Analytical Mechanics: Solutions to Problems in Classical Physics

453 Pages·2014·2.5 MB·English

AnAly ticAl
MechAnics
ANALY TICAL
MECHANICS
Solutions to Problems
in Classical Physics
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We can’t solve problems by using the same kind of
thinking we used when we created them.
Albert Einstein
iii
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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term ”Analytical Mechanics” is somewhat overtaken (out of date).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
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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
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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
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