What Makes the Hardest Equations in Physics So Difficult? (2018)
quantamagazine.org
quantamagazine.org
"From... elementary theories we build up descriptions of more and more complex systems. But in all these efforts we take for granted that we may use any language we wish and as many [languages] as necessary. That is, we choose whatever mathematical formalism is most useful and then interpret the symbols and measurement operations in very highly developed natural language. To a large degree, the simplicity of natural laws arises through the complexities of the languages we use for their expression."
– H. H. Pattee
The whole problem with the Navier-Stokes equations is that the math seems to work extremely well, but we have no way to be sure it actually captures every aspect of reality (given suitably accurate initial inputs). You can use the equations to generate pretty convincing simulations, but they certainly do not always predict the fine-grained behavior of real-world turbulent systems.
Feynman's lectures repeatedly stress that physical laws (and the math that formalizes them) are, at best, idealized approximations of reality. Here's one, but you can google "feynman approximation laws" for more: https://www.feynmanlectures.caltech.edu/I_01.html
I wouldn't conflate intuitions and observations like that. Observations in many physical realms are best described by math, which is used to build up a natural-language approximation for communicating of unintuitive findings.
Note: my math is not very advanced, at least not good enough to understand quantum mechanics
But why does it not work the other way? When the math tells me something very wrong, can't the conctext and meaning show where the math modell is wrong?
As far as I understood, every physical modell is only a limited model of reality, so they all have flaws. Meaning the math can be wrong when applied to reality, which one could spot, with the understanding of reality?
The point is that when learning a new part of physics, it is far more likely that your intuition was wrong as the math was right, even if it's surprising.
Sure, the mathematical model is not a perfect model, but it can still be _very_ good, so if you disagree with it, you're very likely to be wrong.
However, if you count the number of times a physicist uses math to resolve conceptual confusion and count when they use physical principles to fix the math, you will find the former many times larger than the latter.
The possible explanations for the weirdness are all speculation about how to solve a very important problem. But they neither valid interpretations (because they don't explain anything anybody can see) nor about Quantum Mechanics (they are about an open problem of physics, not about the theory the teacher is explaining).
It is important to speculate on how one can solve problems. That's where solutions come from. But this is not the same thing as interpreting results.
- Lewis Fry Richardson, 1922
It's curious to think that a mathematical phenomenon like this can hint at new physics.
[1] Saari, D. and Xia, Z., "Off to Infinity in Finite Time", https://www.ams.org/notices/199505/saari-2.pdf
John Baez also has a nice, accessible series of articles called "Stuggles with the Continuum" [0]. As a side product, it gives a nice perspective on the development of modern physics as a series of attempts to fix these infinities (only to create more subtle one).
[0]:https://johncarlosbaez.wordpress.com/2016/09/08/struggles-wi...
I believe this guy even talks about it a bit in this video made by the Royal Institution (which I really enjoy watching), discussing the possibility of different fundamental building blocks of nature. https://www.youtube.com/watch?v=zNVQfWC_evg
Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier-Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn’t prove his intelligence, what would?
Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal.
Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.
It was meant to be an easy question to see if students understood Newton's third law, but one student filled in the entire test with momentum calculations showing that the boat would actually move forward at X velocity because the sail would essentially redirect some % of the air backwards like a reverse thruster (conservation of momentum). He left the rest of the test blank because he blew the whole time limit on the first question.
The professor was perplexed when grading this student's exam and built a "sailboat" out of a pinewood derby car with a dowel rod mast and aluminum foil sail. He taped a handheld fan to the car, pointed into the sail, and indeed, the car moved forward (this part he demoed to the class as he was telling the story and just before he did it, he took a poll to see how many people thought it would move forward, backward, or stay still - "stay still" won the poll)
The student reportedly got 100% on the test and the professor threw out that question on future exams.
Their boat went about 10% the speed of just pointing the fan backwards.
Yeah, that seems likely. But sometimes it's difficult to draw the line between issues that are "purely theoretical" and issues that make a practical difference. Conceivably at some scale the Navier-Stokes equations match up with molecular dynamics simulations in a way that is illuminating. Then in that situation maybe the problems with the N-S equations become nice indicators of what's going on.
Its almost like they would have to define a theoretic limit similar to 9.808175174 m/s^2 to limit the finite possibilities of outside forces acting on the molecules before the molecules themselves become the outside force, with those that exceed the limits be classified with a custom equation, and those below the threshold fitting nicely in a bow wrapped package.
And just to add more SWAG, is it going to be easier to produce these finding/understandings in space, where theoretic limits are more well defined and we'll be forced to use custom equations within our atmosphere looking for an answer from a perspective that is environmentally more complex?
Its all greek to me...
https://en.m.wikipedia.org/wiki/Navier%E2%80%93Stokes_equati...
Eddys in a river with more eddys creating eddys is a dizzying theory though.
Rather than confront those, cosmologists have chosen to pretend that, while every single thing they can see is plasma (excepting, uniquely, planets), none of it does anything plasma-ish.
Since plasma physics is scale-invariant, freaky phenomena seen in labs should be playing out at stellar, galactic, and super-cluster scale. If they don't, it needs explanation why not.
Huge props to solar physicists, who confront plasma physics, face-to-face, daily.
Really good book, as far as I got through it. The beginning has a nice derivation of Navier-Stokes, which was mew to me.