Classical Mechanics with Mathematica®

Modeling and Simulation in Science, Engineering, and Technology

Series Editor Nicola Bellomo Politecnico di Torino Torino, Italy Editorial Advisory Board M. Avellaneda Courant Institute of Mathematical Sciences New York University New York, NY, USA

P. Koumoutsakos Computational Science & Engineering Laboratory ETH Zurich ¨ Zurich, ¨ Switzerland

K.J. Bathe Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA, USA

H.G. Othmer Department of Mathematics University of Minnesota Minneapolis, MN, USA

A. Bertozzi Department of Mathematics University of California Los Angleles Los Angeles, CA, USA

L. Preziosi Dipartimento di Matematica Politecnico di Torino Torino, Italy

P. Degond ´ Mathematiques pour l’Industrie et la Physique Universit´e P. Sabatier Toulouse 3 Toulouse, France

K.R. Rajagopal Department of Mechanical Engineering Texas A & M University College Station, TX, USA

A. Deutsch Center for Information Services and High Performance Computing Technische Universit¨at Dresden Dresden, Germany

Y. Sone Department of Aeronautical Engineering Kyoto University Kyoto, Japan

M.A. Herrero Garcia Departamento de Matematica Aplicada Universidad Complutense de Madrid Madrid, Spain

For further volumes: http://www.springer.com/series/4960

Antonio Romano

Classical Mechanics with Mathematicar

Antonio Romano Universit`a degli Studi di Napoli Dipartimento di Matematica e Applicazioni Napoli, Italia

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ISSN 2164-3679 ISSN 2164-3725 (electronic) ISBN 978-0-8176-8351-1 ISBN 978-0-8176-8352-8 (eBook) DOI 10.1007/978-0-8176-8352-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012941619 Mathematics Subject Classification (2010): 70–01, 70E05, 70E17, 70H03, 70H05, 70H06, 70H09, 70H15, 70H20, 70H25, 70H30, 76–00 © Springer Science+Business Media New York 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)

Preface

Books on classical mechanics can roughly be divided into two classes. Books of the first class present the subject using differential calculus and analysis, whereas books of the second class discuss the subject by resorting to the advanced language of differential geometry. The former books, though more accessible to readers with a standard background in calculus, do not allow for a deep understanding of the geometric structures underlying modern mathematical modeling of mechanical phenomena. On the other hand, books of the second class, requiring a wide knowledge of differential geometry, are accessible to readers who are already acquainted with the many aspects of mechanics but wish to understand the most modern developments in the subject. The present book aims to bridge the gap between these two classes. Before discussing the contents of the book in detail, I wish to clarify why I decided to follow a historical approach to presenting mechanics. Over the long period (35 years) during which I taught mechanics to students in physics at the University of Naples Federico II, I saw that students could grasp the fundamentals of the subject if they could understand the historical process that led to the comprehension of mechanical phenomena. This process is like a staircase: as you climb the stairs, each successive step gives you an increasingly broader perspective. To take in a wider horizon, you must go up to the next step. Then, I believe, students needed to remain on a given step until they reached the limit of their perspective and felt the urge to climb to the next step. Clearly, students would achieve the broadest possible view if they started on the last step. However, this approach would allow for, at most, a technical comprehension of the subject without a profound understanding of the roots out of which grew the great tree of classical mechanics. In the history of science, mechanics was the first to resort to mathematical models in an attempt to describe the reality around us. Already Leonardo da Vinci (1452– 1519), conscious of this feature of mechanics, stated that mechanics was heaven for mathematics. Later, Galileo Galilei (1564–1642), having discovered many of the fundamental principles of mechanical theory, stated that Nature was written in the language of geometry. The first almost complete picture of the mechanical world is due to Isaac Newton (1642–1727), heir to Galileo and Johannes Kepler v

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Preface

(1571–1630). He expressed his gratitude to these two great minds stating: “If I have seen farther it is by standing on the shoulders of giants.” In proposing his fundamental laws of dynamics, Newton understood that the mathematics of his time was not up to the task of solving the problems posed by his description of the mechanical world. To make explicit some of the consequences of his laws, he laid down, together with Gottfried Wilhelm Leibniz (1646–1716), the foundations of infinitesimal analysis. However, believing that this new approach was too difficult for his contemporaries to understand, he adopted the language of geometry in writing his magnum opus, Philosophiae Naturalis Principia Mathematica. Although Newton’s laws, in principle, describe the physical behavior of any mechanical system, the description appears in such an implicit form that drawing it out is a very difficult task. We could say that the history of mechanics from Newton to our day is the history of the process of making explicit a part of the hidden content of Newton’s laws. In this process of making things explicit, powerful mathematical descriptions of mechanical systems have been discovered that can be applied to many other branches of physics. The first step in this process was made by Leonhard Euler (1707–1783), who introduced many notations and definitions still in use today. After formulating the fundamental balance equations of mechanics, he proposed a model of rigid bodies. This model introduced an extension to Newton’s model, which was essentially conceived with respect to material points, in particular for the solar system. Further, Euler formulated the balance equations of perfect fluids, i.e., of particular deformable and extended bodies, laying down the foundations of fluidodynamics. The systematic treatment of a system S of rigid bodies subject to smooth constraints is due to Joseph-Louis Lagrange (1736–1813), who in his famous treatise M´ecanique analytique reduced the analysis of the dynamical behavior of such a system to determining a curve of