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

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**About this book :- **
**Calculus of Variations ** written by
** Hansjörg Kielhöfer **

In the history of the calculus of variations the existence of a minimizer was questioned only in the second half of the 19th century by Weierstraß. We present his famous counterexample to Dirichlet’s principle, which awakens the requirement for an existence theory. This leads to the “direct methods in the calculus of variations.” Here one independent variable has the advantage that the Sobolev spaces and the functional analytic tools can be given without great difficulties in the text or in the Appendix. Some emphasis is put on quadratic functionals, since their EulerLagrange equations are linear. The above-mentioned Dirichlet’s principle offers an elegant way to prove the existence of solutions of (linear) boundary value problems: simply obtain minimizers.

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.

**Book Detail :- **
** Title: ** Calculus of Variations: An Introduction to the One-Dimensional Theory with Examples and Exercises
** Edition: **
** Author(s): ** Hansjörg Kielhöfer
** Publisher: ** Springer International Publishing
** Series: ** Texts in Applied Mathematics
** Year: ** 2018
** Pages: ** 242
** Type: ** PDF
** Language: ** English
** ISBN: ** 978-3-319-71122-5, 978-3-319-71123-2
** Country: ** German

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**About Author :- **

Author ** Hansjörg Kielhöfer ** is German mathematician. He is from Rimsting, Bayern, Germany.

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**Book Contents :- **
**Calculus of Variations ** written by
** Hansjörg Kielhöfer **
cover the following topics.

Preface

Introduction
**1. The Euler-Lagrange Equation**

1.1 Function Spaces

1.2 The First Variation

1.3 The Fundamental Lemma of Calculus of Variations

1.4 The Euler-Lagrange Equation

1.5 Examples of Solutions of the Euler-Lagrange Equation

1.6 Minimal Surfaces of Revolution

1.7 Dido’s Problem

1.8 The Brachistochrone Problem of Johann Bernoulli

1.9 Natural Boundary Conditions

1.10 Functionals in Parametric Form

1.11 The Weierstraß-Erdmann Corner Conditions
**2. Variational Problems with Constraints **

2.1 Isoperimetric Constraints

2.2 Dido’s Problem as a Variational Problem with Isoperimetric Constraint

2.3 The Hanging Chain

2.4 The Weierstraß-Erdmann Corner Conditions under Isoperimetric Constraints

2.5 Holonomic Constraints

2.6 Geodesics

2.7 Nonholonomic Constraints

2.8 Transversality

2.9 Emmy Noether’s Theorem

2.10 The Two-Body Problem

3.1 The Method

3.2 An Explicit Performance of the Direct Method in a Hilbert Space

3.3 Applications of the Direct Method

Solutions of the Exercises

Bibliography

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