# Built on Facts

## Lagrangian of free fall

#### June 13th, 2008 · 4 Comments

Vis consili expers mole ruit sua.
- Quintus Horatius Flaccus

The words of Horace above attain a spare and austere beauty in Latin, but the meaning is carried equally well in English. Force without wisdom falls of its own weight. Two different sets of words carry his millenia-old thoughts to the present time. There are different ways of saying the same thing.

In physics, force permeates the first few semesters of study and continues to be a useful tool in higher physics. But in many respects force is not the most elegant tool in our arsenal. It’s a vector; not only do we have to worry about magnitude but we also have to deal with direction. With some exceptions, vectors are significantly more difficult to deal with than regular numbers. But look at Newton’s First Law, which we all know:

F and a are both vectors. Force in the x-direction produces acceleration in the x-direction, etc. Easy enough for basic problems, but for complicated ones things get rapidly very difficult. Other coordinate systems can make the situation even worse. Non-classical physics is worse still; the concept of a force doesn’t even necessarily have much meaning in quantum mechanics.

But like translating from one language to another, we have alternate ways to work with a given physical situation. Instead of examining the forces, we can look at the energy. This is just a conceptual introduction so we’ll skip all the derivation for now, and present the final result. Define a quantity L, called the Lagrangian. It’s a measure of the energy of the system; specifically it’s the measure of the kinetic energy K minus the potential energy U. Thus by definition L = K - U.

Now pick coordinates: maybe x, y, and z, or θ and φ, dimensionless quantities, or whatever you feel like and in any combination. Pick and choose as many coordinates as it takes to describe the system, and call them qi, for however many i’s you need. It turns out that the following equation completely describes the classical time evolution of the system:

In this differential equation, you have

1. Energy specified by L
2. Coordinates specified by q. The dot above q just means the time derivative of q, to save space.
3. A time relationship connecting them

And that’s all you need. No forces to be seen. No vectors of any kind. You can specify the coordinate system in whatever way most easily describes the configuration of the system without having to worry about whether that same coordinate system also can easily describe the forces within that system. Now let’s pick the easiest example in the world to test it out. In the future we’ll do harder things that really show the power of Lagrangian dynamics, but since this is just dipping our toes into the water we’ll just use a very simple rock falling straight downward.

The potential energy of that rock is just U = mgy, where y is the height above the ground.

The kinetic energy of that rick is just K = (1/2)mv2, where v is the velocity and thus the time derivative of y, which we write as y with a dot over it.

That makes the Lagrangian L = K - U become:

Plug that into the differential equation above with q = y (because as we said before the various q are any coordinate you want), and you get:

Now carry out the y and y-dot differentiation formally, resulting in:

Finally doing the time derivative:

Which lo and behold finally gives us:

The rock has an acceleration g vertically downward, independent of its mass exactly as we expect from Newton’s laws. We never touched a force or a vector. “But Matt,” you say, “isn’t that actually a lot harder than doing things with forces?” Yes it is, for this problem. But this problem is just a toy example to show that this does give us the right answer for a problem we can already do trivially. For instance, try to characterize the motion of a marble placed on top of a basketball using forces. If you can do it at all, you’re a much more patient physicist than me. Using energy methods it’s still not quite a simple calculation but it’s thoroughly tractable. But that’s a story for another day.

### 4 responses so far ↓

• 1 Uncle Al // Jun 13, 2008 at 10:41 am

You derived acceleration but not path. Einstein’s elevator demands the Equivalence Principle. All local bodies identically fall (relativistic bodies and photons not according to Newton) along minimum action parallel paths… if the vacuum is isotropic. A single unexamined exception is consistent with prior observation:

Matt replies: No. The differential equation and its initial conditions completely specify the path.

http://www.ift.unesp.br/gcg/tele.pdf
Chapter 9, Eq. 9.52, p. 75,

“a self dual (anti-self dual) torsion couples only to the left-hand (right-hand) component of the spinor field. In other words, gravitation becomes a chiral interaction, a property that may eventually have important consequences at the microscopic level.”

A chiral vacuum background is a left foot. One can fit it with opposite shoes - opposite parity mass distributions- that fit with different energies. They will vacuum free fall along divergent paths. Put on two left shoes, close your eyes, try to walk a straight line. Local gee might get you minimum action divergent vacuum free fall paths. Somebody should look. (pdf)

Matt: Feel free to give the experiment a try to satisfy yourself, but at this point you might as well be arguing that the sky is green. But hey, I could be wrong. Do the experiment and get it published, and get back to me.

• 2 Carl Brannen // Jun 13, 2008 at 8:47 pm

If the universe was designed with energy (and other symmetry) principles, then making the guess that the underlying symmetry is simple is the way to advance physics further. This has been done by 1000s of physicists for decades, it may not be the best way to further the field. If the universe was designed around simple forces, then one should instead look for a simple version of F=ma.

Even if the universe uses simple equations of motion instead of simple symmetry principles, we will still use symmetry principles to solve those equations of motion. We do this because, as you said, force is a vector and is more complicated than energy.

Symmetry is how mathematicians solve problems. That symmetry is unnaturally effective at solving physics problems can be attributed to the fact that we have solved the easy problems with symmetry. The hard problems are unsolved. As one of my professors used to say, “at night, the drunk looks for his car keys under the street lamp”.

So our current situation, where physics is in a bit of a rut, might be attributed to a universe with simple equations of motion, inhabited by an intelligent species that insists on looking only for simple symmetry principles.

Uncle Al has been pursuing his goal for quite some time. It’s a tough experiment.

Matt replies: It’s very true that physicists can only figure out things that they’re smart enough to figure out. The discovery and characterization of chaos theory by Lorenz and others is one of the few real examples in the last century or so of working with things which are off the trail beaten previously by mathematicians. Quantum mechanics was a much more important discovery, but in some sense the mathematics had already been at least worked on by people like Hilbert.

Very specifically on the question of whether forces or energy (and thus symmetry in time as per Noether’s theorem) is the most fundamental in some sense, there’s a school of thought which says that the Ahronov-Bohm experiment tends to suggest that energy instead of force is the thing which really runs the universe. I couldn’t really pretend to know how accurate that idea is though.

• 3 CCPhysicist // Jun 14, 2008 at 8:17 pm

Which is cooler, Lagrange or Hamilton or Newton?

http://doctorpion.blogspot.com/2008/06/hamilton-lagrange-or-newton.html

• 4 Carl Brannen // Jun 15, 2008 at 4:40 pm

When I was a physics grad student, my major was elementary particles and so I tended to see everything in elementary particles form. And I assumed that it was elementary particles that is the most fundamental. So I see the Aharonov Bohm effect as more of a condensed matter effect than something that says much about the universe. I should explain.

The magnetic field in the Aharanov Bohm effect is the result of a very large number of virtual photons. You could write this up as a sum of a great deal of Feynman diagrams for the interaction of an electron and a photon. In this way, the problem is written entirely as an equation of motion (the Dirac equation), and the QED vertex, plus linear superposition.

But in physics, it is good that there are lots of people working on problems from different points of view. And those people have to have a certain faith in their methods (so that they sacrifice the necessary effort), otherwise a corner would be left unexplored.