Action Principles in Continuum Mechanics?

[bolbteppa]

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?

 

[Greg Bernhardt]

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

 

[vanhees71]

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

 

[bolbteppa]

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.

 

[rezeew]

As a scientist in the field of continuum mechanics, I can understand the desire for a comprehensive book that focuses solely on the use of Lagrangian/Action principles in fluid mechanics and theory of elasticity. Unfortunately, I am not aware of any book that specifically covers these topics in the same depth and approach as Landau's books on mechanics and electromagnetism.

However, I would suggest looking into books that focus on variational principles in continuum mechanics, as these often incorporate the use of Lagrangian/Action principles. Additionally, there are many research papers and articles available that discuss the application of these principles in fluid mechanics and elasticity.

It is also worth noting that while Landau's books are highly regarded and widely used, they are not the only resources available for understanding these concepts. It may be beneficial to explore other textbooks and resources to gain a deeper understanding of Lagrangian/Action principles in continuum mechanics.

Overall, while there may not be a single book that covers these topics in the same way as Landau's books, there are still many valuable resources available for understanding and applying these principles in fluid mechanics and elasticity.