Elements of Mechanics

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Giovanni Gallavotti

The Elements of Mechanics

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Giovanni Gallavotti Dipartimento di Fisica

Universit`a di Roma “La Sapienza” Pl. Moro 2

00185, Roma, Italy

e-mail: giovanni.gallavotti@roma1.infn.it web: http://ipparco.roma1.infn.it

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2007 Giovanni Gallavotti, II Edition c

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III

Giovanni Gallavotti Dipartimento di Fisica

Universit`a di Roma “La Sapienza” Pl. Moro 2

00185, Roma, Italy

e-mail: giovanni.gallavotti@roma1.infn.it web: http://ipparco.roma1.infn.it

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2007 Giovanni Gallavotti, II Edition c

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Preface

Preface to the Second English edition (2007).

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This is Version 1.3: October 9, 2012

In 2007 I 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

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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.

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Preface VII

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 logmathemat-ical 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 of 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∞_{-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.

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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

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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 6= Q preceding Q on the line P Q; and after n + 1 operations one may reach point Qn+1following Q on the line P Q. 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 Qn1 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

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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,

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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 spaceR3 _{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)

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(“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_R3 _by the vector equations P (t)− O = x(t), t ∈ 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 ∈ I. Operationally, velocity is defined by fixing t0 ∈ I, finding the positions P (t0) and P (t0+ ε), and setting

v(t0) =

P (t0+ ε)− P (t0)

ε , (1.2.1)

where the parameter ε > 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∈ I as the derivative of the function t_{→ x(t) at t = t}0.

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 inR3 _{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 _R3_{, with a velocity v = (v}

x, vy, 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+ ε is 0, where ε > 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+ ε, 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 R3 _{as time varies within an interval I is, in its mathematical}

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model, described by a curve in seven-dimensional space: one of its points (t, x, y, z, fx, fy, fz) 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 (fx, fy, fz). 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 R3 _{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, vz), then, denoting such force as f (vx, vy, 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 v0 and the position x0are 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 _R3 _{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.

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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

m a = 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 m1, . . . , 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= X j6=i

f_j→i, (1.3.5)

where fj→iare such that

(a) fj→i=−fi→j, j, i = 1, 2, ..., N, i6= j;

(b) fj→iis parallel to Pj− Pi, i.e., to the line joining the positions Pi and Pj of the i-th and j-th points;

(c) f_j→idepends solely upon the positions and velocities of the it-h and j-th points and on time:

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This assumption corresponds to a precise empirical fact: it is possible to define operationally what should be understood by fj→i “the force exerted by the point Pj on the point Pi”.

For instance, the force f_j→i could be measured as follows: one measures, in the given inertial frame of reference, the force fi, acting on i and then one measures, after removing the point j from the system, the new force acting on the i-th point, obtaining the result f_i(j); then one sets

f_j→i= fi− fi(j). (1.3.7)

The Third Principle of Mechanics arises from the experimental observation that f_j→i=_−f_i→j, that f_j→iis parallel to Pj− Pi, 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 mm′_/kr2_{, 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_/r2_{, 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 physimathemati-cal phenomena, that the mathematimathemati-cal 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

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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

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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.

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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 ξ _{→ f(ξ) is, given} ξ_{∈ R, which we shall suppose to be of class C}∞_{, associating with every point} ξ on the line_{R the component f(ξ) of the force acting on the point when it} happens to occupy the position ξ.

A “motion” of the point mass, observed as t varies in an interval I, is a function t_{→ x(t), t ∈ I, of class C}∞_{(I) such that}

m ¨x(t) = f (x(t)), ∀ t ∈ I (2.1.1)

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

m ˙x ¨x = ˙x f (x), (2.1.2)

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

η→ T (η)def= 1 2m η 2_, _ξ → V (ξ)def= − Z ξ f (ξ′) dξ′, (2.1.3) it is d dtT ( ˙x) = m ˙x ¨x, d dtV (x) =−f(x) ˙x (2.1.4)

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so that Eq. (2.1.2) becomes d

dt(T ( ˙x) + V (x)) = 0 (2.1.5)

This implies a constant E can be associated with every motion t → x(t), t_{∈ I, depending on the motion under consideration and such that}

T ( ˙x(t)) + V (x(t)) = E, _{∀ t ∈ I.} (2.1.6) The expressions T ( ˙x) and V (x) are respectively called the “kinetic energy” and the “potential energy” and Eq. (2.1.6) has to be read as follows: “in every motion developing under the action of a force with potential energy V , the sum of the kinetic energy and potential energy is a constant”. This constant is given the name “total energy” of the considered motion. The “qualitative theory” of Eq. (2.1.1) is concerned with the analysis of the properties of the motion verifying Eq. (2.1.1), which are valid independently of the choice of f , at least for vast classes of functions f . The energy conservation is a first example of a qualitative property.

