Giovanni Gallavotti

The Elements of Mechanics

Ipparco Editore, 2007

2

Giovanni Gallavotti
Dipartimento di Fisica
Universita di Roma “La Sapienza”
Pl. Moro 2
00185, Roma, Italy
e-mail: giovanni.gallavotti@roma1 .infn.it
web: http:/ /ipparco.roma1 .infn.it

© 2007 Giovanni Gallavotti, II Edition

© 1983 Springer-Verlag, I Edition: ISBN 0-387-11753-9, ISBN 3-540-11753-9
A Daniela per amore infinito
4

Giovanni Gallavotti
Dipartimento di Fisica
Universita di Roma “La Sapienza”
Pl. Moro 2
00185, Roma, Italy
e-mail: giovanni.gallavotti@roma1 .infn.it
web: http:/ /ipparco.roma1 .infn.it

© 2007 Giovanni Gallavotti, II Edition

© 1983 Springer-Verlag, I Edition: ISBN 0-387-11753-9, ISBN 3-540-11753-9
Preface

P r e f a c e t o t h e S e c o n d E n g li s h e d i t i o n ( 2 0 0 7 ) . ©

This is Version 1.3: June 15, 2010

In 20071 recovered the Copyright. This is a new version that follows closely
the first edition by Springer-Verlag. I made very few changes. Among them
the Gauss’ method, already inserted in the second Italian edition, has been
included here. Believing that my knowledge of the English language has im-
proved since the late ’970’s I have changed some words and constructions.
This version has been reproduced electronically (from the first edition) and
quite a few errors might have crept in; they are compensated by the corrections
that I have been able to introduce. This version will be updated regularly and
typos or errors found will be amended: it is therefore wise to wait sometime
before printing the file; the versions will be updated and numbered. The ones
labeled 2.* or higher will have been entirely proofread at least once.
As owner of the Copyright I leave this book on my website for free down-
loading and distribution. Optionally the colleagues who download the book
could send me a one line message (saying “downloaded” , at least): I will be
grateful. Please signal any errors, or sources of unhappiness, you spot.
On the web site I also put the codes that generate the non trivial figures
and which provide rough attempts at reproducing results whose originals are
in the quoted literature. Discovering the phenomena was a remarkable achieve-
ment: but reproducing them, having learnt what to do from the original works,
is not really difficult if a reasonably good computer is available.
Typeset with the public Springer-Latex macros.
Giovanni Gallavotti Roma 18, August 2007

Copyright owned by the Author

6 Preface

Preface to the first English edition

The word ” elements” in the title of this book does not convey the impli-
cation that its contents are ” elementary” in the sense of ” easy” : it mainly
means that no prerequisites are required, with the exception of some basic
background in classical physics and calculus.
It also signifies ” devoted to the foundations” . In fact, the arguments chosen
are all very classical, and the formal or technical developments of this century
are absent, as well as a detailed treatment of such problems as the theory
of the planetary motions and other very concrete mechanical problems. This
second meaning, however, is the result of the necessity of finishing this work
in a reasonable amount of time rather than an a priori choice.
Therefore a detailed review of the ” few” results of ergodic theory, of the
” many” results of statistical mechanics, of the classical theory of fields (elas-
ticity and waves), and of quantum mechanics are also totally absent; they
could constitute the subject of two additional volumes on mechanics.
This book grew out of several courses on “Meccanica Razionale” , i.e.,
essentially, Theoretical Mechanics, which I gave at the University of Rome
during the years 1975-1978.
The subjects cover a wide range. Chapter 2, for example, could be used in
an undergraduate course by students who have had basic training in classical
physics; Chapters 3 and 4 could be used in an advanced course; while Chapter
5 might interest students who wish to delve more deeply into the subject, and
fit could be used in a graduate course.
My desire to write a self-contained book that gradually proceeds from
the very simple problems on the qualitative theory of ordinary differential
equations to the more modem theory of stability led me to include arguments
of mathematical analysis, in order to avoid having to refer too much to existing
textbooks (e.g., see the basic theory of the ordinary differential equations in
§2.2-§2.6 or the Fourier analysis in §2.13, etc.).
I have inserted many exercises, problems, and complements which are
meant to illustrate and expand the theory proposed in the text, both to avoid
excessive size of the book and to help the student to learn how to solve theoret-
ical problems by himself. In Chapters 2-4, I have marked with an asterisk the
problems which should be developed with the help of a teacher; the difficulty
of the exercises and problems grows steadily throughout the book, together
with the conciseness of the discussion.
The exercises include some very concrete ones which sometimes require
the help of a programmable computer and the knowledge of some physical
data. An algorithm for the solution of differential equations and some data
tables are in Appendix O and Appendix P, respectively.
The exercises, problems, and complements must be considered as an im-
portant part of the book, necessary to a complete understanding of the theory.
 Preface 7

