Classical mechanics is a chief example of the scientific method organizing a "complex" collection of information into theoretically rigorous, unifying principles; in this sense, mechanics represents one of the highest forms of mathematical modeling. This textbook covers standard topics of a mechanics course, namely, the mechanics of rigid bodies, Lagrangian and Hamiltonian formalism, stability and small oscillations, an introduction to celestial mechanics, and Hamilton–Jacobi theory, but at the same time features unique examples—such as the spinning top including friction and gyroscopic compass—seldom appearing in this context. In addition, variational principles like Lagrangian and Hamiltonian dynamics are treated in great detail. Using a pedagogical approach, the author covers many topics that are gradually developed and motivated by classical examples. Through `Problems and Complements' sections at the end of each chapter, the work presents various questions in an extended presentation that is extremely useful for an interdisciplinary audience trying to master the subject. Beautiful illustrations, unique examples, and useful remarks are key features throughout the text. <em>Classical Mechanics: Theory and Mathematical Modeling may serve as a textbook for advanced graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference or self-study guide for applied mathematicians and mathematical physicists. Prerequisites include a working knowledge of linear algebra, multivariate calculus, the basic theory of ordinary differential equations, and elementary physics.
Cornerstones Series Editors Charles L. Epstein, University of Pennsylvania, Philadelphia Steven G. Krantz, Washington University, St. Louis
Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus
Emmanuele DiBenedetto
Classical Mechanics Theory and Mathematical Modeling
Emmanuele DiBenedetto Department of Mathematics Vanderbilt University Nashville, TN 37240 USA
[email protected]
ISBN 978-0-8176-4526-7 e-ISBN 978-0-8176-4648-6 DOI 10.1007/978-0-8176-4648-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010936443 Mathematics Subject Classification (2010): 34B25, 34C75, 37J05, 37J10, 37K05, 37K10, 37N05, 37N10, 70A05, 70B05, 70B10, 70E05, 70E15, 70E17, 70E18, 70H03, 70H05, 70H14, 70H15, 70H25, 70J10, 70J25, 70J30, 70J35, 76A02, 36D05, 70G75 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper www.birkhauser-science.com
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvii 1
GEOMETRY OF MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Trajectories in R3 and Intrinsic Triads . . . . . . . . . . . . . . . . . . . . . 2 Areolar Velocity and Central Motions . . . . . .