Konstantin K. Likharev
Essential Graduate Physics
Lecture Notes and Problems
Beta version
Essential Graduate Physics
Lecture Notes and Problems
Beta version, December 2013
with later problem additions and error corrections
© Copyright Konstantin K. Likharev
Open access to this material is provided online, at two mirror sites:
http://commons.library.stonybrook.edu/egp/
and
https://sites.google.com/site/likharevegp/
under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0
International License
– see http://creativecommons.org/licenses/by-nc-sa/4.0/
The above copyright and license do not cover the figures adapted from published
books and journal papers (with copyright holders listed in figure captions) and publicly-accessible Web sites (with URLs cited in figure captions). Using these materials beyond the “fair use”1 boundaries requires obtaining permissions of the
respective copyright holders.
1 To the best of the author’s knowledge, the fair use clause of the U.S. copyright law (see, e.g., http://www.copyright.gov/fls/fl102.html) allows viewing, browsing, and/or downloading copyrighted materials for temporary, non-commercial, personal purposes, without requesting the copyright holder’s permission.
Front Matter
ii
K. Likharev
Preface
This is a series of lecture notes and problems on “Essential Graduate Physics”, consisting of the following four parts:
CM: Classical Mechanics (for a 1-semester course),
EM: Classical Electrodynamics (2 semesters),
QM: Quantum Mechanics (2 semesters), and
SM: Statistical Mechanics (1 semester).
The parts share a teaching style, structure, and (with a few exceptions) notation, and are interlinked by extensive cross-referencing. I believe that due to this unity, the notes may be used for teaching these courses not only in the (preferred) sequence shown above but in almost any order – or in parallel.
Each part is a two-component package consisting of:
(i) Lecture Notes chapter texts,2 with a list of exercise problems at the end of each chapter, and (ii) Exercise and Test Problems with Model Solutions files.
The series also includes this front matter, two brief reference appendices, MA: Selected Mathematical Formulas (16 pp.) and CA: Selected Physics Constants (2 pp), and a list of references.
The series is a by-product of the so-called core physics courses I taught at Stony Brook University from 1991 to 2013. Reportedly, most physics departments require their graduate students to either take a set of similar courses or pass comprehensive exams based on an approximately similar body of knowledge. This is why I hope that my notes may be useful for both instructors and students of such courses, as well as for individual learners.
The motivation for composing the lecture notes (which had to be typeset because of my horrible handwriting) and their distribution to Stony Brook students was my desperation to find textbooks I could actually use for teaching. First of all, the textbooks I could find, including the most influential Theoretical Physics series by Landau and Lifshitz, did not match my class audiences, which included experiment-oriented students, some PhD candidates from other departments, some college graduates with substandard undergraduate background, and a few advanced undergraduates. Second, for the rigid time restrictions imposed on the core physics courses, most available textbooks are way too long, and using them would mean hopping from one topic to another, picking up a chapter here and a section there, at a high risk of losing the necessary background material and logical connections between course components – and students’ interest with them. On the other hand, many textbooks lack even brief discussions of several traditional and modern topics that I believe are necessary parts of every professional physicist’s education.3,4
2 The texts are saved as separate .pdf files of each chapter, optimized for two-page viewing and double-side printing; merged files for each part and the series as a whole, convenient for search purposes, are also provided.
3 To list just a few: statics and dynamics of elastic and fluid continua, basic notions of physical kinetics, turbulence and deterministic chaos, physics of reversible and quantum computation, energy relaxation and dephasing of open quantum systems, the van der Pol method (a.k.a. the Rotating-Wave Approximation, RWA) in classical and quantum mechanics, physics of electrons and holes in semiconductors, the weak-potential and tight-binding approximations in the energy band theory, optical fiber electrodynamics, macroscopic quantum effects in Front Matter
iii
K. Likharev
The main goal of my courses was to make students familiar with the basic notions and ideas of physics (hence the series’ title), and my main effort was to organize the material in a logical sequence the students could readily follow and enjoy, at each new step understanding why exactly they need to swallow the next knowledge pill. As a backside of such a minimalistic goal, I believe that my texts may be used by advanced undergraduate physics students as well. Moreover, I hope that selected parts of the series may be useful for graduate students of other disciplines, including astronomy, chemistry, mechanical engineering, electrical, computer and electronic engineering, and material science.
At least since Confucius and Sophocles, i.e. for the past 2,500 years, teachers have known that students can master a new concept or method only if they have seen its application to at least a few particular situations. This is why in my notes, the range of theoretical physics methods is limited to the approaches that are indeed necessary for the solution of the problems I had time to discuss, and the introduction of every new technique is always accompanied by an application example or two.
Additional exercise problems are listed at the end of each chapter of the lecture notes; they may be used for homework assignments. Individual readers are strongly encouraged to solve as many of these problems as possible.5
Detailed model solutions of the exercise problems (some with additional expansion of the lecture material), and several shorter problems suitable for tests (also with model solutions), are gathered in six separate files – one per semester. These files are available for both university instructors and individual readers – free of charge, but in return for a signed commitment to avoid unlimited distribution of the solutions – see p. vii below. For instructors, these files are available not only in the Adobe Systems’
Portable Document Format (*.pdf) but also in the Microsoft Office 1997-2003 format (*.doc) free of macros, so that the problem assignments and solutions may be readily grouped, edited, etc., before their distribution to students, using either virtually any version of Microsoft Word or independent software tools – e.g., the public-domain OpenOffice.org.
I know that my texts are far from perfection. In particular, some sacrifices made at the topic selection, always very subjective, were extremely painful. (Most regretfully, I could not find time for even a brief introduction to general relativity.6) Moreover, it is almost certain that despite all my effort and the great help from SBU students and teaching assistants, not all typos/errors have been weeded out.
This is why all remarks (however candid) and suggestions by the readers would be highly appreciated; they may be sent to klikharev@gmail.com. All significant contributions will be gratefully acknowledged in future editions of the series.
Bose-Einstein condensates, Bloch oscillations and Landau-Zener tunneling, cavity QED, and the Density Functional Theory (DFT). All these topics are discussed, if only concisely, in these notes.
4 Recently, several high-quality graduate-level teaching materials became available online, including M. Fowler’s Graduate Quantum Mechanics Lectures (http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/home.html), R.
Fitzpatrick’s text on Classical Electromagnetism (farside.ph.utexas.edu/teaching/jk1/Electromagnetism.pdf), B.
Simons’ lecture notes on Advanced Quantum Mechanics (www.tcm.phy.cam.ac.uk/~bds10/aqp.html), and D.
Tong’s lecture notes on several topics (www.damtp.cam.ac.uk/user/tong/teaching.html).
5 The problems that require either longer calculations or more creative approaches (or both) are marked by asterisks.
6 For an introduction to that subject, I can recommend either its review by S. Carroll, Spacetime and Geometry, Addison-Wesley, 2003, or a longer text by A. Zee, Einstein Gravity in a Nutshell, Princeton U. Press, 2013.
Front Matter
iv
K. Likharev
Disclaimer
Since these materials are available free of charge, it is hard to imagine somebody blaming their author for deceiving “customers” for his commercial gain. Still, I would like to go a little bit beyond the usual litigation-avoiding claims,7 and offer a word of caution to potential readers, to preempt their possible later disappointment.
