[HN Gopher] Physics for Mathematicians - Introduction
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Physics for Mathematicians - Introduction
Author : irsagent
Score : 95 points
Date : 2024-02-08 18:52 UTC (4 hours ago)
(HTM) web link (nicf.net)
(TXT) w3m dump (nicf.net)
| sydbarrett74 wrote:
| I am neither a physicist nor a mathematician, but this looks like
| an awesome undertaking that will benefit both communities! :)
| tippytippytango wrote:
| This is really cool. Can also serve as a Rosetta Stone for
| physicists wanting to better understand the language of
| mathematicians.
| Y_Y wrote:
| 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.
|
| Anyone interested in coming at physics from a mathematics
| perspective should read Arnold's mechanics book.
| cycomanic wrote:
| Why is the measure not a satisfactory answer?
|
| https://en.m.wikipedia.org/wiki/Measure_(mathematics)
| Koshkin wrote:
| Unfortunately, even though it is said to be a
| "generalization" of these things, mathematical measure theory
| has nothing to do with physical units of measure or
| dimensional analysis.
| antognini wrote:
| Terence Tao wrote a nice blog post about this:
| https://terrytao.wordpress.com/2012/12/29/a-mathematical-for...
| dang wrote:
| A couple of past discussions:
|
| _A mathematical formalization of dimensional analysis
| (2012)_ - https://news.ycombinator.com/item?id=37517118 -
| Sept 2023 (54 comments)
|
| _A mathematical formalisation of dimensional analysis_ -
| https://news.ycombinator.com/item?id=5018357 - Jan 2013 (19
| comments)
| random3 wrote:
| One concern is that measures, out of the box, have issues in 3+
| dimensions. Concretely due to paradoxes such as Banach-Tarski,
| that arise from the Zermelo Fraenkel (ZF) + Axiom of Choice
| (AC) = ZFC axiomatic formulation for set theory.
|
| Since things need to conserve in pyhsics, one has to account
| for this issue and doing so is harder than it may seem as AC is
| part of the "fabric" of most mathematics which, at large,
| chooses to ignore the problem.
| nyrikki wrote:
| IMHO that is the result of Gibbs style vectors and the cross
| product only being validated in R^3
|
| Lie groups and geometric algebra remove a lot of problems.
|
| It also applies to differential calculus and ML methods like
| back propagation and gradient decent.
|
| Gibbs style vectors and the cross are convenient as they tend
| to match our visual intuitions.
|
| But lots of the 'physics isn't real math' claims just don't
| understand how the algebra arises from the system.
| nyrikki wrote:
| A meter is a displacement vector with a basis vectorthe length
| of the path travelled by light in a vacuum during a time
| interval of 1/299,792,458 of a second.
|
| In physics the length of the basis vector is set to 1 if
| possible which is called 'natural units'
|
| But the SI system is the domain of Metrology, not physics.
| 4ad wrote:
| Meters, seconds, joules, etc, are torsors.
| ysofunny wrote:
| euler is the last titan of pure raw 'classic' mathematics because
| gauss was a pretty strong 'theoretical' physicist.
|
| how have the mathematical contributions of quantum physics
| affected mathematics? have they??
|
| maybe the field that's really lagging in recognizing the
| implications of "recent" scientific revolution (QM) is
| philosophy?
|
| 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.
| rck wrote:
| Spivak (of differential geometry fame) wrote a book with this
| precise title:
|
| https://archive.org/details/physics-for-mathematicians-mecha...
|
| It's a very interesting take on classical mechanics.
| nicf wrote:
| I own a copy of that book, and I also highly recommend it! (The
| full title is "Physics for Mathematicians: Mechanics I", but
| sadly we're now never going to get a "Mechanics II".) It has a
| very different goal than my notes --- he's more interested in
| building up classical mechanics very, very carefully from first
| principles --- but it's a very fun journey if you have the time
| to spend on it.
| scionthefly wrote:
| 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.
| nomemory wrote:
| I was more math oriented during my studies, and I hated physics
| (couldn't openly admit that). I still don't get a lot of the
| physics I was taught, but I did juggle my way out of it using
| math, learning some formulas and getting a passing grade. Deep
| inside I admire physicist more, because for them the things
| that never clicked for me are natural.
| paulpauper wrote:
| imho, high-level physics is harder than pure math. With math
| you can specialize and focus on some formulas or areas of
| interest, but this is not really possible with physics. With
| physics you have to know all the areas of math very well--
| group theory, differential equations, differential geometry,
| etc. You have to have know all the math well and all the
| physics from Maxwell and beyond. It's just much more material
| involved. To be on the frontier of physics is essentially
| pure math, plus hundreds of years of physics.
| michaelrpeskin wrote:
| Computational Physics Ph.D. here...I don't know about that.
| I have written lots of code (not just using off-the-shelf
| packages) to solve Hamiltonian mechanics and Quantum
| reactive scattering. OMG, I spent about 30 minutes going
| through the Hamiltonian mechanics chapter from the point of
| view of a mathematician and I got lost about half way
| through. I feel like in my fairly long career I learned
| just enough of the math to make it work, but don't really
| understand the math at a fundamental level like I do the
| physics.
| max_ wrote:
| For those looking for alternatives, Leonard Suskid's "Theoretical
| Minimum" books in 2 Volumes are way more accessible and easier to
| read.
| paulpauper wrote:
| but those are way more superficial though
| seydar wrote:
| > 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}.
|
| 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?
|
| 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.
| dawnofdusk wrote:
| >Is there a translation of this into category theory terms?
|
| It's essentially the same as the relation between covariance
| and contravariance in category theory.
| mjburgess wrote:
| where on earth is this quote from?
| dawnofdusk wrote:
| Skimmed some of the articles, particularly those nearer to my
| field. Seems like a generally good set of informal notes.
|
| Random comments:
|
| >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.
|
| I have never heard this terminology, I have only heard
| Schrodinger's picture vs. Heisenberg's picture.
|
| >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.
|
| Perhaps the only enlightening comment I have ever heard about the
| tautological 1-form/symplectic approach to Hamiltonian mechanics.
| nicf wrote:
| > I have never heard this terminology, I have only heard
| Schrodinger's picture vs. Heisenberg's picture.
|
| I wrote the QM article a very long time ago at this point, and
| I actually can't reconstruct at the moment why I used those two
| names! I've also heard Schrodinger and Heisenberg much more
| frequently. Might be worth an edit.
| paulpauper wrote:
| This seems way too advanced for an intro. imho you'd be better
| off with textbooks. this assumes you are very strong in math
| nicf wrote:
| 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.
|
| 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: https://nicf.net/tutoring/
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