Fermat's Principle
Why does light bend when it enters water? Pierre de Fermat discovered that light always takes the path that takes the Least Time. This is why it "shortcuts" through the denser medium.
Worked Examples
Example 1: The Lifeguard Problem
A lifeguard at point A on the sand needs to reach a drowning swimmer at point B in the water. The lifeguard runs faster on sand than they swim in water. Where should they enter the water?
- Goal: Minimize Time \(T = \frac{\text{Distance on Sand}}{\text{Speed on Sand}} + \frac{\text{Distance in Water}}{\text{Speed in Water}}\).
- This requires setting up a coordinate system and using the derivative of the time function to find the entry point. This leads directly to Snell's Law.
The Bridge to Quantum Mechanics
The Principle of Least Time is the classical ancestor of the Feynman Path Integral. In Quantum Mechanics, a particle doesn't just take one path; it takes all possible paths at once. However, the paths that are far from the "optimum" path (the path of least action) cancel each other out through destructive interference. Only the paths near the optimum survive. This is why big objects (like baseballs) seem to follow one single path, while tiny particles (like electrons) behave like spread-out waves.