A History of Mechanics

by René Dugas

Overview

"A remarkable work which will remain a document of the first rank for the historian of mechanics." — Louis de Broglie
In this masterful synthesis and summation of the science of mechanics, Rene Dugas, a leading scholar and educator at the famed Ecole Polytechnique in Paris, deals with the evolution of the principles of general mechanics chronologically from their earliest roots in antiquity through the Middle Ages to the revolutionary developments in relativistic mechanics, wave and quantum mechanics of the early 20th century.
The present volume is divided into five parts: The first treats of the pioneers in the study of mechanics, from its beginnings up to and including the sixteenth century; the second section discusses the formation of classical mechanics, including the tremendously creative and influential work of Galileo, Huygens and Newton. The third part is devoted to the eighteenth century, in which the organization of mechanics finds its climax in the achievements of Euler, d'Alembert and Lagrange. The fourth part is devoted to classical mechanics after Lagrange. In Part Five, the author undertakes the relativistic revolutions in quantum and wave mechanics.
Writing with great clarity and sweep of vision, M. Dugas follows closely the ideas of the great innovators and the texts of their writings. The result is an exceptionally accurate and objective account, especially thorough in its accounts of mechanics in antiquity and the Middle Ages, and the important contributions of Jordanus of Nemore, Jean Buridan, Albert of Saxony, Nicole Oresme, Leonardo da Vinci, and many other key figures.
Erudite, comprehensive, replete with penetrating insights, AHistory of Mechanics is an unusually skillful and wide-ranging study that belongs in the library of anyone interested in the history of science.

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ISBN-13:	9780486173375
Publisher:	Dover Publications
Publication date:	11/07/2012
Series:	Dover Books on Physics
Sold by:	Barnes & Noble
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Pages:	688
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A History of Mechanics

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By René Dugas, J. R. Maddox

Dover Publications, Inc.

Copyright © 1988 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-17337-5

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CHAPTER 1

HELLENIC SCIENCE

1. ARISTOTELIAN MECHANICS.

For lack of more ancient records, history of mechanics starts with Aristotle (384-322 B. C.) or, more accurately, with the author of the probably apocryphal treatise called Problems of Mechanics ([TEXT NOT REPRODUCIBLE IN ASCII]). This is, in fact, a text-book of practical mechanics devoted to the study of simple machines.

In this treatise the power of the agency that sets a body in motion is defined as the product of the weight or the mass of the body—the Ancients always confused these concepts—and the velocity of the motion which the body acquires. This law makes it possible to formulate the condition of equilibrium of a straight lever with two unequal arms which carry unequal weights at their ends. Indeed, when the lever rotates the velocities of the weights will be proportional to the lengths of their supporting arms, for in these circumstances the powers of the two opposing powers cancel each other out.

The author regards the efficacy of the lever as a consequence of a magical property of the circle. "Someone who would not be able to move a load without a lever can displace it easily when he applies a lever to the weight. Now the root cause of all such phenomena is the circle. And this is natural, for it is in no way strange that something remarkable should result from something which is more remarkable, and the most remarkable fact is the combination of opposites with each other. A circle is made up of such opposites, for to begin with it is made up of something which moves and something which remains stationary...."

In this way Problems of Mechanics. reduces the study of all simple machines to one and the same principle. "The properties of the balance are related to those of the circle and the properties of the lever to those of the balance. Ultimately most of the motions in mechanics are related to the properties of a lever."

To Aristotle himself, just as much in his Treatise on the Heavens, ([TEXT NOT REPRODUCIBLE IN ASCII]) as in his Physics, concepts belonging to mechanics were not differentiated from concepts having a more general significance. Thus the notion of movement included both changes of position and changes of kind, of physical or chemical state. Aristotle's law of powers, which he called [TEXT NOT REPRODUCIBLE IN ASCII] or [TEXT NOT REPRODUCIBLE IN ASCII], is formulated in Chapter V of Book VII of his Physics in the following way.

"Let the motive agency be α, the moving body β, the distance travelled γ and the time taken by the displacement be δ. Then an equal power, namely the power α, will move half of β along a path twice γ in the same time, or it will move it through the distance γ in half the time δ. For in this way the proportions will be maintained."

