The history of the formation of analytical mechanics. The principles of classical mechanics Which was the main classical mechanics

Basic concepts

Classical mechanics operates with several basic concepts and models. Among them should be highlighted:

Basic Laws

Galileo's principle of relativity

The basic principle on which classical mechanics is based is the principle of relativity, formulated on the basis of empirical observations by G. Galileo. According to this principle, there are infinitely many frames of reference in which a free body is at rest or moves with a constant speed in absolute value and direction. These frames of reference are called inertial and move relative to each other uniformly and rectilinearly. In all inertial frames of reference, the properties of space and time are the same, and all processes in mechanical systems obey the same laws. This principle can also be formulated as the absence of absolute reference systems, that is, reference systems that are somehow distinguished relative to others.

Newton's laws

Newton's three laws are the basis of classical mechanics.

Newton's second law is not enough to describe the motion of a particle. Additionally, a description of the force is required, obtained from consideration of the essence of the physical interaction in which the body participates.

Law of energy conservation

The law of conservation of energy is a consequence of Newton's laws for closed conservative systems, that is, systems in which only conservative forces act. From a more fundamental point of view, there is a relationship between the law of conservation of energy and the homogeneity of time, expressed by Noether's theorem.

Beyond the applicability of Newton's laws

Classical mechanics also includes descriptions of the complex motions of extended non-point objects. Euler's laws provide an extension of Newton's laws to this area. The concept of angular momentum relies on the same mathematical methods used to describe one-dimensional motion.

The equations of rocket motion expand the concept of velocity when an object's momentum changes over time to account for such effects as mass loss. There are two important alternative formulations of classical mechanics: Lagrange mechanics and Hamiltonian mechanics. These and other modern formulations tend to bypass the concept of "force", and emphasize other physical quantities, such as energy or action, to describe mechanical systems.

The above expressions for momentum and kinetic energy only valid if there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for a wire carrying current is violated if it does not include the contribution of the electromagnetic field to the momentum of the system expressed in terms of the Poynting vector divided by c 2 , where c is the speed of light in free space.

Story

ancient time

Classical mechanics originated in antiquity mainly in connection with the problems that arose during construction. The first of the sections of mechanics to be developed was statics, the foundations of which were laid in the works of Archimedes in the 3rd century BC. e. He formulated the rule of the lever, the theorem on the addition of parallel forces, introduced the concept of center of gravity, laid the foundations of hydrostatics (Archimedes force).

Middle Ages

new time

17th century

18th century

19th century

In the 19th century, the development of analytical mechanics takes place in the works of Ostrogradsky, Hamilton, Jacobi, Hertz, and others. In the theory of vibrations, Routh, Zhukovsky, and Lyapunov developed a theory of the stability of mechanical systems. Coriolis developed the theory of relative motion by proving the acceleration theorem. In the second half of the 19th century, kinematics was separated into a separate section of mechanics.

Particularly significant in the 19th century were advances in continuum mechanics. Navier and Cauchy in general form formulated the equations of the theory of elasticity. In the works of Navier and Stokes, differential equations of hydrodynamics were obtained taking into account the viscosity of the liquid. Along with this, there is a deepening of knowledge in the field of hydrodynamics of an ideal fluid: the works of Helmholtz on vortices, Kirchhoff, Zhukovsky and Reynolds on turbulence, and Prandtl on boundary effects appear. Saint-Venant developed a mathematical model describing the plastic properties of metals.

Newest time

In the 20th century, the interest of researchers switched to nonlinear effects in the field of classical mechanics. Lyapunov and Henri Poincaré laid the foundations for the theory of nonlinear oscillations. Meshchersky and Tsiolkovsky analyzed the dynamics of bodies of variable mass. Aerodynamics stands out from continuum mechanics, the foundations of which were developed by Zhukovsky. In the middle of the 20th century, a new direction in classical mechanics is actively developing - the theory of chaos. The issues of stability of complex dynamical systems also remain important.

Limitations of classical mechanics

Classical mechanics gives exact results for the systems we encounter in Everyday life. But her predictions become incorrect for systems approaching the speed of light, where it is replaced by relativistic mechanics, or for very small systems where the laws of quantum mechanics apply. For systems that combine both of these properties, relativistic quantum field theory is used instead of classical mechanics. For systems with very big amount components, or degrees of freedom, classical mechanics also cannot be adequate, but methods of statistical mechanics are used.

