Theoretical Physics 1

Transcription

1 Theoretical Physics 1

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3 Wolfgang Nolting Theoretical Physics 1 Classical Mechanics 123

4 Wolfgang Nolting Inst. Physik Humboldt-UniversitRat zu Berlin Berlin, Germany ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing Switzerland 2016 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

5 General Preface The seven volumes of the series Basic Course: Theoretical Physics are thought to be textbook material for the study of university-level physics. They are aimed to impart, in a compact form, the most important skills of theoretical physics which can be used as basis for handling more sophisticated topics and problems in the advanced study of physics as well as in the subsequent physics research. The conceptual design of the presentation is organized in such a way that Classical Mechanics (volume 1) Analytical Mechanics (volume 2) Electrodynamics (volume 3) Special Theory of Relativity (volume 4) Thermodynamics (volume 5) are considered as the theory part of an integrated course of experimental and theoretical physics as is being offered at many universities starting from the first semester. Therefore, the presentation is consciously chosen to be very elaborate and self-contained, sometimes surely at the cost of certain elegance, so that the course is suitable even for self-study, at first without any need of secondary literature. At any stage, no material is used which has not been dealt with earlier in the text. This holds in particular for the mathematical tools, which have been comprehensively developed starting from the school level, of course more or less in the form of recipes, such that right from the beginning of the study, one can solve problems in theoretical physics. The mathematical insertions are always then plugged in when they become indispensable to proceed further in the program of theoretical physics. It goes without saying that in such a context, not all the mathematical statements can be proved and derived with absolute rigour. Instead, sometimes a reference must be made to an appropriate course in mathematics or to an advanced textbook in mathematics. Nevertheless, I have tried for a reasonably balanced representation so that the mathematical tools are not only applicable but also appear at least plausible. v

6 vi General Preface The mathematical interludes are of course necessary only in the first volumes of this series, which incorporate more or less the material of a bachelor program. In the second part of the series which comprises the modern aspects of theoretical physics, Quantum Mechanics: Basics (volume 6) Quantum Mechanics: Methods and Applications (volume 7) Statistical Physics (volume 8) Many-Body Theory (volume 9), mathematical insertions are no longer necessary. This is partly because, by the time one comes to this stage, the obligatory mathematics courses one has to take in order to study physics would have provided the required tools. The fact that training in theory has already started in the first semester itself permits inclusion of parts of quantum mechanics and statistical physics in the bachelor program itself. It is clear that the content of the last three volumes cannot be part of an integrated course but rather the subject matter of pure theory lectures. This holds in particular for Many-Body Theory which is offered, sometimes under different names as, e.g., Advanced Quantum Mechanics, in the eighth or so semester of study. In this part, new methods and concepts beyond basic studies are introduced and discussed which are developed in particular for correlated many particle systems which in the meantime have become indispensable for a student pursuing master s or a higher degree and for being able to read current research literature. In all the volumes of the series Basic Course: Theoretical Physics, numerous exercises are included to deepen the understanding and to help correctly apply the abstractly acquired knowledge. It is obligatory for a student to attempt on his own to adapt and apply the abstract concepts of theoretical physics to solve realistic problems. Detailed solutions to the exercises are given at the end of each volume. The idea is to help a student to overcome any difficulty at a particular step of the solution or to check one s own effort. Importantly these solutions should not seduce the student to follow the easy way out as a substitute for his own effort. At the end of each bigger chapter, I have added self-examination questions which shall serve as a self-test and may be useful while preparing for examinations. I should not forget to thank all the people who have contributed one way or an other to the success of the book series. The single volumes arose mainly from lectures which I gave at the universities of Muenster, Wuerzburg, Osnabrueck, and Berlin in Germany, Valladolid in Spain and Warangal in India. The interest and constructive criticism of the students provided me the decisive motivation for preparing the rather extensive manuscripts. After the publication of the German version, I received a lot of suggestions from numerous colleagues for improvement, and this helped to further develop and enhance the concept and the performance of the series. In particular I appreciate very much the support by Prof. Dr. A. Ramakanth, a long-standing scientific partner and friend, who helped me in many respects, e.g. what concerns the checking of the translation of the German text into the present English version.

7 General Preface vii Special thanks are due to the Springer company, in particular to Dr. Th. Schneider and his team. I remember many useful motivations and stimulations. I have the feeling that my books are well taken care of. Berlin, Germany May 2015 Wolfgang Nolting

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9 Preface to Volume 1 The first volume of the series Basic Course: Theoretical Physics presented here deals with Classical Mechanics, a topic which may be described as analysis of the laws and rules according to which physical bodies move in space and time under the influence of forces. This formulation already contains certain fundamental concepts whose rigorous definitions appear rather non-trivial and therefore have to be worked out with sufficient care. In the case of a few of these fundamental concepts, we have to even accept them, to start with, as more or less plausible facts of everyday experience without going into the exact physical definitions. We assume a material body to be an object which is localized in space and time and possesses an (inertial) mass.the concept is still to be discussed. This is also valid for the concept of force. The forces are causing changes of the shape and/or in the state of motion of the body under consideration. What we mean by space is the three-dimensional Euclidean space being unrestricted in all the three directions, being homogeneous and isotropic, i.e. translations or rotations of our world as a whole in this space have no consequences. The time is also a fact of experience from which we only know that it does exist flowing uniformly and unidirectionally. It is also homogeneous which means no point in time is a priori superior in any manner to any other point in time. In order to describe natural phenomena, a physicist needs mathematics as language. But the dilemma lies in the fact that theoretical mechanics can be imparted in a proper way only when the necessary mathematical tools are available. If theoretical physics is started right in the first semester, the student is not yet equipped with these tools. That is why the first volume of the Basic Course: Theoretical Physics begins with a concise mathematical introduction which is presented in a concentrated and focused form including all the material which is absolutely necessary for the development of theoretical classical mechanics. It goes without saying that in such a context not all mathematical theories can be proved or derived with absolute stringency and exactness. Nevertheless, I have tried for a reasonably balanced representation so that mathematical theories are not only ix