In some sense they are even more important than the propositions selected
for the proofs, since they illustrate several aspects and several examples and
counterexamples that emerge from the proofs or that are naturally associated
with them.
I have separated the proofs from the text: this has been done to facilitate
reading comprehension by those who wish to skip all the proofs without los-
ing continuity. This is particularly true for the more mathematically oriented
sections. Too often students tend to confuse the understanding of a mathemat-
ical proposition with the logical contortions needed to put it into an objective,
written form. So, before studying the proof of a statement, the student should
meditate on its meaning with the help (if necessary) of the observations that
follow it, possibly trying to read also the text of the exercises and problems
at the end of each section (particularly in studying Chapters 3-5).
The student should bear in mind that he will have understood a theorem
only when it appears to be self-evident and as needing no proof at all (which
means that its proof should be present in its entirety in his mind, obvious and
natural in all its aspects and, if necessary, describable in all details). This level
o f understanding can be reached only slowly through an analysis of several
exercises, problem, examples, and careful thought.
I have illustrated various problems of classical mechanics, guided by the
desire to propose always the analysis of simple rather than general cases. I
have carefully avoided formulating ” optimal” results and, in particular, have
always stressed (by using them almost exclusively) my sympathy for the only
” functions” that bear this name with dignity, i.e., the C TO-functions and the
elementary theory of integration (” Riemann integration” ).
I have tried to deal only with concrete problems which could be ” construc-
tively” solved (i.e., involving estimates of quantities which could actually be
computed, at least in principle) and I hope to have avoided indulging in purely
speculative or mathematical considerations. I realize that I have not been en-
tirely successful and I apologize to those readers who agree with this point
of view without, at the same time, accepting mathematically non rigorous
treatments.
Finally, let me comment on the conspicuous absence of the basic elements
of the classical theory of fluids. The only excuse that I can offer, other than
that of non pertinence (which might seem a pretext to many), is that, perhaps,
the contents of this book (and of Chapter 5 in particular) may serve as an
introduction to this fascinating topic of mathematical physics.
The final sections, §5.9-§5.12, may be of some interest also to non stu-
dents since they provide a self-contained exposition of Arnold's version of the
Kolmogorov-Arnold-Moser theorem.
This book is an almost faithful translation of the Italian edition, with the
addition of many problems and §5.12 and with §5.5, §5.7, and §5.12 rewritten.
I wish to thank my colleagues who helped me in the revision of the
manuscript and I am indebted to Professor V. Franceschini for providing (from
his files) the very nice graphs of §5.8.
8 Preface

I am grateful to Professor Luigi Radicati for the interest he showed in

inviting me to write this book and providing the financial help from the Italian
printer P. Boringhieri.
The English translation of this work was partially supported by the
” Stiftung Volkswagenwerk” through the IHES.

Giovanni Gallavotti
Roma, 27 December 1981
Contents

1 P h en om en a R ea lity and m o d e ls ...................................................... 1

1.1 Statem ents........................................................................................ 1
1.2 An example of a Model .................................................................. 3
1.3 The Laws of Mechanics.................................................................... 5
1.4 General Thoughts on M odels.......................................................... 8

2 Q u alitative A sp e cts o f O ne-D im en sion al M o t i o n .................... 11

2.1 Energy Conservation........................................................................ 11
2.2 General Properties of Motion. U niqueness.................................. 13
2.2.1 Problems for §2.2.................................................................. 16
2.3 General Properties of Motion. Existence ...................................... 18
2.3.1 Problem s................................................................................ 21
2.4 General Properties of Motion. Regularity..................................... 22
2.4.1 Exercises and Problems ...................................................... 26
2.5 Local and Global Solutions of Differential Equations................ 26
2.5.1 Exercises and P roblem s...................................................... 31
2.6 More on Differential Equations. Autonomous Equations .......... 32
2.6.1 Exercises and P roblem s...................................................... 35
2.7 One-Dimensional Conservative Periodic and Aperiodic Motions 36
2.7.1 Exercises and P roblem s...................................................... 39
2.8 Equilibrium: Stability in the Absence of Friction........................ 40
2.8.1 Exercises and P roblem s...................................................... 43
2.9 Stability and F riction ...................................................................... 43
2.9.1 Exercises and P roblem s...................................................... 46
2.10 Period and Amplitude: Harmonic Oscillators.............................. 47
2.10.1 Exercises and Problems ...................................................... 50
2.11 The Damped oscillator: Euler’s Formulae.................................... 52
2.11.1 Exercises and Problems ...................................................... 55
2.12 Forced Harmonic Oscillations in Presence of Friction ................ 56
2.13 Fourier’s series for C “ -Periodic Functions.................................. 60
2.13.1 Exercises and Problems ...................................................... 63
10 Contents

