The story of Lagrangian mechanics, in a way, starts with the brachistochrone problem. In 1696, one of the Bournoulli brothers challenged the mathematicians of the world with this problem: Given two points A and B, what curve will result in an object rolling from A to B in the shortest amount of time? Hint: it's not a straight line! The story goes that Isaac Newton solved this problem within a day, and invented a whole new branch of calculus (called the calculus of variations) to do so! Newton was just that kind of guy.
I'm not going to solve the entire problem here, as it's rather difficult, but I'll outline the general idea. See, the amount of time that the object takes to roll is equal to distance divided by velocity. Of course, the velocity is not always constant, so we have to think about in calculus terms.
T = ∫ dx/v
For those who are afraid of math, don't worry. This is just a fancy way of saying time (T) is distance (dx) divided by velocity (v). The difficult part is that the velocity is a complicated function involving both the shape of the curve and how far we've gone on that curve. And there are infinitely many possible curves. How do we pick the best one? Newton's answer came from the calculus of variations. Unfortunately, this is impossible to show without assuming my readers have a ton of calculus background. I'll spare you!Bournoulli had his own answer to the problem, and it used quite a different method. He used something called Fermat's principle, which states that the path light takes from A to B is the path that takes the least amount of time. Of course, light isn't a normal object that can roll down inclines, but it can be slowed down when it goes through certain types of materials. Just as on object can be slowed down by rolling it up a cliff, light can be slowed down by the appropriate use of materials. If you set it up right, light will travel along the solution to the brachistochrone problem!
It turns out that the solution is a cycloid, which is the same curve traced out by a nail that gets stuck in a tire.
How does this all relate to Lagrangian mechanics? Just like Fermat's principle, Lagrangian mechanics states that the path an object takes from A to B also minimizes something. That quantity which is minimized is called "action".
S = ∫ L dt
Here, S represents the action. Action is equal to the Lagrangian (L) multiplied by time (dt). The Lagrangian is a special quantity that will tell you almost everything you need to know about a physical system in order to predict its future. In a simple system, the Lagrangian is equal to the kinetic energy minus the potential energy.
Let's consider a simple example. I throw a ball up and catch it. The ball is moving from point A (my hand) to point B (my other hand). To minimize action, the ball will "try" to have a high potential energy for as long as possible. To do so, it will soar up into the air. But it will not go too high, because then it would have to move fast to come down in time to land in my other hand. The faster it moves, the higher its kinetic energy, and the higher its action. Therefore, the ball will go up high, but not too high. The result is a parabolic path.
You can see why Lagrangian mechanics seem to imbue physical laws with a sense of purpose. The ball "tries" to minimize action as it goes from here to there. It even seems to foresee into the future, carefully plotting the course that will have the least action in the long run. Well, perhaps it only has foresight because we've already told the ball exactly where it will end up--in my hand at point B.
I think this philosophical side is part of the reason that physicists originally thought up Lagrangian mechanics. Perhaps it would disappoint the original physicists to know that action is not necessarily minimized, except in the simplest of cases. Sometimes it's maximized, or it's on a "saddle point". Try philosophizing that.
When we actually work with Lagrangian mechanics, we don't use this fanciful thought process at all. We use the calculus of variations and the Euler-Lagrange equation. For a simple problem, this will result in the equation F = ma. The equations are the real motivation behind Lagrangian mechanics, not the philosophy, but the philosophy is fun to consider.
Let's consider a simple example. I throw a ball up and catch it. The ball is moving from point A (my hand) to point B (my other hand). To minimize action, the ball will "try" to have a high potential energy for as long as possible. To do so, it will soar up into the air. But it will not go too high, because then it would have to move fast to come down in time to land in my other hand. The faster it moves, the higher its kinetic energy, and the higher its action. Therefore, the ball will go up high, but not too high. The result is a parabolic path.
You can see why Lagrangian mechanics seem to imbue physical laws with a sense of purpose. The ball "tries" to minimize action as it goes from here to there. It even seems to foresee into the future, carefully plotting the course that will have the least action in the long run. Well, perhaps it only has foresight because we've already told the ball exactly where it will end up--in my hand at point B.
I think this philosophical side is part of the reason that physicists originally thought up Lagrangian mechanics. Perhaps it would disappoint the original physicists to know that action is not necessarily minimized, except in the simplest of cases. Sometimes it's maximized, or it's on a "saddle point". Try philosophizing that.
When we actually work with Lagrangian mechanics, we don't use this fanciful thought process at all. We use the calculus of variations and the Euler-Lagrange equation. For a simple problem, this will result in the equation F = ma. The equations are the real motivation behind Lagrangian mechanics, not the philosophy, but the philosophy is fun to consider.
4 comments:
Hmmm…I remember seeing references to the calculus of variations, but I've never sat down and learned it. Maybe I should look those up again…
"The ball "tries" to minimize action"
I think it is a more intersting statement to say that the universe tries to minimise action. Anyway, your post on the philosophy of lagrangian mechanics got at least two people interested. :P
Also, could you give an example where the action is on a saddle point? sounds interesting..
Well, I can't think of a specific example, but I can tell you what it means.
When you're trying to minimize a function, you're basically looking for the deepest valley of a curve. When you're trying to maximize the function, you're looking for the tallest hill. What these two points have in common is that they're locally flat. Lagrangian mechanics only requires that the solution is on one of these locally flat points.
However, there's a different kind of point, a "saddle point" that is also locally flat. Let's say we have a saddle-shaped surface. If we only look to the left and right, we're on the highest point. But if we only look forwards and backwards, we're on the lowest point. So a saddle point is like a combination of a minimum and maximum.
The thing is, this can only happen in a system with multiple dimensions or multiple particles, which is why I can't come up with a simple example.
There's an interesting science fiction story in one of Gardner Dozois's Year's Best SF collections about a race of aliens who see the Langrangian as their "natural" way of looking at physics, and find our local "F=ma" methods complicated and counter-intuitive.
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