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I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.).

I have this book from Saletan and Jose, but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics.

Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing.

Cedric H.
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9 Answers9

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I can't believe nobody's mentioned Arnol'd's book "Mathematical Methods for Classical Mechanics" - it covers everything you ask for in the first paragraph quite elegantly (though sometimes somewhat tersely).

j.c.
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    From the mathematical perspective this is obvious, but I was slightly afraid that physicists view this book differently. +1 – BBischof Nov 03 '10 at 14:10
  • I find it hard to read, I was looking for something intermediate that would allow me to benefit from Arnold's book more easily. – Cedric H. Nov 03 '10 at 14:13
  • Unfortunately, most working physicists don't spend a lot of time thinking in terms of symplectic geometry, so you'll most likely only find such books in the math literature. I'd love to be corrected though! Perhaps some introductory books on symplectic geometry may cover applications to mechanics... – j.c. Nov 03 '10 at 14:19
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    Arnol'd's book is nice. Along the same lines, but more intimidating, there is Abraham and Marsden, Foundations of Mechanics. – Matt Reece Nov 04 '10 at 22:46
  • I am here to know if I have the prerequisites to start learning from Arnold. The following is what I have a good understanding of. Classical mechanics at the level of Kleppner and Kolenkow, calculus at the level of G.B. Thomas( 12th ed. ), Linear Algebra, high school math and classical electricity and magnetism at the level of University Physics. I am interested to take Differential equations( beyond what Thomas offered ), Complex analysis, Electrodynamics by Griffith and Classical mechanics by Arnold for the next 6 months. – R004 Sep 07 '17 at 05:10
  • @roof I'm not sure the comments to this question are the best place to ask, but I would suggest just taking a look at it and seeing how much you understand. I suspect you may need to study a bit more mathematics (perhaps calculus on manifolds) before you can get everything out of Arnold's book, but it never hurts to try to push yourself to understand something above your head (especially if you're thinking about doing research someday). – j.c. Sep 08 '17 at 00:20
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Structure and Interpretation of Classical Mechanics (table of contents) certainly deserves mention. It might not have as much differential geometry as you'd like, though they have a followup article titled Functional Differential Geometry.

nibot
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My favourite for pure classical mechanics is generally the book by Goldstein which includes the Lagrangian and Hamiltonian methods (although I'm not sure about symplectic geometrical and mathematical foundations).

If you want a firm look at curved spacetime manifolds (for vectors, one forms, tangent bundles etc.) I would recommend Carroll's Spacetime and Geometry but it deals with the mathematical backing for General Relativity, which is classical mechanics, but on a curved spacetime.

Heather
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  • Yes, Goldstein is great, but I am looking for more details on symplectig geometry. I'll have a look at the other one. – Cedric H. Nov 03 '10 at 12:48
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Pure beaten gold, any edition, paperback. It never leaves you. Lagrangian approach.
See the reviews on Amazon L D Landau (Author), E.M. Lifshitz (Author) http://www.amazon.com/Mechanics-Third-Course-Theoretical-Physics/dp/0750628960

sigoldberg1
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    I find that landau is a much better book to study from than a book to learn from. He has lots of very nice examples that are worked out in exquisite detail, but for a first look at the material it is much too... dry. – crasic Nov 03 '10 at 04:56
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    @crasic The question is explicitly asking for "a book about "advanced" classical mechanics" – Flint72 Apr 13 '14 at 22:57
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As already mentioned the standard introductory books in hamiltonian geometrical (point-) mechanics are Foundations of Mechanics by Abraham and Marsden and Arnolds Mathematical Methods of Classical Mechanics. Another standard book is: "Classical Mathematical Physics" by Walter Thirring.

You may also have a look at "Symmetry in mechanics: a gentle, modern introduction" by Stefanie Frank Singer, which bridges the gap between standard college courses on classical point mechanics and books like those mentioned above.

Another interesting book for geometrical hamilton mechanics is "Introduction to Symmetry and Mechanics" by J. Marsden, which gives a nice introductory overview to that topic.

For more examples from a geometrical point of view you may also consult "Global aspects of classical integrable systems" by Cushman and Bates.

One sould also mention "Mathematical Aspects of Classical and Celestical Mechanics" by Arnold, Kozlov and Neishtadt.

For a less advanced (and less rigorous approach) with very much examples you may have a look at the german book "Klassische Mechanik" by F. Kuypers.

student
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This isn't explicitly about just mechanics as it tries to hit a lot of different areas in physics but it covers the material you are asking for:

http://www.amazon.com/Differential-Forms-Applications-Physical-Sciences/dp/0486661695

It will absolutely provide you with a firm basis in the geometrical considerations you mentioned in your question (tangent bundle, cotangent bundle, 1-form, 2-form, etc.).

unclejamil
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Although not specifically answering the needs of Cedric H. (symplectic geometry) I find Introduction to Dynamics by Percival & Richards one of the best (and simplest) introductions to Lagrangian and Hamiltonian dynamics, particularly canonical transformations and so on. I point out this book because it is probably not so well known.

perplexity
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Maybe no symplectic geometry or forms here, but this book has a LOT to offer:

http://www.people.fas.harvard.edu/~djmorin/book.html

Eric Zaslow
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Classical Mechanics: Systems of Particles and Hamiltonian Dynamics by Walter Greiner.

This is a very good book for the same reasons that all the books belonging to the series of books written by Greiner are good. They are clear, they do not shy away from mathematics (they are written for people who want to pursue theoretical physics) and they have many examples.

TheQuantumMan
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