2.14 Nonlinear Oscillations. The Pendulum and its Forced

Oscillations. Existence of Small Oscillations................................ 64
2.14.1 Exercises and P roblem s...................................................... 69
2.15 Damped Pendulum: Small Forced Oscillations............................ 70
2.15.1 Problem s................................................................................ 73
2.16 Small Damping: Resonances .......................................................... 74
2.16.1 Exercises and Problems ...................................................... 77
2.17 An Application: Construction of a Rigorously Periodic
Oscillator in the Presence of Friction. The Anchor
Escapement, Feedback Phenomena .............................................. 78
2.17.1 Exercises................................................................................ 82
2.18 Compatibility Conditions for the Anchor Escapement .............. 83
2.19 Encore on Anchor Escapement: Stability of the Periodic motion 87
2.19.1 Problem s................................................................................ 92
2.20 Frictionless Forced Oscillations: Quasi-Periodic Motions .......... 92
2.20.1 Exercises and Problems ...................................................... 96
2.21 Quasi-Periodic Functions. Multi Periodic Functions. Tori and
the Multidimensional Fourier T heorem ........................................ 99
2.21.1 Exercises and P roblem s....................................................... 107
2.22 Observables and Their Time Averages........................................... 108
2.22.1 Exercises and P roblem s....................................................... 112
2.23 Time Averages on Sequences of Times known up to Errors.
Probability and Stochastic Phenomena......................................... 114
2.23.1 Exercises and P roblem s....................................................... 123
2.24 Extremal Properties of Conservative Motion: Action and
Variational Principle ........................................................................ 126
2.24.1 Exercises and P roblem s....................................................... 135

3 System s w ith M a n y D egrees o f F reedom . T h e o ry o f the

constraints. A n a lytica l M e ch a n ics ................................................... 141
3.1 Systems of Points ............................................................................ 141
Exercises ............................................................................................ 144
3.2 Work. Linear and Angular Momentum ........................................ 144
Exercises............................................................................................. 150
3.3 The Least Action P rin cip le............................................................. 151
3.4 Introduction to the Constrained Motion T h e o ry ......................... 153
3.4.1 Exercises................................................................................. 156
3.5 Ideal Constraints as Mathematical Entities ................................. 157
3.5.1 Problem s................................................................................. 166
3.6 Real and Ideal Constraints ............................................................. 168
3.6.1 Exercises and P roblem s....................................................... 175
3.7 Kinematics of Quasi-constrained Systems. Reformulation of
Perfection Criteria for Approximate Conservative Constraints .176
3.7.1 Exercises and P roblem s....................................................... 185
3.8 A Perfection Criterion for Approximate Constraints................... 186
 Contents 11

3.8.1 Problem s................................................................................. 196

3.9 Application to Rigid Motion. Konig’s Theorem ..........................198
3.9.1 Exercises and P roblem s....................................................... 207
3.10 General Considerations on the Theory of Constraints ..............208
3.11 Equations of Hamilton and Lagrange.Analytical Mechanics . . . 211
3.11.1 Exercises, Problems and Complements............................. 227
3.12 Completely Canonical Transformations: Their Structure..........233
3.12.1 Problems and Complements ............................................... 240

4 Special Mechanical Systems ............................................................245

4.1 Systems of Linear Oscillators ........................................................245
4.1.1 Exercises................................................................................. 249
4.2 Irrational Rotations on l-Dimensional Tori..................................250
4.3 Ordered Systems of Oscillators. Phenomenological Discussion
and Heuristic Formulation of the Model of the Perfect Elastic
Body (String, Film, and S o lid )......................................................252
4.4 Oscillator Chains and the Vibrating String ................................258
4.5 The Vibrating String as a Limiting Case of a Chain of
Oscillators. The Case of Vanishing g and h. Wave Equation . . . 264
4.5.1 Exercises................................................................................. 269
4.6 Vibrating String: General Case. Dirichlet Problem in [0, L] . . . 271
4.7 Elastic Film. The Dirichlet Problem in Q C R 2 and General
Considerations on the Waves .......................................................... 278
4.8 Anharmonic Oscillators. Small Oscillations and Integrable
Systems .............................................................................................. 284
4.8.1 Problem s................................................................................. 290
4.9 Integrable Systems. Central Motions withNon vanishing
Areal Velocity. The Two-Body P rob lem ....................................... 291
4.9.1 Problem s................................................................................. 296
4.10 Kepler’s Marvelous L a w s ................................................................. 298
4.10.1 Exercises and Problems ...................................................... 302
4.11 Integrable Systems. Solid with a Fixed Point .............................. 307
4.11.1 Problems and Complements ............................................... 317
4.12 Integrable Systems. Geodesic Motion on the Surface of an
Ellipsoid and OtherSystems ........................................................... 325
4.12.1 Exercises andP roblem s.........................................................331
4.13 Some Integrability Criteria. Introduction: Geometric
Considerations and Preliminary Definitions................................. 333
4.14 Analytically Integrable Systems. Frequency of Visits and
E rgodicity........................................................................................... 341
4.14.1 Exercises and Problems ...................................................... 351
4.15 Analytic Integrability Criteria. Complexity of Motions and
Entropy .............................................................................................. 353
4.15.1 Exercises and P roblem s....................................................... 360
12 Contents

