Teodorescu P.P. Mechanical Systems, Classical Models Volume I: Particle Mechanics

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MECHANICAL SYSTEMS, CLASSICAL MODELS

and 1916, appear

the Special and the General Relativity, respectively, which constitute

new steps in the process of knowledge of the motion of mechanical systems.

We notice that these new models (which tried to maintain the principles of classical

mechanics and to adapt them, (for instance, the principle of inertia) have been verified

by many important experiments (

the crucial experiments in General Relativity). But

there exist still some contradictions (for instance,

the horologes’ paradox in Special

Relativity), so that it arises the necessity of some improved models. It is also possible

that the actual theory of relativity do correspond to velocities comparable to the velocity

of light in vacuum, but not to velocities close to this

superior limit velocity; it is possible

that at such velocities more complex models be necessary. There have been conceived

also models in which the velocities are always greater than the velocity of light in

vacuum, which is thus an

inferior limit velocity. We notice that for velocities v

negligible with respect to the velocity

c of light in vacuum (vc ), the results of

relativistic mechanics tend to the classical ones; there exists thus a matching of the two

mathematical models of mechanics, which is necessary for the viability of the improved

model (the relativistic model).

We mention also that there have been created mathematical models, leading to

quantum mechanics (non-relativistic and relativistic models) to describe other

mechanical phenomena, connected to bodies with “microscopic” character (elementary

particles a.s.o.), in opposition to those connected to bodies with “macroscopic”

character, with which we have to do usually. As a matter of fact, relativistic and

quantum mechanics are the two basic non-classical models of mechanics. Other

mathematical models led to

undulatory mechanics, statistical mechanics, invariantive

mechanics

, nanomechanics a.s.o.

As we have seen above, the usefulness of the notion of mathematical model has been

put into evidence, as well as the great variety of such models, constructed to can study

mechanical phenomena, which – in general – are enough complex. We notice also the

so-called

generalized models of deformable continuous media.

1.2.3 Division of classical mechanics

Synthetizing the above exposure, we may state the object of study of mechanics.

Mechanics deals with the study of the equivalence of systems of forces and with the

study of motion of mechanical systems subjected to the action of these forces

Methodically and conventionally, and taking into account its historical development

too, we may admit that classical mechanics has three great divisions.

Statics deals with the study of equivalence of systems of forces (in particular, the

equivalence to zero) acting upon a mechanical system. Statics studies a particular case

of motion, that case in which a mechanical system, subjected to the action of certain

forces, remains in rest (or has a uniform and rectilinear motion) with respect to a given

frame of reference (considered as an inertial one).

Kinematics considers the motion of mechanical systems without taking into account

the forces which act upon them; hence, the object of study of kinematics is the

geometry

of motion

. Sometimes, kinematics is called also the geometry of mechanics

(

phoronomy).

Dynamics deals with the motion of mechanical systems subjected to the action of

given forces; in fact, this is the object of study of mechanics.

Newtonian model of mechanics

These divisions of classical mechanics allow a systematization of our study; in the

frame of those divisions, the problems of

mechanical vibrations, of stability of

equilibrium

and of motion, of optimal trajectories etc. did develop very quickly.

Chronologically, first appeared statics (in antiquity), then dynamics (at the same time as

the Italian Renaissance); kinematics arises only in the XIXth century.

From a methodical point of view, it is more convenient to consider separately

classical models (Newtonian) and non-classical models (relativistic, quantum (the last

one can be non-relativistic or relativistic), undulatory, statistical, invariantive etc.)

mechanics

. For deformable continuous media (in opposition to what – usually – is

called

general mechanics, containing classical models of mechanics) it is necessary a

separate study; the mathematical models of such mechanical systems must be completed

either they are classical models or non-classical ones. In this case too one can put into

evidence statics, kinematics and dynamics.

Actually, the searching in the frame of classical and non-classical models of

mechanics or in the domain of mechanics of deformable media is developing very

much, in multiple directions, comprising new methods of computation, analytical and

numerical ones. One of the tendencies of today is represented by the extension and the

generalization of the models of mechanics. We mention thus the search in the direction

axiomatization of classical mechanics. A particular impetus takes the theory of

continuous deformable mechanical systems, either by solving boundary value problems

or by

coupling the corresponding problems to other non-mechanical phenomena (a

thermal field, an electromagnetic field etc.).