Observations. The energy conservation goes back at least to Huygens; after-wards, it was used by J. and D. Bernoulli together with the law of conservation of linear momentum (Descartes) (see [48], p. 105 and following).

Eq. (2.1.6) implies an expression for the velocity: ˙x(t) =± _m2(E− V (x(t)))12_, _t

∈ I (2.1.7)

This relation, which will be used and discussed in§2.6, allows the reduc-tion of the determinareduc-tion of the evolureduc-tion law t → x(t), t ∈ I, “time law”, to an area-computation problem for a planar figure, “quadrature”. In fact, supposing ˙x > 0, it yields: t = Z x(t) x(0) dξ q 2 m(E− V (ξ)) dξ (2.1.8) when I_{⊃ [0, t].}

Hence, the area under the graph of the curve with equation ξ_{→ T (ξ) =} (2

m(E− V (ξ)))−

1

2 above the interval [x(0), x(t)] is the time that the point

needs to reach x(t), starting from x(0) at time 0 with positive speed and energy E, at least for small t (i.e., as long as ˙x > 0).

Newton “reduced to quadratures” the simplest problems of motion without explicitly using energy conservation ([37], for instance Book I, Propositions XXXIX, XLI, LIII, LVI, etc.).

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2.2 Uniqueness 13

2.2 General Properties of Motion. Uniqueness

In the preceding§2.1, a motion developing, under the action of a force f, in a time interval I was supposed to be given. We can ask which further properties of a particular motion allow us to select it from among all motions which, in the same time interval I, take place under the action of the same force.

One can even preliminarily ask whether, given an interval I, there exist any motions, i.e., C∞ _{solutions of Eq. (2.2.1) thought of as an equation for} t_{→ x(t), t ∈ I.}

In view of the importance of such questions, before proceeding in the analysis of Eq. (2.1.1), some attention will be devoted to the general problem of the existence, uniqueness, and regularity of the solutions of differential equations in _Rd_.

Eq. (2.1.1), thought of as a “second-order” differential equation in_R1_{, is} equivalent to a “first-order” equation inR2_{: it suffices to write it as}

˙x(t) = y(t), ˙y(t) = f (x(t)), (2.2.1) where Eq. (2.2.1) is an equation for the unknown C∞_{function t}_{→ (x(t), y(t))} defined on I and with values in_R2_.

More generally, consider an arbitrary “s-th order” differential equation in Rd_{, s = 0, 1, . . ., like} ds_x(t) dts = f ( ds−1_x(t) dts−1 , . . . , dx(t) dt , x(t), t), (2.2.2)

with t∈ I, where f is an Rd_{-valued C}∞_{function defined on}_Rd_{× R}d_{× R and} t → x(t) is an unknown Rd_{-valued C}∞ _{function on I. The latter equation} may be thought of as a first-order equation inRd _{by setting}

dx(t) dt = y1, dy(t)₁ dt (t) = y2, . . . dy(t)_s−2 dt = ys−1, dy(t)_s−1 dt = f (ys−1(t), . . . y1(t), x(t), t) (2.2.3) and then considering Eq. (2.2.3) as an equation for the C∞ _{function t} _→ (x(t), y1(t), . . . .ys−1(t)) defined on the interval I and with values in Rd × . . .× Rd ₌_Rd s_.

Eq. (2.2.2) is the most general differential equation that will be met in this book. By virtue of the preceding remark, it will then suffice, for our purposes, to study first-order differential equations inRd _{having the form}

˙x(t) = F(x(t), t), t_{∈ I,} (2.2.4)

It will be convenient to introduce a precise convention about what a dif-ferential equation is or about what one of its solutions is.

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1 Definition. Given anRd_{-valued function F}_{∈ C}∞₍_Rd_{× R), the expression} (2.2.4), denoted, for short, ˙x = F(x, t), will be called a “differential equation on Rd _{associated with F”.}

A “C(k) _{solution”, k > 1, of Eq. (2.2.4) on the interval I, closed or open or} semi open, will be a C(k) _{function which turns Eq. (2.2.4) into an identity} when substituted into it.1 _{A “solution” of Eq. (2.2.4) for t}

∈ I is a C∞ solution. The solutions of Eq. (2.2.4) will often be called “motions”.

Let us first examine the uniqueness problem for the solutions of Eq. (2.2.4). 1 Proposition. Let (ξ, t)_{→ F(ξ, t) be an R}d_{-valued C}∞_{function on} _Rd_x

R. Given a > 0, b > 0, t0 ∈ R, let t → x(t) be a C(1) solution of Eq. (2.2.4) on J = [t0− a, t0+ b]:

(i) the function t_{→ x(t) is in C}∞_(J);

(ii) if t→ y(t) is another solution of Eq. (2.2.4) on J and if y(t0) = x(t0), then x(t) = y(t),∀ t ∈ J.

Observations.