5 Stability P ro p e rtie s for D issipative and C onservative

System s ..................................................................................................... 365
5.1 A Mathematical Model for the Illustration of Some Properties
of Dissipative Systems ..................................................................... 365
5.2 Stationary Motions for a Dissipative G y roscop e........................369
5.2.1 Exercises................................................................................. 373
5.3 Attractors and Stability................................................................... 374
5.3.1 Exercises................................................................................. 381
5.4 The Stability Criterion of Lyapunov ............................................382
5.4.1 Exercises................................................................................. 386
5.5 Application to the Model of §5.1. The Notion of Vague
Attractivity of a Stationary Point ................................................ 389
5.5.1 Exercises................................................................................. 406
5.6 Vague-Attractivity Properties.The Attractive M an ifold ............408
5.6.1 A: Preliminary Considerations and an Equivalent
Problem................................................................................... 413
5.6.2 B: Some Useful Estimates of Derivatives............................414
5.6.3 C: Definition of the Approximate Surfaces........................ 415
5.6.4 D: Proof that the Approximate Surfaces are Well
Defined.....................................................................................416
5.6.5 E: Alternative Proof of the Existence of nt : Its
Uniqueness for t Small and Estimates of Its Derivatives
for t Small............................................................................... 416
5.6.6 F: Check of the Validity of Eq. (5.6.49)for nt, 0 < t < t+ 419
5.6.7 G: Proof of the Existence of the Limitas t ^ +<» of
nnt for t € [0, t+].....................................................................420
5.6.8 H: Independence of the Limit as n ^ +<» of nnt from
n and t £ [0, t + ] ..................................................................... 422
5.6.9 I: Attractivity of a (n TO) ........................................................ 423
5.6.10 L: Order of Tangency.............................................................423
5.6.11 M: Regularity in a ................................................................. 425
5.6.12 N: General Case..................................................................... 427
5.6.13 Exercises ................................................................................ 428
5.7 An Application: Bifurcations of the Vaguely Attractive
Stationary Points into Periodic Orbits. The Hopf Theorem. . . . 431
5.7.1 Exercises and Problems ...................................................... 438
5.8 On the Stability Theory for Periodic Orbits and More
Complex Attractors (Introduction) .............................................. 440
5.8.1 A. Example 1: The “Lorenz Model” ................................... 444
5.8.2 B. Example 2: Navier-Stokes equations on a
two-dimensional torus with a fivemode truncation..........446
5.8.3 C. Example 3: Navier-Stokes equations on a
two-dimensional torus with seven modes............................452
5.8.4 Problems and Complements ............................................... 454
5.9 Stability in Conservative Systems: Introduction ......................... 458
 Contents 13

5.10 Formal Theory of Perturbations. Hamilton-Jacobi Method. . . . 464

5.10.1 Exercises and P roblem s....................................................... 476
5.11 Some Simple Properties of Holomorphic Functions. Analytic
Theorems for the Implicit Functions ............................................. 479
5.11.1 Problems and E xercises....................................................... 486
5.12 Perturbations of Trajectories. Small Denominators Theorem . . 487
5.12.1 Problem s................................................................................. 511

6 A p p e n d ic e s ............................................................................................... 519
6.1 A: The Cauchy-Schwartz Inequality............................................... 519
6.2 B: The Lagrange-Taylor E xpansion............................................... 520
6.3 C : C TO-Functions with Bounded Support and Related
Functions............................................................................................. 521
6.4 D: Principle of the Vanishing Integrals......................................... 522
6.5 E: Matrix Notations. Eigenvalues and Eigenvectors. A List of
some Basic Results in Algebra ...................................................... 523
6.6 F: Positive-Definite Matrices. Eigenvalues and Eigenvectors.
A List of Basic Properties .............................................................. 525
6.7 G: Implicit Functions Theorems .................................................... 527
6.8 H: The Ascoli-Arzela Convergence Criterion ............................... 534
6.9 I: Fourier Series for Functions in C ([0, L } ) ................................. 536
6.10 L: Proof of Eq. (5.6.20) .................................................................. 537
6.11 M: Proof of Eq. (5.6.63).................................................................. 539
6.12 N: Analytic Implicit Functions ...................................................... 540
6.13 O: Finite-Difference M e th o d ........................................................... 544
6.14 P: Astronomical Data ...................................................................... 546
6.15 Q: Gauss Method for Planetary O rb its......................................... 548
6.16 S: Definitions and Symbols ............................................................. 565
6.17 T: Suggested Books and Com plem ents......................................... 566

R e fe r e n c e s ......................................................................................................... 569

I n d e x ................................................................................................................... 573
1

