vi Contents Chapter 3. Second-Order ODE and the Calculus of Variations 63 3.1. Tangent Vectors and the Tangent Bundle 63 3.2. Second-Order Differential Equations 66 3.3. The Calculus of Variations 68 3.4. The Euler-Lagrange Equations 70 3.5. Conservation Laws for Euler-Lagrange Equations 73 3.6. Two Classic Examples 75 3.7. Derivation of the Euler-Lagrange Equations 78 3.8. More General Variations 80 3.9. The Theorem of E. Noether 81 3.10. Lagrangians Defining the Same Functionals 82 3.11. Riemannian Metrics and Geodesics 85 3.12. A Preview of Classical Mechanics 86 Chapter 4. Newtonian Mechanics 91 4.1. Introduction 91 4.2. Newton’s Laws of Motion 92 4.3. Newtonian Kinematics 96 4.4. Classical Mechanics as a Physical Theory 99 4.5. Potential Functions and Conservation of Energy 106 4.6. One-Dimensional Systems 111 4.7. The Third Law and Conservation Principles 118 4.8. Synthesis and Analysis of Newtonian Systems 122 4.9. Linear Systems and Harmonic Oscillators 124 4.10. Small Oscillations about Equilibrium 126 Chapter 5. Numerical Methods 133 5.1. Introduction 133 5.2. Fundamental Examples and Their Behavior 144

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