Aristotle imposed a simple restriction on the application of this rule—a small power should not be able to move too heavy a body, "for then one man alone would be sufficient to set a ship in motion."

This same law of powers reappears in Book III of the Treatise on the Heavens. Its application to statics may be regarded as the origin of the principle of virtual velocities which will be encountered much later.

In another place Aristotle made a distinction between natural motions and violent motions.

The fall of heavy bodies, for example, is a natural motion, while the motion of a projectile is a violent one.

To each thing corresponds a natural place. In this place its substantial form achieves perfection—it is disposed in such a way that it is subject as completely as possible to influences which are favourable, and so that it avoids those which are inimical. If something is moved from its natural place it tends to return there, for everything tends to perfection. If it already occupies its natural place it remains there at rest and can only be torn away by violence.

In a precise way, for Aristotle, the position of a body is the internal surface of the bodies which surround it. To his most faithful commentators, the natural place of the earth is the concave surface which defines the bottom of the sea, joined in part to the lower surface of the atmosphere, the natural place of the air.

Concerning the natural motion of falling bodies, Aristotle maintained in Book I of the Treatise on the Heavens that the "relation which weights have to each other is reproduced inversely in their durations of fall. If a weight falls from a certain height in so much time, a weight which is twice as great will fall from the same height in half the time."

In his Physics (Part V), Aristotle remarked on the acceleration of falling heavy bodies. A body is attracted towards its natural place by means of its heaviness. The closer the body comes to the ground, the more that property increases.

If the natural place of heavy bodies is the centre of the World, the natural place of light bodies is the region contiguous with the Sphere of the Moon. Heavenly bodies are not subject to the laws applicable to terrestrial ones—every star is a body as it were divine, moved by its own divinity.

We return to terrestrial mechanics. All violent motion is essentially impermanent. This is one of the axioms which the Schoolmen were to repeat—Nullum violentum potest esse perpetuum. Once a projectile is thrown, the motive agency which assures the continuity of the motion resides in the air which has been set in motion. Aristotle then assumes that, in contrast to solid bodies, air spontaneously preserves the impulsion which it receives when the projectile is thrown, and that it can in consequence act as the motive agency during the projectile's flight.

This opinion may seem all the more paradoxical in view of the fact that Aristotle remarked, elsewhere, on the resistance of the medium. This resistance increases in direct proportion to the density of the medium. "If air is twice as tenuous as water, the same moving body will spend twice as much time in travelling a certain path in water as in travelling the same path in air."

Aristotle also concerned himself with the composition of motions. "Let a moving body be simultaneously actuated by two motions that are such that the distances travelled in the same time are in a constant proportion. Then it will move along the diagonal of a parallelogram which has as sides two lines whose lengths are in this constant relation to each other." On the other hand, if the ratio between the two component distances travelled by the moving body in the same time varies from one instant to another, the body cannot have a rectilinear motion. "In such a way a curved path is generated when the moving body is animated by two motions whose proportion does not remain constant from one instant to another."

These propositions relate to what we now call kinematics. But Aristotle immediately inferred from them dynamical results concerning the composition of forces. The connection between the two disciplines is not given, but as Duhem has indicated, it is easily supplied by making use of the law of powers—a fundamental principle of aristotelian dynamics. In particular, let us consider a heavy moving body describing some curve in a vertical plane. It is clear that the body is actuated by two motions simultaneously. Of these, one produces a vertical descent while the other, according to the position of the body on its trajectory, results in an increase or a decrease of the distance from the centre. In Aristotle's sense, the body will have a natural falling motion due to gravity, and will be carried horizontally in a violent motion. Consider different moving bodies unequally distant from the centre of a circle and on the same radius. Let this radius, in falling, rotate about the centre. Then it may be inferred that for each body the relation of the natural to the violent motion remains the same. "The contemplation of this equality held Aristotle's attention for a long time. He appears to have seen in it a somewhat mysterious correlation with the law of the equilibrium of levers."

