Hi, this is the author. I've been coming back to this project off and on over the past few years but I often think of these articles as mostly something I'm writing for myself, so I'm really happy to see that some other people might be getting something out of them! I'd definitely love to hear if anyone knows anything I got wrong or can think of a way any particular explanation might be made better.<p>I should also take this chance to mention that I work as a private tutor and I have openings for students! Much more info here: <a href="https://nicf.net/tutoring/" rel="nofollow">https://nicf.net/tutoring/</a>
Spivak (of differential geometry fame) wrote a book with this precise title:<p><a href="https://archive.org/details/physics-for-mathematicians-mechanics-spivak" rel="nofollow">https://archive.org/details/physics-for-mathematicians-mecha...</a><p>It's a very interesting take on classical mechanics.
I’d really love to see a rigorous explanation of renormalization, plus why it doesn’t work for gravity, with no handwaving. As a non-physicist, that has always been the point where I hit a wall on traditional treatments of QFT.
Skimmed some of the articles, particularly those nearer to my field. Seems like a generally good set of informal notes.<p>Random comments:<p>>when the states evolve in time and the observables don’t we are using Liouville’s picture; when the observables evolve in time and the states don’t we are using Hamilton’s picture.<p>I have never heard this terminology, I have only heard Schrodinger's picture vs. Heisenberg's picture.<p>>This means that, very unlike on a Riemannian manifold, a symplectic manifold has no local geometry, so there’s no symplectic analogue of anything like curvature.<p>Perhaps the only enlightening comment I have ever heard about the tautological 1-form/symplectic approach to Hamiltonian mechanics.
I've never gotten a satisfactory explanation of what sort of mathematical object a physical unit (meter, kilo, second etc) is. There are plenty of bones of contention between maths and physics, but this one bothers me the most.<p>Anyone interested in coming at physics from a mathematics perspective should read Arnold's mechanics book.
> The presence of the negative signs in (1) may seem surprising at first, but this is due to the fact that (1) is describing the effect of a passive change of units rather than an active change of the object {x}.<p>This is where the limits of my brain were reached. Is there a translation of this into category theory terms? Is this where category theory could help formalize units in physics?<p>However, his paragraph after that is pretty interesting, which I read as sort of treating units as variables since you couldn't combine them, and he only has length, mass, and time for these examples. But then there's an exponent piece? Okay now I'm lost again.
Okay...I think this might be interesting. I've seen and read a lot of "math for dumb physicists" works, which as a physicist...yeah, I see their point. This could help me understand the math wizards a little better.
For those looking for alternatives, Leonard Suskid's "Theoretical Minimum" books in 2 Volumes are way more accessible and easier to read.
This introduction must assume that the reader already understands physics deeply, right? For instance, the page on Hamiltonian mechanics stated that force is the derivative of momentum with respect to time. I can't imagine how one will understand the intuition behind the definition without having already learned at least college-level physics.
Florian Scheck's textbooks in the Springer Graduate Texts in Physics series could also serve as a bridge. Though very challenging for the non-mathematician I've grown quite fond of "Mechanics: From Newton's laws to Deterministic Chaos" and plan to read the other Scheck books in the series as well.
A bit on a tangent, I have been always curious as to how much of the modern theoretical physics is just math. That is, how little print space the standard treatment of physical theory could be compressed into if all the purely mathematical stuff is assumed known (or delegated to a separate text).
There is always the classical lecture notes by Dolgachev on this. <a href="https://dept.math.lsa.umich.edu/~idolga/physicsbook.pdf" rel="nofollow">https://dept.math.lsa.umich.edu/~idolga/physicsbook.pdf</a>
euler is the last titan of pure raw 'classic' mathematics because gauss was a pretty strong 'theoretical' physicist.<p>how have the mathematical contributions of quantum physics affected mathematics? have they??<p>maybe the field that's really lagging in recognizing the implications of "recent" scientific revolution (QM) is philosophy?<p>finally, I wonder how will the schizm in mathematics that is the IUT (mochizuki's theory) will finally pan out. apparently euler also left stuff behind that took over 70 years to be understood so I ain't holding my breath.