The principle of least action suggests an alternative way to derive the equations of motion for a moving object. Instead of using Newton’s idea of , one can instead try to find the path that minimizes the following integral
where KE = kinetic energy and PE = potential energy,
The reason for searching the path that minimizes that integral is simple; That’s the same path that the actual object will follow in real life. Isn’t this amazing? Nature is a “natural” optimization machine.
This idea is central in classical mechanics and the search for a path that minimizes a certain integral is something that I keep on seeing all over the place. So, here’s two suggested readings on this principle.
First, a lecture by Richard Feynman on the the least action principle and, for the people that want to look at more math, I recommend these notes by Daniel Baumann of DAMTP in Cambridge. Also, check his notes on the concepts in theoretical physics which seems like a fun class.