Aristotle believed in the impossibility of a vacuum (Physics, Book IV, Chapter XI) on the grounds that, in a vacuum, no natural motion, that is to say no tendency towards a natural place, would be possible. Incidentally this idea led him to formulate a principle analogous to that of inertia, and to justify this in the same way as that used by the great physicists of the XVIIIth Century.

"It is impossible to say why a body that has been set in motion in a vacuum should ever come to rest; why, indeed, it should come to rest at one place rather than at another. As a consequence, it will either necessarily stay at rest or, if in motion, will move indefinitely unless some obstacle comes into collision with it."

Aristotle's ideas on gravitation and the figure of the Earth merit our attention, if only because of the influence which they have had on the development of the principles of mechanics. First we shall quote from the Treatise on the Heavens (Book II, Chapter XIV). "Since the centres of the Universe and of the Earth coincide, one should ask oneself towards which of these heavy bodies and even the parts of the Earth are attracted. Are they attracted towards this point because it is the centre of the Universe or because it is the centre of the Earth? It is the centre of the Universe towards which they must be attracted.... Consequently heavy bodies are attracted towards the centre of the Earth, but only fortuitously, because this centre is at the centre of the Universe."

If the Earth is spherical and at the centre of the World, what happens if a large weight is added to one of the hemispheres? The answer to this question is the following. "The Earth will necessarily move until it surrounds the centre of the World in a uniform way, the tendencies to motion of the different parts counterbalancing one another." Duhem points out that the centre, [TEXT NOT REPRODUCIBLE IN ASCII], that in every body is attracted to the centre of the Universe, was not defined in a precise way by Aristotle. In particular, Aristotle did not identify it with the centre of gravity, of which he was ignorant.

In this same treatise Aristotle repeatedly enumerates the arguments for the spherical figure of the Earth. He distinguishes a posteriori arguments, such as the shape of the Earth's shadow in eclipses of the Moon, the appearance and disappearance of constellations to a traveller going from north to south, from a priori arguments, of which he says—

"Suppose that the Earth is no longer a single mass, but that, potentially, its different parts are separated from each other and are placed in all directions and attracted similarily towards the centre. Then let the parts of the Earth which have been separated from each other and taken to the ends of the World be allowed to reunite at the centre; let the Earth be formed by a different procedure—the result will be exactly the same. If the parts are taken to the ends of the World and are taken there similarily in all directions, they will necessarily form a mass which is symmetrical. Because there will result an addition of parts which are equal in all directions, and the surface which defines the mass produced will be everywhere equidistant from the centre. Such a surface will therefore be a sphere. But the explanation of the shape of the Earth would not be changed in any way if the parts which form it were not taken in equal quantities in all directions. In fact, a larger part will necessarily push away a smaller one which it finds in front of it, for both have a tendency towards the centre and more powerful weights are able to displace lesser ones."

To Aristotle, heaviness does not prove rigorously that the Earth will be spherical, but only that it will tend to be so. On the other hand, for the surface of water, this is obvious if it is admitted that "it is a property of water to run towards the lowest places," that is, towards places which are nearest the centre.

Let βεγ be an arc of a circle with centre α; the line αδ is the shortest distance from α to βγ. "Water will run towards δ from all sides until its surface becomes equidistant from the centre. It therefore follows that the water takes up the same length on all the lines radiating from the centre. It then remains in equilibrium. But the locus of equal lines radiating from a centre is a circumference of a circle. The surface of the water, βεγ, will therefore be spherical."

Adrastus (360-317 B. C.), commenting on Aristotle, made the preceding proof precise in the following terms. "water will run towards the point δ until this point, surrounded by new water, is as far from α as β and γ. Similarily, all points on the surface of the water will be at an equal distance from α. Therefore the water exhibits a spherical form and the whole mass of water and Earth is spherical."

Adrastus supplemented this proof with the following evidence, which was destined to become classical.

"Often, during a voyage, one cannot see the Earth or an approaching ship from the deck, while sailors who climb to the top of a mast can see these things because they are much higher and thus overcome the convexity of the sea which is an obstacle."

