Eigenstuff

The Lagrangian

I was already “familiar” with Lagrangian at high school from reading Feynman speeches. How and why it works was a mystery to me, but I loved the idea of having a minimal principle anyways. Since it was like nothing I had known before, I thought it must about some physics I didn’t know by then, maybe quantum mechanics.

Weirdly (and curiously), I took Quantum Mechanics course before Classical Mechanics. I closely watched whether if a minimum principle will pop-up somewhere. There was matter waves, Schrödinger equation, Heisenberg Uncertainty Principle, and a new term “Hamiltonian” I had recently heard of. A bunch of craze came out as a result of probabilistic interpretation, but there was no Lagrangian.

$\left (\frac{p^2}{2m} + V \right ) \psi = E \psi$

Well, time passed, I became a sophomore and finally had Classical Mechanics course. At that time, all mechanics I knew was pure laws of Newton. I was a complete newbie to the real Lagrangians, it was still a strange idea to have a minimal principle. After all they look entirely different.

$F = ma$
Minimize this: $\int (T-V) dt$

Hell, they don’t even have a symbol in common! I wanted to see how two can be equivalent, so I looked around for Classical Mechanics books, dig the pages and tried to work out all steps to derive Lagrangian from Newton mechanics for myself. After some days of struggle, I convinced myself that, albeit they look different, they’re mathematically same. “And that’s it!” I said, there was no mystery at all, Lagrangian mechanics is Newtonian mechanics! I didn’t take the least action principle seriously, because I concluded it was merely a mathematical trick.

Meanwhile, I kept practicing the “art” of Lagrangians. The more I played with it, the more sweet it seemed. I had already noticed some minor differences between two mechanics. In LM there wasn’t any general method for solving problems that involve non-conservative forces, but I no longer needed to mess with free-body diagrams, the triumph was I don’t have to think about internal forces. It felt nice: just pick up a coordinate system you like and write down the Lagrangian accordingly; rest is math.

One of the days I was playing with Lagrangian, a pal who was a sophomore mechanical engineer bumped into me, asking what I’ve been doing. I showed him papers fiddling around and said “learning Lagrangian mechanics”. I was working on a simple problem, something like a ball rolling on an inclined plane, I guess. I had written some Euler-Lagrange equations and trying to solve the equations that came out. “Don’t let it fool you, they’re the same thing as Newton laws”. He has studied only NM so far, so naturally asked me: “Where is the force?”.
So I wrote the equation for one-dimensional case

$\frac{d}{dt}\frac{\partial (mv^2/2 - V(x))}{\partial v} - \frac{\partial (mv^2/2 - V(x))}{\partial x} = 0$

Since force is $-\frac{dV}{dx}$ , it follows that $\frac{d}{dt}mv - F = 0$ which is $F=ma$ .

“Yea, I see” he mumbled, “from potential”. Unlike sophomore mechanical engineers usually do, I had no force-diagrams, and I still had the solution! I guess he was impressed how simply this thing works. Luckily, he didn’t ask what will happen if $V$ depends on $v$ also, or worse, what if we can’t define a potential for our force!

Albeit it doesn’t fit well for practical things -which engineers usually need- it was elegant.

Soon, I realized that my assumption about almightiness of Newton might be wrong. If we take Newton laws as fundamental, Lagrangian’s just a mathematical trick. But how do we know which one is the fundamental? And if we take Lagrangian as the fundamental one -so that Newton’s laws become just a mathematical trick-, why does Nature prefer a minimum principle? Does Lagrangian have a meaning? Well, I didn’t even know whether if these questions made sense. I was looking at two theories predicting exactly same things for same situations. If there’s no way to distinguish the theories experimentally, questions such as “but which is the ultimate truth?” are philosophical rather than scientific. Since I was a natural science -and not a philosophy- student, I swept these questions under the rug!

I kept asking people whether if they know the “meaning” of minimizing action. Heck, I tried school mates, graduate students, CM teacher, forums, tooth fairy… Nope. Luckily, one of these days I was fiddling with the stuff that was beyond my knowledge, I bumped into a very remarkable, original equation in Feynman’s Nobel lecture. I have never seen such a thing before, but it turned out that it was equivalent to Schrödinger equation.

…What Dirac said was the following: There is in quantum mechanics a very important quantity which carries the wave function from one time to another, besides the differential equation but equivalent to it, a kind of a kernal, which we might call $K(x^\prime, x)$ , which carries the wave function $\varphi(x)$ known at time $t$ , to the wave function $\varphi(x^\prime)$ at time, $t+\epsilon$

$K(x^\prime, x)$ is analogous to $e^{i \epsilon L(\frac{x^\prime-x}{\epsilon},x)/\hbar}$

…I simply put them equal, taking the simplest example where the Lagrangian is $m{\dot x}^2/2 - V(x)$ but soon found I had to put a constant of proportionality $A$ in, suitably adjusted. When I substituted $Ae^{i \epsilon L/\hbar}$ for $K$ to get

$\varphi(x^\prime, t+\epsilon) = \int A e^{i \epsilon L(\frac{x^\prime-x}{\epsilon},x)/\hbar} \varphi(x, t) dx$

and just calculated things out by Taylor series expansion, out came the Schrödinger equation. So, I turned to Professor Jehle, not really understanding, and said, “well, you see Professor Dirac meant that they were proportional.” Professor Jehle’s eyes were bugging out - he had taken out a little notebook and was rapidly copying it down from the blackboard, and said, “no, no, this is an important discovery. You Americans are always trying to find out how something can be used. That’s a good way to discover things!” So, I thought I was finding out what Dirac meant, but, as a matter of fact, had made the discovery that what Dirac thought was analogous, was, in fact, equal. I had then, at least, the connection between the Lagrangian and quantum mechanics, but still with wave functions and infinitesimal times.