1.2.4 Short history

From the earliest time A.C., man has been put in connection with mechanical

phenomena; empirical knowledge allowed him to set up tools and mechanical devices,

as it can be seen also from the drawings and the basso-rilievos of Assyria and ancient

Egypt.

The beginnings of mechanics in the ancient eve date from VI-Vth centuries A.C.,

together with the oldest notings. Mo Tsy, in ancient China, gives in his writings some

notions of mechanics, concerning time, motion and force. By Anaximander (610-546

A.C.), Anaximenes (585-525 A.C.), Heraclit of Efes (540-470 A.C.), as well as by

Pythagoras’ school (571-477 A.C.) we meet referrings about matter in motion. Thales at

Miletus (640-546 A.C.) took much interest in astronomy and Herodotus (484-425 A.C.)

tells us that he has even succeeded in predicting an eclipse. Anaxagoras of Clazomenae

(499-427 A.C.) admits that there are some forces, which do not allow celestial bodies to

fall one upon the other. The arguments about motion (Achilles and the tortoise, the

arrow, the stadium, the dichotomy), which are the paradoxes of Zeno of Elea (490-430

A.C.), contribute to the stimulation of studies about space, time, problems of continuity

and discontinuity a.s.o. Œnopides of Chios (b. 465 A.C.) was one of the leading

astronomers of his time; he is thought to have learned the science of the stars and the

obliquity of the ecliptic from the priests and temple astronomers of Egypt. He is said to

have invented the cycle of 59 years for the return of the coincidence of the solar and

lunar years, giving the length of the solar year as 365 days and somewhat less than 9

hours. Parmenides of Elea (b. 460 A.C.) taught at Athens in the middle of the 5th

century A.C., and among his theories on the Universe was the one that the Earth is a

sphere; Meton, Phaeinus and Euctemon dealt with mathematical astronomy in the same

MECHANICAL SYSTEMS, CLASSICAL MODELS

century. Archytas of Tarentum (430-360 A.C.) discovers the pulley, the screw as well as

some mechanical tools to set up curves; he applied mathematics in any noteworthy way

to mechanics. The first written works about mechanics are assigned to him. It is said

that Eudoxus of Cnidus (408-355 A.C.), at one time a pupil of Plato (430-349 A.C.),

introduced the study of spherics (mathematical astronomy) into Greece and made

known the length of the year as he had found it given in Egypt, from where he brought

also the theory of motion of the planets; he found that the diameter of the Sun is nine

times that of the Earth. Aristotle of Stageira (384-322 A.C.) wrote a voluminous work

on mechanical problems; he puts in evidence the relative character of motion and intuits

the principle of virtual displacements in the study of equilibrium of levers. To him we

owe the first known definition of continuity: “A thing is continuous when of any two

successive parts the limits at which they touch are one and the same and are, as the

word implies, held together”. Democritus of Abdera (460-370 A.C.) is considered to be

the founder of the atomic theory of the ancient philosophy, which asserted: “the original

characteristics of matter are functions of quantity instead of quality, the primal elements

being particles homogeneous in quality, but heterogeneous in form”. Other atomists

have been Epicur (341-270 A.C.) and Carus Lucretius (99-55 A.C.); they gave incipient

results concerning the falling of bodies and the structure of matter.

The greatest mathematical centre of ancient times was neither Crotona nor Athens,

but Alexandria; here it was, on the site of the ancient town of Rhacotis, in the Nile

Delta, that Alexander the Great (356-323 A.C.) founded a city worthy to bear his name.