This is NOT a course of theoretical physics – at least in the contemporary sense of the term Though much of the included material is similar to that in textbooks on “theoretical physics”
(most notably in the famous series by L. Landau and E. Lifshitz), this lecture note series is different from them by its focus on the basic concepts and ideas of physics, their relation to experimental data, and most important applications – rather than on sophisticated theoretical techniques. Indeed, the set of theoretical methods discussed in the notes is limited to the minimum necessary for quantitative understanding of the key notions of physics and for solving a few (or rather about a thousand :-) core problems. Moreover, because of the notes’ shortness, I have not been able to cover some key fields of theoretical physics, most notably the general relativity and the quantum field theory – beyond some introductory elements of quantum electrodynamics in QM Chapter 9. If you want to work in modern theoretical physics, you need to know much more than what this series teaches!
Moreover, this is NOT a textbook – at least not the usual one
A usual textbook tries (though most commonly fails) to cover virtually all aspects of the addressed field. As a result, it is typically way too long for being fully read and understood by students during the time allocated for the corresponding course, so that the instructors are forced to pick up selected chapters and sections, frequently losing the narrative’s logic lines. In contrast, these notes are much shorter (about 200 pages per semester), enabling their thorough reading – perhaps with just a few later sections dropped, depending on the reader’s interests. I have tried to mitigate the losses due to this minimalistic approach by providing extensive further reading recommendations on the topics I had no time to cover. The reader is highly encouraged to use these sources (and/or the corresponding chapters of more detailed textbooks) on any topics of their special interest.
Then, what these notes ARE and why you may like to use them – I think
By tradition, graduate physics education consists of two main components: research experience and advanced physics courses. Unfortunately, the latter component is currently under pressure in many physics departments, apparently because of two reasons. On one hand, the average knowledge level of the students entering graduate school is not improving, so that bringing them up to the level of contemporary research becomes increasingly difficult. On the other hand, the research itself is becoming more fragmented, so that the students frequently do not feel an immediate need for a broad physics 7 Yes Virginia, these notes represent only my personal opinions, not necessarily those of the Department of Physics and Astronomy of Stony Brook University, the SBU at large, the SUNY system as a whole, the Empire State of New York, the federal agencies and private companies that funded my group’s research, etc. No dear, I cannot be held responsible for any harm, either bodily or mental, their reading may (?) cause.
Front Matter
v
K. Likharev
knowledge base for their PhD project success. Some thesis advisors, trying to maximize the time they could use their students as a cheap laboratory workforce, do not help.
I believe that this trend toward the reduction of broad physics education in graduate school is irresponsible. Experience shows that during their future research career, a typical current student will change their research fields several times. Starting from scratch in a new field is hard – terribly hard in advanced age (believe me :-). However, physics is fortunate to have a stable hard core of knowledge, which many other sciences lack. With this knowledge, students will always feel in physics at home, while without it, they may not be able even to understand research literature in the new field, and would risk being reduced to auxiliary work roles – if any at all.
I have seen the main objective of my Stony Brook courses to give an introduction to this core of physics, at the same time trying to convey my own enchantment by the unparalleled beauty of the concepts and ideas of this science, and the remarkable logic of their fusion into a wonderful single construct. Let me hope that these notes relay not only the knowledge as such but also at least a part of this enchantment.
Acknowledgments
I am extremely grateful to my faculty colleagues and other readers who commented on certain sections of the notes; here is their list (in the alphabetic order):8
A. Abanov, P. Allen, D. Averin, S. Berkovich, P.-T. de Boer, M. Fernandez-Serra,
R. F. Hernandez, P. Johnson, T. Konstantinova, A. Korotkov, V. Semenov, F. Sheldon, E. Tikhonov, O. Tikhonova, X. Wang.
(Evidently, these kind people are not responsible for the remaining deficiencies.)
The Department of Physics and Astronomy of Stony Brook University was very responsive to my kind requests of certain time ordering of my teaching assignments, that was beneficial for note writing and editing. The department, and the university as a whole, also provided a very friendly general environment for my work there for almost three decades.
A large part of my scientific background and experience, reflected in these materials, came from my education (and then research work) in the Department of Physics of Moscow State University.
And last but not least, I would like to thank my wife Lioudmila for several good bits of advice on aesthetic aspects of note typesetting, and more importantly for all her love, care, and patience – without them, this writing project would be impossible.
Konstantin K. Likharev
https://you.stonybrook.edu/likharev/
8 I am very much sorry that I have not kept proper records from the beginning of my lectures at Stony Brook, so I cannot list all the numerous students and TAs who had kindly attracted my attention to typos in earlier versions of these notes. Needless to say, I am very grateful to them all as well.
Front Matter
vi
K. Likharev
Problem Solution Request Templates
Requests should be sent to either klikharev@gmail.com or konstantin.likharev@stonybrook.edu in either of the following forms:
- an email from a valid university address,
- a scanned copy of a signed letter – as an email attachment.
Approximate contents:
A. Request from a Prospective Instructor
Dear Dr. Likharev,
I plan to use your lecture notes and problems of the Essential Graduate Physics series, part(s)
<select: CM, EM, QM, SM>, in my course <title> during <semester, year> in the <department, university>. I would appreciate sending me the file(s) Exercise and Test Problems with Model Solutions of that part(s) of the series in the <select: .pdf, both .doc and .pdf> format(s).
I will avoid unlimited distribution of the solutions, in particular their posting on externally searchable websites. If I distribute the solutions among my students, I will ask them to adhere to the same restraint.
I will let you know of any significant typos/deficiencies I may find.
Sincerely, <signature, full name, university position, work phone number>
B. Request from an Individual Learner
Dear Dr. Likharev,
I plan to use your lecture notes and problems of the Essential Graduate Physics series, part(s)
<select: CM, EM, QM, SM>, for my personal education. I would appreciate sending me the file(s) Exercise and Test Problems with Model Solutions of that part(s) of the series.
I will not share the material with anyone, and will not use it for passing courses that are officially based on your series.
I will let you know of any significant typos/deficiencies I may find.
Sincerely, <signature, full name, present home address (in English), acting phone number>
Front Matter
vii
K. Likharev
Notation
Abbreviations
Fonts
Symbols
Eq. any formula (e.g., equality) F,
F scalar variables9
time differentiation operator ( d/ dt)
Fig. figure
F, F vector variables
spatial differentiation vector ( del)
Sec. section
ˆ
F F ˆ
, scalar operators
approximately equal to
c.c. complex conjugate
ˆF F ˆ
, vector operators
~ of the same order as
h. c. Hermitian conjugate
F matrix
proportional to
Fjj’ matrix element
equal to by definition (or evidently)
scalar (“dot-”) product
Parts of the series
vector (“cross-”) product10
CM: Classical Mechanics
time averaging
EM: Classical Electrodynamics
statistical averaging
QM: Quantum Mechanics
[ , ] commutator
SM: Statistical Mechanics
{ , } anticommutator
Appendices
n unit vector
MA: Selected Mathematical Formulas
CA: Selected Physical Constants
Prime signs
The prime signs (′, ″, etc) are used to distinguish similar variables or indices (such as j and j′ in the matrix element above), rather than to denote derivatives.
