E-Book Content
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