"Local" vs "global" ways to solve physics problems

Inspired by a recent post of Ross McKenzie, I thought it would be fun to try to write a popularly accessible piece about the enormously successful, wholly remarkable  theory that most people have never heard of, density functional theory.

To get there will require a couple of steps.  First, it's important to appreciate that sometimes, thanks to the mathematical structure of the universe, it is possible to think about and solve physics problems with two seemingly very different approaches - call them "local" and "global".  In the local approach, we write down equations that describe the underlying problem in great detail, and by carefully working out their solution, we arrive at an answer.  In the global approach, we come at the problem from an overview perspective of considering possible solutions and figuring out which one is correct.

For example, let's think about a light ray propagating from point P (in air) to point Q (in water), as shown in the figure (courtesy wikipedia).  It turns out that light travels at a speed \(c/n\) in a medium, where \(c\) is the speed of light in vacuum, and \(n\) is the "index of refraction" that depends on the material and the frequency of the light.  (This is already short-hand for solving the complicated problem of electromagnetic radiation and its interactions with a material containing charges, something that Feynman wrote about elegantly in this book, based on these lectures.)  The "local" approach would be to write down the equations describing the electromagnetic light waves, and solve these, including the description of the air, the water, and their interface.  The result we would find is so simple and compact that we teach it to freshmen, Snell's Law:  \(n_{1}\sin(\theta_{1}) = n_{2}\sin(\theta_{2})\), where the angles are defined in the figure.

The "global" way to solve this problem (and again arrive at Snell's Law) was found by Fermat (yes, the one with the "last" theorem).  He didn't have the option of solving the microscopic equations governing the radiation, since he died two hundred years before Maxwell published them.  Instead, Fermat knew that light seems to travel in straight lines within a given medium.  Therefore, he considered all the possible paths that a light ray could take from P to Q (such as the blue and green alternatives shown in the modified figure), trying to figure out which combination of straight segments (and hence which angles) were picked out by nature.  The answer he posited was that the correct path for the light is the one that minimizes the overall time taken by the light in going from P to Q.   This does give Snell's Law as a consequence, and seems to hint at a deeper organizing principle or structure at work than just "we solved complex equations with tricky boundary conditions, and Snell's Law fell out".  (These days, if a student is asked to derive the Snell's Law from Fermat's Principle of Least Time, they would use calculus to do so, since that plus coordinate geometry provides a clear way to right down an expression for the transit time and a way to minimize that function.  Fermat couldn't do that, as modern calculus didn't exist at the time, though he was among the people thinking along those lines.  He was pretty sharp.)

Next up:  another example of a "global" approach, the Action Principle.