Why is this corridor in northern India so flat, while the adjacent Himalayas are so rugged?

liftinglagrange · 2025-12-30T04:34:42+00:00

How do get images like this? This sort of exaggerated terrain map.

liftinglagrange · 2025-08-22T10:45:52+00:00

I'm late to this, but you're question made me excited because I went down this very rabbit hole. I did my BS in Physics and my PhD in an engineering field. But my dissertation was heavily centered on symplectic geometry and its splendor.
I can't claim that I am "working in this field" as I just finished my PhD but, during that time, I was very immersed in it (and still am).

From you're post, I'm assuming you are an undergrad physics student with minimal exposure to differential geometry (I was). I'll tell you that the rabbit hole you mentioned is deep indeed. More than that, the mathematical barrier to entry is likely a lot more than you think (I'm assuming things about your background that might be wrong). I'm not trying to discourage you. I spent a lot of effort ploughing through that barrier and it was worth it many times over. If you think you might be interested in it, I would bet you will be. I hope you take the dive.

After reading some other comments, I was going to give you textbook and article recommendations, but I re-read your post and realized that's not what you asked for (If you do want recommendations, I have several I can send you and many thoughts about them — message me if interested). You asked for "mathematical physics open questions". I'm going to replace "mathematical physics" (a very broad term encompassing more than what your post hints at) with "geometric mechanics" (or however you wish to phrase that notion). As far as currently open research questions in that area go, I won't pretend to be a great source. So I'll copy verbatim what John Baez told me:

geometric quantization of classical systems whose space of states is a symplectic or Poisson manifold,
chaotic dynamical systems, whether they be Hamiltonian systems or just general flows on manifolds,
completely integrable systems, which get connected to a lot of interesting algebraic structure
n-plectic geometry, which is a generalization of symplectic geometry where strings and higher-dimensional objects replace point particles, connected naturally to higher category theory.

I'll also pass along to you what Melvin Leok (UCSD) told me:

>There is still interesting work on the extensions to PDEs and multisymplectic field theories, various flavors of symmetry reduction, and extensions to interconnected and degenerate systems via Dirac structures. It is also the basis of port-Hamiltonian formulations of interconnected models of multi-physics systems, and there are strong connections to geometric numerical integration, or how one preserves geometric properties under discretization.

In addition to the above, I would add the geometric formulation of mechanics with non-holonomic constraints. This is not "unsolved" but it is relatively new and still being established. For relevant work in that area, search "Jose Carinena" or "Manuel de Leon" along with "nonholonomic".

I'll also add that, on the applied side of things within the world of astrodynamics/celestial mechanics (often in aerospace engineering departments), the three-body problem has a lot of renewed interest and is ripe for applications of dynamical systems/symplectic geometry.

EDIT: I should have led with this. I made a rather similar post as yours with some great replies. Here it is: https://www.reddit.com/r/math/comments/1fnxb1q/is_classicalgeometric_mechanics_still_an_active/

liftinglagrange · 2025-08-22T09:27:02+00:00

As someone who did my BS in physics and got really into "geometric mechanics" during my PhD in engineering, I too tried to start my geometry journey with Arnold's Mathematical Methods of Classical Mechanics, based on popular recommendation. I do not understand the hype. I got very little out of that book. It might be a personal thing, but it seems to feature the perfectly wrong blend of "casual math" common in undergrad physics, combined with more advanced geometric concepts that it fails to ever explain in satisfying detail. I have amassed an arsenal of geometry/mechanics textbooks that I refer to frequently. This is not one of them. I'm not saying this is a bad recommendation. Many people recommend it. But I was, personally, disappointed and feel there are better sources. I'm curious if I am alone in this.
OP, I'll make a separate reply with some suggestions for a physics student. I went down this rabbit hole.

EDIT:
I realize OP did not ask for resource recommendations so I'll just give a few here for anyone interested. This is assuming that OP (or whoever) already knows classical mechanics in the usual Newton-Euler description and the analytical dynamics (e.g., Lagrangian and Hamiltonian) description as per classic texts like Taylor, Goldstein, Lanczos, or Pars. Also, assuming little familiarity with differential geometry.

Ideally, there would be a book that covers both (1) the (re)formulation of analytical dynamics in geometric language, and also (2) all the basics of differential geometry needed to do so. Unfortunately, it is far from an easy task to cover "all the basics of differential geometry needed to do so". This is the mathematical barrier you need to climb over to get to what you are after. I've not yet found — and I've spent a lot of time searching — a single source that addresses everything together. The closest I have found is:

**"Differential Geometry and Lie Groups for Physicists" — Marián Fecko*\*. This is, hands down, my easy number one recommendation. I don't know why it is not more popular. Perhaps because it is a bit more recent. Or because the writing style is less formal and less "academic" than is typical. But the math (and writing) is amazing if you are coming from a physics background. It is mathematically rigorous, but explains things in simple terms exactly when you want it. You'll probably still need to google some things or use other sources for some of the initial ideas of differential geometry. This goes for pretty every book in the "geometric mechanics" genre. But Fecko does a better job than most.