Formulas
The most general and/or important formulas are highlighted with blue frames and short titles on the margins.
Numbering
Chapter numbers are dropped in all references to formulas, figures, footnotes, and problems within the same chapter.
9 The same letter, typeset in different fonts, typically denotes different variables.
10 On a few occasions, the cross sign is used to emphasize the usual multiplication of scalars.
Front Matter
viii
K. Likharev
General Table of Contents
Exercise
CM: Classical Mechanics
Pages Problems
Table of Contents
4
–
Chapter 1. Review of Fundamentals
14
14
Chapter 2. Lagrangian Analytical Mechanics
14
11
Chapter 3. A Few Simple Problems
22
26
Chapter 4. Rigid Body Motion
32
36
Chapter 5. Oscillations
38
22
Chapter 6. From Oscillations to Waves
30
26
Chapter 7. Deformations and Elasticity
38
23
Chapter 8. Fluid Mechanics
30
25
Chapter 9. Deterministic Chaos
14
5
Chapter 10. A Bit More of Analytical Mechanics
16
10
CM TOTAL: 252
198
Additional file (available upon request):
Pages Problems
Exercise and Test Problems with Model Solutions 340
198 + 48 = 246
Exercise
EM: Classical Electrodynamics
Pages Problems
Table of Contents
4
–
Chapter 1. Electric Charge Interaction
20
19
Chapter 2. Charges and Conductors
68
47
Chapter 3. Dipoles and Dielectrics
28
30
Chapter 4. DC Currents
16
15
Chapter 5. Magnetism
42
29
Chapter 6. Electromagnetism
38
30
Chapter 7. Electromagnetic Wave Propagation
68
39
Chapter 8. Radiation, Scattering, Interference, and Diffraction
38
28
Chapter 9. Special Relativity
56
42
Chapter 10. Radiation by Relativistic Charges
40
15
EM TOTAL: 418
295
Additional file (available upon request):
Pages
Problems
Exercise and Test Problems with Model Solutions
464
295 + 54 = 349
Front Matter
ix
K. Likharev
Exercise
QM: Quantum Mechanics
Pages Problems
Table of Contents
4
–
Chapter 1. Introduction
28
15
Chapter 2. 1D Wave Mechanics
76
43
Chapter 3. Higher Dimensionality Effects
64
40
Chapter 4. Bra-ket Formalism
52
34
Chapter 5. Some Exactly Solvable Problems
48
48
Chapter 6. Perturbative Approaches
36
31
Chapter 7. Open Quantum Systems
50
14
Chapter 8. Multiparticle Systems
52
31
Chapter 9. Introduction to Relativistic Quantum Mechanics
36
21
Chapter 10. Making Sense of Quantum Mechanics
16
–
QM TOTAL: 462
277
Additional file (available upon request):
Pages
Problems
Exercise and Test Problems with Model Solutions 518
277 + 70 = 347
Exercise
SM: Statistical Mechanics
Pages Problems
Table of Contents
4
–
Chapter 1. Review of Thermodynamics
24
16
Chapter 2. Principles of Physical Statistics
44
32
Chapter 3. Ideal and Not-So-Ideal Gases
34
29
Chapter 4. Phase Transitions
36
18
Chapter 5. Fluctuations
44
21
Chapter 6. Elements of Kinetics
38
15
SM TOTAL: 224
131
Additional file (available upon request):
Pages
Problems
Exercise and Test Problems with Model Solutions
232
131 + 23 = 154
Appendices
Pages
MA: Selected Mathematical Formulas
16
CA: Selected Physical Constants
2
References
Pages
A partial list of books used at work on the series
2
Front Matter
x
Konstantin K. Likharev
Essential Graduate Physics
Lecture Notes and Problems
Beta version
Open online access at
http://commons.library.stonybrook.edu/egp/
and
https://sites.google.com/site/likharevegp/
Part CM:
Classical Mechanics
Last corrections: August 22, 2022
A version of this material was published in 2017 under the title
Classical Mechanics: Lecture notes
IOPP, Essential Advanced Physics – Volume 1, ISBN 978-0-7503-1398-8,
with model solutions of the exercise problems published in 2018 under the title
Classical Mechanics: Problems with solutions
IOPP, Essential Advanced Physics – Volume 2, ISBN 978-0-7503-1401-5
However, by now this online version of the lecture notes, as well as the
problem solutions available from the author by request, have been much better corrected
About the author:
https://you.stonybrook.edu/likharev/
© K. Likharev
CM: Classical Mechanics
Table of Contents
Chapter 1. Review of Fundamentals (14 pp.)
1.0. Terminology: Mechanics and dynamics
1.1. Kinematics: Basic notions
1.2. Dynamics: Newton laws
1.3. Conservation laws
1.4. Potential energy and equilibrium
1.5. OK, can we go home now?
1.6. Self-test problems (14)
Chapter 2. Lagrangian Analytical Mechanics (14 pp.)
2.1. Lagrange equation
2.2. Three simple examples
2.3. Hamiltonian function and energy
2.4. Other conservation laws
2.5. Exercise problems (11)
Chapter 3. A Few Simple Problems (22 pp.)
3.1. One-dimensional and 1D-reducible systems
3.2. Equilibrium and stability
3.3. Hamiltonian 1D systems
3.4. Planetary problems
3.5. Elastic scattering
3.6. Exercise problems (26)
Chapter 4. Rigid Body Motion (32 pp.)
4.1. Translation and rotation
4.2. Inertia tensor
4.3. Fixed-axis rotation
4.4. Free rotation
4.5. Torque-induced precession
4.6. Non-inertial reference frames
4.7. Exercise problems (36)
Chapter 5. Oscillations (38 pp.)
5.1. Free and forced oscillations
5.2. Weakly nonlinear oscillations
5.3. Reduced equations
5.4. Self-oscillations and phase locking
5.5. Parametric excitation
5.6. Fixed point classification
5.7. Numerical approaches
5.8. Higher-harmonic and subharmonic oscillations
5.9. Relaxation oscillations
5.10. Exercise problems (22)
Table of Contents
Page 2 of 4
CM: Classical Mechanics
Chapter 6. From Oscillations to Waves (30 pp.)
6.1. Two coupled oscillators
6.2. N coupled oscillators
6.3. 1D waves
6.4. Acoustic waves
6.5. Standing waves
6.6. Wave decay and attenuation
6.7. Nonlinear and parametric effects
6.8. Exercise problems (26)
Chapter 7. Deformations and Elasticity (38 pp.)
7.1. Strain
7.2. Stress
7.3. Hooke’s law
7.4. Equilibrium
7.5. Rod bending
7.6. Rod torsion
7.7. 3D acoustic waves
7.8. Elastic waves in thin rods
7.9. Exercise problems (23)
Chapter 8. Fluid Mechanics (30 pp.)
8.1. Hydrostatics
8.2. Surface tension effects
8.3. Kinematics
8.4. Dynamics: Ideal fluids
8.5. Dynamics: Viscous fluids
8.6. Turbulence
8.7. Exercise problems (25)
Chapter 9. Deterministic Chaos (14 pp.)
