back to a "sitting-up" position, you essentially add energy to the pendulum system in phase and frequency with the full-length pendulum's oscillation.  
I'm sure there are some pictures of this somewhere :-)
EDIT: I should have said, as Peter Shor did, that this makes use of gravitational force.
A: This phenomenon is independent of any force other than gravity and those of the system child and the swing. It is simply due to conservation of momentum and what is known as parametric resonance. When the child at rest moves his legs, conservation of momentum implies that his body moves in the opposite direction and appropriate timing of this motion adds energy to the system by leveraging gravity. 
The cause of this is independent of whether the hinge is fixed or not in the horizontal direction. The inertia of the chain is enough (the chain will not form a straight line, therefore the force on the hinge will be closer to the vertical than the line connecting the hinge and the swing. This is a small effect and I don't know if it can be leveraged in the real world.) to delay the motion of the hinge enough so that a restoration torque due to gravity is formed.
A: As Peter Shor writes, there is external force. Imagine the swing was on rollers: it would swing back and forth as the child swings forth and back. And the horizontal position of the mass center would not move.
The process is conversion of muscular energy into kinetic energy, then via the swing attachment, to potential energy. And loop: produce kinetic energy again when it reaches 0 (and potential energy is maximum).
A: 
Starting a Swing from Rest
BRYAN F. GORE
Central Washington State College
Ellensburg, Washington 98926
(Received 3 August 1970)
When the upright occupant of a swing at rest falls backward and catches himself in a reclining position, the swing seat moves forward both during and after the occupant's rotation. This is in apparent contradiction to the backward motion of the occupant's center of mass (c.m.) expected during his rotation.¹ Nevertheless, the unmarked c.m. does move backwards. Its motion is simply masked by the forward motion of the seat due to the occupant's rotation. This may be easily seen if the position of the seat is marked at the end of the rotation and compared with its equilibrium position after friction has stopped the motion.
While efficient pumping of a swing is done so that the tangential force (associated with the torque changing the occupant's rotation about his c.m.) is always parallel to the swing's motion, the first pump takes place during a time shorter than a quarter of the swing's period. Therefore, the swing's motion does not reverse between imposition of the torques starting and stopping the occupant's rotation, and the tangential force due to the stopping torque opposes the motion imparted by the starting torque. It is straightforward to show that after the first pump the center of mass of the occupant, then displaced from its equilibrium position, will be moving slowly toward it.
The tangential impulse imparted to the c.m. during the pump has two components, one due to gravity and the other imparted by the torques. Since the tangential component of the rope's tension is proportional to the torque about the c.m., its contribution to the tangential impulse is proportional to the net angular impulse about the c.m. (assuming a constant moment of inertia). At the conclusion of the pump, rotation is halted, so the net angular impulse is zero. Consequently, the total tangential impulse is just that contributed by gravity. A slow motion toward the equilibrium position is thus predicted, which is just what is observed.
¹ B. F. Gore, Amer. J. Phys. 38, 378 (1970).


Transcribed and edited (by adding a missing closing parenthesis, splitting a long paragraph into two, and adding an initialism hint after center of mass) by me.
Source: https://doi.org/10.1119/1.1986146

A: Moving your weight back is equivalent to moving the pivot forward, like having a ball on a string and moving your hand forward, then, when it moves forward, moving your hand back, but opposite, like tha ball leaning back instead of your hand moving forward.
What I don't understand is how you can propel it forward by moving your legs forward instead of back.
