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Calculus of Variations by Hansjörg Kielhöfer
(An Introduction to the One-Dimensional Theory with Examples and Exercises)

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Calculus of Variations by Hansjörg Kielhöfer , Rimsting, Bayern, Germany. 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.

Calculus of Variations by Hansjörg Kielhöfer cover the following topics.

  • Preface

  • 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. Direct Methods in the Calculus of Variations
    3.1 The Method
    3.2 An Explicit Performance of the Direct Method in a Hilbert Space
    3.3 Applications of the Direct Method

  • Appendix
    Solutions of the Exercises

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