9.1. Chaos in maps
9.2. Chaos in dynamic systems
9.3. Chaos in Hamiltonian systems
9.4. Chaos and turbulence
9.5. Exercise problems (5)
Chapter 10. A Bit More of Analytical Mechanics (16 pp.)
10.1. Hamilton equations
10.2. Adiabatic invariance
10.3. The Hamilton principle
10.4. The Hamilton-Jacobi equation
10.5. Exercise problems (10)
* * *
Additional file (available from the author upon request):
Exercise and Test Problems with Model Solutions (198 + 48 = 246 problems; 340 pp.) Table of Contents
Page 3 of 4
CM: Classical Mechanics
This page is
intentionally left
blank
Table of Contents
Page 4 of 4
CM: Classical Mechanics
Chapter 1. Review of Fundamentals
After a brief discussion of the title and contents of the course, this introductory chapter reviews the basic notions and facts of the non-relativistic classical mechanics, that are supposed to be known to the reader from their undergraduate studies. 1 Due to this reason, the discussion is very short.
1.0. Terminology: Mechanics and dynamics
A more fair title for this course would be Classical Mechanics and Dynamics, because the notions of mechanics and dynamics, though much intertwined, are still somewhat different. The term mechanics, in its narrow sense, means the derivation of equations of motion of point-like particles and their systems (including solids and fluids), solution of these equations, and interpretation of the results.
Dynamics is a more ambiguous term; it may mean, in particular:
(i) the part of physics that deals with motion (in contrast to statics);
(ii) the part of physics that deals with reasons for motion (in contrast to kinematics); (iii) the part of mechanics that focuses on its two last tasks, i.e. the solution of the equations of motion and discussion of the results.2
Because of this ambiguity, after some hesitation, I have opted to use the traditional name Classical Mechanics, with the word Mechanics in its broad sense that includes (similarly to Quantum Mechanics and Statistical Mechanics) studies of dynamics of some non-mechanical systems as well.
1.1. Kinematics: Basic notions
The basic notions of kinematics may be defined in various ways, and some mathematicians pay much attention to alternative systems of axioms and the relations between them. In physics, we typically stick to less rigorous ways (in order to proceed faster to solving particular problems) and end debating any definition as soon as “everybody in the room” agrees that we are all speaking about the same thing –
at least in the context they are being discussed. Let me hope that the following notions used in classical mechanics do satisfy this criterion in our “room”:
1 The reader is advised to perform (perhaps after reading this chapter as a reminder) a self-check by solving a few problems of those listed in Sec. 1.6. If the results are not satisfactory, it may make sense to start with some remedial reading. For that, I could recommend, e.g., J. Marion and S. Thornton, Classical Dynamics of Particles and Systems, 5th ed., Saunders, 2003; and D. Morin, Introduction to Classical Mechanics, Cambridge U., 2008.
2 The reader may have noticed that the last definition of dynamics is suspiciously close to the part of mathematics devoted to differential equation analysis; what is the difference? An important bit of philosophy: physics may be defined as an art (and a bit of science :-) of describing Mother Nature by mathematical means; hence in many cases the approaches of a mathematician and a physicist to a problem are very similar. The main difference between them is that physicists try to express the results of their analyses in terms of the properties of the systems under study , rather than the functions describing them, and as a result develop a sort of intuition (“gut feeling”) about how other similar systems may behave, even if their exact equations of motion are somewhat different – or not known at all. The intuition so developed has an enormous heuristic power, and most discoveries in physics have been made through gut-feeling-based insights rather than by plugging one formula into another one.
© K. Likharev
CM: Classical Mechanics
(i) All the Euclidean geometry notions, including the point, the straight line, the plane, etc.3
(ii) Reference frames: platforms for observation and mathematical description of physical phenomena. A reference frame includes a coordinate system used for measuring the point’s position (namely, its radius vector r that connects the coordinate origin to the point – see Fig. 1) and a clock that measures time t. A coordinate system may be understood as a certain method of expressing the radius vector r of a point as a set of its scalar coordinates. The most important of such systems (but by no means the only one) are the Cartesian (orthogonal, linear) coordinates 4 rj of a point, in which its radius vector may be represented as the following sum:
3
r n r ,
(1.1)
Cartesian
j j
coordinates
j1
where n1, n2, and n3 are unit vectors directed along the coordinate axis – see Fig. 1.5
r z
3
n
point
3
r
0
n
Fig. 1.1. Cartesian coordinates of a point.
r y
n
2
2
1
r x
1
(iii) The absolute (“Newtonian”) space/time,6 which does not depend on the matter distribution.
The space is assumed to have the Euclidean metric, which may be expressed as the following relation between the length r of any radius vector r and its Cartesian coordinates: 3
r
2
2
r
2
r ,
(1.2)
j
Euclidean
j1
metric
while time t is assumed to run similarly in all reference frames. These assumptions are critically revised in the relativity theory (which, in this series, is discussed only starting from EM Chapter 9.) 3 All these notions are of course abstractions: simplified models of the real objects existing in Nature. But please always remember that any quantitative statement made in physics (e.g., a formula) may be strictly valid only for an approximate model of a physical system. (The reader should not be disheartened too much by this fact: experiments show that many models make extremely precise predictions of the behavior of the real systems.) 4 In this series, the Cartesian coordinates (introduced in 1637 by René Descartes, a.k.a. Cartesius) are denoted either as either { r 1, r 2, r 3} or { x, y, z}, depending on convenience in each particular case. Note that axis numbering is important for operations like the vector (“cross”) product; the “correct” (meaning generally accepted) numbering order is such that the rotation n1 n2 n3 n1… looks counterclockwise if watched from a point with all rj > 0 – like the one shown in Fig. 1.
5 Note that the representation (1) is also possible for locally-orthogonal but curvilinear (for example, polar/cylindrical and spherical) coordinates, which will be extensively used in this series. However, such coordinates are not Cartesian, and for them some of the relations given below are invalid – see, e.g., MA Sec. 10.
6 These notions were formally introduced by Sir Isaac Newton in his main work, the three-volume Philosophiae Naturalis Principia Mathematica published in 1686-1687, but are rooted in earlier ideas by Galileo Galilei, published in 1632.
Chapter 1
Page 2 of 14
CM: Classical Mechanics
(iv) The (instant) velocity of the point,
Velocity
dr
v( t)
r ,
(1.3)
dt
and its acceleration:
dv
Acceleration
a( t)
v r .
(1.4)
dt
(v) Transfer between reference frames. The above definitions of vectors r, v, and a depend on the chosen reference frame (are “reference-frame-specific”), and we frequently need to relate those vectors as observed in different frames. Within Euclidean geometry, the relation between the radius vectors in two frames with the corresponding axes parallel at the moment of interest (Fig. 2), is very simple:
Radius
vector’s
r
'
0
in
r
0
in
r
.
(1.5)
0
'
0
in
transformation
point
r
in 0'
r in 0
Fig. 1.2. Transfer between two reference frames.
'
0
r
0
0 in 0'
If the frames move versus each other by translation only (no mutual rotation!), similar relations are valid for the velocities and accelerations as well:
v
'
0
in
v
0
in
v
,
(1.6)
0
'
0
in
a
'
0
in
a 0
in
a
.
