*Natural Extrema* Series — 1. Preface

《自然极值》系列——1.前言 苏剑林 · 2010-11-27 · Mathematics physics extremum nature

Note: after the mid-term exams, the coursework has picked up and free time has shrunk, so updates to Scientific Spaces have slowed down as well. Still, BoJone will try to keep posting content and share the joy of learning with everyone.

A continuous function ?(x) on the closed interval [a,b], with its maximum marked by a red point and its minimum by a blue point

Over the past week and this one, BoJone has been summarizing and organizing some things he's learned about physics and extrema, and has put them together into an article called Natural Extrema. So starting today, and for most of the time between now and December, Scientific Spaces will be talking about and discussing the topic of "extrema" — I hope readers will enjoy this material. Of course, I'm not a professional researcher, still less an experienced teacher of physics and mathematics — one might even say I'm still "wet behind the ears" — so mistakes are inevitable. I only hope fellow enthusiasts won't hesitate to point them out, and that this rough "brick" I'm tossing out might draw out some beautiful "jade" in response. more

Natural Extrema mainly covers the following topics:

1. Fermat's principle (the optics part)
2. The axiom of equilibrium states (the principle of minimum potential energy)
3. The Fermat point problem (an application of the above two principles)
4. The brachistochrone problem
5. The catenary problem
6. A preliminary analysis of extrema (deriving a basic formula of the calculus of variations)

All of the above will be presented in as elementary a way as possible. (Why "as elementary as possible"? One important reason is that I don't actually know how to present it at a more advanced level — ha ha ^_^)

Preface:

In mathematics, finding extrema is probably one of the problems we deal with most often. For a mathematician, finding an extremum usually means feeding the function into a computer, taking its derivative or partial derivatives (the Lagrange multiplier method we all learn), setting them to zero, and then brute-forcing the result out. Admittedly, in many situations this approach is necessary — it fits well with an "efficiency-oriented" society. But for those who love physics, or who are keen on seeking out the beauty of science, this kind of processing feels especially dry and monotonous. At the same time, a great many calculations are meant to solve real-world problems, and if the method of solution can be brought back into the realm of physics, the result is clearly far more delightful. Just as we would rather measure the volume of a pear-shaped light bulb the way Edison did — by filling it with water — than grind through a tedious integral the way Ampère might have.

For this reason, we all hope to find, within nature itself, certain scientific facts — things that might be called "axioms" or "principles" — and turn them into tools for solving problems in mathematics and physics, or even develop them into full, solid theories in their own right. And so BoJone has tried his hand at writing Natural Extrema, to introduce just such a process.