Tuesday, January 1, 2013

The Principle of Least Action

Richard Feynmann tells the story of how his high school science teacher introduced him to the Principle of Least Action, and then he goes on to show how your can use it to calculate a trajectory. (Along the way he shows you how to do Calculus of Variations.) It's a strange method because it makes much of the difference between potential and kinetic energies. We know that energy is conserved, so the sum of potential and kinetic must be a constant. But what could be the physical significance of the difference of those two energies?

The path taken by a projectile in a natural trajectory, we are told, is the path which minimizes the action, which is the time integral of the difference of energies. How does the projectile choose the correct path out of all possible paths?

When we learn about Snell's Law, we see another example of how a path gets chosen which turns out to minimize a certain quantity. In the case of light passing from one medium to another, we find that when we calculate the time to get from point A to point B, that light chooses the path which gets there in the least possible time.

This seems a bit miraculous, but then when we analyze the path in terms of the wave theory of light, we see that it makes a lot of sense. Light travels in a direction perpendicular to the wavefronts at all times, and this basically keeps parallel rays in lock-step with each other. It's not so much that the light chooses the path of least time; rather, that all nearby paths are synchronous, so there is no benefit to shunting onto a side path and then cutting back in at the last minute. That's how it works for light.

That doesn't explain how it works for a ball thrown into the air, but it gives us a hint as to how we might do the calculation. The action integral, which is cumulative over time, should be the same for all nearby paths; and in particular, it should be the same for a parallel path. This is a little different from what Feynmann does in the Lecutres; he takes the starting and ending points as given constraints, and lets the path vary in-between. What I'm going to do is take a parallel path and force the action to be equal between the two paths. It turns out you can get a useful physical result this way.

Here's how it works. You take a ball travelling horizontally at 10 meters per second. Then you take a second ball, slightly higher up, and assume that the two balls are, like the case of the light rays, riding the same wavefront. The direction of travel must be at all times perpendicular to the wavefront, and the wavefronts must stay in synch with each other. And the propagation of the wavefront is caluclated by using the action integral. Let's see what happens.

I wonder if you can see that I've chosen these two paths so that they have the same action? The top path has a little more potential energy, so I've given it just enough additional kinetic energy to keep the difference to be a constant.

There's only one problem with my little picture. The "wavefronts" do not remain perpendicular to the direction of propagation. But that's because projectiles don't maintain a flat trajectory. We can fix up the picture by curving the trajectories so the two paths stay in lock-step with each other:

Now the trajectories line up with each other, and the "action" calculated along each line stays in step. We just had to curve the paths. It shouldn't be too hard for you to calculate the radius of curvature; it comes to 10 meters in this case.

And it's not too hard to verify that 10 meters is actually the correct radius of curvature for the path of a baseball at 10 m/sec. (You can see it most easily by calculating v-squared-over-r). So the Principle of Least Action actually gives us a correct result for a real physics problem. And we didn't even have to use Calculus of Variations.

I left us off a few days ago with a discussion about cloud chamber trajectories, and I had promised to make some kind of an argument as to how the wave theory of matter could justify those straight-line paths of ionization. The point of today's digression was to lay the groundwork for the quantum mechanical case. In quantum mechanics, we have the wave function, and the idea is that the wave function guides the particle to find the path of least action. When we return, I'm going to try and apply the same type of analysis as I used today on the baseball, to see what happens if we use electrons instead.

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