Under Ptolemy’s Soter (Ptolemy the Preserver) benevolent reign (323-283 A.C.),

Alexandria became the centre not only of the world’s commerce but also of its literary

and scientific activity; here was established the greatest of the world’s ancient libraries

and its first international university. Of all the great names connected with Alexandria,

that of Euclid (320-270 A.C.) is the best known; he was the most successful textbook

writer that the world has ever known, over one thousand editions of his geometry

having in print since 1482. Euclid’s greatest work is known as the “Elements”, and

represents the basis of the geometry bearing his name. We notice the interesting

contributions of Euclid to statics; he wrote also “Phoenomena”, dealing with the

celestial sphere and containing twenty-five geometric propositions. Eratosthenes of

Cyrene (274-194 A.C.) dealt with the measurement of the diameter of the Earth and

found 7850 miles, that is 50 miles less than the polar diameter, as we know it; he

considered also the problem of distances in the solar system, i.e., the distances from the

Earth to the Moon and to the Sun, respectively. Archimedes (287-212 A.C.) was born

and died at Syracuse in Sicily; he may be considered as to be the founder of statics.

Archimedes discovers various simple devices and defines the centre of gravity.

Archimedes is the most important researcher of ancient mechanics; his work “On the

equilibrium of plane surfaces and on centres of gravity” is comparable to Euclid’s

“Elements”. After Ctesibios (approx. 200 or 180 A.C.), Hypsicles of Alexandria

(approx. 180 A.C.), Hypparchus of Rhodes (190-120 A.C.), Filon of Byzantium (second

half of IInd century A.C.) and Poseidonius of Rhodes (135-84 A.C.), the most important

name is that of Heron of Alexandria (first century A.C.); he invented the pneumatic

device, commonly known as

Heron’s fountain, a simple form of the steam engine, and

wrote on pneumatics and mechanisms. Of the Romans, no one is more prominent than

Marcus Vitruvius Pollio, commonly known as Vitruvius (50 A.C.-20 A.D.), who

published the textbook “De architectura”.

Newtonian model of mechanics

At the beginning of Christian era (from now on, all data are A.D.), Ptolemy

(Claudius Ptolemaeus) (70-147) did for astronomy what Euclid did for plane geometry,

Apollonius Pergaeus (Pamphylia) (approx. 225 A.C.) for conics and Nicomachus of

Gerasa (approx. 100 A.C.) for arithmetic; his greatest work, commonly known as the

“Almagest” (with the Arabic “al” (the)) contains much information about the history of

ancient astronomy as well as his concept on the geometric stellar system. Ma Kim (IIIrd

century) created in China mechanisms with geared wheels. Pappus of Alexandria

(approx. beginning of IVth century) gives a synthesis of known results in mathematics,

astronomy and mechanics in his great work “Mathematical Collections”; we find here

the famous theorems concerning centres of gravity of bodies of revolution (taken over

by the Swiss Paul (original first name Habakuk) Guldin (1577-1643)). To Proclus of

Byzantium (412-485) (surnamed the Successor, because he was looked upon as the

successor of Plato in the field of philosophy) are due many comments on mechanics.

Joannes Philiponus of Alexandria (known also as Joannes Grammaticus) (VIth century)

develops the notion of impetus (because of which the motion continues without any

external action). The Arabic science is that which takes again and amplifies these results

till the XIIth century; we mention thus Mohammed ibn Mohammed ibn Tarkhan ibn

Auzlag, Abu Nasr al Farrabi (of Farab, in Turkestan) (870-951), who wrote “About the

eternal motion of the celestial sphere”, Al-Hosein ibn Abdallah ibn al-Hosein (or

Hasan) ibn Ali Abu Ali al-Sheich al-Rais (known in Christian Europe as Avicenna)

(980-1037), born in Safar, not far from Bokhara (Uzbekistan), philosopher and

physician, which considered the motion of matter as fundamental, Mohammed ibn

Ahmed ibn Mohammed ibn Roshd Abu Velid (commonly called Averroes in the Middle

Age) (1126-1198) of Cordova, to whom we due interesting comments on Aristotle, and

Nured-din al-Betruji Abu Abdallah (called Alpetragius by the Christians) (XIIth

century) simplified the cosmic system of Ptolemy and applies the “impetus” to the

motion of celestial bodies. Rabbi Moses ben Maimum (called also Maimonides) (1135-

1204), a Jewish writer native of Cordova, physician to the sultan, is an astronomer of

prominence, and to him is due a Jewish calendar.