10 x Preface to Volume 1 readily applicable but also at least appear plausible. Thereby only that much mathematics is offered which is necessary to proceed with the presentation of theoretical physics. Whenever in the presentation one meets new mathematical barriers, a corresponding mathematical insertion appears in the text. Therefore, mathematical discourses are found only at the positions where they are directly needed. In this connection, the numerous exercises provided are of special importance and should be worked without fail in order to evaluate oneself in self-examination. This volume on classical mechanics arose from respective lectures I gave at the German Universities in Muenster and Berlin. The animating interest of the students in my lecture notes has induced me to prepare the text with special care. This volume as well as the subsequent volumes is thought to be a textbook material for the study of basic physics, primarily intended for the students rather than for the teachers. It is presented in such a way that it enables self-study without the need for a demanding and laborious reference to secondary literature. I had to focus on the essentials, presenting them in a detailed and elaborate form, sometimes consciously sacrificing certain elegance. It goes without saying that after the basic course, secondary literature is needed to deepen the understanding of physics and mathematics. I am thankful to the Springer company, especially to Dr. Th. Schneider, for accepting and supporting the concept of my proposal. The collaboration was always delightful and very professional. A decisive contribution to the book was provided by Prof. Dr. A. Ramakanth from the Kakatiya University of Warangal (India). Many thanks for it! Berlin, Germany May 2015 Wolfgang Nolting

11 Contents 1 Mathematical Preparations Elements of DifferentialCalculus Set ofnumbers Sequence of Numbers and Limiting Values Series and Limiting Values Functionsand Limits Continuity Trigonometric Functions Exponential Function and Logarithm DifferentialQuotient Rules of Differentiation TaylorExpansion Limiting Values of Indeterminate Expressions ExtremeValues Exercises Elements of IntegralCalculus Notions First Rules of Integration Fundamental Theorem of Calculus The Techniqueof Integration Multiple Integrals Exercises Vectors ElementaryMathematical Operations Scalar Product Vector (Outer, Cross) Product Higher Vector Products Basis Vectors Component Representations Exercises xi

12 xii Contents 1.4 Vector-ValuedFunctions Parametrizationof Space Curves Differentiationof Vector-ValuedFunctions Arc Length MovingTrihedron Exercises Fields Classification ofthe Fields Partial Derivatives Gradient Divergenceand Curl (Rotation) Exercises Matrices and Determinants Matrices CalculationRules for Matrices Transformationof Coordinates(Rotations) Determinants CalculationRules for Determinants Special Applications Exercises CoordinateSystems Transformationof Variables,Jacobian Determinant Curvilinear Coordinates CylindricalCoordinates SphericalCoordinates Exercises Self-ExaminationQuestions Mechanics of the Free Mass Point Kinematics Velocity and Acceleration Simple Examples Exercises Fundamental Laws of Dynamics Newton s Laws of Motion Forces InertialSystems, Galilean Transformation Rotating Reference Systems, Pseudo Forces (Fictitious Forces) Arbitrarily Accelerated Reference Systems Exercises Simple Problems of Dynamics Motionin the HomogeneousGravitationalField LinearDifferentialEquations

13 Contents xiii Motion with Friction in the Homogeneous GravitationalField Simple Pendulum ComplexNumbers Linear Harmonic Oscillator Free Damped Linear Oscillator Damped Linear Oscillator Under the Influence of an ExternalForce Arbitrary One-Dimensional Space-Dependent Force Exercises Fundamental Concepts and Theorems Work, Power,and Energy Potential Angular Momentum and Torque (Moment) Central Forces Integrationofthe EquationsofMotion Exercises PlanetaryMotion Exercises Self-ExaminationQuestions Mechanics of Many-Particle Systems ConservationLaws Principle of Conservation of Linear Momentum (Centerof Mass Theorem) Conservation of Angular Momentum Conservationof Energy Virial Theorem Two-Particle Systems Relative Motion Two-Body Collision Elastic Collision Inelastic Collision PlanetaryMotionas a Two-ParticleProblem Coupled Oscillations Exercises Self-ExaminationQuestions The Rigid Body Model of a Rigid Body Rotation Around an Axis Conservationof Energy Angular-Momentum Law Physical Pendulum Steiner s Theorem

14 xiv Contents Rolling Motion AnalogyBetween TranslationalandRotational Motion Inertial Tensor Kinematics of the Rigid Body Kinetic Energyof the Rigid Body Propertiesof the Inertial Tensor Angular Momentum of the Rigid Body Theoryof the SpinningTop Euler s Equations Euler s Angles Rotations Around Free Axes Force-FreeSymmetric SpinningTop Exercises Self-ExaminationQuestions A Solutions of the Exercises Index