(1) This proposition applied to Eq. (2.2.2) via Eq. (2.2.3) tells us that two C(s) _{solutions of an s-th order differential equation in}

Rd _{for t}

∈ J coincide if and only if at time t0∈ J (“initial time”) they have the same first (s − 1) derivatives (“equal initial data”). When Eq. (2.2.2) is the equation governing a physical motion in _Rd_{, it is s = 2; this means that the motion is uniquely} determined, if existing at all, by its initial position x(t0) and by its initial velocity ˙x(t0), i.e., as one says, by its initial “act of motion” ˙x(t0).

(2) It would appear that it might be interesting or important to know if, by specifying properties of the solutions of Eq. (2.2.2) other than the just-mentioned initial data at some initial time, the solution verifying such prop-erties is uniquely determined 2_{, if existing at all. The uniqueness criterion} that we chose above for illustration purposes, Proposition 1, has been se-lected only because it quickly leads to a simple answer and because it is one of the uniqueness criteria which are most useful in many applications. (3) From the proof it will appear that if F had been only supposed to be of class C(_{k), k}_{≥ 1, then uniqueness would have followed in an equal way. The} regularity of t→ x(t), t ∈ J, could also be deduced in this case, but one would only obtain that t_{→ x(t) is a C}(k+1) _function.

Proof. By integrating both sides of Eq. (2.2.4) and by setting x0= x(t0) = y(t0), we get: x(t) = x0+ Z t t0 F(x(τ ), τ ) dτ, t_{∈ J,} (2.2.5) 1

We shall see that every C(k)_{solution, k > 1, is automatically a C}∞_{solution, if F}_{∈ C}∞_. 2_{For instance, we can ask the following question. Consider Eq. (2.2.2) with s = 2 and}

lei t1, t2be two times and x1, x2∈ Rdbe two positions. Is the motion [solution of Eq.

(2.2.2)] leading from x1 to x2 as time elapses from t1 to t2 (assuming that one such

motion, at least, exists) unique? We shall see that the answer to this question will, in general, be no.

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2.2 Uniqueness 15

and, similarly, since also t→ y(t) is a solution of Eq. (2.2.4): y(t) = x0+ Z t t0 F(y(τ ), τ ) dτ, t_{∈ J.} (2.2.6) Hence, x(t)− y(t) = Z t t0 (F(x(τ ), τ )− F(y(τ), τ)) dτ. (2.2.7) To prove (ii) the procedure that will be followed is very interesting since it obviously goes beyond the particular result that we wish to obtain.

Informally, the argument is the following: the difference |x(t) − y(t)| is, by Eq. (2.2.7), about |t − t0| |F(x(t), t) − F(y(t), t)|, if t ∼ t0; however, the increment |F(x(t), t) − F(y(t), t)| is proportional, by Lagrange’s theorem, to the increment of the argument of F, i.e., to C|x(t) − y(t)|, where C is an estimate of the first derivatives of F. Hence, Eq. (2.2.7) implies that _{|x(t) −} y(t)_{| and C|t − t}0| |x(t) − y(t)| are about equal if t ∼ t0, and this, in turn, implies that _{|x(t) − y(t)| = 0 for t dose to t}0 because for t ∼ t0, one has C_{|t − t}0| < 1.

To estimate the integrand of Eq. (2.2.7) let S _{⊂ R}d _{be a sphere with so} large a radius that it contains all the values x(τ ), y(τ ),_{∀ τ ∈ J, and let}

MS = max ξinS,τ ∈J d X i,j=1 ∂F(i) ∂ξj (2.2.8)

where F(i)_{(ξ, t) is the i-th component of the vector F(ξ, t) = (F}(1)_{(ξ, t), . . . ,} F(d)_{(ξ, t))}

∈ Rd_{. Then, from Taylor’s formula:}

|F(x(τ), τ) − F(y(τ), τ)| ≤ MS|x(τ) − y(τ)|. (2.2.9) Inserting this inequality into Eq. (2.2.7), yields

|x(t) − y(t)| ≤ MS Z t

t0

|x(τ) − y(τ)| dτ (2.2.10)

Let M (t) = maxt0≤τ ≤t|x(τ) − y(τ)|, t ∈ [t0.t0+ b]; then Eq. (2.2.10) implies

|x(t) − y(t)| ≤ MSM (t)|t − t0|, ∀ t ∈ [t0, t0+ b].

Since M (t) is monotonic nondecreasing and since this inequality holds for all t∈ [t0, t0+ b], one easily finds that

M (t)_{≤ M}S|t − t0| M(t), ∀ t ∈ [t0, t0+ b] (2.2.11) which implies M (t) = 0 for_{|t − t}0| < M_S−1, t∈ [t0, t0+ b].

