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Learning Advanced Mechanics 
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#1
Jun1708, 09:26 AM

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PF Gold
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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? 


#2
Jun1708, 09:33 AM

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What are the topics you are planning to learn? 


#3
Jun1708, 09:42 AM

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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. 


#4
Jun1708, 09:45 AM

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PF Gold
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Learning Advanced Mechanics



#5
Jun1708, 10:13 AM

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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. 


#6
Jun1708, 11:43 AM

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Jun1708, 12:22 PM

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#8
Jun1708, 12:32 PM

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PF Gold
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#9
Jun1708, 12:38 PM

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#10
Jun1708, 12:54 PM

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#11
Jun1708, 01:24 PM

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The Lagrangian [itex]L \left( q , \dot{q} \right)[/itex] is a realvalued 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 realvalued function on the cotangent bundle. The generalized momenta [itex]p[/itex] are covectors in the cotangent spaces. 


#12
Jun1708, 01:49 PM

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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.) 


#13
Jun1708, 02:19 PM

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Hi robphy,
Could you tell me a little more about Woodhouse's book? What are the things it covers? 


#14
Jun1708, 03:26 PM

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"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) threedimensional 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 headon 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=S0M...20198531974%22 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: http://www.physicsforums.com/showthread.php?t=176933 


#15
Jun1708, 10:17 PM

P: 52

V.I. Arnold seems to relate mechanics to differential geometry alot? I haven't read the books yet. Anyone?



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