Classical mechanics is widely used because, firstly, it is much simpler and easier to apply than the theories listed above, and, secondly, it has great possibilities for approximation and application for a very wide class of physical objects, starting from the usual, such as a spinning top or a ball, to large astronomical objects (planets, galaxies) and very microscopic ones (organic molecules).

Although classical mechanics is generally compatible with other "classical" theories such as classical electrodynamics and thermodynamics, there are some inconsistencies between these theories that were found in the late 19th century. They can be solved by methods of more modern physics. In particular, the equations of classical electrodynamics are not invariant under Galilean transformations. The speed of light enters them as a constant, which means that classical electrodynamics and classical mechanics could only be compatible in one chosen frame of reference associated with the ether. However, experimental verification did not reveal the existence of the ether, which led to the creation of the special theory of relativity, in which the equations of mechanics were modified. The principles of classical mechanics are also inconsistent with some of the claims of classical thermodynamics, leading to the Gibbs paradox, according to which it is impossible to accurately determine the entropy, and to the ultraviolet catastrophe, in which absolutely black body must radiate an infinite amount of energy. To overcome these incompatibilities, quantum mechanics was created.

Notes

Internet links

Video lecture 1. Physics: Classical mechanics (autumn 1999)// Massachusetts Lectures Institute of Technology: 8.01

Literature

Arnold V.I. Avets A. Ergodic problems of classical mechanics. - RHD, 1999. - 284 p.
B. M. Yavorsky, A. A. Detlaf. Physics for high school students and those entering universities. - M .: Academy, 2008. - 720 p. -( Higher education). - 34,000 copies. - ISBN 5-7695-1040-4
Sivukhin D.V. General course of physics. - 5th edition, stereotypical. - M .: Fizmatlit, 2006. - T. I. Mechanics. - 560 p. - ISBN 5-9221-0715-1
A. N. MATVEEV Mechanics and the Theory of Relativity. - 3rd ed. - M .: ONYX 21st century: World and Education, 2003. - 432 p. - 5000 copies. - ISBN 5-329-00742-9
C. Kittel, W. Knight, M. Ruderman Mechanics. Berkeley Physics Course. - M .: Lan, 2005. - 480 p. - (Textbooks for universities). - 2000 copies. - ISBN 5-8114-0644-4

“Think of the benefit that good examples bring us, and you will find that the memory of great people is no less useful than their presence”

Mechanics is one of the most ancient Sciences. It arose and developed under the influence public practice requests and also thanks to abstracting activity of human thinking. Even in prehistoric times, people created buildings and observed the movement of various bodies. Many laws mechanical movement and balance of material bodies were known by mankind through repeated repetitions, purely experimentally. This socio-historical experience, passed down from generation to generation, and was the the source material on the analysis of which mechanics as a science developed. The emergence and development of mechanics was closely associated with production, With needs human society. “At a certain stage in the development of agriculture,” writes Engels, “and in certain countries (raising water for irrigation in Egypt), and especially along with the emergence of cities, large buildings and the development of handicrafts, developed and Mechanics. Soon it also becomes necessary for shipping and military affairs.

First surviving manuscripts and scientific messages in the field of mechanics belong ancient scholars of Egypt and Greece. The oldest papyri and books, in which studies of some of the simplest problems of mechanics have been preserved, relate mainly to various problems. statics, i.e. the doctrine of balance. First of all, here it is necessary to name the works of the outstanding philosopher ancient greece(384-322 BC), who introduced the name into scientific terminology Mechanics for a wide field of human knowledge, in which the simplest movements of material bodies, observed in nature and created by man during his activities, are studied.

Aristotle was born in the Greek colony of Stagira in Thrace. His father was a physician to the Macedonian king. In 367, Aristotle settled in Athens, where he received a philosophical education at the Academy of the famous idealist philosopher in Greece. Plato. In 343 Aristotle took over teacher of Alexander the Great(Alexander the Great said: “I honor Aristotle on a par with my father, since if I owe my life to my father, then I owe Aristotle everything that gives her a price”), later the famous commander ancient world. His philosophical school, called the school peripatetics, Aristotle founded in 335 in Athens. Some philosophical provisions of Aristotle have not lost their significance to the present day. F. Engels wrote; "The ancient Greek philosophers were all born elemental dialecticians, and Aristotle, the most universal head among them, has already explored all the essential forms of dialectical thinking." But in the field of mechanics, these broad universal laws of human thinking did not receive a fruitful reflection in the works of Aristotle.

