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I'm a mathematician that would like to read some books of physics. However, trying to read some texts I'm getting confused many times because the lack of mathematical rigor, at least at a formal level.

I am thus looking for rigorous differential geometric treatments of physical topics, wherever applicable. The only ones that I've found are the books/notes of Leon A. Takhtajan, but I would like to know if the are some more books at a similar level of rigour. In particular, I would like some recommendations on rigorous differential geometric treatments of classical mechanics.

Song of Physics
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Masacroso
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    it really depends on what you're going for here. For example, there's the classic tome by Abraham and Marsden on classical mechanics, Hawking Ellis or Wald for GR, just to name some. How much detail are you looking for? Also, there are some differential geometry books which treat applications to physics, for example the book by W.D.Curtis on differential geometry and theoretical physics. There's also a pretty recent text by Godinho, Natario on Riemannian geometry and applications to classical mechanics and some SR and GR. – peek-a-boo Jul 02 '22 at 12:50
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    Possible duplicates: https://physics.stackexchange.com/q/1601/2451 , https://physics.stackexchange.com/q/15002/2451 and links therein. – Qmechanic Jul 02 '22 at 14:23
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    I actually like such questions (and am very sympathetic because I've been in this exact situation), but here the question is pretty broad. Classical/Relativistic mechanics is a pretty large area to seek suggestions for (it's very roughly $1/4$-$1/3$ of an undergraduate physics degree), and there are similar questions as linked above (which has some good suggestions). To make the question better, I would suggest including your physics background, and some of the texts you've read (so people know what not to recommend), and what kind of text you're looking for... – peek-a-boo Jul 02 '22 at 14:41
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    e.g. do you want a harsh monograph type text (Abraham Marsden), or a more self-study type of text which introduces the basics/textbook type questions in the diff geo formalism (e.g the Godinho, Natario text above, where you learn mostly by doing the exercises... kind of like Spivak's CoM; another text for students is Bamberg&Sternberg). Are you looking for certain topics in specific? etc etc I have come across several texts (some I've read carefully, some in passing, but each clarifying a certain aspect of a topic. so at this level, it's pretty broad I feel). – peek-a-boo Jul 02 '22 at 14:44
  • ok, I specified more, just restricted to classical mechanics. I would like to understand the notions of e.g. classical mechanics from the point of view of differential geometry. Then, by example, variations can be seen as paths in an infinite dimensional manifold – Masacroso Jul 02 '22 at 18:12
  • If you are learning physics for the first time, I think you should start with less mathematical rigorous textbooks like Halliday, Feynman Lectures to get the intuition and practical usage. Trying to do physics without a good deal of intuition is almost always a bad idea. – Socrates Jul 14 '22 at 18:03
  • Similar question at Math SE. – Massimo Ortolano Jul 16 '22 at 21:04

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Perhaps one of the nicest differential geometric accounts on classical mechanics, written by a mathematician and for mathematicians is,

It contains several historical and philosophical nuggets that would also give the working mathematician a rough idea about how physicists think about and approach a problem. Some other great books are,

Hope these books can be of help to you :)

Song of Physics
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