Phenomena Reality and models

1.1 Statements

The results of physical experiments are determined by observations based

on the measurement of various entities, i.e. the association of well defined
sequences of numbers with well defined sequences of events.
The physical entities are “operationally defined” . This means that they
are defined in terms of the operations used to construct the numbers that
provide their “measure” .
For instance, the sequence of operations necessary to measure the “dis­
tance” between two given points P and Q in space consists in choosing a
particular ruler and placing it on the straight line joining points P and Q,
starting from P . Taking the endpoint of the ruler as the new starting point,
the procedure is repeated n times until the endpoint of the ruler is superim-
posed on Q. If the distance P Q is not an exact multiple of the length of the
ruler, one may, after n such operations, reach a point Qn = Q preceding Q on
the line P Q ; and after n + 1 operations one may reach point Qn+ 1 following
Q on the line PQ . Then one takes a new ruler “ten times shorter” and puts it
on QnQ trying to match, as before, the second endpoint with Q. When this
turns out to be impossible, one can, as in the first case, define a new point
Qni on QnQ and, then, take a third ruler ten times shorter than the second
and repeat the operation.
Thus, inductively, a number n + 0.n1n2 .. . (in decimal representation) is
built which, by definition, is the measure of the distance between P and Q.
The above sequence of operations appears well defined but, in fact, a care-
ful analysis shows that it does not have the prerequisites to be considered a
2 1 Phenomena Reality and models

mathematically precise definition. What, for instance is “space” , what is a

“point” , what is a “ruler” ? Is it possible to “divide” a ruler into parts, and
infinitely often?
The physicist is not too concerned (or, rather, not at all concerned) with
such aspects of the question: he considers a physical entity well defined when-
ever the empirical procedure necessary for its measurement is clear.
A measurement procedure is considered to be clear when every observer
is led to the same result when measuring the same physical entity. It should
be stressed, however, that this is an empirical criterion perpetually subject to
critique; thus physical entities which today are considered to be well defined
may no longer be so in the future.
Hence, the physicist, from his observations of nature, obtains a set of num-
bers corresponding to the performance of some operations which are consid-
ered to be “objectively defined” . Trying to organize such numbers coherently,
the physicist often formulates “models” .
In the attempt to organize coherently such numbers, the physicist formu-
lates “models” : i.e. he associates well-defined mathematical structures with
his measurements, and he tries to establish a (small) number of mathematical
relationship among them. From such relationships new ones logically follow,
which reinterpreted through the model, used inversely, may serve to predict
new relations between various empirical measurements.
The belief in the existence of good models motivated Galileo to write:
“Philosophy is written in the great book which is always open before our eyes
(I mean the universe) but it cannot be understood unless one first learns the
language and distinguishes the characters in which it is written. It is a mathe­
matical language and the characters are triangles, circles and other geometri-
cal figures, without which it cannot be understood by the human mind; without
them one would vainly wonder through a dark labyrinth’ .1
A mathematical model is considered satisfactory whenever it does not lead
to contradictions with the experiments. If a contradiction occurs, the physicist
dismisses the model as “wrong” ; nevertheless, the mathematical construction
built with it remains valid and is witness to an imperfect representation of
nature.
Strictly speaking there is no model which is not wrong: only models that
have not yet been shown to be wrong exist. However, all “serious” models (such
as the dynamics of point masses, the theory of relativity, quantum mechanics,
electromagnetism, thermodynamics, statistical mechanics, etc.) have led, and
still lead, to the formulation of extremely interesting mathematical problems.
Furthermore, it often happens that the analysis of the mathematical properties
of a “wrong” model helps in the formulation of the new “more elaborate”
model that the physicist tries to set up as a substitute.
A link between phenomena reality and mathematics can therefore be es-
tablished as just described, through what has been called “a model” . However,

1 G.Galilei, II Saggiatore, p. 232, [20].

1.2 Example of a Model 3

it would be impossible to give a precise mathematical definition of the notion

of a model because it is a rather empirical notion which can only be well
understood through the analysis of several concrete cases.

