Advanced Mechanics

Classical Mechanics is the oldest and best understood part of physics. This does not mean that it is cast in marble yet, a museum piece to be admired from a distance. Instead, mechanics continues to be an active area of research by physicists and mathematicians. Every few years, we need to re-evaluate the purpose of learning mechanics and look at old material in the light of modern developments. Once you have learned basic mechanics (Newton's laws, the solution of the Kepler problem) and quantum mechanics (the Schrodinger equation, hydrogen atom) it is time to go back and relearn classical mechanics in greater depth. It is the intent of this book to take you through the ancient (the original meaning of <"classical>") parts of the subject quickly: the ideas started by Euler and ending roughly with Poincare. We then take up the developments of twentieth century physics that have largely to do with chaos and discrete time evolution (the basis of numerical solutions). Along the way you will learn about elliptic functions and their connection to the Arithmetico-Geometric-Mean; Einstein's calculation of the perihelion shift of Mercury; that spin is really a classical phenomenon; how Hamilton came very close to guessing wave mechanics when he developed a unified theory of optics and mechanics; how Riemannian geometry is useful to understand the impossibility of long range weather prediction; why the maximum of the potential is a stable point of equilibrium in certain situations; the similarity of the orbits of particles in atomic traps and of the Trojan asteroids; about Julia sets and the Mandelblot; what Feigenbaum constants are and how Newton's iterations help establish the Kolmogorov-Arnold-Moser theorem. By the end you should be ready to absorb modern research in mechanics.
"Rajeev's book provides a fresh look at the venerable subject of Classical Mechanics. Mathematical results on integrability, geodesic flows and the three-body problem, usually left out of physics textbooks, are introduced in a compact and elegant way and connected to recent developments on chaotic dynamics. Instructors will enjoy it as a textbook for an advanced Classical Mechanics course or as a complement to a more basic textbook in an introductory graduate course."
-- Yannick Meurice, University of Iowa

"Sarada Rajeev has produced a masterpiece which is as steeped in tradition as it is modern in outlook. The book is a minimalist's dream, unleashing on the unsuspecting reader a treasure trove of novel results within its slender frame. The informal yet stimulating narrative, the excellent choice of topics, and the lucid explanations truly make the book a connoisseur's delight."
-- V. V. Sreedhar, Chennai Mathematical Institute

"Rajeev approaches his choice of topics and organization from the point of view of one who is intimately familiar with modern physics. Although he starts from very elementary classical mechanics, he nevertheless does not shy away from sophisticated mathematical tools, such as elliptic functions, which he develops from scratch. This book should be of interest to many physics graduate students and researchers in physics, as well as to many mathematicians."
-- Leonard Gross, Cornell University

1. The Variational Principle ; 2. Conservation Laws ; 3. The Simple Pendulum ; 4. The Kepler Problem ; 5. The Rigid Body ; 6. Geometric Theory of Ordinary Differential Equations ; 7. Hamilton's Principle ; 8. Geodesics ; 9. Hamilton-Jacobi Theory ; 10. Integrable Systems ; 11. The Three Body Problem ; 12. The Restricted Three Body Problem ; 13. Magnetic Fields ; 14. Poisson and Symplectic Manifolds ; 15. Discrete Time ; 16. Dynamics in One Real Variable ; 17. Dynamics On The Complex Plane ; 18. KAM Theory