Beginning with the XIIIth century, appear the great European schools, where are

translated the most important works of the Antiquity and of the Middle Age; the ideas

of the time are thus developed. The scholasticism accommodates many of these ideas to

the needs of the Catholic Church. A critical interpretation of Aristotle is made, among

others, by Siger of Brabant (1235-1281), Thomas d’Aquino (1226-1274) and Richard of

Middletown (beginning of the XIVth century). The nominalist current reflects certain

liberties of thinking and position with respect to the church, initiating methods of

research based on experiments; we mention thus, Pierre Abélard (Petrus Abaelardus)

(1079-1142), Roger Bacon (1214-1294), the most prominent scholar in England, a man

of erudition and of prophetic vision, and – especially – Jordan of Namur (Jordanus de

Saxonia, Jordanus Nemorarius) (XIIIth century), who studies the motion of heavy

bodies along a curve and leads a true school of mechanics. Beginning with the end of

the XIIIth century till the XVth century appear the antischolastics; among them we

mention Jean Buridan (1300-1360), Nicolas Oresme (1323-1382), Thomas Bradwardine

(1290-1349), representative of the school of mechanics of Oxford, Filippo Bruneleschi

(1377-1446), Florentine sculptor and architect, Biagio Pelacani (Blasius of Parma) (d.

1416) and the German cardinal Nikolaus Krebs (Nicolaus Cusanus) (1401-1464).

The XVth century represents the beginning of the Renaissance; research is made on

experimental basis and statics becomes a complete discipline. Niklas Koppernigk

MECHANICAL SYSTEMS, CLASSICAL MODELS

(Nicholaus Copernicus of Thorn on the Vistula) (1473-1543) published his heliocentric

concept on the world “De revolutionibus orbium coelestium” (Nürnberg, 1543) (The

orbits of the planets are circles, the Sun being in the centre); other works are published

by Leone Battista Alberti (1404-1472) and by Giorgio Valla (1447-1500). The most

prominent figure of this epoch is Leonardo da Vinci (1452-1519), who said that

“mechanics is the paradise of mathematical knowledge, because – through the agency of

it – one bears the fruits of mathematics”; he studies the laws of the free falling and of

friction, introduces the notion of moment and applies the principle of virtual

displacements. To the progress of mechanics in this period contributed also Jules César

Scaliger (1484-1558), who adopts the denomination of

motion for impetus, Girolamo

Cardan (1501-1576), Nicolo Fontana (Tartaglia) (1499- or 1501-1557), Frederico

Commandin di Urbino (1509-1575) and Giovanni Battista Benedetti (1530-1590). The

XVIth and XVIIth centuries bring a new raising of mechanics; the technologies in

various branches of production bear a particular development and bring into life the

premises of the great industry in the future. The Academies of Sciences do appear

(Accademia dei Lincei in Rome (1590), Accademia del Cimento in Florence (1651),

Royal Society in London (1622-1663), Académie des Sciences in Paris (1666) etc.),

contributing to the promotion of theoretical and experimental research. The

mathematical model of classical mechanics is completed in the XVIIth century. Simon

Stevin (1548-1620), Flemish mathematician and physicist, solves the problem of the

inclined plane and enounces the rule of composition of forces, Giordano Bruno (1548-

1600) brings arguments against Aristotle, while Luca Valerio (1552-1618) determines a

great number of centres of gravity. Galileo Galilei (1564-1642) is one of the founders of

the modern dynamics. To him are due the principle of inertia and the principle of initial

conditions, the formulation of which represents a revolutionary step in the development

of mechanics, and thus is put an end to Aristotelic conceptions; he states the so-called

“golden rule of mechanics”, that is: how much is won in force is lost in velocity, and

studies the motion of a projectile in vacuum. His most important ideas are contained in

“Dialogo di Galileo Galilei delle due massimi sistemi del mondo, il Tolemaico e il

Copernicano” (1632) and in “Discorsi e dimonstrazioni mathematiche intorno a due

nuove scienze attenanti alla meccanica e i movimenti locali” (Leyda, 1638). Francis

Bacon (1561-1626) considers the motion as a property of the matter and criticizes the

scholastic conceptions in “Novum organum” (1620). Evangelista Torricelli (1608-

1647), disciple of Galileo, develops the theory of motion of heavy bodies and of the

stability of equilibrium. Johannes Kepler (1571-1630) starts from observations of Tycho

