Contents Foreword ix Acknowledgments xiii Lecture 1. The Brachistochrone 1 Lecture 2. The Fundamental Problem. Extremals 7 Appendix 2: The Fundamental Lemma 15 Lecture 3. The Insuﬃciency of Extremality 19 Appendix 3: The Principle of Least Action 26 Lecture 4. Important First Integrals 29 Lecture 5. The du Bois-Reymond Equation 35 Appendix 5: Another Fundamental Lemma 40 Lecture 6. The Corner Conditions 41 Lecture 7. Legendre’s Necessary Condition 51 Appendix 7: Yet Another Lemma 55 Lecture 8. Jacobi’s Necessary Condition 57 Appendix 8: On Solving Jacobi’s Equation 65 vii

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