Los Alamos Electronic Archives: physics/9909035 arXiv:physics/9909035 v1 19 Sep 1999 CLASSICAL MECHANICS HARET C. ROSU
[email protected] a a p dp dt (r) V graduate course c 1999 H.C. Rosu Copyright Le´ on, Guanajuato, Mexico v1: September 1999. 1 CONTENTS 1. THE “MINIMUM” PRINCIPLES ... 3. 2. MOTION IN CENTRAL FORCES ... 19. 3. RIGID BODY ... 32. 4. SMALL OSCILLATIONS ... 52. 5. CANONICAL TRANSFORMATIONS ... 70. 6. POISSON PARENTHESES... 79. 7. HAMILTON-JACOBI EQUATIONS... 82. 8. ACTION-ANGLE VARIABLES ... 90. 9. PERTURBATION THEORY ... 96. 10. ADIABATIC INVARIANTS ... 111. 11. MECHANICS OF CONTINUOUS SYSTEMS ... 116. 2 1. THE “MINIMUM” PRINCIPLES Forward: The history of “minimum” principles in physics is long and interesting. The study of such principles is based on the idea that the nature acts always in such a way that the important physical quantities are minimized whenever a real physical process takes place. The mathematical background for these principles is the variational calculus. CONTENTS 1. Introduction 2. The principle of minimum action 3. The principle of D’Alembert 4. Phase space 5. The space of configurations 6. Constraints 7. Hamilton’s equations of motion 8. Conservation laws 9. Applications of the action principle 3 1. Introduction The empirical evidence has shown that the motion of a particle in an inertial ~ = d~ system is correctly described by Newton’s second law F p/dt, whenever possible to neglect the relativistic effects. When the particle happens not to be forced to a complicated motion, the Cartesian coordinates are sufficient to describe the movement. If none of these conditions are fulfilled, rather complic