Brahe (1546-1601) on Mars and in “De revolutionibus orbium coelestium” enounces his

three famous laws, which replace the Copernican motion of planets, supposed to be

circular and uniform, by a motion on an ellipse. Other contributions to the construction

of the mathematical model of mechanics are due to Tomaso Campanella (1568-1639),

to Paul (Habakuk) Guldin (1577-1643), who founds again the theorems of Pappus, to

Marin Mersenne (1588-1648), who introduces Galileo’s work in Paris, to René

Descartes (1596-1650), who studies the collision of bodies and enounces the theorem of

conservation of momentum, and is the first to put into evidence the infinitesimal

character of virtual displacements, to Pierre Fermat (1601-1665), to whom belongs the

“theorem of minimum time”, to Gilles Personne de Roberval (1602-1675), who dealt

with levers and balances and research in statics and kinematics, to Giovanni Alfonso

Borelli (1608-1679), a monk who laid the bases of biomechanics, to John Wallis (1616-

1703), who dealt with the collision of inelastic bodies, to Jacques Rohault (1620-1675),

Newtonian model of mechanics

to Pierre Varignon (1654-1722), who develops the theory of moments, to Christian

Huygens (1629-1695), who studies the physical pendulum, discovers the isochronism of

the cycloidal pendulum and invents the clock with pendulum, to Cristoph Wren (1652-

1723), to Nicole Malbranche (1638-1715) and to Gottfried Wilhelm Leibniz (1646-

1716), who considers the “vis viva” to be the basic mechanical characteristic of motion.

But the scientist who attached his name to the first mathematical model of

mechanics, the model of classical mechanics, has been Sir Isaac Newton (1642-1727),

illustrious English mathematician, physicist and astronomer. He defined the notion of

mass, generalized the notion of force and formulated mechanics’ laws in “Philosophiae

Naturalis Principia Mathematica” (London, 1686-87), published under pressure from his

good friend Edmund Halley (1656-1742), the astronomer, fighting for priority with

Robert Hooke (1635-1703); Newton discovered the law of universal attraction,

verifying thus the laws of motion of planets around the Sun, given by J. Kepler. But

Newton sketched also an ample plane for the development of the science thus created;

today, classical mechanics is developing in the large frame of the three fundamental

laws as a theoretical and experimental science, which must permanently introduce new

concepts and confront theoretical constructions to reality.

The XVIIIth century represents the beginning of the industrial revolution, leading to

an intensive development of classical mechanics; the first superior technical schools are

founded, for instance “l’École Polytechnique” of Paris (1794), which – due to the

activity of Gaspard Monge (1746-1818) and Louis Poinsot (1777-1859) – played an

important rôle in the development of mechanics. Jacques (Jacob) Bernoulli (1654-1705)

dealt with dynamics of mechanical systems subjected to constraints and gave the

solution to the problem of the oscillation centre; some corrections are brought by

Guillaume François de l’Hospital (1661-1704). Jean (Johannes) Bernoulli (1667-1748)

gives a correct formulation to the principle of virtual displacements, while his son

Daniel Bernoulli (1700-1782) studies the conservation of the “vis viva” principle. Jacob

Hermann (1678-1733) studies the dynamics of constrained systems, Colin Mclaurin

(1698-1746) is the first to enounce, in 1742, the second law of Newton in the form

(1.1.89), while Pierre Louis Moreau de Maupertuis (1698-1759) enounces in 1744 the

principle of least action. A special contribution is due to Leonhard Euler (1707-1783),

one of the most prominent scientists in mathematics and mechanics of the century, who

introduced the notions of material point, moment of inertia and moment of “amount of

motion” (moment of momentum); he studies the motion of the rigid solid, gives the

differential equations of motion of the rigid with a fixed point and integrates them in a

particular case. He also enounces, in 1748, the second law of Newton in the form

(1.1.89). Mikhail Vasilievich Lomonosov (1711-1765) extends the “principle of

conservation of matter” to the “conservation of motion”. Samuel Koenig (1712-1757)

polemizes with Maupertuis about the priority on the principle of least action; Euler and

François Marie Arouet (Voltaire) (1694-1778) are also participants to this dispute.