Archimedes owns a large number technical inventions, including the simplest water-lifting machine (archimedean screw), which has found application in Egypt for draining cultivated lands flooded with water. He showed himself as military engineer while defending his hometown of Syracuse (Sicily). Archimedes understood the power and great importance for humanity of accurate and systematic scientific research, and proud words are attributed to him: Give me a place to stand on and I will move the earth."

Archimedes was killed by the sword of a Roman soldier during the massacre arranged by the Romans during the capture of Syracuse. Tradition says that Archimedes, immersed in the consideration of geometric figures, said to a soldier who approached him: "Do not touch my drawings." The soldier, seeing in these words an insult to the power of the victors, cut off his head, and the blood of Archimedes stained his scientific work.

famous ancient astronomer Ptolemy(II century AD - there is evidence that Ptolemy (Claudius Ptolemaeus) lived and worked in Alexandria from 127 to 141 or 151. According to Arabic legend, he died at the age of 78.) in his work " The Great Mathematical Construction of Astronomy in 13 Books"developed a geocentric system of the world, in which the apparent movements of the firmament and planets were explained on the assumption that the Earth is motionless and is at the center of the universe. The entire firmament makes a complete revolution around the Earth in 24 hours, and the stars participate only in diurnal movement, keeping its relative position unchanged; planets, moreover, move relative to the celestial sphere, changing their position relative to the stars. The laws of the apparent motions of the planets were established by Ptolemy to such an extent that it became possible to predict their positions relative to the sphere of the fixed stars.

However, the theory of the structure of the universe, created by Ptolemy, was erroneous; it led to extraordinarily complex and artificial schemes of the motion of the planets and in a number of cases could not fully explain their apparent movements relative to the stars. Particularly large discrepancies between calculations and observations were obtained when predicting solar and lunar eclipses made for many years to come.

Ptolemy did not adhere strictly to the methodology of Aristotle and conducted systematic experiments on the refraction of light. Physiological-optical observations Ptolemy have not lost their interest to date. The angles of light refraction found by him during the transition from air to water, from air to glass and from water to glass were very accurate for its time. Ptolemy remarkably combined strict mathematician and subtle experimenter.

In the era of the Middle Ages, the development of all sciences, as well as mechanics, was strongly slowed down. Moreover, during these years the most valuable monuments of science, technology and art of the ancients were destroyed and destroyed. Religious fanatics wiped out all the achievements of science and culture from the face of the earth. Most of the scientists of this period blindly adhered to the scholastic method of Aristotle in the field of mechanics, considering all the provisions contained in the writings of this scientist to be unconditionally correct. The geocentric system of the world of Ptolemy was canonized. Speech against this system of the world and the main provisions of the philosophy of Aristotle were considered a violation of the foundations scripture, and researchers who decided to do this were announced heretics. “The priesthood killed the living in Aristotle and immortalized the dead,” wrote Lenin. Dead, empty scholasticism filled the pages of many treatises. Ridiculous problems were posed, and exact knowledge was persecuted and withered. A large number of works on mechanics in the Middle Ages were devoted to finding " perpetuum mobile”, i.e. perpetual motion machine operating without receiving energy from outside. These works, for the most part, contributed little to the development of mechanics (Mohammed well expressed the ideology of the Middle Ages, saying: "If the sciences teach what is written in the Koran, they are superfluous; if they teach otherwise, they are godless and criminal"). “The Christian Middle Ages left nothing to science,” says F. Engels in Dialectics of Nature.

The intensive development of mechanics began in renaissance from the beginning of the 15th century in Italy, and then in other countries. In this era, especially great progress in the development of mechanics was achieved thanks to the work (1452-1519), (1473-1543) and Galilee (1564-1642).

Famous Italian painter, mathematician, mechanic and engineer, Leonardo da Vinci engaged in research on the theory of mechanisms (he built an elliptical lathe), studied friction in machines, investigated the movement of water in pipes and the movement of bodies along inclined plane. He was the first to recognize the extreme importance of the new concept of mechanics - the moment of force relative to a point. Investigating the balance of forces acting on the block, he established that the role of the shoulder of force is played by the length of the perpendicular dropped from the fixed point of the block to the direction of the rope carrying the load. The equilibrium of the block is possible only if the products of forces and the lengths of the corresponding perpendiculars are equal; in other words, the equilibrium of the block is possible only under the condition that the sum of the static moments of forces relative to the weight gain point of the block will be equal to zero.