We shall say no more about aristotelian mechanics. However inadequate they may seem now, these intuitive theories have their origin in the most everyday observations, precisely because they take the passive resistances to motion into account. To an unsophisticated observer, a horse pulling a cart seems to behave according to the law of powers, in the sense that it develops an effort which increases regularly with the speed. In order to break away from Aristotle's ideas and to construct the now classical mechanics, it is necessary to disregard the various ways in which motion may be damped, and to introduce these explicitly at a later stage as frictional forces and as resistances of the medium. However it may be, aristotelian doctrines provided the fabric of thought in mechanics for nearly two thousand years, so that even Galileo, who was to become the creator of modern dynamics, made his first steps in science in commenting Aristotle, and proved in his early writings to be a faithful Peripatetic; which, it may be said in passing, in no way diminishes his glory as a reformer, on the contrary, it only adds to it.

(Continues...)

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Table of Contents

PREFACE
PART ONE THE ORIGINS
CHAPTER ONE. - HELLENIC SCIENCE
1. Aristotelian mechanics
2. The Statics of Archimedes
CHAPTER II. - ALEXANDRIAN SOURCES AND ARABIC MANUSCRIPTS
1. "The "mechanics" of Hero of Alexandria"
2. Pappus' theories of the inclined plane and of the centre of gravity
3. The fragments attributed to Euclid in arabic writings
4. The book of Charistion
CHAPTER III. - THE XIIIth CENTURY. THE SCHOOL OF JORDANUS
1. "Jordanus of Nemore and "gravitas secundum situm"
2. "The anonymous author of "Liber Jordani de ratione ponderis." The angular lever. The inclined plane"
CHAPTER IV. - THE XIVth CENTURY. THE SCHOOLS OF BURIDAN AND ALBERT OF SAXONY. NICOLE ORESME AND THE OXFORD SCHOOL
1. "The doctrine of "impetus" (John Buridan)"
2. The sphericity of the earth and the oceans-Albert of Saxony and the aristotelian tradition
3. Albert of Saxony's theory of centre of gravity
4. Albert of Saxony's kinematics. The acceleration of falling bodies
5. The discussion of action at a distance
6. Nicole Oresme-a disciple of Buridan
7. Oresme's rule in kinematics. (Uniformly accelerted motion.)
8. Oresme as a predecessor of Copernicus
9. The Oxford School
10. The tradition of Albert of Saxony and of Buridan
CHAPTER V. - XVth AND XVIth CENTURIES. THE ITALIAN SCHOOL. BLASIUS OF PARMA. THE OXFORD TRADITION. NICHOLAS OF CUES AND LEONARDO DA VINCI. NICHOLAS COPERNICUS. THE ITALIAN AND PARISIAN SCHOOLMEN OF THE XVIth CENTURY. DOMINIC SOTO AND THE FALL OF BODIES
1. Blasius of Parma and his treatise on weights
2. The Italian tradition of Nicole Oresme and the Oxford School
3. "Nicholas of Cues (1404-1464) and the doctrine of "impetus impressus"
4. Leonardo da Vinci's contribution to mechanics
5. Nicholas Copernicus (1472-1543). His system of the world and his ideas on attraction
6. John Fernel (1497-1558) and the figure of the earth
7. Italian scholasticism in the XVIth century
8. Parisian scholasticism in the XVIth century
9. The attack of the humanists
10. Dominic de Soto (1494-1560) and the laws of falling bodies
CHAPTER VI. - XVIth CENTURY (continued) THE ITALIAN SCHOOL OF NICHOLAS TARTAGLIA AND BERNARDINO BALDI
1. Nicholas Tartaglia
2. Jerome Cardan (1501-1576)
3. Julius-Caesar Scaliger and Buridan's doctrine
4. Bento Pereira (1535-1610). The classical reaction
5. "The "Mechanicorum Liber" of Guido Ubaldo (1545-1607)"