Hence, x(t0+M_S−1) = y(t0+M_S−1), if t0+M_S−1< t0+b, and the argument can be repeated, replacing t0 by t0+ MS−1, to show that M (t) = 0 for t ∈ [t0, t0+ 2MS−1] if t0+ 2MS−1< t0+ b, etc., so that M (t) = 0 for t∈ [t0, t0+ b]. For t∈ [t0− a, t0], one proceeds likewise.3

3 _{Alternatively, Eq. (2.2.10) could be iterated n times to yield, if µ = max}

|x(τ) − y(τ)|, τ∈ [t0− a, t0+ b]:

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To check (i), i.e., that t → x(t) is a C∞ _{function on J, remark that} if t → x(t) is a C(1)_{(J) function, then Eq. (2.2.4) implies that t} _{→ ˙x(t)} is in C(1)(J), being a composition of a C∞ function with a C(1) function; furthermore, by differentiating Eq. (2.2.4):

¨ x(t) = d X i=1 ∂F ∂ξi (x(t), t)_{· ˙x}(i)+∂F ∂t(x(t), t) (2.2.12) which, in turn, implies that t_{→ ¨x(t) is a C}(1)_{function by the same argument} as above. Then, by differentiating Eq. (2.2.12), one finds that x. . .(t) is a C(1)

function on J, etc. mbe

2.2.1 Problems for §2.2

1. If t → x(t), t ≥ 0, solves ˙x = f (x) and x(0) = x(T ) for some T > 0, then x(t) = x(t + T ),∀ t > 0; assume f ∈ C∞₍_Rd_{). Would this also be true if f}_{∈ C}1₍_Rd_{)? (Hint: Use}

uniqueness).

2.The property of the preceding problem is not valid when the differential equation right-hand side is explicitly time dependent (i.e., ˙x = f (x, t), and ∂f /∂t6= 0, the “non autonomous case”). Find an example.

3.Let f (x, t) be such that f (ξ, t) = f (ξ, t + T ) for some T > 0 and for all ξ∈ Rd_{. Suppose}

that t → x(t) is a solution of ˙x = f (x, t) such that for some integer m > 0, one has x(0) = x(mT ), then x(t)≡ x(t + mT ), ∀ t ≥ 0. (Hint: Use uniqueness.)

4. Consider the equation ˙x(t) = ℓ(t) x(t) with ℓ ∈ C∞_{(R). Show that if t → x(t) and}

t→ y(t) are two solutions for t ∈ J and if x(t) 6≡ 0, there exists a constant A such that y(t)≡ Ax(t), ∀ t ∈ J.

5. If the function ℓ of the Problem 4 is periodic with period T > 0 and t→ x(t) 6≡ 0, is one of its solutions then also t → x(t + T ) is a solution. Hence, ∃λ 6= 0 such that x(t + T ) = λx(t). Show that λ > 0. (Hint: Otherwise either λ = 0 and x(T ) = 0, hence x(t)≡ 0 (by uniqueness on [0, +∞)), or λ < 0 and there would be t ∈ (0.T ] where x(t) = 0: hence, again, x(t) = 0 by uniqueness.)

6.The most general solution t→ y(t), t ∈ R+, of the equation in Problem 4, with ℓ periodic

with period T has the form y(t) = Aλt/T_{z(t), where z}_{∈ C}∞₍_R₊_{) is T -periodic.}

7.∗_{Consider the equation ˙x = L(t)x in}_Rd_{, where t}_{→ L(t), t ∈ R, is a d× d-matrix valued}

C∞ _{function. Consider d solutions x}(1)_{, . . . , x}(d) _{for t} _{∈ I = [a, b] and call them}

“inde-pendent” if∃ t0∈ I such that the d vectors x(1)(t0), . . . , x(d)(t0) are linearly independent.

Show that, if t∈ I, then also x(1)_{(t), . . . , x}(d)_{(t) are linearly independent whenever they}

are such for t = t0 and, furthermore, any solution t→ y(t), t ∈ I, can be represented as

y(t) =Pd_j=1Ajx(j)(t),∀ t ∈ I. (Hint: If for t = t, the d vectors were not independent,

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2.2 Uniqueness 17

one could find constants A, . . . , Ad, not all equal to zero, such thatPdj=1Ajx(j)(t) = 0;

hence, by linearity and uniqueness, Pd_j=1Ajx(j)(t) = 0,∀ t ∈ I which contradicts the

independence for t = t0.)

8. Show that Problem 7 implies that, given d solutions t→ x(1)_{(t), . . . , x}(d)_{(t), t}_{∈ I, to}

˙x = L(t)x, the matrix W (t) (“Wronskian matrix” of x(1)_{, . . . , x}(d)_{) defined by}

Wij(t) = x(i)j (t), i, j = 1, 2, . . . , d, t∈ I

has a determinant w(t) non vanishing for t∈ I if and only if ∃t0∈ I such that w(t0)6= 0.

(Hint. By linear algebra, this is just another way of phrasing Problem 7: d vectors are linearly independent if and only if the “determinant of their components” is not zero.) 9.Using the determinant differentiation rule, by rows, show that

d dtw(t)≡ d dtdet W (t) = _Xd i=1 ℓij(t) w(t); hence, ifPd_i=1ℓij(t) = ℓ(t), one has w(t) = w(t0) e

R

t t0ℓ(τ )dτ.

