Understanding Hamiltonian Field Equations and Their Applications in Field Theory

[MManuel Abad]

Hi there, physics lovers. I'm studying field theory. So far, so well. I got it with the lagrangian density. I understood it. But then I DIDN'T FIND stuff about the Hamiltonian density. I couldn't find anything in Landau-Lifgarbagez series, and that makes me worry. I've been looking in the internet and I couldn't find ANYTHING about it. I found some stuff in the Goldstein, but not as explained and as extended as I'd like. Besides, I found in the Goldstein what could be considered the equivalent canonical equations for this Hamiltonian density, but in some other reference (which I don't remember) I found these equations very very different. That is kind of driving me crazy. Could you please help and tell me:

- What is the Hamiltonian density?

- How can I derive the Canonical field equations and what are they? (if there's such a thing, as there is in particle mechanics)

- Adventages and disadventages of the Hamiltonian density formulation?

- Why not many people use it? I mean, through the internet I found stuff about Klein-Gordon fiels and that. They find the lagrangian density, they use the lagrangian field equations and find the equations for the field. Then they find the Hamiltonian density... and that's all. ¿Why don't they use the hamiltonian field equations (again, if there's such a thing) for finding the equations of motion?

Please, I'm kind of desperate. This is not homework (I'm just in the fourth semester), it's something I'm studying on my own.
Greetings.

 

[A. Neumaier]

I'm studying field theory. So far, so well. I got it with the lagrangian density. I understood it. But then I DIDN'T FIND stuff about the Hamiltonian density. I couldn't find [...] ANYTHING about it.

- Adventages and disadventages of the Hamiltonian density formulation?

The main disatvantage is that you can hardly find ANYTHING in the literature about it. The main reason is that for fields, the Lagrangian approach is much more elementary that the Hamiltonian approach, since things are essentially the same as for particle mechanics.

A Hamiltonian approach to field theory needs advanced concepts - namely nonstandard Poisson brackets, but then it shows its value.

If you tell me about your math background and where you want to use Hamiltonian field methods, I can perhaps say more.

 

[MManuel Abad]

Hi, thanks for replying!

Well, I'm just studying the very basics of field theory, so I'm not using the Hamiltonian density approach soon. A professor is kind of my tutor, and is giving me this subjects to study (I'm also studying Special and General Relativity). I've not taken a class on Quantum Mechanics, not to say in QFT (I think that in QFT is where Lagrangian and Hamiltonian densities are most used, aren't they?).

My mathematical background is, I hate to confess, not so wide:

- Calculus (On R; and Vector Calculus)
- Linear Algebra
- ODEs (Ordinary Differential Equations)
- An introductory course on mathematical analysis
- Currently I'm taking a course on Complex Variables.

I've already taken a course on Analytical Mechanics, but just for discrete systems.

Do you think it's enough for understanding the answers to my questions? What else should I study? And, if there's little literature about the subject, where can I find info??

 

[A. Neumaier]

I've not taken a class on Quantum Mechanics, not to say in QFT (I think that in QFT is where Lagrangian and Hamiltonian densities are most used, aren't they?).

What else should I study? And, if there's little literature about the subject, where can I find info??

OK; so I'll give you some pointers to classical Hamiltonian mechanics of conservative field theories. You'll probably know enough to study Morrison's paper
http://www.usna.edu/Users/math/rmm/Papers/MorrisonReza.pdf

You would need some differential geometry to understand the book ''Mechanics and Symmetry'' by Marsden and Ratiu; but this will give you a systematic approach. Marsden has also written much more advanced stuff.

From there go back and forth in time using the references and scholar.google.com

 

[MManuel Abad]

Thank you so much! This was of great help! :D Yeah, I do admire Marsden. I have as a goal to fully comprehend his book on "Foundations of Classical Mechanics". The little I understand of it is very beautiful! :)

Thank you very much, again :D

 

[brocks]

There is a book by Calkin called "Lagrangian and Hamiltonian Mechanics" that is intended for undergraduates. I haven't read it, so maybe someone else can tell you if it's what you need.

 

[MManuel Abad]

Thanks, dude. I've been looking for it, but it seems it's just an exercise book (yeah... weird for a subject on physics)... But thanks anyway :)

 

[brocks]

One of us is mistaken. As I said, I haven't read it, but I did look through it at a bookstore last year, and unless my memory is totally shot, it was a textbook. Of course it had exercises at the end of the chapters, but no more than most textbooks.

This online book also has a fairly long chapter on Hamiltonian mechanics:

http://mitpress.mit.edu/SICM/book-Z-H-4.html#%_toc_start

 

[MManuel Abad]

Good Lord, I think you're right!

I looked for it in Amazon and I found it. Then I clicked in the hyperlink "Look inside" and I saw just exercises, not a textbook. But you know, in Amazon and that stuff, they always eliminate some pages because of author rights and stuff. So, thinking it more slowly, I guess you're right. Well, I'll look for it. But in the web version you gave, I think there's no text about Lagrangian and Hamiltonian field theory. Thank you so much :)