Saturday, April 12, 2008

Cosmic purpose and the Lagrangian

In my recent studies of physics, we learned about Lagrangian equations of motion. Now, unless you study physics, these equations would seem rather dry. They help me solve problems like the double pendulum, but they do so with purely mathematical equations. So as much as I know you'd all like to see the math, I will instead talk about the philosophy of Lagrangian dynamics.

"Philosophy?" you exclaim with bewilderment. Well, this subject was not entirely unprovoked. My text book has a paragraph, where it starts waxing metaphysical about how in Lagrangian dynamics, nature tries to achieve a certain purpose (emphasis not mine). I thought this paragraph was just hilarious.

To understand this point, I must explain what this "Lagrangian" thing is. Basically, Lagrangian mechanics is an alternative to Newtonian mechanics.

Newtonian mechanics is based on the equation F = ma. That is, force is equal to mass times acceleration. If I push something, it changes how fast it's going. So if we know exactly what forces are acting on an object, we can, in theory, determine where it will go. Of course, in practice, this is very difficult. What if the force depends on the location of the object (which is in turn affected by the force)? After some point, the math becomes hard. I don't mean this in the sense of, "Math is hard! Let's go shopping!" I mean that some of this math is impossible to solve, or near impossible. And if you're lucky, you can prove that it's impossible. That is why physicists try to simplify everything, wherever possible.

One of the ways to simplify Newtonian mechanics is by using Lagrangian mechanics instead. Lagrangian mechanics is more advanced (which is why they don't teach it in introductory physics), but mathematically cleaner. Instead of F = ma, Lagrangian mechanics states that action is minimized. Now, before the Daoists start jumping all over that one, I have to qualify the word "action." "Action" is a technical term for a particular quantity, the details of which are not covered in the scope of this post. To determine the path an object will take, you consider every possible path, and find the one path in which this quantity, action, is minimized. It sounds complicated, but it actually makes a variety of problems much simpler. Lagrangian mechanics is equivalent to Newtonian mechanics--it will always give you the same result, and you can prove it.

Back to my textbook. My textbook simply states that while the Newtonian and Lagrangian are mathematically equivalent, they are not philosophically equivalent. These philosophical implications have historically had profound influence on the development of mechanics.

So what are these philosophical differences? According to Newtonian mechanics, the world is viewed in terms of cause and effect. You push something, it moves. According to Lagrangian mechanics, the laws of physics are produced because nature is trying to achieve a purpose--to minimize action. An object has many possible paths, but the actual path is the one with the least action.

What are the implications for philosophy? Does it mean that the universe is governed by cause and effect? Or is it trying to fulfill a purpose? Or is it both, or neither?

If you asked me, this is an indication that such questions are irrelevant. It doesn't matter whether there is cause and effect, or whether there is a cosmic purpose. Physics behaves the same either way. If you can change the answers to these deep philosophical questions by simply moving around some mathematical equations, what meaning do these answers have?

Another layer of complication is the addition of another alternative to Newtonian mechanics: Hamiltonian Mechanics. As far as I can tell, Hamiltonian mechanics have no philosophical implications whatsoever. They're just a bunch of mathematical equations that make things simpler for physicists. Cosmic purpose is fleeting.

Has Lagrangian Mechanics piqued your interest? I wrote more.


Anonymous said...

Aw, now I want to see the math!

I'm rather intrigued by this idea of "minimizing action", but I have a question. I can imagine a situation where the path with minimal action from point A to point B passes through some intermediate point C, but the action is very high for the path from A to C and low between C and B. On the other hand, other paths from A have very low action at the start but ultimately require more action toward the end, making the path through C the path of minimal action.

In this case, does Lagrangian mechanics say the path through C that I've described is the path that will be taken, or some other path? It seems that "choosing" to follow that path would indicate a sort of global knowledge about the possibilities, and I can certainly see why the philosophical side of things looks more goal-oriented. I am uncomfortable with this interpretation, though. Am I understanding correctly?

miller said...

Hmm, maybe I'll write something up about Lagrangians.

I had to think about your question for a while, because the way it's formulated is purely mathematical, with little reference to things like paths between point A and B. But I think you're partly right and partly wrong.

The path will go through point C, yes. But it wouldn't quite make sense to say the action from A to C is "high". High compared to other possible paths from A to C? Note that the action from A to C is also minimized.

Anonymous said...

I guess I'd have to see how "action" is defined/calculated. Please do write another post on it!

Anonymous said...

What if there are three paths from A to B of equal distance, and one of the paths (Path 1) requires less action at the beginning of an object's journey between the points than the other two(Path 2 and Path 3), but requires more and more action to continue near the end of its journey so that the action required to travel path 1 is greater than the action required to travel path 2 or path 3. If the object travels along one of the other paths to "minimize action", according to the Lagrangian model, then isn't prescient knowledge implied?

miller said...

Yes, in a way. But this shouldn't be very surprising, since you've already specified that it's going to point B in the future. How does it "know" it will end up at B? Well, you assumed that it would!