Note: After the midterm exams, the curriculum has become more intensive and free time has decreased; therefore, the updates to Scientific Space have slowed down. However, BoJone will try his best to update some content and share the joy of learning with everyone.
Over the past week and this week, BoJone has summarized and integrated some of his studies on physics and extrema, writing them into an article titled "Natural Extremum". Therefore, from today until most of December, Scientific Space will describe and discuss issues regarding "extrema" with everyone. I hope readers will enjoy this content. Of course, I am not a professional researcher, nor an experienced physics or mathematics teacher; one might even say I am just a "wet-behind-the-ears kid." Consequently, errors are inevitable. I only hope that fellow enthusiasts will not hesitate to point them out, and even more so, I hope that the "brick" I throw out can lead to the discovery of beautiful "jade."
"Natural Extremum" mainly includes the following content:
Fermat’s Principle (Optics section)
Equilibrium Axiom (Principle of Minimum Potential Energy)
Fermat Point Problem (Application of the above two principles)
Brachistochrone Problem
Catenary Problem
Preliminary Analysis of Extrema (Deriving a fundamental formula of the calculus of variations)
The above content will be explained as elementarily as possible (Why "as elementarily as possible"? One very important reason is: I don’t know how to explain it using advanced methods, haha ^_^)
Preface:
In mathematics, finding extrema is probably one of the problems we deal with most frequently. For mathematicians, finding an extremum involves inputting a function into a computer, then finding the derivative or partial derivatives (the Lagrange multiplier method we learned), setting them to zero, and then "brute-forcing" the result. Admittedly, in many cases, this mode is necessary as it adapts to an "efficiency-oriented" society. However, for friends who love physics or are keen on finding the beauty of science, this processing procedure appears particularly monotonous and boring. At the same time, many calculation problems are meant to solve practical issues. If the solution can return to the realm of physics, it is clearly a sight to behold. Just as we would rather be like Edison, using the method of filling a pear-shaped bulb with water to measure its volume, rather than performing tedious integration like Upton.
Therefore, we all hope to find some scientific facts from nature (things that can be called "axioms" or "principles") and turn them into tools for solving mathematical and physical problems, or even develop them into substantial and complete theories. Thus, BoJone attempted to write the article "Natural Extremum" to introduce such a process.
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