(1.7)
0
'
0
in
Note that in the case of mutual rotation of the reference frames, the transfer laws for velocities and accelerations are more complex than those given by Eqs. (6) and (7). Indeed, in this case, the notions like v0in 0 ’ are not well defined: different points of an imaginary rigid body connected to frame 0 may have different velocities when observed in frame 0 ’. It will be more natural for me to discuss these more general relations at the end of Chapter 4 devoted to rigid body motion.
(vi) A particle (or “point particle”): a localized physical object whose size is negligible, and the shape is irrelevant to the given problem. Note that the last qualification is extremely important. For example, the size and shape of a spaceship are not too important for the discussion of its orbital motion but are paramount when its landing procedures are being developed. Since classical mechanics neglects the quantum mechanical uncertainties,7 in it, the position of a particle at any particular instant t, may be identified with a single geometrical point, i.e. with a single radius vector r( t). The formal final goal of classical mechanics is finding the laws of motion r( t) of all particles participating in the given problem.
7 This approximation is legitimate when the product of the coordinate and momentum scales of the particle motion is much larger than Planck’s constant ~ 10-34 Js. More detailed conditions of the classical mechanics’
applicability depend on a particular system – see, e.g., the QM part of this series.
Chapter 1
Page 3 of 14
CM: Classical Mechanics
1.2. Dynamics: Newton’s laws
Generally, the classical dynamics is fully described (in addition to the kinematic relations discussed above) by three Newton’s laws. In contrast to the impression some textbooks on theoretical physics try to create, these laws are experimental in nature, and cannot be derived from purely theoretical arguments.
I am confident that the reader of these notes is already familiar with Newton’s laws,8 in one or another formulation. Let me note only that in some formulations, the 1st Newton’s law looks just a particular case of the 2nd law – when the net force acting on a particle equals zero. To avoid this duplication, the 1st law may be formulated as the following postulate:
There exists at least one reference frame, called inertial, in which any free particle (i.e. a 1st Newton
particle fully isolated from the rest of the Universe) moves with v = const, i.e. with a = 0.
law
Note that according to Eq. (7), this postulate immediately means that there is also an infinite number of inertial reference frames – because all frames 0 ’ moving without rotation or acceleration relative to the postulated inertial frame 0 (i.e. having a0in 0 ’ = 0) are also inertial.
On the other hand, the 2nd and 3rd Newton’s laws may be postulated together in the following elegant way. Each particle, say number k, may be characterized by a scalar constant (called mass mk), such that at any interaction of N particles (isolated from the rest of the Universe), in any inertial system, N
N
Total
P p m v
(1.8)
momentum
k
const.
k
k
and its
k 1
k 1
conservation
(Each component of this sum,
p m v ,
(1.9)
Particle’s
k
k
k
momentum
is called the mechanical momentum 9 of the corresponding particle, while the sum P, the total momentum of the system.)
Let us apply this postulate to just two interacting particles. Differentiating Eq. (8) written for this case, over time, we get
p p
.
(1.10)
1
2
Let us give the derivative p (which is a vector) the name of the force F exerted on particle 1. In our 1
current case, when the only possible source of the force is particle 2, it may be denoted as F12: p F .
1
12
Similarly, F p , so that Eq. (10) becomes the 3rd Newton’s law
21
2
F F .
(1.11) 3rd Newton
12
21
law
Plugging Eq. (1.9) into these force definitions, and differentiating the products mkv k, taking into account that particle masses are constants,10 we get that for the k and k’ taking any of values 1, 2, 8 Due to the genius of Sir Isaac, these laws were formulated in the same Principia (1687), well ahead of the physics of his time.
9 The more extended term linear momentum is typically used only in cases when there is a chance of confusion with the angular momentum of the same particle/system – see below. The present-day definition of the linear momentum and the term itself belong to John Wallis (1670), but the concept may be traced back to more vague notions of several previous scientists – all the way back to at least a 570 AD work by John Philoponus.
Chapter 1
Page 4 of 14
CM: Classical Mechanics
m v m a F ,
where k' k..
(1.12)
k
k
k
k
kk '
Now, returning to the general case of several interacting particles, and making an additional (but very natural) assumption that all partial forces F kk’ acting on particle k add up as vectors, we may generalize Eq. (12) into the 2nd Newton’s law
2nd Newton
m a p
law
F F ,
(1.13)
k
k
k
kk '
k
k ' k
that allows a clear interpretation of the mass as a measure of a particle’s inertia.
As a matter of principle, if the dependence of all pair forces F kk’ of particle positions (and generally of time as well) is known, Eq. (13) augmented with the kinematic relations (2) and (3) allows calculation of the laws of motion r k( t) of all particles of the system. For example, for one particle the 2nd law (13) gives an ordinary differential equation of the second order:
mr F(r, t) ,
(1.14)
which may be integrated – either analytically or numerically.
In certain cases, this is very simple. As an elementary example, for local motions with r << r, Newton’s gravity force11
mm'
Newton’s
F G
R
(1.15)
3
gravity law
R
(where R r – r ’ is the distance between particles of masses m and m’)12 may be approximated as Uniform
F g
m ,
(1.16)
gravity field
with the vector g –( Gm’/ R 3)R being constant.13 As a result, m in Eq. (13) cancels, it is reduced to just r g = const, and may be easily integrated twice:
t
t
2
t
r( t) v( t) g dt' v(0) g t v(0), r( t) v( t' ) dt' r(0) g
v( )
0 t r(0)
,
(1.17)
2
0
0
thus giving the generic solution to all those undergraduate problems on the projectile motion, which should be so familiar to the reader.
10 Note that this may not be true for composite bodies of varying total mass M (e.g., rockets emitting jets, see Problem 11), in these cases the momentum’s derivative may differ from Ma.
11 Introduced in the same famous Principia!
12 The fact that the masses participating in Eqs. (14) and (16) are equal, the so-called weak equivalence principle, is actually highly nontrivial, but has been repeatedly verified experimentally with gradually improved relative accuracy, currently reaching ~10-14 – see P. Touboul et al., Phys. Rev. Lett. 119, 231101 (2017).
13 Of course, the most important particular case of Eq. (16) is the gravity field near the Earth’s surface. In this case, using the fact that Eq. (15) remains valid for the gravity field created by a spherically-uniform sphere, we get g = GM
2
E/ R E , where M E and R E are the Earth’s mass and radius. Plugging in their values, M E 5.971024 kg and R E 6.37106 m, we get g 9.82 m/s2. The experimental value of g varies from 9.78 to 9.83 m/s2 at various locations on the surface (due to the deviations of Earth’s shape from a sphere, and the location-dependent effect of the centrifugal “inertial force” – see Sec. 4.5 below), with an average value of approximately 9.807 m/s2.
Chapter 1
Page 5 of 14
Essential Graduate Physics
CM: Classical Mechanics
All this looks (and indeed is) very simple, but in most other cases, Eq. (13) leads to more complex calculations. As an example, let us think about how would we use it to solve another simple problem: a bead of mass m sliding, without friction, along a round ring of radius R in a gravity field obeying Eq. (16) – see Fig. 3. (This system is equivalent to the usual point pendulum, i.e. a point mass suspended from point 0 on a light rod or string, and constrained to move in one vertical plane.) R
initial
0
position,
v = 0
N
intermediate
position
v
final
Fig. 1.3. A bead sliding along a vertical ring.
position, mg
v = ?
