Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.
- Isaac Newton
The moon orbits the earth, held precisely by a delicate balance between gravity and inertia. A piece of lint clings to a shirt, held tightly by a delicate balance between electric charges and fields. Laughter rings through the air, carried by waves of unsettled air. Starlight adorns the sky, carried by waves of unsettled abstractions which we christen fields. A spiral of water and air a thousand miles wide counterpoints a spiral of stars and emptiness a hundred quadrillion miles wide.
This is the Astronomy Picture of the Day for May 17 of this year. One is the Pinwheel Galaxy, the other was Typhoon Rammasun. The physical forces holding them together are entirely different. The typhoon is an atmospheric phenomenon. Its behavior is driven by the Navier-Stokes equations
which describe the behavior of gases and fluids. For weather, the tensor T and the external forces f are going to be really complicated. Sunlight, geography, thermodynamics, you name it and it’s probably affecting the weather. The result is chaotic and highly nonlinear. That’s why it’s not possible to accurately predict weather very far in advance.
Galaxies, on the other hand, are at least in theory much more simple. They obey Newtonian gravity though a great deal of their mass has proved difficult to detect.
Much simpler, and though strictly speaking it’s only an approximation to the general relativistic reality, it’s so close as to be fine for almost every research purpose. So why can one hideously complicated and one quite simple equation describing two very different processes cause the same pretty spiral? Feynman said the same equations describing different phenomena have the same solutions, but these aren’t even the same equations. Like love songs in Spanish and English, their sameness is conceptual rather than textual. Mathematically speaking the answer lies in the fact that the equations do share particular properties which are not at all obvious. In particular, both equations allow what amounts to central forces and with the conservation of angular momentum these spiral structures obey Hamilton’s principle. I don’t think I could further explain it simply enough for the nonmathematical readers or accurately enough (in the case of the Navier-Stokes equations) to satisfy the mathematicians, so I’ll leave it for another day.
The real lesson is that mathematics works - and works better than we have any right to expect. There’s no reason nature is obligated to follow mathematical rules at all. But not only does it follow mathematics precisely, it follows the most subtle and unexpectedly beautiful mathematics that we know. Unexpected connections appear between things that are otherwise completely different. And that is something awe-inspiring.