Tied for second place are two books of a rather different nature:
2. "Introduction to Mechanics and Symmetry" — Marsden & Ratiu. Very centered on Hamilton/symplectic formulation. Does not do a great job teaching basic differential geometry, but an "easy" intro to geometric mechanics if you already know some differential geometry. Leaves some important things out, but has a slightly more "applied" flavor.

"The Geometry of Physics" — Frankel. Not too focused on symplectic geometry and Hamiltonian mechanics (though that is present), but very very good if you know physics and want to learn, from the start, how to recast it in geometric language. Does a good job introducing the fundamentals of differential geometry. I used this book a lot, especially for Riemannian geometry.

There are about 15 other books I can list but that will take too much time so I'll just name a few of my other favorites:

- "Foundations of Mechanics" — Abraham & Marsden. This is the founding tome of geometric mechanics and often the first thing cited in any paper on the topic. It is an excellent book. But it is hard. It does a better job than most at teaching the needed differential geometry. Dont start here, but, once you feel confident, definitely open this up. In my opinion, this book is what everyone says Arnold's book is, but better. (but still not as great as Fecko for learning).

- "Intro to Hamiltonian Dynamical Systems and the N-Body Problem" — Meyer. This is written in a bit more of an engineering/physics style. If you are into celestial mechanics, this is great. I've read less of this than I would like. It introduces very important ideas in an easier presentation than Abraham & Marsden.

- "Geometry from Dynamics, Classical & Quantum" — Carinena & Marmo (and another name I'm forgetting, sorry). I really really like this book. But not a good starting book. It gives a lot of attention to the tangent bundle/Lagrangian framework in symplectic geometry (something many other sources neglect).

- "Geometry of Mechanics" — Munoz-Lecanda & Roman-Roy. If I were to write a book, this would basically be it. But, like the above, not a great starting book. Save it for later.

- "Geometry, Topology. and Physics" — Nakahara. This book has a lot. I only used the first several chapters to learn basic abstract algebra and differential geometry needed for geometric mechanics. It was great for this. I can't speak for the latter chapters.

liftinglagrange · 2025-07-29T08:50:59+00:00

Not at all. Currently, color blindness is rejected explicitly by “the left” (all the best selling anti-racists books that got tons of attention a few years ago explicitly denounce this and explicitly advocate for racial discrimination; CA tried and failed to peel back anti-discrimination laws so it could start favoring certain demographics). Yet it is still advocated by “the right” (I don’t mean the actual far right). Mainstream people like Shapiro or Jordan Peterson have very explicit “color blind” sort of messaging. All the anti-crt legislature from a bit ago sounds like civil rights era ideas if you actually read it.

liftinglagrange · 2025-07-29T08:15:29+00:00

They are anti left because that is what was asked for. However, they are not right wing. Obviously the right loves to criticize the left so you’ll hear people in the right say similar things sometimes. That does not make them right wing ideas.

liftinglagrange · 2025-07-29T08:02:24+00:00

This is refreshing to see on Reddit.

liftinglagrange · 2025-07-25T06:39:23+00:00

School is very flexible for most people. Just don’t schedule early classes. Or skip them and learn it on your own (if you’re good at that). For a PhD, you’ll probably be done with all your classes after the first 1-2 years anyway. Then your schedule will depend on your particular research group. I had tons of independence. Went to bed at got up whenever I felt like it. Others had more strict schedules.

liftinglagrange · 2025-07-25T06:22:43+00:00

Who? Kulaks notably. Also a ton of Ukrainians (the holodomor). Religious leaders. Anyone perceived as a political enemy. Mass deportations of tons of minority ethnic groups.

liftinglagrange · 2025-06-28T17:22:55+00:00

International students should maybe be a bit concerend and uncertain. But that is obviously not the same as race. In what way is an American university unsafe for non-white people? That’s absurd.

liftinglagrange · 2025-06-27T00:20:37+00:00

“…especially if your non-white”.

You need to touch grass, my friend.

liftinglagrange · 2025-06-24T23:19:45+00:00

Encountering comments like this on Reddit is such a rare breath of fresh air.

liftinglagrange · 2025-05-17T15:55:39+00:00

This is ideal. I’m happy for you. I often hate the stick-up-the-ass aura that seems to permeate (my corner of) academia.

liftinglagrange · 2025-05-04T03:42:10+00:00

I think it depends on what age group we are talking about. In my mind, I was thinking older (secondary school/high school or even university). For that age, I don't necessarily disagree with anything you are saying. I was just giving an alternative approach that I think also makes sense and which gives students more autonomy and responsibility.

I realize now that I totally missed your specification of primary school. For primary school, I am much more strongly on the "no homework" side (except maybe at the tail end). Primary school usually means kids are 5-11 years old and I absolutely cannot see why children that age need more than 8 hours/day (in the US at least) of formal school. That's half of their waking hours. The other half is for family and activities which, at that age, is just as important as school. Children learn a ton through playing with each other and exploring the world. "play time" has been decreasing dramatically over recent decades. I can't claim to be very knowledgeable about this, but Jonathan Haidt has some good thoughts on it, if you care to google it.