1.2 An example of a Model

Consider the historically particularly important and significant case of the

“mechanics of point masses” . Its construction from empirical observations
will be briefly and concretely analyzed, presenting it as a model of one or
several point masses subject to forces.
The first statement (or “axiom” , to use a mathematical term) says that
the point masses are in a three-dimensional Euclidean space R 3 in which any
point can be represented by its three coordinates with respect to an orthogonal
reference system (O; i, j , k). The notation means that O is the origin and i, j , k
are the three orthogonal unit vectors pointing along the x , y , z coordinate axes,
respectively.
Such an idealization has a clear mathematical meaning, but it appears to
be unprovable in mathematical terms: it just renders the following empirical
observation.
In practice, a point in space is determined by measuring (often only in
principle and with the ruler method described in §1.1) its distance from three
orthogonal walls. It is to be remarked that all such operations are ordinarily
considered well defined.
A second statement (or “axiom” ) concerns “time” which, for the physicist,
is the physical entity measured by a “clock” (classically described as a pen-
dulum, although any more modern device will do as well). One assumes that
time is an absolute “entity” : in other words, one states that, at least in prin-
ciple it is possible to associate with every point in space a clock mechanically
identical at every point, and, furthermore, to coordinate ( “synchronize” ) the
clocks.
This means that if P , P ' are two points and t, t' are two chosen time
instants t < t' it is then possible to send a signal from P towards P ' leaving P
at time t and reaching P ' at time t' (as indicated by the local clocks in P and
in P ', respectively); while, vice versa, if t > t', the above operation should be
impossible.
A little thought makes it clear that the operational definition of a “system
of synchronized clocks” is based on the empirical fact that it is possible to
send signals with arbitrary speed. It is also clear that the notion of time is a
phenomenological notion, far from being mathematically well posed.
Accepting the point of view so far discussed, one is led to say that the math-
ematical scheme, or model, representing the space-time continuum,where our
observations take place, consists of a four-dimensional space: each of its points
( x , y , z , t ) represents a point seen in a Cartesian coordinate frame (O; i, j , k)
4 1 Phenomena Reality and models

( “laboratory” ) and observed at the instant t (as measured by the formerly

introduced universal clocks).
Empirically, a point mass is any object which, at least as far as our obser-
vations are concerned, can be assimilated with a point in space (for instance, a
planet or a star in the universe, a stone falling in a ravine, a ship sailing in the
ocean, etc.). Such a point preserves its identity over the course of time; hence,
it is possible to define its trajectory through a function of time t ^ x (t), where
x (t) = (x(t), y(t), z(t)) is the vector whose components are the coordinates of
the point at time t, in the chosen reference frame (O; i, j , k).
Mathematically, a point mass moving in the reference frame (O; i, j , k)
observed as t varies over an interval I is represented as a curve C in R 3 by
the vector equations P (t) — O = x (t), t G I ; and the parameter t has the
interpretation of time (i.e., it is called “time” ).
Given a point mass moving as t varies in I, one can associate with it its
“velocity” at time t G I . Operationally, velocity is defined by fixing t0 G I ,
finding the positions P(to) and P (t0 + e), and setting

P (t0 + e) — P (t0)
v (to) (1.2.1)
e
where the parameter e > 0 is to be chosen “suitably small” (according to well-
defined criteria which, however, depend on the concrete cases). The mathe-
matical model defines the point mass velocity at time t0 G I as the derivative
of the function t ^ x (t) at t = t0.
To complete the mathematical model of a point mass, it is important to
define the “force” acting on it.
Operationally, the force acting at a given instant on the point mass con-
sists of three scalar quantities which together define a vector f (t). The force
acting on the point mass moving in R 3 and observed in the frame (O; i, j , k)
is measured through a “dynamometer” which is an instrument whose use is
convenient to describe in a strongly idealized form. It is, basically, a suitably
built spring which will be imagined as a very thin, light segment with a hook.
Consider a point mass moving in R 3, with a velocity v = (vx,v y, vz)
relative to the reference frame (O; i, j , k) at time t0. To measure the force
acting upon it, hook it to the dynamometer to which the same velocity v has
been imparted and which will be kept fixed during the measurement. Then
try to adjust the spring length and direction so that the acceleration at time
t0 + e is 0, where e > 0 is chosen “suitably small” . (The empirical notion of
acceleration and the corresponding mathematical model of it, as the second
derivative with respect to t of the point position, is discussed along the same
lines as the notion of velocity.)
The force is then the vector f whose direction is that of the dynamometer
at time t0 + e, whose orientation is that parallel to hook but pointing away
from it and whose modulus is the size of the spring elongation.
Summarizing: a point mass subject to forces and observed in a frame
(O; i, j , k) in R 3 as time varies within an interval I is, in its mathematical
1.3 Example of a Model 5

model, described by a curve in seven-dimensional space: one of its points

(t, x, y, z, f x, f y, f z) represents a point mass which at time t has coordinates
(x, y, z ) in (O; i, j, k) and, in the same frame, is subject to a force ( f x, f y, f z).
The curve representing this situation can be parameterized by the parameter
t itself, as t varies in some time interval I ; it shall also be assumed that in
this parametric representation the functions t ^ (x(t),y(t), z(t)) are twice
continuously differentiable so that a mathematical definition of velocity and
acceleration is meaningful.