Suppose we are only interested in the bead’s velocity v at the lowest point, after it has been dropped from the rest at the rightmost position. If we want to solve this problem using only the Newton laws, we have to make the following steps:
(i) consider the bead in an arbitrary intermediate position on a ring, described, for example by the angle θ shown in Fig. 3;
(ii) draw all the forces acting on the particle – in our current case, the gravity force mg and the reaction force N exerted by the ring – see Fig. 3 above
(iii) write the Cartesian components of the 2nd Newton’s law (14) for the bead acceleration: max
= Nx, may = Ny – mg,
(iv) recognize that in the absence of friction, the force N should be normal to the ring, so that we can use two additional equations, Nx = – N sin and Ny = N cos ; (v) eliminate unknown variables N, Nx, and Ny from the resulting system of four equations, thus getting a single second-order differential equation for one variable, for example, :
mR mg sin ;
(1.18)
(vi) use the mathematical identity d 2
/ 2/
d to integrate this equation over once to get
an expression relating the velocity and the angle ; and, finally,
(vii) using our specific initial condition ( 0 at / 2 ), find the final velocity as v
R at
0 .
All this is very much doable, but please agree that the procedure it too cumbersome for such a simple problem. Moreover, in many other cases even writing equations of motion along relevant coordinates is very complex, and any help the general theory may provide is highly valuable. In many cases, such help is given by conservation laws; let us review the most general of them.
Chapter 1
Page 6 of 14
CM: Classical Mechanics
1.3. Conservation laws
(i) Energy conservation is arguably the most general law of physics, but in mechanics, it takes a more humble form of mechanical energy conservation, which has limited applicability. To derive it, we first have to define the kinetic energy of a particle as14
Kinetic
m
energy
2
T
v ,
(1.19)
2
and then recast its differential as15
m
m
dr dv
dp
2
dT d v d v v mv dv m
dr
.
(1.20)
2
2
dt
dt
Now plugging in the momentum’s derivative from the 2nd Newton’s law, dp/ dt = F, where F is the full force acting on the particle, we get dT = F dr. The integration of this equality along the particle’s trajectory connecting some points A and B gives the formula that is sometimes called the work-energy principle:
B
Work-
energy
Δ T T (r ) T (r ) F r
d ,
(1.21)
B
A
principle
A
where the integral on the right-hand side is called the work of the force F on the path from A to B.
The next step may be made only for a potential (also called “conservative”) force that may be represented as the (minus) gradient of some scalar function U(r), called the potential energy. 16 The vector operator (called either del or nabla) of spatial differentiation17 allows a very compact expression of this fact:
Force vs
F
potential
U .
(1.22)
energy
For example, for the uniform gravity field (16),
U mgh const,
(1.23)
where h is the vertical coordinate directed “up” – opposite to the direction of the vector g.
Integrating the tangential component F of the vector F given by Eq. (22), along an arbitrary path connecting the points A and B, we get
B
B
F dr F r
d U (r ) U (r )
,
(1.24)
A
B
A
A
14 In such quantitative form, the kinetic energy was introduced (under the name “living force”) by Gottfried Leibniz and Johann Bernoulli (circa 1700), though its main properties (21) and (27) had not been clearly revealed until an 1829 work by Gaspard-Gustave de Coriolis. The modern term “kinetic energy” was coined only in 1849-1851 by Lord Kelvin (born William Thomson).
15 In these notes, ab denotes the scalar (or “dot-”) product of vectors a and b – see, e.g., MA Eq. (7.1).
16 Note that because of its definition via the gradient, the potential energy is only defined to an arbitrary additive constant. This notion had been used already by G. Leibniz, though the term we are using for it nowadays was introduced much later (in the mid-19th century) by William Rankine.
17 Its basic properties are listed in MA Sec. 8.
Chapter 1
Page 7 of 14
CM: Classical Mechanics
i.e. work of potential forces may be represented as the difference of values of the function U(r) in the initial and final points of the path. (Note that according to Eq. (24), the work of a potential force on any closed path, with rA = rB, is zero.)
Now returning to Eq. (21) and comparing it with Eq. (24), we see that
T (r ) T (r ) U (r ) U (r ), i.e. T (r ) U (r ) T (r ) U (r ) , (1.25)
B
A
A
B
A
A
A
A
so that the total mechanical energy E, defined as
Total
E T U ,
(1.26) mechanical
energy
is indeed conserved:
Mechanical
E(r ) E(r ) ,
(1.27) energy:
A
B
conservation
but for conservative forces only. (Non-conservative forces may change E by either transferring energy from its mechanical form to another form, e.g., to heat in the case of friction, or by pumping the energy into the system under consideration from another, “external” system.)
The mechanical energy conservation allows us to return for just a second to the problem shown in Fig. 3 and solve it in one shot by writing Eq. (27) for the initial and final points:18
m
0
2
mgR
v .
0
(1.28)
2
The (elementary) solution of Eq. (28) for v immediately gives us the desired answer. Let me hope that the reader agrees that this way of problem’s solution is much simpler, and I have earned their attention to discuss other conservation laws – which may be equally effective.
(ii) Linear momentum. The conservation of the full linear momentum of any system of particles isolated from the rest of the world was already discussed in the previous section, and may serve as the basic postulate of classical dynamics – see Eq. (8). In the case of one free particle, the law is reduced to the trivial result p = const, i.e. v = const. If a system of N particles is affected by external forces F(ext), we may write
N
F (ext)
F
F .
(1.29)
k
k
kk'
k 1
If we sum up the resulting Eqs. (13) for all particles of the system then, due to the 3rd Newton’s law (11) valid for any indices k k’, the contributions of all internal forces F kk’ to the resulting double sum on the right-hand side cancel, and we get the following equation:
N
System’s
(ext)
P F
,
where (ext)
(ext)
F
F .
(1.30) momentum
k
evolution
k 1
It tells us that the translational motion of the system as the whole is similar to that of a single particle, under the effect of the net external force F(ext). As a simple sanity check, if the external forces have a zero sum, we return to the postulate (8). Just one reminder: Eq. (30), as its precursor Eq. (13), is only valid in an inertial reference frame.
18 Here the arbitrary constant in Eq. (23) is chosen so that the potential energy is zero at the final point.
Chapter 1
Page 8 of 14
CM: Classical Mechanics
I hope that the reader knows numerous examples of the application of the linear momentum’s conservation law, including all these undergraduate problems on car collisions, where the large collision forces are typically not known so that the direct application of Eq. (13) to each car is impracticable.
