Now that you know a bit about what the Least Action Principle might be it’s time to learn a little bit about what it does (we’ll learn about why it’s true later).

The least action principle is a way of finding out what physical systems will do. That’s pretty useful, when you consider that pretty much anything can be considered as a physical system. Of course it only works on simple systems, but that’s because humans aren’t very patient or clever. We can use it to tell how protons, neutrons and electrons interact, and as far as the modern theory goes that’s what almost everything you dea with every day is made of. All the same we aren’t going to be able to use the Least Action Principle to work out why two people fall in love, or if you can hide from colourblind people by wearing camouflage in front of a cow.

If you throw a ball what happens?

*It goes somewhere.*

How does it get there?

* Don’t know. *

How would you find out?

*Equations?*

Which equations?

*Those ones in the picture?*

Why?

*Least Action Principle?*

Bam.

This is all true. I won’t get into the real gore of what the equations mean just yet, but I will tell you roughly what they mean.

**Physical systems are lazy**. They are lazy but crafty. They are so lazy that they will figure out the easiest way to do something, the fastest, the cheapest. When you drop a ball it goes straight down, it doesn’t mess about on the way (other things don’t do that I know, it’s just an example). Like in the first part of this series when they want to save a drowning dude (or be a photon passing through a glass) they do it in the fastest way possible.

Another way to put this is that things go in straight lines. Like the falling ball, or light from a laser.

But what about the thrown ball you ask? Why doesn’t that go in a straight line?

Well here’s the magic part (which most likely won’t make sense yet). **It does**.

Of course any reasonable person at this point would reply that it doesn’t. And they’d be right. Right except that it depends on what you mean by a straight line. Rather than getting into semantics I’ll give you an everyday example.

Say you Tinie Tempah and you need to get from Miami to Ibiza in a hurry (not for a party, to escape debt collectors).

You’ve got a private plane, obviously, so you can go whatever route you like. You try Google Maps, but it lets you down, just like your luck. So maybe you draw a straight line on your map, fly along that?

Nah, that’s bonkers, and you’re not Dizzie Rascal. You get out your globe, measure out the shortest distance there (a straight line in the curved space, braaap). This is called a **great circle**. It is pretty great, because it’s the fastest way there:

Don’t believe it’s the fastest and cheapest? It’s what commercial airlines do, and they include some of the craftiest and cheapest people you’d ever have the pleasure to meet.

Next week: Maybe actually doing some physics and getting some answers.

Advertisements

%d bloggers like this:

Hmm… I’m not sure if the 2nd equation is correct. I think the differential operators should be acting on L, not S, and the operator order on the right hand side should be reversed.

You’re absolutely right. Then again I never claimed that that was the Euler-Lagrange equation. It’s non-trivial to prove that what’s there doesn’t work (i.e. the derivatives don’t commute with each other and the integral).

The problem is S[q] is the functional we are trying to extremize. Since it is a functional, what is written above wouldn’t make sense. For example, it would not make sense to hit S with a partial derivative with respect to q, when q is the very function that must be varied. Nor would it make sense to hit S with the operator d/dt, since S is not a function of t.

We must instead use functional derivatives. We can derive a very elegant expression that captures the principle of least action.

That equation (which is equivalent to the Euler-Lagrange equation) would be the correct form.

But suppose the commutations I’d mentioned worked. Then the equation would be valid. It’s easy (and I’ll try to work it out nicely in the next post) to show that Euler-Lagrange is the way to do it. Showing that what I gave isn’t valid is actually pretty awkward, if you’re being rigourous. You and I both know how to do variational calculus, that isn’t the point of the post. I don’t want to preach to the choir. On the other hand I greatly appreciate keeping you honest, I’ll update the graphic once I’m back at my workstation.

No worries. Yeah the equations aren’t terribly important to the average reader. All that is relevant (which you rightly explained), is that for every conceivable way a system could evolve, there is an associated quantity called “action”.