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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.

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9 Answers 9

up vote 8 down vote accepted

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.

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When I click on the table of contents link I get: The requested page could not be found. – user29727 Jan 2 '14 at 9:36
Fixed the link. – nibot Jan 5 '14 at 14:33

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:

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.).

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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.

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I am just starting with the book Mechanics: From Newton's Laws to Deterministic-Chaos. It seems to introduce the mathematics later.

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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.

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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).

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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 3 '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 3 '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 3 '10 at 14:19
Arnol'd's book is nice. Along the same lines, but more intimidating, there is Abraham and Marsden, Foundations of Mechanics. – Matt Reece Nov 4 '10 at 22:46

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.

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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 3 '10 at 12:48

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)

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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 3 '10 at 4:56
@crasic The question is explicitly asking for "a book about "advanced" classical mechanics" – Flint72 Apr 13 '14 at 22:57

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

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Thanks, I'll have a look at it. – Cedric H. Nov 3 '10 at 1:56
That's an introductory book, meant for something like an American college sophomore. – Mark Eichenlaub Nov 3 '10 at 4:05

protected by Manishearth Jan 19 '13 at 15:01

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