(iii) The angular momentum of a particle19 is defined as the following vector:20
Angular
momentum:
definition
L r p,
(1.31)
where ab means the vector (or “cross-“) product of the vector operands.21 Differentiating Eq. (31) over time, we get
L r p r .
p
(1.32)
In the first product, r is just the velocity vector v, parallel to the particle momentum p = mv, so that this term vanishes since the vector product of any two parallel vectors equals zero. In the second product, p
is equal to the full force F acting on the particle, so that Eq. (32) is reduced to Angular
momentum:
L τ,
(1.33)
evolution
where the vector
Torque
τ r F,
(1.34)
is called the torque exerted by force F.22 (Note that the torque is reference-frame specific – and again, the frame has to be inertial for Eq. (33) to be valid, because we have used Eq. (13) for its derivation.) For an important particular case of a central force F that is directed along the radius vector r of a particle, the torque vanishes, so that (in that particular reference frame only!) the angular momentum is Angular conserved:
momentum:
L const.
(1.35)
conservation
For a system of N particles, the total angular momentum is naturally defined as
System’s
angular
N
momentum:
L L .
(1.36)
k
definition
k 1
Differentiating this equation over time, using Eq. (33) for each L , and again partitioning each force per k
Eq. (29), we get
N
N
(ext)
L r F τ
where
,
(ext)
τ
(ext)
r
F
(1.37)
k
kk
.
'
k
k
k , k ' 1
k 1
k ' k
The first (double) sum may be always divided into pairs of the type (r k F kk’ + r k’ F k’k). With a natural assumption of the central forces, F kk’ (r k – r k’), each of these pairs equals zero. Indeed, in this case, 19 Here we imply that the internal motions of the particle, including its rotation about its axis, are negligible.
(Otherwise, it could not be represented by a point, as was postulated in Sec. 1.)
20 This explicit definition of the angular momentum (in different mathematical forms, and under the name of
“moment of rotational motion”) has appeared in scientific publications only in the 1740s, though the fact of its conservation (35) in the field of central forces, in the form of the 2nd Kepler law (see Fig. 3.4 below), had been proved already by I. Newton in his Principia.
21 See, e.g., MA Eq. (7.3).
22 Alternatively, especially in mechanical engineering, torque is called the force moment. This notion may be traced all the way back to Archimedes’ theory of levers developed in the 3rd century BC.
Chapter 1
Page 9 of 14
CM: Classical Mechanics
each component of the pair is a vector perpendicular to the plane containing the positions of both particles and the reference frame origin, i.e. to the plane of the drawing in Fig. 4.
)
(ext
F
F k
kk'
r k
/2
)
(ext
F
F k'
h h
k'k
kk'
k'k
r k'
Fig. 1.4. Internal and external forces, and
the internal torque cancellation in a system
of two particles.
0
Also, due to the 3rd Newton’s law (11), these two forces are equal and opposite, and the magnitude of each term in the sum may be represented as Fkk’ hkk’, with equal “lever arms” hkk’ = hk’k.
As a result, each sum (r kF kk’ + r k’F k’k), and hence the whole double sum in Eq. (37) vanish, and it is reduced to a very simple result,
System’s
(ext)
L τ
,
(1.38) angular
momentum:
evolution
which is similar to Eq. (33) for a single particle, and is the angular analog of Eq. (30).
In particular, Eq. (38) shows that if the full external torque (ext) vanishes for some reason (e.g., if the system of particles is isolated from the rest of the Universe), the conservation law (35) is valid for the full angular momentum L even if its individual components L k are not conserved due to interparticle interactions.
Please note again that since the conservation laws may be derived from Newton’s laws (as was done above), they do not introduce anything new to the dynamics of any system. Indeed, from the mathematical point of view, the conservation laws discussed above are just the first integrals of the second-order differential equations of motion following from Newton’s laws. However, for a physicist, thinking about particular systems in the terms of the conserved (or potentially conserved) quantities frequently provides decisive clues on their dynamics.
1.4. Potential energy and equilibrium
Another important role of the potential energy U, especially for dissipative systems whose total mechanical energy E is not conserved because it may be drained to the environment, is finding the positions of equilibrium (sometimes called the fixed points) of the system and analyzing their stability with respect to small perturbations. For a single particle, this is very simple: the force (22) vanishes at each extremum (either minimum or maximum) of the potential energy.23 (Of those fixed points, only the minimums of U(r) are stable – see Sec. 3.2 below for a discussion of this point.) A slightly more subtle case is a particle with an internal potential energy U(r), subjected to an additional external force F(ext)(r). In this case, the stable equilibrium is reached at the minimum of not the function U(r), but of what is sometimes called the Gibbs potential energy 23 Assuming that the additional, non-conservative forces (such as viscosity) responsible for the mechanical energy drain, vanish at equilibrium – as they typically do. (The static friction is one counter-example.) Chapter 1
Page 10 of 14
CM: Classical Mechanics
r
Gibbs’
potential
U r U r F
r '
r
d '
(1.39)
G
ext
,
energy
which is defined, just as U(r) is, to an arbitrary additive constant.24 The proof of Eq. (39) is very simple: in an extremum of this function, the total force acting on the particle,
r
tot
ext
ext
F
F F
U F
' r d ' r U
(1.40)
G
vanishes, as it is necessary for equilibrium.
Physically, the difference U G – U specified by Eq. (39) is the r-dependent part of the potential energy U(ext) of the external system responsible for the force F(ext), so that U G is just the total potential energy U + U(ext), excluding its part that does not depend on r and hence is irrelevant for the analysis.
According to the 3rd Newton’s law, the force exerted by the particle on the external system equals (–
F(ext)), so that its work (and hence the change of U(ext) due to the change of r) is given by the second term on the right-hand side of Eq. (39). Thus the condition of equilibrium, U G = 0, is just the condition of an extremum of the total potential energy, U + U(ext) + const, of the two interacting systems.
For the simplest (and very frequent) case when the applied force is independent of the particle’s position, the Gibbs potential energy (39) is just25
U r U r F
r
.
(1.41)
G
ext
const
As the simplest example, consider a 1D deformation of the usual elastic spring providing the returning force (– x), where x is the deviation from its equilibrium. As follows from Eq. (22), its potential energy is U = x 2/2 + const, so that its minimum corresponds to x = 0. Now let us apply an additional external force F, say independent of x. Then the equilibrium deformation of the spring, x 0 = F/, corresponds to the minimum of not U, but rather of the Gibbs potential energy (41), in our particular case taking the form
x 2
U U Fx
Fx .
(1.42)
G
2
1.5. OK, we’ve got it – can we go home now?
Sorry, not yet. In many cases, the conservation laws discussed above provide little help, even in systems without dissipation. As a simple example, consider a generalization of the bead-on-the-ring problem shown in Fig. 3, in which the ring is rotated by external forces, with a constant angular velocity
, about its vertical diameter.26 In this problem (to which I will repeatedly return below, using it as an 24 Unfortunately, in most textbooks, the association of the (unavoidably used) notion of U G with the glorious name of Josiah Willard Gibbs is postponed until a course of statistical mechanics and/or thermodynamics, where U G is a part of the Gibbs free energy, in contrast to U, which is a part of the Helmholtz free energy – see, e.g., SM
Sec. 1.4. I use this notion throughout my series, because the difference between U G and U, and hence that between the Gibbs and Helmholtz free energies, has nothing to do with statistics or thermal motion, and belongs to the whole physics, including not only mechanics but also electrodynamics and quantum mechanics.
