Let's Talk Physics

A stochasticly updated blog about interesting topics in Physics & Astronomy

The Least Action Principle 1: David Hasselhoff is a Photon

Imagine you’ve gotten a sweet job for the summer as a lifeguard. You sit on your highchair and concentrate on maximum chilling. But wait, the sea erupts in whimpers and gurgles as some unfortunate person gets themselves into aquatic difficulty!

A lifeguard must be both effiecient and gorgeous.

Of course time is critical here and you want to reach your infelicitous swimmer as quickly as possible. You can’t swim as fast as you can run, but then you cant run on water. If you run straight to the sea and swim (purple line) you’ll cover ground swimming (slow) that you could have covered by running (fast). On the other hand if you run as close as you can before swimming (yellow line) you end up taking a pretty indirect path.

A scientific digram of your predicament.

The clever reader may have at this point noticed that there is another choice, If we compromise and take the green path it should be faster than either of these. The question remains, however, what is the green path? In a simple situation like this we can work it out with algebra or geometry or calculus. The easiest way to visualise this (maybe) is to imagine that we know what point we want to go from the beach to the water (which after all is the only choice we have to make) and the to “stretch” the water out until our speed on water is the same as that on land. This is a slightly trick point and worth thinking about. Imagine a bird looking down at the whole scene, who looks only at the water with binoculars. With that in mind look at the diagram:

snell

Now that’s not a formal proof. In fact you can’t even read the answer directly off it, but it’s a nice way to get a feel for what’s happening. If you’re impatient to get the answer you can calculate the total distance as a function of the point where you dive in, and then use old-school calculus to minimise.

But what does this have to do with physics? Well it just so happens that nature is the best lifeguard, and that, as far as modern theory is concerned, any physical process that we understand can now can be expressed as a problem similar to this. The easiest example is that of the path of a beam of light (or a single photon, on average). Light waves travel on the whole,  more slowly in different materials than others, and we call this slowdown the refractive index. It’s just the ratio of how fast the wave would travel in a vacuum to how fast it would travel in our material. For example the refractive index of sea water is roughly 1.33. That means that going the same distance in water as in vacuum takes the light 1.33 times as long.

Maybe you see how this might relate to our lifeguard problem. If the light is to be “efficient”, i.e. take the shortest path, then it will do just as Hasselhoff does and turn when it moves from one substance to the other. This is absolutely what happens (why it happens is a pretty deep question that I may get into in another post). This phenomenon is why lensess work, and pretty cool. It’s governed by something called Snell’s Law which just relates the angles to the speeds, and is a direct consequence of the solution to the lifeguard problem.

None of this might be news to you if you’re a man of the world and guard life or wear glasses but this is the tip of the iceberg on a really philosophically and physically deep topic that I’ll be developing over the next couple of posts.

Just to finish, it’s worth pointing out that the speed of light, and hence the angle of bending depends on it’s colour. Pink Floyd fans will know this.

I’ll see you on the dark side of the moon.

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3 comments on “The Least Action Principle 1: David Hasselhoff is a Photon

  1. Alan
    June 12, 2013

    Why didn’t you draw the green line as a straight path to the swimmer?

    • letstalkshite
      June 12, 2013

      That’s a good question. The green line represents the best (fastest) path to take. It corresponds to a line straight from the lifeguard to the swimmer only when the lifeguard swims exactly as fast as he runs. This is kinda why the idea is weird, naively you might expect the straight line to be the best, but it isn’t, at least not a straight line as you imagine it normally (without rescaling).

  2. Pingback: Least Action Principle 2: I Like Bubble Baths | Let's Talk Physics

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This entry was posted on June 6, 2013 by in Uncategorized.
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