1.3 The Laws of Mechanics

Once it is established what is meant by a point mass subject to forces and

studied in a given frame of reference in R 3 as the time varies in an interval
I (briefly, “a point mass subject to forces” ), it is possible to complete the
mathematical model of the point mechanics. For this purpose, the “laws of
dynamics” and their mathematical interpretation have to be discussed.
Experimentally, given a point mass, a simple relation is observed between
its acceleration a at time t (in a given frame of reference) and the force f acting
on it at that time (observed in the same frame). Such a relation is called the
Second Law of Mechanics and establishes the existence of a constant m > 0,
characteristic of the point mass and independent of the frame of reference
used for the observations, such that:

ma = f . (1.3.1)

This law introduces, via the properties of the differential equations, many
relations among the quantities x, v, t, and such relations can sometimes be
experimentally checked. For instance, if it is known a priori which force will
act on the point mass whenever it is at the point (x, y, z) at time t with velocity
(vx, vy, v z), then, denoting such force as f(v x,v y, vz, x , y , z , t ) = f(v , x, t), the
differential equation

m X = f (X, x, t) (1.3.2)
allows the determination of the motion following an initial state, in which the
velocity v 0 and the position x 0 are given at time t0, at least for a small time
interval around t0 if f is a smooth function, see Chapter 2.
The First Principle of Mechanics postulates the existence of at least one
reference frame (O; i, j, k), called “inertial frame” , in R 3 where a point mass
“far” from the other objects in the universe appears to be subjected to a null
force in (O; i, j, k). Such a frame is experimentally identified with a frame with
origin in a fixed star and with axes oriented towards three more fixed stars.
It is to such a frame that motion is often referred.
Of course the notions of “far” and of “fixed star” are empirical notions
rather than mathematical ones.
6 1 Phenomena Reality and models

Mathematically, the first principle is used to grant to a particular frame

of reference in the space-time continuum a privileged role and to define the
“absolute force” or the “true force” as that acting on the point mass in this
frame. This frame has to be chosen once and for all and is called the “fixed
reference frame” (as opposed to “moving reference frame” ).
It is possible and sometimes convenient to introduce frames whose ori-
gin and axes vary with time with respect to the “fixed” frame (O; i, j , k) :
(0(t); i(r),j (t), k(t)).
Since f = ma, it follows that if the moving frame is in uniform rectilinear
translational motion with respect to the fixed frame, then the force acting
upon the point is the same whether observed in the fixed frame or in the
moving frame: hence, in this moving frame, the “inertia principle” , i.e., the
first principle, is valid: a point mass which is “very far” from the other objects
in the universe is subject to a null force, since the acceleration is the same
in the two frames. All frames in rectilinear uniform motion with respect to a
fixed frame are called “inertial frames” .
The mathematical model of a point mass with mass m subject to forces
and obeying the laws of dynamics is then, simply, a point mass subject to
forces, in the sense of the preceding section, and such that the relation

ma= f (1.3.3)
holds and, furthermore, f is a function of the point velocity, position, and
time; i.e., the following relation holds:

f = f (v , x ,t). (1.3.4)
Clearly, from such a mathematical viewpoint (where f is imagined as given a
priori), the first principle is deprived of its deep physical meaning.
An important extension of the point mass model is a model for the me-
chanics of a “system of N point masses” . Mathematically, such a system con-
sists of N point masses with mass m i ,. . . , mN, in the above sense, satisfying
the Third Principle of Mechanics. This means that it should be possible to
represent the force fi acting on the i-th point as

fi = E fj - j , (1.3.5)
j =j
where fj —i are such that
(a) fj —i = —fi—j , j, i = 1, 2, N , i = j;
(b) fj —i is parallel to Pj —Pi, i.e., to the line joining the positions Pi and Pj
of the i-th and j-th points;
(c) fj —i depends solely upon the positions and velocities of the it-h and j-th
points and on time:

fj —-i — fj —-i (vj , v i, Pj , Pi, t) • (1.3.6)

1.3 Example of a Model 7

This assumption corresponds to a precise empirical fact: it is possible to define

operationally what should be understood by “the force exerted by the
point Pj on the point P j ”.
For instance, the force fj^ j could be measured as follows: one measures,
in the given inertial frame of reference, the force fj, acting on i and then one
measures, after removing the point j from the system, the new force acting
(j)
on the i-th point, obtaining the result fj ; then one sets

fj^ i = fj - fj(j). (1.3.7)