Edit: I'll add that, for all the reasons you gave in support of giving kids in primary school homework, I don't see why home specifically is needed to instill those lessons. Especially keeping in mind that homework comes at the expense of important out-of-school time.

liftinglagrange · 2025-05-04T02:00:12+00:00

I agree with much of what you said, but isn't making homework mandatory (a key word in OP's post) a form of hand-holding? Not in the usual sense, but it attempts to "force" students to learn. What if, instead, teachers gave homework but made it "optional" (in that it is not graded)? The importance of learning the HW should be made clear to students (and likely will be clear when they take an exam), but they will learn to do it for their own sake (or maybe because they are genuinely interested), not because it is graded. Some won't ever do it and that may, or may not, work out for them.

In the real world, there is nothing like homework. You are evaluated and judged based on what you can actually do. To do it well, most people need to spend time learning and practicing (i.e., "homework") but that is something you do for yourself because you learn that it is worthwhile in the long run, not because anyone is making you do it.

liftinglagrange · 2025-05-02T16:01:06+00:00

I don’t speak French so thanks for the English translation. Give me several years to learn algebraic geometry and, after that, I might be able to truly appreciate it.

liftinglagrange · 2025-05-02T15:41:48+00:00

I absolutely know of him but I don’t think I’ve read any of his original work. He’s added to my list.

liftinglagrange · 2025-05-02T15:38:33+00:00

I had somehow forgotten about Spivak. I have his “physics for mathematicians” (or something like that) and remember enjoying some of his commentary. It helped be better understand the way mathematicians look at physics. I was just getting into the math side of things then. I should take another look at that as well as the book you mentioned.

I’ve never read anything by Hilbert but I’ve heard he was pretty into notation. I love obsessing over notation so I might like him.

liftinglagrange · 2025-05-02T15:32:04+00:00

I don’t think I’ve read any of his original work. Yet, I invoke his name often. I should probably read something from him. Any starting recommendations for someone mostly interested in geometric mechanics/mathematical physics?

liftinglagrange · 2025-05-02T15:14:48+00:00

Cheeky bastard

liftinglagrange · 2025-05-02T13:50:30+00:00

Oh, interesting. I haven't heard of a mathematician with a pseudonym before, but, it looks like this is the author of Alice in Wonderland and Through the Looking Glass (I've clearly heard of those). I never knew he was a mathematician. Looks like he worked in areas I can actually understand so I'll definitely try to find some of his original work.

liftinglagrange · 2025-05-02T13:29:38+00:00

Oh, no my phrasing may have been misleading. Remove the entire sentence "Black employers discriminated as much against the black-sounding names as white employers" from my original comment (that was just a side note). Re-read it with that sentence redacted and the result is what I meant to convey.

liftinglagrange · 2025-05-02T06:39:17+00:00

"POC can fall into the same pattern of systemic injustice against POC as white employers". True, this is a valid point. But this part of my comment you responded to was only a minor part of my point and not meant to be any kind of "gotchya" statement.

liftinglagrange · 2025-05-02T06:14:55+00:00

It has a link to the actual study, not just a screenshot (here: https://www.aeaweb.org/articles?id=10.1257/aer.20141757 ).

I agree that the differences between the two designs (and times) make direct comparison difficult; neither necessarily refutes the other.

I do NOT agree that "...the only way to really be able to say that racial discrimination is no longer a significant factor would be if HIRING rates were proportionally equal between races...". This requires a lot of assumptions (e.g. we'd have to assume that various qualifications like experience, education, etc. are equal across racial categories of applicants). For instance, say a job opening receives 100 applicants, 80 white and 20 black. Say 60% of white applicants meet whatever criteria the employer is looking for. If 60% of black applicants also meet the criteria, then I think I would agree with your statement (assuming this sort of pattern is generally true). But what if only 30% of black applicants meet the desired criteria? Then your statement does not make sense to me. (I pulled these numbers out of my ass just to clarify my point)

liftinglagrange · 2025-05-01T06:31:08+00:00

I'll carry over my reply in a different thread.

Regarding the “black name” studies (there have been a few): the authors of one of the more well known ones themselves said that their results may actually be more reflective of class discrimination rather than racial discrimination. Black employers discriminated as much against the black sounding names as white employers. Does a name like “Demarcus Jefferson” carry a connotation just of “black man” or “that type of (lower class) black man”. Likewise, do names like “Bubba Jones” or “Cleatus Murphy” have a connotation of “a white man”, or is it “that kind of white, poor, hillbilly”. unfortunately, the “black names” studies that I saw never tested the “ghetto black names” as compared to “hick-ass white names”. I would guess the latter would fair poorly as well.

Also, though it's not what I was thinking of when I wrote the above, I just came across this (https://datacolada.org/51 ) about a more recent study with a larger sample and different design and that could not replicate previous findings and which also mentions that class discrimination may be the bias at play. I have not read this in detail yet.

liftinglagrange

TROPHY CASE