Advanced Mechanics
From Euler's Determinism to Arnold's Chaos
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Product details:
- Publisher OUP Oxford
- Date of Publication 25 July 2013
- ISBN 9780199670864
- Binding Paperback
- No. of pages184 pages
- Size 245x174x10 mm
- Weight 328 g
- Language English
- Illustrations 18 b/w illustrations 0
Categories
Short description:
This book can be used as a textbook for a graduate course on mechanics or for self-study. There are a variety of problems ranging from exercises that verify parts of the text, moderately difficult calculations, to suggested research projects. Connections to other disciplines of mathematics and physics are emphasized.
MoreLong description:
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 Schrödinger 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 reometry 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.
The book appears to be well suited to serve the intended role of introducing physics students to mechanics and making clear to them the relevance of the subject for modern physics; but it will also be useful to mathematics students to understand that the subject is relevant and alive well beyond the classical realms of applications and/or abstract mathematical developments.
Table of Contents:
The Variational Principle
Conservation Laws
The Simple Pendulum
The Kepler Problem
The Rigid Body
Geometric Theory of Ordinary Differential Equations
Hamilton's Principle
Geodesics
Hamilton-Jacobi Theory
Integrable Systems
The Three Body Problem
The Restricted Three Body Problem
Magnetic Fields
Poisson and Symplectic Manifolds
Discrete Time
Dynamics in One Real Variable
Dynamics On The Complex Plane
KAM Theory
