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What is a good book for Analytical Mechanics?

To be more specific, I would prefer a book that:

  • Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis on obscure definitions such as "virtual displacements").
  • However, it must not assume graduate-level mathematics such as differential geometry.

Does this kind of thing exist?

R S
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  • Possible duplicates: https://physics.stackexchange.com/q/111/2451 and https://physics.stackexchange.com/q/1601/2451 – Qmechanic Jan 08 '13 at 15:26
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    Spivak's Physics for Mathematicians, volume I: Mechanics sounds exactly like what you're after! – Alex Nelson Aug 21 '13 at 21:59
  • Check out Schaum's Theory and Problems of Theoretical Mechanics. It might not do for a comprehensive book, but it certainly does not waste any time. – Mikael Kuisma Aug 12 '16 at 21:42
  • If you really think that Landau & Lifshitz (LL) vol 1 'deals with diff. geometry' (as you say in a comment to aignas's answer), then no, the kind of book you are looking for doesn't exist. LL only use what used to be called 'advanced calculus', and it is not possible to do 'analytical mechanics' with less math than that. (Mind you, I would not recommend LL as a first book on analytical mechanics. OTH, Spivak's book mentioned above is excellent, though you'll have to stop with Lagrangian mech. if you really want to avoid diff. geometry, which does get used in the chapter on Hamiltonian mech.) – linguisticturn Jun 26 '20 at 20:06

2 Answers2

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Community
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Emilio Pisanty
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Why not try Landau and Lifshitz Volume 1 on Classical Mechanics? It's a very good, short and dense text which is time tested and wonderfully written.

Ar do you want something different?

aignas
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  • This book explicitly deals with differential geometry, which I would like to avoid. – R S Jan 09 '13 at 10:59
  • Well not all the chapters. However, you can skip the differential geometry bits and then come back later on, when you feel like doing differential geometry. This will be a very useful book in the future if you plan to study physics later on. – aignas Jan 15 '13 at 13:13
  • @RS One can say many things about Landau and Lifshitz (LL), but that they write 'with high mathematics precision' (as the OP requests) is definitely not one of them. The tone is set in Eq. (4.3), where they write v^2 = (dl/dt)^2=(dl)^2/(dt)^2. I am not sure if this can be made mathematically precise even using hyperreal numbers. – linguisticturn Jun 26 '20 at 13:12
  • More nontrivially, many derivations use non-obvious tricks that LL don't really justify or explain how general they are; you just sort of have to 'assimilate' such reasoning and, in time, hopefully become adept at it. A paradigmatic example of all that is their treatment of the Mathieu equation, (27.8). Finally, numerous moderately nonobvious statements are just asserted without explanation (a hallmark of LL). The LL texts are definitely a treasure trove of insight, but, IMHO, one that is best approached only after first learning much of the material elsewhere. – linguisticturn Jun 26 '20 at 13:12
  • At the same time, if one really thinks that Landau and Lifshitz vol.1 'deals with differential geometry', then the answer to OP's question is that there simply isn't a book of the sort OP is asking for, nor could there ever be. (In fact, the mathematics in LL vol. 1 never goes beyond undergraduate mathematics: what used to be called 'advanced calculus', linear algebra, and differential equations, with a tiny bit of complex analysis.) – linguisticturn Jun 26 '20 at 19:37
  • @RS, isn't contradictory to seek a highly formalistic book on analytical mechanics and at the same time wanting to avoid differential geometry? I have the impression that the formalism of analytical mechanics is precisely based on differential topology and differential geometry. – Albert May 30 '22 at 10:47