Textbook for calculus of variations? Hamiltonian mechanics?

In summary, the conversation discusses the need to learn about Hamiltonian mechanics involving functional and functional derivative, as well as generalized real and imaginary Hamiltonian. The individual has only learned the basics of Hamiltonian mechanics during undergrad and is looking for recommendations on textbooks to learn more about it. Goldstein and Landau&Liftshitz are suggested as potential resources. Variational calculus and Gateaux derivative are also mentioned as important concepts to understand in order to grasp the subject.
  • #1
betelgeuse91
32
0
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?
 
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  • #2
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. [itex] df(x) = f'(x)dx[/itex] and the need to deal with arbitrary variations in the choice of function [itex] f(x) \to f(x) + \delta f(x)[/itex].

Then the variation of a functional (scalar valued function of a function) will involve a variational derivative best expressed as a Gateaux derivative.
[tex] \delta L[f] = \lim_{h\to 0} \frac{G[f+h\delta f] - G[f]}{h}[/tex]
where [itex]G[/itex] 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
 
  • #4
Goldstein is also flawed, concerning non-holnomic constraints. For an intro I recommend Landau&Liftshitz vol. 1.
 

1. What is the purpose of a textbook for calculus of variations?

A textbook for calculus of variations is designed to provide a comprehensive understanding of the principles, methods, and applications of this mathematical field. It covers topics such as functionals, Euler-Lagrange equations, and variational problems, and prepares students for advanced studies in physics, engineering, and other related fields.

2. What is the difference between calculus of variations and traditional calculus?

Calculus of variations is a branch of mathematics that deals with finding the optimal solution for a given functional. Traditional calculus, on the other hand, focuses on finding the optimal value for a function. In calculus of variations, the function itself is the variable, while in traditional calculus, the variable is the independent variable.

3. How is Hamiltonian mechanics related to calculus of variations?

Hamiltonian mechanics is a mathematical framework for describing the dynamics of physical systems, particularly in classical mechanics. It is closely related to calculus of variations, as it uses variational principles to derive the equations of motion for a system. In fact, the Hamiltonian, which is a function used in Hamiltonian mechanics, is a special type of functional used in calculus of variations.

4. What are some real-world applications of calculus of variations?

Calculus of variations has a wide range of applications in various fields, including physics, engineering, economics, and biology. Some examples include determining the optimal path for a spacecraft, finding the optimal shape of an object to minimize air resistance, and optimizing the design of a bridge to support the maximum weight.

5. Is a strong background in traditional calculus necessary to understand calculus of variations?

While a basic understanding of traditional calculus is helpful, it is not necessary to have a strong background in it to understand calculus of variations. However, a solid understanding of concepts such as derivatives and integrals is important, as they are used in the principles and methods of calculus of variations.

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