6. J.-B. Villalpand (1552-1608) and the polygon of sustentation
7. "J.-B. Benedetti (1530-1590). Statics. Figure of the earth. Doctrine of "impetus"
8. Giordano Bruno (1548-1600) and the composition of motion
9. "Bernardino Baldi (1553-1617). Statics and gravity "exviolentia"
CHAPTER VII. - XVIth CENTURY (continued). XVIIth CENTURY. TYCHO-BRAHE AND KEPLER
1. The system due to Tycho-Brahe (1546-1601)
2. Kepler (1571-1631). The general chracter of his contribution
3. The origin of the law of areas
4. Origin of the law of the ellipticity of planetary trajectories
5. Kepler's third law
6. Kepler and the concept of inertia
7. Kepler and the doctrine of attraction
PART II THE FORMATION OF CLASSICAL MECHANICS
CHAPTER I. - STEVIN'S STATICS. SOLOMON OF CAUX
1. The statics of Stevin (1548-1620)
2. Stevin and the principle of virtual work
3. Stevin's hydrostatics
4. Solomon of Caux (1576-1630) and the concept of work
CHAPTER II. - GALILEO AND TORRICELLI
1. Galileo's statics
2. Galileo and the fall of bodies
3. Galileo and the motion of projectiles
4. Galileo and the hydrostatics
5. Galileo and the Copernican system
6. Torricelli's principle
7. Torricelli and the motion of projectiles
8. Torricelli's experiment
9. Torricelli's law flow through an orifice
CHAPTER III. - MERSENNE (1588-1648) AS AN INTERNATIONAL GO-BETWEEN IN MECHANICS. ROBERVAL (1602-1675)
1. The arrival of foreign theories in France. The part played by Mersenne
2. Roberval and compound motion
3. Roberval's treatise on mechanics
4. Roberval and the law of composition of forces
CHAPTER IV. - DESCARTES' MECHANICS. PASCAL'S HYDROSTATICS
1. Descartes' statics
2. Descartes and the fall of heavy bodies
3. Descartes and the conservation of quantities of motion
4. Descartes and the impact of bodies
5. The discussion between Descartes and Roberval on the centre of agitation
6. The quarrel about geostatics
7. Pascal's hydrostatics
"CHAPTER V. - THE LAWS OF IMPACT (WALLIS, WREN, HUYGHENS, MARIOTTE). THE MECHANICS OF HUYGHENS (1629-1697)"
1. The mechanics of Wallis (1616-1703)
2. Wren (1632-1723) and the laws of elastic impact
3. Huyghens (1629-1697) and the laws of impact
4. The plan of Huyghens' fundamental treatise
5. Huyghens and the fall of bodies
6. The isochronism of the cycloidal pendulum
7. The theory of the centre of oscillation
8. The theory of centrifugal force
9. Huyghens and the principle of relativity
10. Mariotte and the laws of impact
CHAPTER VI. - NEWTON (1642-1727)
1. The newtonian method
2. The newtonian concepts
3. The newtonian laws of motion
4. Newton and the dynamical law of composition of forces
5. The motion of a point under the action of a central force
6. Newton's explanation of the motion of the planets
7. The universal attraction
CHAPTER VII. - LEIBNIZ AND LIVING FORCE
1. "The "vis motrix" in the sense of Leibniz"
2. Leibniz and the laws of impact
3. Living and dead forces
CHAPTER VIII. - THE FRENCH-ITALIAN SCHOOL OF ZACCHI AND VARIGNON
1. Zacchi and Saccheri. Lamy and the composition of forces
2. The statics of Varignon (1654-1722)
3. Varignon and Torricelli's law of flow
PART III THE ORGANISATION AND DEVELOPMENT OF THE PRINCIPLES OF CLASSICAL MECHANICS IN THE XVIIIth CENTURY
CHAPTER I. - JEAN BERNOULLI AND THE PRINCIPLE OF VIRTUAL WORK (1717). DANIEL BERNOULLI AND THE COMPOSITION OF FORCES (1726)
1. Jean Bernoulli and the principle of virtual work
2. Daniel Bernoulli and the composition of forces
CHAPTER II. - THE CONTROVERSY ABOUT LIVING FORCES
CHAPTER III. - EULER AND THE MECHANICS OF A PARTICLE (1736)
CHAPTER IV. - JACQUES BERNOULLI AND THE CENTRE OF OSCILLATION (1703). D'ALEMBERT'S TREATISE ON DYNAMICS (1743)