25 Eq. (41) is a particular case of what mathematicians call the Legendre transformations.
26 This is essentially a simplified model of the mechanical control device called the centrifugal (or “flyball”, or
“centrifugal flyball”) governor – see, e.g., http://en.wikipedia.org/wiki/Centrifugal_governor. (Sometimes the Chapter 1
Page 11 of 14
CM: Classical Mechanics
analytical mechanics “testbed”), none of the three conservation laws listed in the last section, holds. In particular, the bead’s energy,
m
E
v 2 mgh ,
(1.43)
2
is not constant, because the external forces rotating the ring may change it. Of course, we still can solve the problem using Newton’s laws, but this is even more complex than for the above case of the ring at rest, in particular because the force N exerted on the bead by the ring now may have three rather than two Cartesian components, which are not simply related. On the other hand, it is clear that the bead still has just one degree of freedom (say, the angle ), so its dynamics should not be too complicated.
This case gives us a clue on how situations like this one can be simplified: if we only could exclude the so-called reaction forces such as N, that take into account external constraints imposed on the particle motion, in advance, that should help a lot. Such a constraint exclusion may be provided by analytical mechanics, in particular its Lagrangian formulation, to which we will now proceed.
Of course, the value of the Lagrangian approach goes far beyond simple systems such as the bead on a rotating ring. Indeed, this system has just two externally imposed constrains: the fixed distance of the bead from the center of the ring, and the instant angle of rotation of the ring about its vertical diameter. Now let us consider the motion of a rigid body. It is essentially a system of a very large number, N >> 1, of particles (~1023 of them if we think about atoms in a 1-cm-scale body). If the only way to analyze its motion would be to write Newton’s laws for each of the particles, the situation would be completely hopeless. Fortunately, the number of constraints imposed on its motion is almost similarly huge. (At negligible deformations of the body, the distances between each pair of its particles should be constant.) As a result, the number of actual degrees of freedom of such a body is small (at negligible deformations, just six – see Sec. 4.1), so that with the kind help from analytical mechanics, the motion of the body may be, in many important cases, analyzed even without numerical calculations.
One more important motivation for analytical mechanics is given by the dynamics of “non-mechanical” systems, for example, of the electromagnetic field – possibly interacting with charged particles, conducting bodies, etc. In many such systems, the easiest (and sometimes the only practicable) way to find the equations of motion is to derive them from either the Lagrangian or Hamiltonian function of the system. Moreover, the Hamiltonian formulation of the analytical mechanics (to be reviewed in Chapter 10 below) offers a direct pathway to deriving quantum-mechanical Hamiltonian operators of various systems, which are necessary for the analysis of their quantum properties.
1.6. Self-test problems
1.1. A bicycle, ridden with velocity v on wet pavement, has no mudguards on its wheels. How far behind should the following biker ride to avoid being splashed over? Neglect the air resistance effects.
device is called the “Watt’s governor”, after the famous James Watts who used it in 1788 in one of his first steam engines, though it had been used in European windmills at least since the early 1600s.) Just as a curiosity: the now-ubiquitous term cybernetics was coined by Norbert Wiener in 1948 from the word “governor” (or rather from its Ancient-Greek original ή) exactly in this meaning because the centrifugal governor had been the first well-studied control device.
Chapter 1
Page 12 of 14
Essential Graduate Physics
CM: Classical Mechanics
1.2. Two round disks of radius R are firmly connected with a coaxial
R
T
cylinder of a smaller radius r, and a thread is wound on the resulting spool.
r
The spool is placed on a horizontal surface, and the thread’s end is being
O
pooled out at angle – see the figure on the right. Assuming that the spool
does not slip on the surface, what direction would it roll?
1.3.* Calculate the equilibrium shape of a flexible heavy rope of
d
length l, with a constant mass per unit length, if it is hung in a
, l
uniform gravity field between two points separated by a horizontal
distance d – see the figure on the right.
g
1.4. A uniform, long, thin bar is placed horizontally on two
similar round cylinders rotating toward each other with the same
angular velocity and displaced by distance d – see the figure on
the right. Calculate the laws of relatively slow horizontal motion of
the bar within the plane of the drawing, for both possible directions
d
of cylinder rotation, assuming that the friction force between the
g
R
R
slipping surfaces of the bar and each cylinder obeys the simple
Coulomb approximation 27 F = N, where N is the normal pressure force between them, and is a constant (velocity-independent) coefficient. Formulate the condition of validity of your result.
1.5. A small block slides, without friction, down a smooth slide
that ends with a round loop of radius R – see the figure on the right.
R
What smallest initial height h allows the block to make its way around h
the loop without dropping from the slide if it is launched with negligible
initial velocity?
g
1.6. A satellite of mass m is being launched from height H over
v0
the surface of a spherical planet with radius R and mass M >> m – see the H
figure on the right. Find the range of initial velocities v
m
0 (normal to the
radius) providing closed orbits above the planet’s surface.
M, R
1.7. Prove that the thin-uniform-disk model of a galaxy describes small sinusoidal (“harmonic”) oscillations of stars inside it, along the direction normal to the disk, and calculate the frequency of these oscillations in terms of Newton’s gravitational constant G and density of the disk’s matter.
27 It was suggested in 1785 by the same Charles-Augustin de Coulomb who has discovered the famous Coulomb law of electrostatics, and hence pioneered the whole quantitative science of electricity – see EM Ch. 1.
Chapter 1
Page 13 of 14
Essential Graduate Physics
CM: Classical Mechanics
1.8. Derive differential equations of motion for small oscillations of two
similar pendula coupled with a spring (see the figure on the right), within their
l
l
common vertical plane. Assume that at the vertical position of both pendula, the
F L
spring is not stretched ( L = 0).
g
m
m
1.9. One of the popular futuristic concepts of travel is digging a straight railway tunnel through the Earth and letting a train go through it, without initial velocity – driven only by gravity. Calculate the train’s travel time through such a tunnel, assuming that the Earth’s density is constant, and neglecting the friction and planet-rotation effects.
2 d
1.10. A small bead of mass m may slide, without friction,
along a light string stretched with force T >> mg, between two T
T
points separated by a horizontal distance 2 d – see the figure on the
m
right. Calculate the frequency of horizontal oscillations of the bead
about its equilibrium position.
g
1.11. For a rocket accelerating due to its working jet motor (and hence spending the jet fuel), calculate the relation between its velocity and the remaining mass.
Hint: For the sake of simplicity, consider the 1D motion.
1.12. Prove the following virial theorem:28 for a set of N particles performing a periodic motion, N
1
T F r ,
k
k
2 k1
where the top bar means averaging over time – in this case over the motion period. What does the virial theorem say about:
(i) a 1D motion of a particle in the confining potential29 U( x) = ax 2 s, with a > 0 and s > 0, and (ii) an orbital motion of a particle in the central potential U( r) = – C/ r?
N
Hint: Explore the time derivative of the following scalar function of time: G t p r .
k
k
k 1
1.13. As will be discussed in Chapter 8, if a body moves through a fluid with a sufficiently high velocity v, the fluid’s drag force is approximately proportional to v 2. Use this approximation (introduced by Sir Isaac Newton himself) to find the velocity as a function of time during the body’s vertical fall in the air near the Earth’s surface.
