"Extrema in Nature" Series — 3. The Equilibrium Axiom

《自然极值》系列——3.平衡态公理 苏剑林 · 2010-11-28 · Mathematics potential energy balance extremum nature

Huangguoshu Waterfall

The law of optics is undoubtedly a beautiful principle, and there is another "axiom" in nature that we encounter everywhere. In everyday life, we always observe the phenomenon of "water flows downhill," which is a consequence of water being subject to Earth's gravitational field (and it is precisely because of this that certain people intent on ending their lives have been able to carry out their acts successfully; of course, we don't need to test this ourselves to verify it). From this we're led to a concept: gravitational potential energy. What does "water flows downhill" mean? It means the height decreases. And what does a lower height mean? It means the gravitational potential energy decreases! In other words, objects in nature tend toward the minimum potential energy. We can understand this from the following angle: a system always tends toward stability, and the higher the energy (potential energy) it possesses, the more unstable it is.more

Having said this, we've arrived at another extremum with clear practical significance. But we still haven't figured out how to turn it into something computable. Indeed, we need one more principle—the "equilibrium axiom"—to build the "bridge connecting physics and mathematics":

When the total potential energy of a system reaches its minimum, the forces on it must be in equilibrium (the net external force is zero).

This is actually quite clear: if the net external force were not zero, there would necessarily be a force driving the system to move toward a state of lower energy. However, the converse of this axiom does not necessarily hold. Of course, this doesn't prevent us from applying it.

A system always tends toward stability, and the higher the energy (potential energy) it possesses, the more unstable it is.more

Addendum:

Regarding the equilibrium axiom, it seems there isn't much more to say, since it's such a familiar part of everyday experience. So in this section, let's give a preliminary summary of Fermat's principle and the equilibrium axiom: both are reflections of phenomena common in daily life. Starting from these two principles and using them to answer certain problems, we're often struck by how elegant and beautiful the results are. Some might say this resembles "acrobatics in a competition," but that's not quite right—physics guides us to think correctly, while mathematics helps us summarize and analyze our conclusions. Mathematics is very scientific, but what's truly magical is physics. Physical science has astonished humanity time and time again. God is an artist, and the world he created is so harmonious. As the saying goes: Chemistry is physics without thought. Mathematics is physics without purpose.

Afterward, in the study of problems such as the catenary and the brachistochrone (later we'll discover that these problems share the same essential nature), Euler and Lagrange developed a vigorous and vibrant branch of mathematics called "the calculus of variations." This was both unexpected and, in a sense, exactly what everyone had been waiting for.