1. Jacques Bernoulli and the centre of oscillation
2. The introductory argument of d'Alembert's Treatise on dynamics
3. D'Alembert and the concept of accelerating force
4. D'Alembert's principle
5. D'Alembert's solution of the problem of the centre of oscillation
6. The priority of Herman and Euler in the matter of d'Alembert's principle
7. D'Alembert and the laws of impact
8. D'Alembert and the principle of living forces
CHAPTER V. - THE PRINCIPLE OF LEAST ACTION
1. Return to Fermat
2. Cartesian objections to Fermat's principle
3. "Leibniz and the path of "least resistance" for light"
4. Maupertuis' law of rest
5. The principle of least action in Maupertuis' sense (1744)
6. The application of the principle of least action to the direct impact of two bodies
7. The principle of least action in Maupertuis' work
8. D'Alembert's condemnation of final causes
9. The polemic on the principle of least action
10. Euler's judgment on the controversy on least action
11. Euler and the law of the extremum of ? mvds
12. Final remark
CHAPTER VI. - EULER AND THE MECHANICS OF SOLID BODIES (1760)
CHAPTER VII. -CLAIRAUT AND THE FUNDAMENTAL LAW OF HYDROSTATICS
1. Clairaut's principle of the duct
2. The condition to be satisfied by the law of gravity to assure the conservation of the shape of a rotating fluid mass
3.
CHAPTER VIII. - DANIEL BERNOULLI'S HYDRODYNAMICS. D'ALEMBERT AND THE RESISTANCE OF FLUIDS. EULER'S HYDRODYNAMICAL EQUATIONS. BORDA AND THE LOSSES OF KINETIC ENERGY IN FLUIDS
1. Return to the hydrodynamics of the XVIIth century
2. Daniel Bernoulli's hydrodynamics
3. D'Alembert and the motion of fluids
4. D'Alembert and the resistance of fluids - His paradox
5. Euler and the equilibrium of fluids
6. Euler and the general equations of hydrodynamics
7. Borda and the losses of kinetic energy in fluids
"CHAPTER IX. - EXPERIMENTS ON THE RESISTANCE OF FLUIDS (BORDA, BOSSUT, DU BUAT). COULOMB AND THE LAWS OF FRICTION"
1. Borda's experiments and newtonian theories
2. The abbè Bossut's expriments
3. Du Buat (1734-1809) : Hydraulics and the resistance of fluids
4. Coulomb's work on friction
CHAPTER X. - LAZARE CARNOT'S MECHANICS
1. Carnot and the experimental character of mechanics
2. The concepts and postulates of Carnot's mechanics
3. Carnot's theorem
"CHAPTER XI. - THE "MÉCANIQUE ANALYTIQUE" OF LAGRANGE"
1. "The content and purpose of Lagrange's "Mécanique analytique"
2. Lagrange's statics
3. Lagrange and the history of dynamics
4. Lagrange's equations
5. The conservation of living forces as a corollary of Lagrange's equations
6. The principle of least action as a corollary of Lagrange's equations
7. "On some problems treated in the "Mécanique analytique"
8. Lagrange's hydrodynamics
PART IV SOME CHARACTERISTIC FEATURES OF THE EVOLUTION OF CLASSICAL MECHANICS AFTER LAGRANCE
FOREWARD
CHAPTER I. - LAPLACE'S MECHANICS (1799)
1. Laplace and the principles of dynamics
2. "The general mechanics compatible with an arbitrary relation between the "force" and the velocity"
3. Laplace and the significance of the law of universal gravitation
CHAPTER II. - FOURIER AND THE PRINCIPLE OF VIRTUAL WORKS (1798)
CHAPTER III. - THE PRINCIPLE OF LEAST CONSTRAINT (1829)
CHAPTER IV. - RELATIVE MOTION: RETURN TO A PRINCIPLE OF CLAIRAUT. CORIOLIS' THEOREMS. FOUCAULT'S EXPERIMENTS
1. Return to a principle of Clairaut (1742)
2. Coriolis' first theorem
3. Coriolis' second theorem
4. The experiments of Foucault (1819-1868)
CHAPTER V. - POISSON'S THEOREM (1809)
1. Poisson's theorem and brackets
2. The Lagrange-Poisson square brackets
