vi Contents §3.4. Geodesics 62 §3.5. Recommended reading 63 §3.6. Exercises 64 Chapter 4. Basic Generalizations 69 §4.1. Higher-order derivatives 69 §4.2. Case study: The cantilever beam 71 §4.3. Multiple unknown functions 77 §4.4. Lagrangian mechanics 79 §4.5. Case study: The spherical pendulum 85 §4.6. Hamiltonian mechanics 88 §4.7. Ray optics 93 §4.8. Double integrals 96 §4.9. Recommended reading 101 §4.10. Exercises 101 Chapter 5. Constraints 107 §5.1. Types of constraints 107 §5.2. Lagrange multipliers 114 §5.3. Isoperimetric constraints 117 §5.4. Case study: Queen Dido’s problem 120 §5.5. Case study: Euler’s elastica 123 §5.6. Holonomic constraints 129 §5.7. Case study: A sliding rod 133 §5.8. Recommended reading 136 §5.9. Exercises 136 Chapter 6. The Second Variation 139 §6.1. Introduction 139 §6.2. Legendre’s condition 142 §6.3. Jacobi’s condition 147 §6.4. Case study: The catenoid revisited 160
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