10.In the context of Problem 8, suppose that the matrix function t → L(t), t ∈ R, is periodic with period T > 0, i.e., t→ ℓij(t), i, j = 1, . . . , d are T−periodic functions. Let

x(1)_{, . . . , x}(d)_{be d linearly independent solutions for t > 0. Then there exist d}2 _constants

A(i)_j , i, j = 1, . . . , d, such that

x(i)(t + T ) =

d

X

j=1

A(i)_j x(j)(t), t≥ 0. Show that det W (T )/ det W (0) = w(T )/w(0) = det A6= 0.

11.Suppose that the matrix A is similar, via a real nonsingular matrix S, to a real diagonal matrix Λ, Λij= λiδij, i, j = 1, . . . , d: SAS−1= Λ. In the context of Problem 10, define

y(i)(t) =

d

X

j=1

Sijx(j)(t).

Show that y(1)_{, . . . .y}(d)_{are linearly independent solutions, λ}

1, . . . , λd6= 0, and

y(i)(t + T ) = λiy(i)(t), t≥ 0

12.Suppose that A is a matrix similar to a diagonal matrix Λ via a complex nonsingu-lar matrix S. Show that y(1)_{, . . . , y}(d)_{, defined as in the preceding problem, are complex}

solutions of ˙x = L(t)x and that y(i)_{(t + T ) = λ}

iy(i)(t),∀ t ≥ 0. (For applications, recall

that from linear algebra (see Appendix E), a sufficient condition for the similarity between A and a diagonal matrix Λij = λiδij is that the roots λ1, . . . , λdof the secular equation

det(A− λ) = 0 are pairwise different.)

13.Given the assumptions of Problems 10,11 and supposing λ1, . . . , λd> 0, show that the

most general solution to ˙x = L(t)x has the form x(t) =

d

X

j=1

αjλt/Tj z(j)(t)

where the functions z(1)_{, . . . , z}(d)_{are d C}∞_{functions periodic with period T , and α}₁_{, . . . , α}_d

are arbitrary constants. (Hint: Let z(i)_{(t) = λ}−t/T i y(i)(t).)

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14.Suppose that for every nonzero complex number λ, there exists a C∞ _{function t}_→

γ(t), t∈ R, such that γ(t + t′_{) = γ(t)γ(t}′_{), γ(0) = 1, γ(T ) = λ}−1_{, γ(t)}_{6= 0 ∀ t ∈ R; then}

the conclusions of Problem 13 would hold, replacing λ−t/T _{by γ(t), without the assumption}

λj> 0, j = 11, . . . , d, under the only assumption det A6= 0. See also the following problem.

15.Let λ∈ C, λ−1 _{= ̺ (cos θ + i sin θ, ̺ > 0, θ}_{∈ [0, 2π]. Define γ(t) = ̺}t/T_(cos t Tθ +

i sin_Ttθ). Show that γ(0) = 1, γ(t)γ(t + t′_{) = γ(t + t}′_{), γ(T ) = λ}−1_{, γ(t)}_{6= 0, ∀ t ∈ R (e.g.,}

(−1)t/T _{= cos} t

Tπ + i sin t Tπ).

Observations to Problems 8-15.

We shall see that there always exist d linearly independent solutions to ˙x = L(t)x. However, the existence of S is a restrictive condition. When such an S does not exist, it is possible to show that the most general solution to ˙x = L(t)x, with L periodic with period T > 0 and C∞_{, can be written in the form}

x(t) = p X j=1 δ(j)−1_X k=0 αjkλt/Tj tkz(j)(t),

where Pp_j=1δ(j) = d, and δ(j), λj are suitably chosen, and t→ z(j)(t), t≥ 0, are C∞

functions periodic with period T and possibly complex valued (when λj are not positive

and λt/T_j is interpreted as explained in Problem 15), and αjk are arbitrary constants (see

[38], for instance, Vol. 1, pp. 63-68, ).

16.Consider a differential equation ¨x + a(t) ˙x + b(t)x = 0, t ∈ R, a, b ∈ C∞₍_{R). After}

reducing it to a first-order system of two differential equations inR2_{, interpret the results}

of Problems 7-15 in terms of its solutions. Show first that the matrix W (t) associated with this system is expressed in terms of two of its solutions t→ x(1)_{(t) and t}_{→ x}(2)_{(t) as}

W (t) =

_x(1)_(t) _˙x(1)_(t)

x(2)_(t) _˙x(2)_(t)

and ˙w(t) = a(t)w(t).

17.* Extend Problem 16 to the case of the sth-order differential equation inR: ds_x dts + s−1 X j=0 aj(t)d j_x dtj, t∈ R.