CHAPTER VI. - ANALYTICAL DYNAMICS IN THE SENSE OF HAMILTON AND JACOBI
1. Hamilton's optics. Its double interpretation in terms of emision and wave propagation
2. The dynamical law of varying action in Hamilton's sense
3. The significance of the hamiltonian dynamics
4. Jacobi's criticism
5. Jacobi's fundamental theorem
6. The canonical equations and Jacobi's multiplier
7. Geometrisation of the principle of least action
CHAPTER VII. - NAVIER'S EQUATIONS
1. The molecular hypothesis
2. Equilibrium of fluids
3. The molecular forces in the motion of a fluid
4. Remark on the origin of the general equations of elasticity
CHAPTER VIII. - CAUCHY AND THE FINITE DEFORMATION OF CONTINUOUS MEDIA
CHAPTER IX. - HUGONIOT AND THE PROPAGATION OF MOTION IN CONTINUOUS MEDIA
1. Nature of the problem
2. Compatibility of two solutions. Velocity of propagation of one solution in another. Hugoniot's theorem
3. Discontinuities in the propagation of motion
"CHAPTER X. - HELMHOLTZ AND THE ENERGETIC THESIS DISCUSSION OF THE NEWTONIAN PRINCIPLES (SAINT-VENANT, REECH, KIRCHHOF, MACH, HERTZ, POINCARÉ, PAINLEVÉ, DUHEM)"
1. Helmholtz and the energetic thesis
2. Barré de Saint-Venant
3. "Reech and the "School of the thread"
4. Kirchhoff and the logistic structure of mechanics
5. Mach
6. Hertz
7. Poincaré - Criticism of the principles and discussion of the notion of absolute motion
8. Poincaré and the energetic thesis
9. Painlevé and the princple of causality in mechanics
10. Duhem and the evolution of mechanics
11. Conclusion of this chapter
PART V THE PRINCIPLES OF THE MODERN PHYSICAL THEORIES OF MECHANICS
FOREWARD
CHAPTER I. - SPECIAL RELATIVITY
A. PRESENTATION
1. Immediate antecedents of the special theory of relativity
2. Michelson's experiment and Lorentz's hypothesis of contraction
3. The Lorentz transformation
4. Introduction to Einstein's electrodynamics
5. Definition of simultaneity
6. Relativity of lengths and times
7. Transformation of the coordinates of space and time
8. Contraction of lengths and correlative dilation of times
9. Composition of velocities
10. Transformation of Maxwell's equations in the vacuum. Electrodynamic relativity
11. Transformation of Maxwell's equations including convection currents
12. Dynamics of the slowly accelerated electron
13. Space-time in the sense of Minkowski
B. ANALYSIS AND INTERPRETATION
2a. On Michelson's experiment
3a. Dynamics of the electron in Poincaré's sense
3b. From Lorentz to Einstein
4a. The ether made superfluous
5a. Difficulties of Einstein's notion of simultaneity
6a. "Field of validity of the principle of special relativity - "Galilean" systems of reference"
7a. On different mathematical ways of obtaining the Lorentz transformation
8a. Pseudo-paradoxes in special relativity
9a. Composition of velocities and Fizeau's experiment
12a. Return to the dynamics of variable mass in Painlevé's sense
13a. On the meaning of space-time
CHAPTER II. - GENERALISED RELATIVITY
A. PRESENTATION
1. Statement of the principle of generalized relativity
2. Remark on the mathematical tools of generalised relativity
3. The equations of motion of a free particle in a gravitational field
4. Equations of the gravitational field in the absence of matter
5. General form of the equations of gravitation
6. Reversion to Newton's theory in the first approximation
7. The conduct of measurements of space and time in a static gravitational field. Deviation of light rays. Displacement of the perihelion of the planets
8. The spatially closed universe
9. Gravitation and electricity
B. ANALYSIS AND INTERPRETATION
&nb
6a. On the geometrisation of classical mechan