Action, A simple example

Imagine that we have a circular table with a small hole in the center. Imagine that we have a ball attached to a string. The ball is in the table and the string threads thru the hole. You sit beneath the table and hold on to the string. I give the ball some tangential velocity and it rolls about on the table in a circle whose radius depends on what you do with the string.

We assume no friction and that the string is indefinitely twistable. When you pull on the string you add energy to the ball and it moves faster even though it travels in a smaller circle. Energy is not conserved but angular momentum is. The quantity that stays the same here is action which is not generally conserved.