Light can do calculus? What?!

[Diocletian]

My friends and I were having an interesting discussion today. When light is radiated through a tank of water, it "bends" because it has a natural affinity to finding the path of least resistance to the end of the tank and since the travel through water would take more time than travel through land, the bending cuts down a little on the time spent.

 

We were wondering: how does light "know" the path of least resistance? The only explanation I can think of is that light takes the derivative of each path and thus selects the shortest one. But that's ridiculous; light can't do calculus, so how does it instinctively go down the correct line? Another guess was that light tries each path until it finds the right one; but that's equally unrealistic, because light would spend an astronomical amount of time doing that and would thus counteract its own purpose of finding the quickest route.

 

Perhaps there's a scientific explanation somewhere there that I missed.

[swansont]

That's how light behaves; it's called the principle of least time or Fermat's principle. (Wikipedia link)

 

Calculus is part of the model that we use to describe the behavior, but is a construct of human intellect. Light does not do calculus as it transits a medium any more than a rock does calculus when it is dropped, and the distance fallen is the double integral of the acceleration.

[D H]

We were wondering: how does light "know" the path of least resistance? The only explanation I can think of is that light takes the derivative of each path and thus selects the shortest one. But that's ridiculous; light can't do calculus, so how does it instinctively go down the correct line? ... Perhaps there's a scientific explanation somewhere there that I missed.

You can ask the same question of a flexible rope or chain whose ends are attached to two fixed points. Regardless of the location of the two end points and regardless of how much slack is in the rope, the rope will form a section of a catenary. This curve is the minimal energy configuration of the rope. How does the hanging rope "know" to follow the minimal energy configuration? Does the rope do calculus to form itself into a catenary?

 

What you are looking for is called the principle of least action. The development of this principle has a long history, starting with Newton and Leibniz. Physicists now use Hamilton's principle: A dynamic system follows the path that represents an extremum of the time integral of the difference between kinetic and potential energy. This extremum is almost always a minimum.

 

Hamilton's principle is widely used in classical and quantum mechanics. In classical mechanics, Hamilton's principle bypasses the need to represent all of the forces acting on a system. Suppose you have some classical problem readily soluble by means of Hamilton's principle. If you can represent the forces in the system and if you can solve the problem using the Newtonian force formulation, the resulting solution will be the same as that arrived by applying Hamilton's principle. Hamiltonian and Newtonian mechanics are mathematically equivalent.

 

You can look at Hamilton's principle as magical: Some magical agent does the calculus. The early developers of the principle essentially viewed it as magical: God does the calculus. There is no need for God here; you can also look at at Hamilton's principle as an abstraction of some deeper but much more difficult to manipulate principles. For example, consider the case of your light beam. In quantum electrodynamics, a photon simultaneously takes all possible paths from point A to point B. Sum all of these possible paths and one path will arise as the "winner" by virtue of interference. This winning path is the time minimizing path.

[Severian]

You might want to read this:

 

http://en.wikipedia.org/wiki/Path_integral_formulation