Alexis Claude Clairaut (1713-1765) deals with the relative motion and with the motion

of the Moon. Jean le Rond d’Alembert (1717-1783) states in his “Traité de dynamique”

(Paris, 1743) a general method to study the discrete mechanical systems (d’Alembert’s

principle). Paul d’Arcy (1725-1779) enounces the theorem of conservation of the

kinetic moment for a conservative system (1747), Charles-Augustin de Coulomb (1736-

1806) does important research on sliding and rolling friction and Pierre Simon de

Laplace (1749-1827) publishes his famous “Traité de mécanique celeste” (Paris, 1799-

1825), in five volumes, where important cosmogonic hypotheses are made. “La

MECHANICAL SYSTEMS, CLASSICAL MODELS

mécanique analytique” (Paris, 1788) by Joseph-Louis Lagrange (1736-1813) had a great

influence for the subsequent directions of development of mechanics; in this volume,

which contains the famous equations bearing Lagrange’s name, are studied discrete

mechanical systems, without figures, using only methods of mathematical analysis. He

deals with the study of small motions of a mechanical system around a stable position of

equilibrium; we notice also his results on “ideal constraints” and the introduction of the

so-called Lagrange’s multipliers. Fundamental contributions to the development of

analytical methods in mechanics are brought by Wiliam Rowan Hamilton (1805-1865),

who finds the canonical equations of analytical mechanics bearing his name and

enounces the most important variational principle of mechanics and by E.J. Routh

(1831-1907), the equations of which are a generalization of Lagrange’s and Hamilton’s

ones. Carl Gustav Jacob Jacobi (1809-1882), Siméon-Denis Poisson (1781-1840) and

Joseph Liouville (1809-1882) have valuable contributions concerning the integration of

the canonical system; Peter Gustav Lejeune-Dirichlet (1805-1859), Jules Henry

Poincaré (1854-1912), Adolf Hurwitz (1859-1919) and Aleksandr Mikhailovich

Lyapunov (1857-1918) deal with the problem of stability of motion. Gustav Gaspard

Coriolis (1792-1843) discovers the complementary acceleration, used by Jean Bernard

Léon Foucault (1819-1868), who puts in evidence the motion of rotation of the Earth,

by means of the pendulum bearing his name. George Atwood (1746-1807) verifies the

laws of the bodies’ falling, using the device to which was given his name, and Lazare

Nicolas Carnot (1753-1823) gave the first analytical formulation of the principle of

virtual displacements and enounced the theorem of dissipation of energy in case of

collision. Carl Friedrich Gauss (1777-1855) elaborates a method to calculate the

elliptical orbits of planets. Jacques Philippe Marie Binet (1786-1856) deals with the

action of central forces and Mikhail Vasilievich Ostrogradskiĭ (1801-1862) enounces a

generalization of Hamilton’s principle, independently of him, and develops the

mechanics of systems with unilateral constraints. André Marie Ampère (1775-1836)

studies many problems of kinematics, introducing this denomination. Jean Baptiste

Joseph Fourier (1768-1830) tries to give a general demonstration to the principle of

virtual work. Jean Victor Poncelet (1788-1867) connects the notion of work to that of

energy. Rodrigues B. Olinde (1794-1851) deals with the principle of least action, Jacob

Steiner (1796-1863) studies moments of inertia, while Nicolas Léonard Sadi Carnot

(1796-1832) calculates the mechanical equivalent of heat. Eugen Dühring (1833-1921),

Felix Klein (1849-1925) and Ernst Mach (1839-1916) contribute to a criticism of

classical mechanics, and Vladimir Ilich Ulyanov (Lenin) (1870-1924) is against them.

Ivan Vsevolod Meshcherskiĭ (1895-1935), Konstantin Eduardovich Tsiolkovskiĭ (1857-

1935) and Tullio Levi-Civita (1873-1941) elaborate basic works concerning the

mechanics of bodies of variable mass. To William-John Macquorn Rankine (1820-

1872) is due the exact expression of kinetic energy, eliminating the notion of “vis viva”.

