Contents

Foreword i

Acknowledgments xi

Lecture 1. The Brachistochrone

Lecture 2. The Fundamental Problem. Extremals

Appendix 2: The Fundamental Lemma 1

Lecture 3. The Insuﬃciency of Extremality 1

Appendix 3: The Principle of Least Action 2

Lecture 4. Important First Integrals 2

Lecture 5. The du Bois-Reymond Equation 3

Appendix 5: Another Fundamental Lemma 4

Lecture 6. The Corner Conditions 4

Lecture 7. Legendre’s Necessary Condition 5

Appendix 7: Yet Another Lemma 5

Lecture 8. Jacobi’s Necessary Condition 5

Appendix 8: On Solving Jacobi’s Equation 6

v

Foreword i

Acknowledgments xi

Lecture 1. The Brachistochrone

Lecture 2. The Fundamental Problem. Extremals

Appendix 2: The Fundamental Lemma 1

Lecture 3. The Insuﬃciency of Extremality 1

Appendix 3: The Principle of Least Action 2

Lecture 4. Important First Integrals 2

Lecture 5. The du Bois-Reymond Equation 3

Appendix 5: Another Fundamental Lemma 4

Lecture 6. The Corner Conditions 4

Lecture 7. Legendre’s Necessary Condition 5

Appendix 7: Yet Another Lemma 5

Lecture 8. Jacobi’s Necessary Condition 5

Appendix 8: On Solving Jacobi’s Equation 6

v