(Unlike the previous case, Lagrangian approach was not derived from the older laws –with purely mathematical tricks, but came out by miracle!)

Just like Professor Jehle, my eyes were bugging out too. They look entirely different. I mean, how could it be Schrödinger equation?? (it seems I haven’t learned a bit from my previous encounter with Euler-Lagrange equations!) I was aching to work out the series expansion for myself, so after struggling for a while, with the help of some educated ones, it was OK.

$\varphi(x^\prime, t+\epsilon) = \int A e^{i \epsilon (m(\frac{x^\prime-x}{\epsilon})^2/2 - V(x))/\hbar} \varphi(x, t) dx$

Assuming that we have a “stable” potential, and throwing away higher order terms of $\epsilon$ , we get $e^{-i \epsilon V(x) /\hbar} = 1 - i \epsilon V(x^\prime)/\hbar$ . Defining $\xi = x-x^\prime$ and expanding $\varphi(x,t)$ around $x = x^\prime$ .

$= A (1 - i \epsilon V(x^\prime)/\hbar) \int e^{i m\xi^2/(2 \hbar \epsilon)} (\varphi(x^\prime, t) + \xi \frac{\partial \varphi(x^\prime, t)}{\partial x} + \frac{\xi^2}{2} \frac{\partial^2 \varphi(x^\prime, t)}{\partial x^2}) d\xi$

$= A (1 - i \epsilon V(x^\prime)/\hbar) \sqrt{\frac{2 \hbar \epsilon \pi}{im}} (\varphi(x^\prime, t) + \frac{\hbar \epsilon}{2im} \frac{\partial^2 \varphi(x^\prime, t)}{\partial x^2})$

Throwing away the term which involves second order term $\epsilon^2$

$= A \sqrt{\frac{2 \hbar \epsilon \pi}{im}} (\varphi(x^\prime, t) + (- i \epsilon V(x^\prime)/\hbar)\varphi(x^\prime, t) + \frac{\hbar \epsilon}{2im} \frac{\partial^2 \varphi(x^\prime, t)}{\partial x^2})$

As Feynman pointed out, this’s where $A$ comes in. When we make $A$ such that coefficient of $\varphi(x^\prime, t)$ is equal to 1:

$\varphi(x^\prime, t+\epsilon) = \varphi(x^\prime, t) + (- i \epsilon V(x^\prime)/\hbar)\varphi(x^\prime, t) + \frac{\hbar \epsilon}{2im} \frac{\partial^2 \varphi(x^\prime, t)}{\partial x^2}$

$i \hbar \frac{\varphi(x^\prime, t+\epsilon) - \varphi(x^\prime, t)}{\epsilon} = V(x^\prime)\varphi(x^\prime, t) + -\frac{\hbar^2}{2m} \frac{\partial^2 \varphi(x^\prime, t)}{\partial x^2}$

Finally, “out comes Schrödinger equation”!

Ouch.

$\varphi(x^\prime, t+\epsilon) = \int \sqrt{\frac{2 \hbar \epsilon}{im\pi}} e^{i \epsilon L(\frac{x^\prime-x}{\epsilon},x)/\hbar} \varphi(x, t) dx$

I wasn’t sure if this monster was practically useful, but at least, there I had the Lagrangian, again. Further explanation came out of the blue, when I was studying electrodynamics from Feynman Lectures. The next page was supposed to be another lecture on electrodynamics, but I bumped into a lecture with the title: “The Principle of Least Action”. And that was it.

Here is how it works: Suppose that for all paths, $S$ is very large compared to $\hbar$ . One path contributes a certain amplitude. For a nearby path, the phase is quite different, because with an enormous $S$ , even a small change in $S$ means a completely different phase –becasue $\hbar$ is so tiny. So nearby paths will normally cancel their effects out in taking the sum –except for one region, and that is when a path and a nearby path all give the same phase in the first approximation (more precisely, the same action within $\hbar$ )). Only those paths will be the important ones. So in the limiting case in Planck’s constant $\hbar$ goes to zero, the correct quantum mechanical laws can be summarized by simply saying: “Forget about all these probability amplitudes. The particle does go on a special path, namely, that one for which $S$ does not vary in the first approximation.” That’s the relation between the principle of least action and quantum mechanics.

The action turned out to be proportional with frequency of a kind of stop watch advancing as the particle propagates! Though I haven’t found the “ultimate truth” of Nature, I learned why the Lagrangian has to be small.

This entry was posted on Saturday, August 19th, 2006 at 11:54 am and is filed under Physics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “The Lagrangian”

tramadol Says:
April 4th, 2007 at 8:58 am
It is a very interesting story. Thanks!

The Lagrangian

One Response to “The Lagrangian”

Leave a Reply

Eigenquote