Poisson Brackets and Constraints...

[Freeman]

Hello,

 

I didn't know where to put this (since it's not really fitting anywhere else), but I am kind of learning variational calculus and more specifically Classical Field Theory (Hamiltonian mechanics and the like) and I need a good book (or even a explanation) that explains second class constraints really well. I am familiar with Lagrangian and Hamiltonian mechanics, but I am a little rusty on the Poisson bracket (especially when using vectors with indices!).

 

The reason I ask is because I was talking about a field theory (I can't remember it now, it was just a toy model) with my professor in his office hours, and he goes to the chalk board and says "Well, bing bing bing, you have this as a second-class constraint and zoooop you have this Poisson bracket and bing bing bing it doesn't vanish and looks really nasty. You'll be dealing with delta functions and more fun." And I sat there dazed as he did this; something told me I needed to read up a bit on it.

 

Again, any help would be greatly appreciated!

[ajb]

The modern way to treat theories with constraints, like gauge theories is via BRST methods.

 

I suggest "Quantization of Gauge Systems " by Henneaux &. Teitelboim.

 

They deal with gauge symmetries in mechanics and field theories using both the canonical BRST and the antifield formulism. Both approches require supergeometry with is just about explained enough in the book.

 

 

I must warn you that the subject is quite heavy.

[Freeman]

Is there something a little more "user friendly" than Henneaux? It appears more heavy than I can handle at the moment.

[ajb]

you had a look at the book?

 

It is the only one that springs to my mind. Theories with constraints are hard, maybe you just have to bite the bullet!

 

I'll have a think and see if I cna find a more gental book.

[Freeman]

I told my professor "Uh...I think I should look this up for reference." He says "Well, there's only one text that comes to mind, but it's a rather hard introduction it's a book by Henneaux"!

 

So I went to the library, and I checked it out, and it is well beyond something as simple as I'd like.

[ajb]

Like I said, it is the only book that I can think of.

 

The first few chapters will give you the introduction you want.

 

Before you look at theories with constraints are you happy with Hamiltonian mechanics, Poisson Brackets and symplectic geometry? You should be comfortable with these notions before looking at constraints.

 

If you have any direct questions I will try to help.