Textbook for calculus of variations? Hamiltonian mechanics?

[betelgeuse91]

I need to learn about Hamiltonian mechanics involving functional and functional derivative...
Also, I need to learn about generalized real and imaginary Hamiltonian...
I only learned the basics of Hamiltonian mechanics during undergrad and now those papers I read show very generalized version and I wish to know which book I should read...!

Could you recommend me some textbooks to learn those?

 

[jambaugh]

Let's see... Goldstein is the defacto standard on Classical Mechanics, both Lagrangian and Hamiltonian. The basics of variational calculus is on wikipedia to a good extent. I don't have a good text but you can sit down with the web and blank sheets of paper and work through examples.

In hind sight I'd say the subject reduces to learning the meaning of the Gateaux derivative and corresponding differential. Then recognize that when dealing with function valued variables there are two levels of differential... the differential representing the variation of the value of the function as the independent variable varies. i.e. $df(x) = f'(x)dx$ and the need to deal with arbitrary variations in the choice of function $f(x) \to f(x) + \delta f(x)$.

Then the variation of a functional (scalar valued function of a function) will involve a variational derivative best expressed as a Gateaux derivative.
$$\delta L[f] = \lim_{h\to 0} \frac{G[f+h\delta f] - G[f]}{h}$$
where $G$ is a functional i.e .a scalar valued function of a function-valued variable.

Once you then extrapolate your differential calculus of many variables i.e. vectors to differential calculus of functions treated as elements of a vector space, the variational calculus unfolds in a natural way. Between this post and that understanding is then a simple matter of months of exploring examples.

Go to it.
JB

 

[alan2]

I find Goldstein tedious. I like these.

https://www.amazon.com/dp/0486432610/?tag=pfamazon01-20

https://www.amazon.com/dp/0486696901/?tag=pfamazon01-20

 

[vanhees71]

Goldstein is also flawed, concerning non-holnomic constraints. For an intro I recommend Landau&Liftshitz vol. 1.