Conservative forces, that is!

Hold a basketball in your hand and move it around. When you move it up, you do work against gravity. You have to put energy into the system to get it to gain height. Relax your arm, and the force of gravity pulls it downward, converting that potential energy into kinetic energy and speeding up the ball's downward motion until you stop it with your hand.

You can wiggle the ball up, down, and all around. It doesn't matter what path the ball takes, if it starts and ends in the same place, the total energy transfer done by gravity is zero. Forces with this property of path-independence are called conservative forces. Nonconservative forces are those where path does make a difference. Friction is the classic example. Move from point A to point B and you'll have to do a certain amount of work and transfer that particular amount of energy in order to accomplish the motion. Move it back, and you don't get the energy back. In fact the energy required to move from A to B will depend entirely on your path. Sliding a heavy box over to a position two feet to its left will require a lot more energy if you first move it ten feet to its right.

Conservative forces are more interesting, more fundamental, and more mathematically tractable. Given the equation for a force F we'd like to know if it's conservative or not. We do it this way:

That tells us to take the force F, evaluate its curl, and if the result is 0 the force is conservative. The calculus definition of curl is a little complicated, but intuitively it's pretty easy. If the lines of force don't circulate, the curl is zero. Here's an example of a curl-free vector field:

The arrows represent the strength and direction of the force at a particular point. You can take my word for it that none of the bends and curves actually constitute circulation. Here's one that does:

See that central swirl? If you go around it counterclockwise you'll be getting energy from it. But it will take energy to go around it clockwise. Since the energy to return to your starting point varies with the path, it's not conservative. And energy varies with the path because there's circulation.

Though you might not know it there, that statement contains a pretty huge chunk of physics. Don't worry, we're bound to cover it all eventually!

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If your curl is non-zero then a preferred direction is defined. There's a lot of fun to be had when symmetry is broken. If your founding postulates coveniently demand isotropy but the universe had other ideas, observation will repeatedly trump theory.

So conservatives (attempt to) gain energy by moving against the observed flow of time?

No wonder the old witches and other forces of darkness were regularly described as dancing widdershins!

Ah yes, velocity fields and conservative forces - the stuff we use to make cool visual effects for movies :)