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AxiomOfChoice
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Can someone please tell me what the best book for learning calculus of variations is?
Landau said:At what level, for what purposes? The physical, computational way, or the mathematically rigorous way?
For the computational approach I would say Goldstein has a pretty clear explanation.AxiomOfChoice said:I first encountered calculus of variations in my graduate mechanics class, and we did a few problems with it, but I never really understood it completely. (I understand that it's one way to derive the Euler-Lagrange equations.)
https://www.amazon.com/dp/0486414485/?tag=pfamazon01-20 is a great classic text (Dover, cheap), see Google books to browse through it. It is theoretical, but with a lot of physics applications (and a clear lay out of Noethers theorem, which I couldn't really follow in one of my physics classes).Is there a text, adequate for self-study, that lays out the rigorous mathematical framework and then goes on to apply the theory to physical problems, like deriving the Euler-Lagrange equations or showing that the shortest path between two points in the plane is a straight line?
Cantab Morgan said:I learned to love the subject from Gelfand and Fomin.
The purpose of studying Calculus of Variations is to find the optimal solution to a functional, which is a mathematical expression that assigns a value to a function or a set of functions. This is useful in various fields such as physics, engineering, and economics, where finding the optimal path or shape is crucial.
Calculus of Variations deals with the optimization of functions, while traditional Calculus deals with finding the maximum and minimum values of functions. In Calculus of Variations, we look for the function that minimizes or maximizes a functional, which is a mathematical expression that involves functions instead of just numbers.
Calculus of Variations has many applications in physics, engineering, and economics. Some examples include finding the path of least resistance for a river, determining the shape of a hanging cable under its own weight, and optimizing the trajectory of a rocket.
The basic concepts in Calculus of Variations include functionals, variation, and the Euler-Lagrange equation. A functional is a mathematical expression that assigns a value to a function or a set of functions. Variation is the change in a functional when the function is varied. The Euler-Lagrange equation is a necessary condition for finding the function that minimizes or maximizes a functional.
Some techniques used in solving problems in Calculus of Variations include the Euler-Lagrange equation, the method of variations of parameters, and the principle of least action. These techniques use concepts such as derivatives, integrals, and boundary conditions to find the optimal solution to a functional.