Hermann Ludwig Ferdinand von Helmholtz (1821-1894) generalizes the principle of

least action to non-mechanical phenomena. Sonya Krukowsky (Sophia Kovalevsky)

(1850-1891) deals with the motion of a rigid solid with a fixed point. Valuable

contributions are due to James Clerk Maxwell (1831-1879), William Thomson (lord

Kelvin) (1842-1907), Eötvös Loránd (1848-1919), Isaac Todhunter (1820-1884), Karl

Pearson (1857-1936), Paul Émile Appel (1855-1930), Heinrich Rudolph Hertz (1857-

1894), Paul Painlevé (1863-1933), Sergei Alekseevich Chaplygin (1869-1942) and

Georg Hamel (1877-1955). Gheorghe Vrănceanu (1900-1979) constructs

non-

holonomic geometries

, suitable to the study of non-holonomic mechanical systems.

Newtonian model of mechanics

Max Karl Ernst Ludwig Planck (1858-1947) gets up in 1900 to the conclusion that

the change of energy between radiation and substance (the radiation problem of the

black body) is discontinuous, in quanta, and

quantum theory is thus born. Erwin

Schrödinger (1887-1961), Paul Adrien Maurice Dirac (1902-1984) and Werner Karl

Heisenberg (1901-1976) worked (in 1925-26) to the bases of

quantum mechanics.

Starting from the experiments of Albert Abraham Michelson (1852-1931) and Edward

Williams Morley (1838-1923), and from the interpretations of Hendrick Anton Lorentz

(1853-1928) and of others, Albert Einstein (1879-1955) puts, by his paper “Zur

Elektrodynamik der bewegter Körper” (Annalen der Physik, 1905), the bases of the

Special Relativity, and, by the paper “Die Grundlagen der allgemeinen

Relativitätstheorie” (Annalen der Physik, 1916), the bases of the

General Relativity;

Hermann Minkovski (1864-1909), H. Poincaré and M. Planck contribute to the

development of these theories. Josiah Dixon Willard Gibbs (1839-1903) and Ludwig

Eduard Boltzmann (1844-1906) put, by their studies, the bases of

statistical mechanics.

To Louis Victor Pierre Raymond prince de Broglie (1892-1987) is due the birth of

undulatory mechanics. The name of Octav Onicescu (1892-1983) states for the

invariantive mechanics.

Nowadays, research in the frame of classical and non-classical models of mechanics

or in the frame of continuous deformable media is very much developed, in many

directions. One of the actual tendencies is represented by the extension and

generalization of mechanical models. Efforts have been made by Clifford

Ambrose Truesdell (1919-2000), Walter Noll (b. 1925) and Bernard Coleman for the

axiomatization of classical mechanics. In the so-called frame of general mechanics rise

studies in vibration, stability, dynamical systems and optimal control. A particular

development is noticed in the theory of continuous mechanical systems, by solving

many boundary value problems, as well as by coupling the respective mechanical

phenomena with other non-mechanical ones (corresponding to a thermic or

electromagnetic field etc.). Due to this enormous development, it is very difficult to

review, even summarily, the most important researchers dealing with these problems.

So that we preferred to mention only scientists who became classic and to mention only

research till the first decades of the XXth century.

2. Dimensional analysis. Units. Homogeneity. Similitude

Dimensional analysis deals with the study of relations which describe mechanical,

physical, chemical phenomena a.s.o.; this study is based on the property of dimensional

homogeneity, which must be verified by all relations obtained on a rational or empiric

way. To do this, we will put into evidence the physical quantities which appear in the

study of mechanical systems, as well as the corresponding units. We consider also the

properties of homogeneity and similitude and the possibility of using them.

2.1 Physical quantities. Units

In what follows, we introduce the notion of physical quantity (in particular, the

notion of mechanical quantity), as well as the basic and derived quantities and units. We

may thus define the systems of units.