The Third Principle of Mechanics arises from the experimental observation
that fj^i = —fi ^ j , that fj^i is parallel to Pj —Pj, that the total force acting
on a singe point mass is the sum of the forces exerted on it by the other
system points (in the sense of vectors addition) if observed in an inertial
frame of reference, and, finally, that fj^i depends only upon the positions
and velocities of the points involved and, possibly, on time.
Physics often places still more requirements and restrictions upon the laws
of force which can be used to give a more detailed specification of a mechani-
cal system model. However, they do not have a general character comparable
to the three principles but, rather, are statements explaining which laws of
force are to be considered a good model under given circumstances. For in­
stance, two point masses “without structure” (this is, again, an empirical
notion which we refrain from elucidating) attract each other with a force of
intensity m m '/k r2, where r is the distance between the points, m and m' are
their masses, and k is a universal constant. If the structure of the two points
can be summarized by saying that they have an “electric charge e” (a new em­
pirical notion), the mutual force will be the vector sum of the above-described
gravitational force and of a repulsive force with intensity k'e2/ r 2, where k' is
another universal constant.
The principles of mechanics already place enough restrictions upon the na­
ture of the forces admissible in mechanical problems: therefore it is convenient
and interesting to examine their implications before passing to the analysis
of special models obtained by concretely specifying the “force laws” , i.e., the
functions giving the forces in terms of the points positions and velocities and
of time.
It should be stressed, and this is a general comment on the mathemati-
cal models for physical phenomena, that the mathematical model is always
“poorer” than the physical reality that it tries to imitate. For instance in the
above mathematical model for mechanics, the first principle loses its meaning.
Another example, implicit in the above discussion, is the following.
To give an operational meaning to the notions of position, speed, force, etc.,
it must be possible to repeat “identical” experiments several times (e.g., see
the position measurement in §1.1 by repeating the measurement operations.
However, time inexorably flows away, and this is impossible. Physically, this
difficulty is avoided by the “principle of homogeneity of space-time” which
8 1 Phenomena Reality and models

says that experiments starting at any time in any space location will yield
the same results if the points involved are in the same relative positions and
situations.
In the mathematical model for mechanics just described, the necessity of
understanding the above problems does not arise, nor do many other similar
problems which the reader will easily think of.
Usually it is possible to complicate the models in order to imbue them with
any given number of physical facts: but an analysis of this type of questions
would lead us beyond the scope of this book.
In any case, a decision is always needed on where to put a stop to the
process of model improvement, which would otherwise hopelessly continue ad
infinitum. We must recall that we have the more down-to-earth, and more
interesting, problem of obtaining some concrete prediction algorithms for our
observations of nature.

1.4 General Thoughts on Models

In this book more abstract schematization processes concerning empirically

observed phenomena will be met (e.g., when we discuss the notion of an
“observable” or of a “vibrating string” ). In such cases, however, the details
of the construction of the mathematical model will not be repeated: a very
common practice based on the idea that the very words used to designate
well-defined mathematical objects will implicitly define the model.
It is such a practice, or better, its imperfect understanding, which some-
times causes misunderstandings between physicists and mathematicians and
provokes allegations of non-rigorous use of mathematics.
It is important to realize that when the physicist speaks in mathematical
terms he is by no means attributing to them the same rigid meaning that a
mathematician would assume for them. Rather he is using this language to
help himself in the formulation of a model which, once well defined, he shall
rigorously treat (since he believes, or at least hopes, that the book of nature
is written in mathematical characters).
Possibly logically non rigorous steps or apparently wild mathematical ap-
proximations in a physicist's argument should always be interpreted as further
complications or, better, refinements of the model that the physicist is trying
to build.
In the hectic development of research, a physicist often modifies a model
while using it, or he modifies the mathematical meaning of the objects and
entities which belong to the model without changing their names (otherwise,
a dictionary would not suffice). He does this because his main interest is in
the construction of models and only secondarily in its mathematical theory,
often considered trivial for his needs.
To avoid excessively pedantic discussions, we shall adhere, in the following,
to the well-established practice of avoiding the physical analysis necessary to
1.4 Thoughts 9

the construction of a model and shall leave it to the reader to imagine such an
analysis via the suggestive names used for the various mathematical entities
(with the exception of a few important cases). In any case, this book is devoted
to the mathematical, rather than physical aspects, of mechanical problems.
Bibliographical Comment. It is very useful to study at least the defi-
nition and the laws of motion in the Philosophiae Naturalis Principia Mathe-
matica by I. Newton, [37], to understand exactly the Newtonian formulation
of mechanics and its modernity. To avoid “reading too much” , i.e., to avoid
interpreting these immortal pages in too modern a way, it is a good idea to
read the paper Essays on the history of mechanics by C. Truesdell, pp. 85-137
([48]). The reading of the first two chapters of the work by E. Mach, [31],)
will be a very useful and stimulating complement to the first three chapters
of this book.
2

Qualitative Aspects of One-Dimensional

Motion

2.1 Energy Conservation

Consider a point mass, with mass m, on the line R and subject to a force law
depending uniquely on its position. Therefore, a force law £ ^ f (£) is, given
£ t R , which we shall suppose to be of class C , associating with every point
£ on the line R the component f (£) of the force acting on the point when it
happens to occupy the position £.
A “motion” of the point mass, observed as t varies in an interval I , is a
function t ^ x(t), t t I , of class C TO( I ) such that

mx ( t) = f (x(t)), Vt t I (2.1.1)

The “energy conservation theorem” follows by multiplying Eq. (2.1.1), side

by side, by x(t):

mX X = f (x), (2.1.2)

omitting, as will often be done, the explicit mention of the t-dependence.

Then, defining the functions,

def 1
n ^ T (n) —m ri2, (2.1.3)
2 1’
it is

d d
T (x) = m;x x, V (x) = —f (x) ać (2.1.4)
dt dt