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2018 | Book

The Euler-Lagrange Equation Read chapter Read first chapter

Calculus of Variations

An Introduction to the One-Dimensional Theory with Examples and Exercises

Author: Hansjörg Kielhöfer

Publisher: Springer International Publishing

Book Series : Texts in Applied Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This clear and concise textbook provides a rigorous introduction to the calculus of variations, depending on functions of one variable and their first derivatives. It is based on a translation of a German edition of the book Variationsrechnung (Vieweg+Teubner Verlag, 2010), translated and updated by the author himself. Topics include: the Euler-Lagrange equation for one-dimensional variational problems, with and without constraints, as well as an introduction to the direct methods. The book targets students who have a solid background in calculus and linear algebra, not necessarily in functional analysis. Some advanced mathematical tools, possibly not familiar to the reader, are given along with proofs in the appendix. Numerous figures, advanced problems and proofs, examples, and exercises with solutions accompany the book, making it suitable for self-study.

The book will be particularly useful for beginning graduate students from the physical, engineering, and mathematical sciences with a rigorous theoretical background.

Table of Contents

Frontmatter
Chapter 1. The Euler-Lagrange Equation
Abstract
In order to give the functionals
$$ J(y) = \int ^b_a F(x,y, y') dx $$
a domain of definition, we need to introduce suitable function spaces. First of all we require that the Lagrange function or Lagrangian,
$$ F : [a, b] \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}, {\qquad }\text {is continuous.} $$
Here \([a, b] = \{x|a\le x\le b\}\) is a compact interval in the real line.
Hansjörg Kielhöfer
Chapter 2. Variational Problems with Constraints
Abstract
Many early variational problems like Dido’s problem or the problem of the hanging chain have constraints in a natural way: Maximize the area with given perimeter or minimize the potential energy of a hanging chain with given length. These constraints belong to the class of isoperimetric constraints, and they are of the same type as the functional to be maximized or minimized.
Hansjörg Kielhöfer
Chapter 3. Direct Methods in the Calculus of Variations
Abstract
The Euler-Lagrange calculus was created to determine extremals of functionals. If the solution of the Euler-Lagrange equation is unique among all admitted functions, then physical or geometric insights into the problem might lead to the conclusion that it is indeed the desired extremal. In addition, the second variation provides necessary and also sufficient conditions on extremals.
Hansjörg Kielhöfer
Backmatter
Metadata
Title
Calculus of Variations
Author
Hansjörg Kielhöfer
Copyright Year
2018
Electronic ISBN
978-3-319-71123-2
Print ISBN
978-3-319-71122-5
DOI
https://doi.org/10.1007/978-3-319-71123-2

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