2.3 General Properties of Motion. Existence

An existence problem for the solutions of Eq. (2.2.4), hence of Eq. (2.2.2), naturally associated with the uniqueness property given in Proposition 1, §2.2, is solved by the following proposition:

2 Proposition. Let F be an_Rd_{-valued function in C}∞₍_Rd

×R). Let x0∈ Rd and t0∈ R. Let S(ξ0, ̺) be the closed ball inRd with center ξ0 and radius ̺. Let θ > 0. There exists T̺,θ > 0 and a solution of Eq. (2.2.4), i.e., ˙x = F(x, t), defined for t_{∈ [t}0− T̺,θ, t0+ T̺,θ] and of class C∞ such that:

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2.3 Existence 19

Furthermore, if one defines: M̺,ξ0,t0,θ def = max ξ_∈S(ξ0,̺) t∈[t0−θ,t0+θ] |F(ξ, t)| ≡ M (2.3.2)

one can choose

T̺,θ= ̺

̺ + θM θ. (2.3.3)

Observations.

(1) By Proposition 1, _{§(2.2), it is enough to show the existence of a C}(1) solution verifying Eq. (2.3.1).

(2) The proof that follows is “constructive” in the sense that it provides a sequence t→ x(n)_{(t), t}_{∈ [t}

0− T̺,θ, t0+ T̺,θ], of functions approximating (as n → ∞) the solution and, at the same time, it provides an estimate of the approximation error defined as max|x(t) − x(n)_(t)_{|, where the maximum is} taken on the interval [t0− T̺,θ, t0+ T̺,θ].

(3) It is often useful, in applications, not to follow the solution scheme pro-posed by the following proof of Proposition 2. It might, in fact, be more convenient to use ad hoc procedures based on the particular features of the F under analysis in a concrete case. Usually, with such procedures one finds much better error estimates than the ones following from general methods, where one cannot take into account some special properties of the equations (e.g., symmetry properties, Hamiltonian form, etc.).

(4) To understand informally the bound on the magnitude of the interval of existence consider first that, during the proof, it appears necessary to have an a priori control of how far x(t) can travel away from the initial position ξ0. The continuity of F guarantees the boundedness of the maximum of_{|F(ξ, t)|,} for, say, ξ _{∈ S(ξ}0, ̺), t ∈ [t0 − θ, t0+ θ]. It follows that during the whole time interval [t0− T̺,θ, t0+ T̺,θ], the point x(t) stays inside S(ξ0, ̺) because ˙x(t) = F(x(t), t) and the right-hand side of this relation does not exceed M , Eq. (2.3.2): notice, in fact, that T̺,θ has been chosen, just to achieve this effect, smaller than both θ and ̺M−1 _{(i.e., T}_̺,θ _{= (θ}−1_{+ ̺}−1_{M )}−1 _{so that} M T̺,θ < ̺).

(5) The interval [t0− T̺,θ, t0+ T̺,θ] is certainly not optimal, at least because the choice of the set S(ξ0, ̺)× [t0− θ, t0+ θ], where the maximum of |F| is considered, was arbitrary. A better existence interval could be obtained using this arbitrariness and optimizing the result over the possible sets on which one takes the maximum. Also, once the existence of a solution verifying Proposi-tion 2 has been established, one could apply ProposiProposi-tion 2 and ProposiProposi-tion 1 to the equation with initial datum x(t0+T̺,θ) at the initial time t0+T̺,θ, thus continuing it beyond T̺,θ. However one cannot hope, in general, for an infinite existence interval containing_R+: this can be seen through counterexamples. The simplest among them is provided by the equation ˙x = x2_{, x(0) = 1, in}

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Proof. Rather than studying C(1)_{solutions of ˙x = F(x, t) verifying the initial} conditions (2.3.1), look forRd_{-valued C}(0)_([t

0− T̺,θ, t0+ T̺,θ]) solutions of the equation: x(t) = ξ0+ Z t t0 F(x(τ ), τ ) dτ. (2.3.4) Every C(0)_([t

0− T̺,θ, t0+ T̺,θ]) function verifying Eq. (2.3.4) is a C(1) solution to the original equation also verifying Eq. (2.3.1), and vice versa. For t _{∈ [t}0− T̺,θ, t0+ T̺,θ] define the sequence of Rd-valued functions t→ x(n)_{(t), n = 0, . . ., through the following recursive scheme:}

x(0)(t) = ξ0, x(1)(t) = ξ0+ Z t t0 F(x(0)(τ ), τ ) dτ, . . . x(n)(t) = ξ0+ Z t t0 F(x(n−1)(τ ), τ ) dτ, (2.3.5)

and remark that each such function is in C∞₍_{R) and it s natural to try taking} the limit as n→ +∞. The existence, uniformly in t ∈ [t0− T̺,θ, t0+ T̺,θ], of

lim

n→∞x

(n)_{(t) = x(t)} _(2.3.6)

should imply that the limit function will also be continuous. Existence and uniformity of the limit is obtained by rewriting it as

x(0)(t) + ∞ X k=1

(x(k)(t)_{− x}(k−1)(t)) (2.3.7)

and deducing that if

µk = max t∈[t0−T̺,θ,t0+T̺,θ]|x (k)_(t) − x(k+1)(t)_|, then (2.3.8) ∞ X k=0 µk < +∞ (2.3.9)

This will mean that the series of Eq. (2.3.7) is uniformly convergent for t_∈ [t0− T̺,θ, t0+ T̺,θ]: hence, the same will hold for the limit of Eq. (2.3.6).

