# Prosaic discussion of basic lagrangian field theory

1. Jul 13, 2007

### Sojourner01

I'm currently trying to pre-familiarise myself with the course on lagrangian dynamics I'll be taking in the upcoming year, by reading the course notes supplied. I'm somewhat getting the hang of it, but I could really do with some more indepth discussion about the whys and wherefores. Could someone suggest a good introductory book with plenty of clear discussion? Problems and examples are secondary at this point, though of course if they make the reasoning clearer they're hunky dory. I've heard various people mention Goldstein; could someone give the full title to make it a little bit easier for me to find?

2. Jul 13, 2007

### robphy

Some suggestions:

Mechanics (Landau / Lifshitz)
http://www.amazon.com/Course-Theoretical-Physics-Mechanics/dp/0750628960

Introduction to Analytical Dynamics (Woodhouse)
http://www.amazon.com/Introduction-Analytical-Dynamics-science-publications/dp/0198531982

Lagrangian Interaction (Doughty) was enlightening
http://www.amazon.com/Lagrangian-Interaction-Introduction-Relativistic-Electrodynamics/dp/0201416255

Structure and Interpretation of Classical Mechanics (Sussman / Wisdom) might also be enlightening
http://mitpress.mit.edu/SICM/

This looks like it might be fun:
http://www.authorhouse.com/BookStore/ItemDetail~bookid~32221.aspx
(I'll have to see if our library can get it.)

3. Jul 14, 2007

### Meir Achuz

Put "Herbert Goldstein" into Amazon.com

4. Jul 14, 2007

### Shahin

I think the Gantmajer´s book is one of the best, and it is, of course, of a russian guy.

5. Jul 14, 2007

### smallphi

Many of the books mentioned above are classics or 'modern' but I doubt a beginner can pick up significant amount of understanding from them.

A good short book (only 109 pages and free download) that focuses on Lagrangians and Hamiltonians with amazingly succinct and straight to the point definitions (totally confusing in other books) with plenty of examples and exercises with answers is

Robert Dewar, Classical Mechanics:
http://freescience.info/go.php?pagename=books&id=1431

Last edited: Jul 14, 2007
6. Jul 15, 2007

### Sojourner01

Hmph, Dewar's work is fine right up until halfway down page 4. Like most other books I've seen, there's no explanation whatsoever of why the definitions given are the case.

"The condition for functional independence of the m constraints is that the rank of the matrix [whatever] must be its maximal possible value, m."

Why? If I knew the mathematics so well, I wouldn't need this book, would I?

7. Jul 15, 2007

### robphy

I agree that Dewar's notes might be a little advanced...which might be expected of something concise. However, to be fair, the complete quote is:

"The condition for functional independence of the m constraints is that
there be m nontrivial solutions of eq. (1.3), i.e. that the rank of the matrix
$$\partial f_j({\mathbf q})/\partial q_i$$ be its maximal possible value, m."

where eq. (1.3) was
$$\sum_{i=1}^n \displaystyle\frac{\partial f_j({\mathbf q})}{\partial q_i} {\rm d}q_i \equiv \displaystyle\frac{\partial f_j({\mathbf q})}{\partial {\mathbf q}} {\rm d}{\mathbf q} =0$$

The use of the term rank was to help restate the main sentence with a little more mathematics. On a first or second pass, one could gloss over those finer mathematical details.

The Woodhouse text might be a little more your speed.
In it, he addresses what he has dubbed as the first and second "fundamental confusions of calculus".

Doughty's text also treats the more advanced "Lagrangian field theory", which you used in the title of this thread.

For something online, you might like:
Richard Fitzpatrick's Analytical Classical Dynamics: An intermediate level course
http://farside.ph.utexas.edu/teaching/336k/336k.html

Last edited: Jul 15, 2007
8. Jul 15, 2007

### Sojourner01

Thanks for your help, all. I think I'm sturggling here partly due to my fairly limited mathematical equipment - my department are fairly lazy when it comes to maths, so we get very little formalism. As an example, I don't have a clue where the discussion of matrices came from as it's never been introduced to me to use them in this kind of situation before. I find this negligence most frustrating, as it leaves me largely unable to pursue independent study since by necessity most books are very heavy on jargon, whereas my course is very light on it and thus I don't understand what they're talking about, even thugh the concept may actually be quite simple.

9. Jul 15, 2007

### robphy

Maybe it's a good idea to invest in a good book on mathematical methods, like
Boas, Mathematical Methods in the Physical Sciences (3rd),
http://www.amazon.com/Mathematical-Methods-Physical-Sciences-Mary/dp/0471198269

The now-cheaper 2nd edition is a less-pricey alternative:
http://www.amazon.com/Mathematical-Methods-Physical-Sciences-2nd/dp/0471044091

In my experience, lots of things first appear to come from thin air... Sometimes, you have to push your "I believe" button and move on. I think that it is very difficult to learn something "linearly", proceeding only when each step in a presentation is fully understood from start to finish. Hopefully, someday you'll see what was going on (by taking another course [on, possibly, a completely different topic] or by teaching it to someone or by reading on your own [following your own personal plan of study]).

10. Jul 15, 2007

### Dr Transport

11. Jul 25, 2007