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Action Principles in Continuum Mechanics?

  1. Jul 5, 2014 #1
    Is there any book that does what Landau does in Fluid Mechanics and Theory of Elasticity, only using a Lagrangian/Action-principles the whole way through?

    I can really only find brief tiny descriptions like this one in books on other topics, is there nothing that does for fluids/elasticity like what Landau does for mechanics and em?
     
  2. jcsd
  3. Aug 14, 2014 #2
    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
     
  4. Aug 15, 2014 #3

    vanhees71

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    The only source I know is

    A. Sommerfeld, Lectures on Theoretical Physics

    It's anyway a marvelous theoretical-physics book series (6 volumes). Particularly vol. 6 about partial differential equations is great.

    I once worked this out for myself for a student seminar on hydrodynamics. Unfortunately I have this only in German. Perhaps even this helps a bit, because there are many formulae.

    http://theory.gsi.de/~vanhees/faq/hydro/hydro.html
    http://theory.gsi.de/~vanhees/faq-pdf/hydro.pdf
     
  5. Aug 19, 2014 #4
    I'm very appreciative of your notes, thank you. I will use google translate to refer to them when I sit down to do this properly. I looked in Sommerfeld and from memory found Lanczos gives a better presentation of the same material, but I'll check it again.

    Regarding fluid mechanics, my main concern is that one apparently it's been proven that can't derive Navier-Stokes or any viscous fluid dynamics from an action principle (i.e. friction, a velocity dependent potential), so at best we can do chapter 1 of Landau via action principles + Noether's theorem and no more... However this book

    www.amazon.com/Hamilton-Type-Principle-Fluid-Dynamics-Magnetohydrodynamics/dp/3211249648/[/URL]

    claims to do it, and apparently explains the flaw in the old approach, page 16 - 17:

    [url]http://books.google.ie/books?id=ONGsaXO1VToC&lpg=PA18&ots=3Le8YeDRaS&dq=Angel%20Fierros%20Palacios&pg=PA16#v=onepage&q=Angel%20Fierros%20Palacios&f=false[/url]

    While it might be cranky, the only review I can find

    [url]http://onlinelibrary.wiley.com/doi/10.1002/andp.200610224/abstract[/url]

    gives the book a bad review but doesn't even mention this important issue, it gives out about the book for absolutely ridiculous reasons tbh so it's not a credible source.

    The other approach is differential forms, which apparently can derive Navier-Stokes nice enough, just have to find a nice presentation.

    Anybody have any thoughts on these, it's be great to read them.
     
    Last edited by a moderator: May 6, 2017
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