To estimate µk we can refer to Eq. (2.3.5) to obtain for k = 2, 3, . . .,

x(k)(t)− x(k−1)_{(t) =}Z t t0

F(x(k−1)(τ ), τ )− F(x(k−2)_{(τ ), τ )}_dτ _(2.3.10) Through Lagrange’s theorem in the form

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2.3 Existence 21 |F(ξ, τ) − F(η, τ)| ≤ L |ξ − η|, ∀ ξ, Bh ∈ S(ξ0, ̺), ∀ τ ∈ [t0− T̺,θ, t0+ T̺,θ] (2.3.11) where L = max ξ∈S(ξ0,̺) t∈[t0−T̺,θ,t0+T̺,θ] d X i,j=1 ∂F(i) ∂ξj (ξ, t) (2.3.12)

Eqs. (2.3.10) and (2.3.11) imply:

|x(k)_(t)

− x(k−1)_(t)_{| ≤ L}Z [t0,t]

|x(k−1)_{(τ )}_{− x}(k−2)_{(τ )}_{| dτ} _(2.3.13) ∀ k = 2, 3, . . . provided we preliminarily check that for all k = 0, 1, . . ., the functions t− → x(k)_{(t), t}_{∈ [t}

0− T̺,θ, t0+ T̺,θ], take their values in S(ξ0, ̺). This last property is proved inductively starting from Eq. (2.3.5): keeping in mind the choice of T̺,θ (chosen, as essentially stated in observation (4), just in such a way to make this property true) suppose, inductively, that |x(h)_(t)

− ξ0| ≤ ̺, ∀ h = 0, . . . , k − 1; it is a property which holds for k = 1. To check that_|x(k)_(t)

− ξ0| ≤ ̺ remark that Eqs. (2.3.5) and (2.3.3) give |x(k)_(t) − ξ0| ≤ Z [t0,t] dτ|F(x(k−1)_{(τ ), τ )}_{| ≤ M} ̺,ξ0,θ|t − t0| < ̺ (2.3.14)

Eq. (2.3.13), follows because Eq. (2.3.14) with k = 1 yields for t _{∈ [t}0 − T̺,θ, t0+ T̺,θ], |x(k)(t)_{− x}(k−1)(t)_{| ≤ L}k−1 Z [t0,t] dτ1 Z [t0,τ1] dτ2. . . × Z [t0,τk−2] dτ_k−1_|x(1)(τ_k−2)_{− ξ}0| ≤ Lk−1_Tk−1 ̺,θ (k− 1)! ̺ (2.3.15)

since T̺,θ ≥ |t − t0|. Eq. (2.3.15) shows the convergence of the series of Eq. (2.3.9) and, therefore, the limit of Eq. (2.3.6) exists uniformly for t _{∈ [t}0− T̺,θ, t0+ T̺,θ] and defines a function t→ x(t) on this interval with values in S(ξ0, ̺). It satisfies Eq. (2.3.4) as it is seen by taking the n→ ∞ limit in Eq. (2.3.5) and by using the uniformity of the limit of Eq. (2.3.6) to exchange the

integration with the limit. mbe

2.3.1 Problems

1.Give a lower estimate for the magnitude of T̺,θ, the amplitude of the existence interval

as in Proposition 2, for the following second-order equations, assuming x(0) = 0, ˙x(0) = 1 or x(0) = 1, ˙x(0) = 0 as initial data at t0= 0:

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¨

x = x, x = x + x¨ 3, ¨x = x− ˙x + x3, ¨x =− ˙x2, ¨x ==− sin x.

Also estimate sup̺,θT̺,θ from below. (Hint: Reduce the equation to first order and then

apply Proposition l.)

2.Solve the equation ¨x = x with initial datum x(0) = 1, ˙x(0) = 0.

3.Solve the equations ˙x =−x2_{, ˙x = cos x, ˙x = (cos x)}2_{with initial datum x(0) = 1.}

4.Solve the equation ˙x = x + y, ˙y =−x + 2y with initial datum x(0) = 0, y(0) = 1. 5.Using the “quadrature method”, solve the equation ¨x = 4(x3_{− x), x(0) = 0, ˙x =}√_{2 (see}

§2.1, final comment).

6.As in Problem 5 for ˙x =−(4x3_{+ 6x}2_{− 2), x(0) = 0, ˙x(0) =}√_2.