MECHANICAL SYSTEMS, CLASSICAL MODELS

2.1.1 Physical quantities

Considering a set of physical objects of the same nature from the point of view of a

certain property, we reach the notion of

physical quantity. A process of abstraction

defines this notion by: i) the statement of an

equivalence relation, which allows to

distribute the respective objects in

equivalence classes; ii) the statement of a relation of

order

between these equivalence classes, to can appreciate if the objects belonging to

one class are “greater” or “smaller” than the objects belonging to another class, from the

point of view of the considered property; iii) the statement of a

criterion of comparison

by which one can estimate “how many times” an object of a class is greater or smaller

than an object belonging to another class, from the point of view of the same property.

A physical quantity

M , which appears in the study of a mechanical system (the

velocity of a particle, the intensity of a force etc.), is a mechanical quantity.

The direction of a force, considered as a physical object, does not represent a

property to allow the introduction of the notion of physical quantity; indeed, one can

obtain classes of parallel forces, but one cannot state a relation of order. But the

property of intensity of a force permits the introduction of a physical quantity; the

corresponding property can be put into evidence with the aid of a dynamometer.

By the comparison criterion, to any class of equivalence is attached a real number.

The function

, which states the correspondence between the classes of equivalence

and the set

\ of real numbers, must be strictly increasing, while the ratio between two

quantities of the same nature must remain the same; it follows that the function

must

be linear, hence of the form

+() (0)

xkxf,

> 0k

(1.2.1)

One can assign arbitrary numbers (corresponding to the arbitrary constants

k and (0)

)

only to two equivalence classes; for the other classes, the numbers, which must be

attached, result from (1.2.1). These arbitrary constants may correspond to a

conventional unit equal to

zero and to a conventional intensity equal to unity. In physics

(in particular, in mechanics) the zero convention is made for a same class of

equivalence, which is naturally imposed. For instance, if upon a dynamometer does not

act any force, then its deformation is equal to zero; we can say that a force of null

intensity is acting. Conventionally, one considers also the class to which the number one

is assigned; a physical object of this class is called a

unit U of the respective physical

quantity. The real number attached to a class of equivalence is an abstract (non-

dimensional) number, called the

numerical value n of the physical quantity M

considered; this value, obtained by an operation of measurement (on an experimental

way, with a certain degree of approximation), will be given by

the basic equation of

measurement

MnU

. (1.2.2)

2.1.2 Quantities and basic units. Quantities and derived units

The unit U , corresponding to a certain physical quantity, must be really reproduced

with the best precision possible (depending on the technical possibilities at a certain

moment), in the form of a

gauge. For certain quantities (length, mass, time) one can

Newtonian model of mechanics

easily construct units (metre, kilogram, second); these quantities can be measured

directly. For other quantities (velocity, acceleration etc.) one cannot easily realize units,

a direct measurement being difficult to do. In the last case, a quantity

M is measured

indirectly, measuring directly quantities of a different nature with numerical values

, ,...nn ; a relation of the form

(

)

,,...Mfnn

(1.2.3)

takes place. We must judiciously choose the units, so that coefficients depending on

these units do not appear.

The numerical values

n ,

n of two physical quantities

,MM of the same nature

can be added only if they proceed from measurements with the same units

U ; in this

case

==+=+

121 2

MnUM M nUnU,

nn n.

(1.2.4)

Also

(1.2.4')

If we measure the same physical quantity

M with different units

U and

U , then

the corresponding numerical values

n and

n are linked by the relation

11 22

MnU nU

(1.2.5)

Certain relations take place between the physical quantities which appear in nature.

We may thus choose a relatively restrained number of independent physical quantities,

so that the other quantities be expressed by certain relations depending on the first ones;

thus, using the above relations, one can express the units of all physical quantities with

the aid of a restrained number of units. The physical quantities thus chosen are called

basic (primitive) quantities, the corresponding units being basic (primitive) units. All

the other quantities and units are

derived quantities or units, respectively.

Usually, one takes as basic quantities the physical ones which naturally appear, that

is quantities corresponding to space, mass and time, for instance, in case of mechanical

systems; such quantities are called also

primary quantities, the other ones being

secondary quantities. But one can choose as basic quantities also other quantities than

the primary ones. For instance, the force can be also a basic quantity, useful in case of

mechanical systems.

We notice that a relation describing a physical phenomenon in which appear

quantities the units of which are fixed (basic and derived units) contains a

proportionality coefficient, and any other relation deriving from it will contain the same

coefficient.