A twice weekly blog about interesting topics in Physics & Astronomy
We’re getting a little closer to seeing now what action means, and why it needs to be minimised, so to develop the idea a bit further before pinning it down properly we’ll discuss a couple of classic problems.
First of all we have a favourite problem of mine, the Delta Airlines Problem. Long long ago, when men were men and the internet was something that happened with telephones, a business could arrange with the telephone company to have its intra-office networks linked, via the phone line, to each other to create one large inter-office network. Now our client, Delta Airlines, has offices near three big airports, Chicago, Denver and Houston:
Now the billing for this was a little funny. While the phone lines already existed, the phone company thought that the fairest way to charge was a to look at what it would have cost to build the network from scratch, just to connect the offices. Of course they didn’t want to be unfair, so they used straight lines between all the offices like so:
(You might argue at this point that only two links are needed, but removing a link would double the time it took information to get between the two adjacent offices. Also this way agrees with the physics.)
Now we are charged with coming up with a way to save our client some money (and making a bit of money for ourselves in the process). At first sight it doesn’t look like there’s much we can do, short of relocating.
One day you’re sitting in the bath, making yourself a bubble beard and you have a eureka moment.
What if you added an extra, pretend, office right in the middle. A little tent office, with a telephone inside. What then you ask?
This way is shorter. You can do the trigonometry yourself. What we’ve done here is solve anther minimisation problem. Nature, as I’m trying to explain, is all about minimising something. The action. We want the least action. For example you can do this exact problem with soap bubbles. They have surface tension and try to minimise surface area. If you have a board with pegs where the houses are, and you sit back into the bath and link the pegs by bubbles, and give it a shake you’ll get just the same solution as for the telephone lines.
Sometimes it goes the other way and we want to maximise distance. This works for free charge in a conductor (it distributes itself mostly around the surface), or polite people in an elevator, or Randall Munroe’s urinal problem. For an exercise draw a square, pretend it’s an elevator, and then add people one by one and try to work out where they’ll stand.
We haven’t quite gotten to actually pinning down the Least Action Principle yet, but we’re well on our way. I’m also trying to keep the rigour and maths to a minimum, although I’ll be introducing more in the next couple of posts as our ideas get clearer.