7.Find two linearly independent solutions for the equation in Problem 4. 8.* Compute w(t) for the equation in Problem 4 (see Problem 8,§(2.2).

9.* Let t→ L(t) be a d × d-matrix-valued C∞ _{function on} _{R. Show that the equation}

˙x(t) = L(t)x(t) admits d linearly independent solutions defined for|t| ≤ T with T small enough. (Hint: Let x(i)_{be the solution with initial data x}(i)

j (0) = δi,j, i, j = 1, . . . d. Then

evaluate an existence interval for such initial data.)

10.* Compute T1,1 for the equation in Problem 9 when |t0| < σ and ξ0 is arbitrary,

ξ0 = x(t0); for the symbols, see Proposition 1. Show that |ξ0| T1,1 can be taken to be

independent of t0and ξ0at a given σ > 0. Deduce from this that every solution to ˙x = L(t)x

can be extended to a solution defined for t∈ R.

11.Let L be a d× d matrix and consider the equation ˙x = Lx in Rd_{. Suppose that L has}

d pairwise distinct real eigenvalues (see Appendix E for the eigenvalue notion) λ1, . . . , λd.

Let v1)_{, . . . , v}(d)_{be the respective real linearly independent eigenvectors (see Appendix E).}

Show that the functions t→ eλit_v(i)_{are d linearly independent solutions. Show that any}

solution t→ x(t) has the form x(t) =

d

X

j=1

αjeλjtv(j), with (α1, . . . , αd)∈ Rd.

2.4 General Properties of Motion. Regularity.

In proving Proposition 2 it was found that C(1) _{solutions of ˙x = F(x, t), F}_∈ C∞₍_Rd_{× R), are necessarily C}∞ _{solutions. This is the simplest regularity} property shown by the solutions of such differential equations. Other regularity properties of the solutions will be now analyzed.

In applications it often happens that the right-hand side of Eq. (2.2.4) depends on parameters α _{∈ R}m _{and that, furthermore, it is important to} know how the solutions change as the initial data ξ0 and the parameters α vary in _Rd _and

Rm_{, respectively. A first answer to this question is provided} by the following proposition.

3 Proposition. Let ξ, t, α → F(ξ, t, α) be a C∞₍_Rd

× R × Rm_{) function} taking its values in Rd_{, and consider the equation}

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2.4 Regularity Regularity 23

x(t) = ξ0+ Z t

t0

F(x(τ ), τ, α0) dτ (2.4.1) as an equation for the continuous function t_{→ x(t) parameterized by ξ}0, t0, α0 ∈ Rd

× R × Rm_{. Given ̺, θ, a > 0 and (ξ, t, α)} ∈ Rd

× R × Rm_{, there exists} T > 0 such that:

(i) Eq. (2.4.1) admits a solution for every (ξ0, t0, α0) close enough to (ξ, t, α) such that|ξ −ξ0| <̺₂, |t−t0| < θ₂,|α −α0| < a. Such solution will be denoted t→ St(ξ0; t0, α0) and it is defined for t∈ [t0− T, t0+ T ].

(ii) The function St(ξ0; t0, α0), defined for |ξ − ξ0| <

̺

2, |t − t0| < θ

2, |α − α0| < a, |t − t0| ≤ T (2.4.2) takes its values inside the ball S(ξ; ̺) with center ξ and radius ̺ and it is a C∞ function of its arguments.

(iii) The value T can be taken as:

T = ̺

2(̺ + θ max_{|F(ξ, t, α)|)}θ (2.4.3)

where the maximum is considered on the set_{|ξ−ξ| <} ̺₂,_{|t−t| <}θ

2,|α−α| < a. Observations.

(1) Eq. (2.4.1) is equivalent to

˙x(t) = F(x(t), t, α0), x(t0) = ξ0 (2.4.4) and, therefore, the above proposition provides a regularity theorem for the solutions of Eq. (2.4.4) as functions of the initial data, of the initial time, of time itself, and of the parameters α on which F may possibly depend. The set (2.4.2) and the key estimate (2.4.3) should not be taken too seriously as they are not optimal: they merely show an example of the type of concreteness that can be attained in the formulation of a regularity criterion (see, also, observation 4, p. 19).

(2) Let β = (β1, . . . , βd+m+2)≡ ((ξ0)1, . . . , (ξ)d, (α0)1, . . . , (α0)m, t, t0) and x(t) = (x1(t), . . . , xd(t)) = St(ξ0; t0, α0)

≡ (St(ξ0; t0, α0)1, . . . , (St(ξ0; t0, α0)d)

(2.4.5) Formal differentiation of Eq. (2.4.4) with respect to βi, i = 1, 2, . . . , m + d, gives d dt ∂x(t) ∂βi = d X h=1 ∂F ∂ξh (x(t), t, α0) ∂xh(t) ∂βi + d X h=1 ∂F ∂αk (x(t), t, α0) ∂αk ∂βi (2.4.6)