Hi, I will be teaching myself advanced mechanics over the next few weeks. Is it better to start with a modern book using all the new mathematics (manifolds etc.) or with an older one like Goldstein or Landau first?
Goldstein is the perfect mechanics book for someone who is only interested in learning quantum mechanics. Far better, for an actual mechanics text, is Segel's "Mathematics Applied to Continuum Mechanics", or even "The Classical Field Theories" (Encyclopedia of Physics, vol III part 1), or anything by Noll, Truesdell, or that school of thought.
Ok, this depends on how you learn best. I learned this topic simply from college notes and very little advanced math was involved. The derivation of the Euler, Lagrange equations, Hamilton's equations, Poison bracket's, Noether's theorem, etc. can all be explained with just first year's university math. As you can see here, The Hamilton–Jacobi equation is a trivial consequence of Hamilton's equations. So, wouldn't worry and just study the subject. Do plenty of exercises to make sure you really understand the topic at a deep level. If you don't do that, you can have a false sense of understanding.
I used Goldstein in my mechanics class and it seemed to me like it had a lot of modern mathematics. It covered tensors, groups, Lie Groups, etc. I think the most recent edition of Goldstein is pretty new actually. It was very mathematically rigorous IMO and I don't see any reason not to use it just because there might exist more modern math.
I'm not really interested in rigor. I just wanted to know if the newer language of manifolds makes the theory more transparent, and if it was better to start with that instead of older standard books like Goldstein.
OK, then disregard my last post. I've never studied manifolds except in GR and pure mathematics. I guess I am interested in your question also... where are manifolds applied in classical mechanics?
I think they come in when we talk about the phase spaces of mechanical systems. I've been told that Hamiltonian mechanics is deeply connected with the geometry of symplectic manifolds (which is no more than a word to me right now).
Configuration space is a differentiable manifold. The Lagrangian [itex]L \left( q , \dot{q} \right)[/itex] is a real-valued function on the tangent bundle. The generalized coordinate [itex]q[/itex] labels which point in the manifold and the generalized velocities [itex]\dot{q}[/itex] are tangent vectors in the tangent spaces at these points. The Hamiltonian [itex]H \left( q , p \right)[/itex] is a real-valued function on the cotangent bundle. The generalized momenta [itex]p[/itex] are covectors in the cotangent spaces.
In my opinion, some attention to the underlying geometrical structures makes the subject more digestible. Certainly, you can go overboard with abstractions and rigor and not see how to do a calculation. Introduction to Analytical Dynamics (by N. M. J. Woodhouse) is a nice book. (Goldstein was my first advanced mechanics text... and I didn't really like it.)
http://www.amazon.ca/Introduction-Analytical-Dynamics-N-Woodhouse/dp/0198531982 says: "This book is an introduction to Lagrangian and Hamiltonian mechanics primarily for mathematics undergraduates. Although the approach is traditional and coordinate based, it incorporates some of the insights and new perspectives of modern geometric treatments of mechanics. The book is intended for advanced undergraduates or graduate students and assumes familiarity with linear algebra, the chain rule for partial derivatives, and (to a lesser extent) three-dimensional vector mechanics. The aims are to give a confident understanding of the chain of argument that leads from Newton's laws through Lagrange's equations and Hamilton's principle to Hamilton's equations and canonical transformations; to confront head-on the points that mathematicians in particular find most awkward and confusing; to give practice in problem solving; and to elucidate the techniques that will reappear in later courses on relativity and quantum theory." http://books.google.com/books?id=S0MsGQAACAAJ&dq="0198531974" http://www.gamca.sk/~kubo/doc/notes/mechanics.pdf http://www.worldcat.org/wcpa/oclc/13861051 provides the TOC: 1. Frames of Reference 2. Lagrangian Mechanics 3. Rigid Bodies 4. Hamiltonian Mechanics 5. Impulses 6. Oscillations Notes Index (I don't have easy access to my copy right now.) see also: https://www.physicsforums.com/showthread.php?t=176933
Opinions on Arnold's texts? V.I. Arnold seems to relate mechanics to differential geometry alot? I haven't read the books yet. Anyone?