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July 14, 2004 INTRODUCTION TO LAGRANGIAN AND HAMILTONIAN MECHANICS Alain J. Brizard Department of Chemistry and Physics Saint Michael’s College, Colchester, VT 05439 Contents 1 Introduction to the Calculus of Variations 1.1 1 Fermat’s Principle of Least Time . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Euler’s First Equation . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Euler’s Second Equation . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Application of Fermat’s Principle . . . . . . . . . . . . . . . . . . . 7 Geometric Formulation of Ray Optics . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Frenet-Serret Curvature of Light Path . . . . . . . . . . . . . . . . 9 1.2.2 Light Propagation in Spherical Geometry . . . . . . . . . . . . . . . 11 1.2.3 Geodesic Representation of Light Propagation . . . . . . . . . . . . 13 1.2.4 Eikonal Representation . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Brachistochrone Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 2 Lagrangian Mechanics 21 2.1 Maupertuis-Jacobi Principle of Least Action . . . . . . . . . . . . . . . . . 21 2.2 Principle of Least Action of Euler and Lagrange . . . . . . . . . . . . . . . 23 2.2.1 Generalized Coordinates in Configuration Space . . . . . . . . . . . 2