AnAly ticAl MechAnics ANALY TICAL MECHANICS Solutions to Problems in Classical Physics (cid:9)(cid:15)(cid:1)(cid:14)(cid:53)(cid:13)(cid:5)(cid:18)(cid:3)(cid:8)(cid:5)(cid:19)(cid:53)(cid:82)(cid:53)(cid:4)(cid:1)(cid:14)(cid:9)(cid:5)(cid:12)(cid:53)(cid:18)(cid:1)(cid:4)(cid:21) Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140612 International Standard Book Number-13: 978-1-4822-3940-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. 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Albert Einstein iii TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk PREFACE As the story goes, not everything new is also useful, and not every- thing old is also obsolete. In theoretical physics, one of the most convincing examples in this respect is offered by Analytical Mechan- ics. Created by Jean Bernoulli (1654-1705), Pierre Louis Moreau de Maupertuis (1698-1759), Leonhard Euler (1707-1783), Jean le Rond D’Alembert (1717-1783), Joseph Louis Lagrange (1736-1813), Karl Gustav Jacobi (1804-1851), William Rowan Hamilton (1805-1865), Jules Henri Poincar´e (1854-1912), and other prominent minds, An- alytical Mechanics proved to be a very useful tool of investigation not only in Newtonian Mechanics, but also in almost all classical and modern branches of physics: Electrodynamics, Quantum Field The- ory, Theory of Relativity, etc. It can be stated, without exaggeration, that Analytical Mechanics is essential in understanding Theoretical Physics, being a sine qua non condition of a profound training of a physicist. Due to its large field of applications, we dare to say that the term ”Analytical Mechanics” is somewhat overtaken (out of date). One of the essential properties of Analytical Mechanics is that it uses abstract, mathematical techniques as methods of investigation. By its object, Analytical Mechanics is a physical discipline, while its methods belong to various branches of mathematics: Analytical and Differential Geometry, Analysis, Differential Equations, Tensor Calcu- lus, Calculus of Variations, Algebra, etc. That is why a physicist who studies analytical formalism must have an appropriate mathematical training. It is widely spread the idea that it is more important to learn and understand the practical applications of physics, than the theo- ries. In our opinion, connection between Analytical Mechanics and the important chapter devoted to its applications is similar to that between physical discoveries and engineering: if the discovery of elec- tricity, electromagnetic waves, nuclear power, etc., haven’t been put into practice, they would have remained within the laboratory frame. The purpose of this collection of solved problems is intended to give the students possibility of applying the theory (Lagrangian and v Hamiltonian formalisms for both discrete and continuous systems, Hamilton-Jacobi method, variational calculus, theory of stability, etc.) to problems concerning several chapters of Classical Physics. Some problems are difficult to solve, while others are easy. One chapter (the third), as a whole, is dedicated to the gravitational plane pendu- lum, the problem being solved by all possible analytical formalisms, including, obviously, the Newtonian approach. This way, one can eas- ily observe similarities and differences between various analytical ap- proaches, and their specific efficiency as well. When needed, some theoretical subjects are developed up to some extent, in order to offer the student possibility to follow solutions to the problems without appealing to other reference sources. This has been done for both discrete and continuous physical systems, or, in analytical terms, systems with finite and infinite degrees of freedom. A special attention is paid to basics of vector algebra and vec- tor analysis, in Appendix B. Notions like: gradient, divergence, curl, tensor, together with their physical applications, are thoroughly de- veloped and discussed. This collection of solved problems is a result of many years of teaching Analytical Mechanics, as the first course of theoretical phy- sics, to the students of the Faculty of Physics. There are many ex- cellent textbooks dedicated to applied Analytical Mechanics for both students and their instructors, but we modestly pretend to offer an original view on distribution of the subjects, the thorough analysis of solutions to the problems, and an appropriate choice of applications in various branches of Physics: Mechanics of discrete and continuous sys- tems, Electrodynamics, Classical Field Theory, Equilibrium and small oscillations, etc. IASI, February 2014 The authors vi CONTENTS CHAPTER I. FUNDAMENTALS OF ANALYTICAL MECHANICS ......................................... 1 I.1. Constraints...............................................1 I.1.1. Classification criteria for constraints .................. 2 I.1.2. The fundamental dynamical problem for a constrained particle..............................................7 I.1.3. System of particles subject to constraints ............. 9 I.1.4. Lagrange equations of the first kind..................11 I.2. Elementary displacements...............................12 I.2.1. Generalities.........................................12 I.2.2. Real, possible and virtual displacements ............. 13 I.3. Virtual work and connected principles ................... 19 I.3.1. Principle of virtual work.............................19 I.3.2. Principle of virtual velocities ........................ 22 I.3.3. Torricelli’s principle ................................. 23 CHAPTER II. PRINCIPLES OF ANALYTICAL MECHANICS........................................26 II.1. D’Alembert’s principle..................................26 II.1.1. Configuration space ................................ 28 II.1.2. Generalized forces..................................29 II.2. Hamilton’s principle....................................35 CHAPTER III. THE SIMPLE PENDULUM PROBLEM ........................................... 47 III.1. Classical (Newtonian) formalism.......................47 III.2. Lagrange equations of the first kind approach..........68 III.3. Lagrange equations of the second kind approach........72 III.4. Hamilton’s canonical equations approach...............78 III.5. Hamilton-Jacobi method...............................80 III.6. Action-angle variables formalism.......................85 CHAPTER IV. PROBLEMS SOLVED BY MEANS OF THE PRINCIPLE OF VIRTUAL WORK.........92 vii CHAPTER V. PROBLEMS OF VARIATIONAL CALCULUS.........................................110 V.1. Elements of variational calculus........................110 V.1.1. Functionals. Functional derivative.................110 V.1.2. Extrema of functionals ............................ 120 V.2. Problems whose solutions demand elements of variational calculus...........................................125 1. Brachistochrone problem............................125 2. Catenary problem...................................129 3. Isoperimetric problem...............................132 4. Surface of revolution of minimum area...............136 5. Geodesics of a Riemannian manifold.................139 CHAPTER VI. PROBLEMS SOLVED BY MEANS OF THE LAGRANGIAN FORMALISM ............. 156 1. Atwood machine....................................156 2. Double Atwood machine.............................158 3. Pendulum with horizontally oscillating point of suspension................................163 4. Problem of two identical coupled pendulums.........172 5. Problem of two different coupled pendulums.........178 6. Problem of three identical coupled pendulums ....... 203 7. Problem of double gravitational pendulum...........210 CHAPTER VII. PROBLEMS OF EQUILIBRIUM AND SMALL OSCILLATIONS .......................... 227 CHAPTER VIII. PROBLEMS SOLVED BY MEANS OF THE HAMILTONIAN FORMALISM........265 CHAPTER IX. PROBLEMS OF CONTINUOUS SYSTEMS...........................................313 A. Problems of Classical Electrodynamics .............. 313 B. Problems of Fluid Mechanics........................332 C. Problems of Magnetofluid Dynamics and Quantum Mechanics...............................360 APPENDICES...............................................377 REFERENCES...............................................439 viii