A revolutionary revolution in the views on the structure of the universe was carried out by a Polish scientist who, as figuratively written on his monument in Warsaw, "stopped the Sun and moved the Earth." new, heliocentric system of the world explained the movement of the planets, based on the fact that the Sun is a fixed center, around which all the planets move in circles. Here are the original words of Copernicus, taken from his immortal work: “What appears to us as the movement of the Sun does not come from its movement, but from the movement of the Earth and its sphere, with which we revolve around the Sun, like any other planet. So, the Earth has more than one movement. The apparent simple and retrograde motions of the planets are not due to their motion, but to the motion of the Earth. Thus, one movement of the Earth is sufficient to explain so many apparent inequalities in the sky.

In the work of Copernicus, the main feature of the motion of the planets was revealed and calculations were made related to the predictions of solar and lunar eclipses. The explanations of the apparent return motions of Mercury, Venus, Mars, Jupiter, and Saturn relative to the sphere of the fixed stars have acquired clarity, distinctness, and simplicity. Copernicus clearly understood the kinematics of the relative motion of bodies in space. He writes: “Every perceived change in position occurs due to the movement of either the observed object or the observer, or due to the movement of both, if, of course, they are different from each other; for when the observed object and the observer move in the same way and in the same direction, no movement is noticed between the observed object and the observer.

Truly scientific Copernican theory made it possible to obtain a number of important practical results: to increase the accuracy of astronomical tables, to reform the calendar (introducing a new style), and to determine the length of the year more strictly.

Works of the brilliant Italian scientist Galilee were fundamental to the development speakers.
Dynamics as a science was founded by Galileo, who discovered many very important properties equally accelerated and equally slow motions. The foundations of this new science were set forth by Galileo in a book entitled "Conversations and Mathematical Proofs Concerning Two New Branches of Science Relating to Mechanics and Local Motion." In chapter III, on dynamics, Galileo writes: “We create new science, whose subject matter is extremely old. In nature, there is nothing ancient movement, but it is precisely with regard to it that philosophers have written very little significant. Therefore, I have repeatedly studied its features by experience, which are quite deserving of this, but until now either unknown or unproven. So, for example, they say that the natural motion of a falling body is accelerated motion. However, the extent to which the acceleration increases has not yet been indicated; as far as I know, no one has yet proved that the spaces traversed by a falling body at the same time intervals are related to each other as successive odd numbers. It was also noticed that the thrown bodies or projectiles describe a certain curved line, but no one indicated that this line is a parabola.

Galileo Galilei (1564-1642)

Before Galileo, forces acting on bodies were usually considered in a state of equilibrium and the action of forces was measured only by static methods (lever, scales). Galileo pointed out that force is the cause of the change in speed, and thus established dynamic method comparison of forces. Galileo's research in the field of mechanics is important not only for the results that he managed to obtain, but also for his consistent introduction to mechanics. experimental movement research method.

So, for example, the law of isochronism of pendulum oscillations at small angles of deflection, the law of motion of a point along an inclined plane were investigated by Galileo through carefully staged experiments.

Thanks to the works of Galileo, the development of mechanics is firmly associated with the demands technology, And scientific experiment systematically introduced as fruitful research method phenomena of mechanical movement. Galileo in his conversations directly says that observing the work of the “first” masters in the Venetian arsenal and talking with them helped him understand “the causes of phenomena that were not only amazing, but also seemed at first completely unbelievable.” Many provisions of Aristotle's mechanics were specified by Galileo (as, for example, the law on the addition of motions) or very ingeniously refuted by purely logical reasoning (refutation by setting up experiments was considered insufficient at that time). We present here Galileo's proof to characterize the style. refuting Aristotle's position that heavy bodies on the surface of the Earth fall faster, and light bodies fall more slowly. The reasoning is given in the form of a conversation between a follower of Galileo (Salviati) and Aristotle (Simplicio):

« Salviati: ... Without further experience, by a brief but convincing reasoning, we can clearly show the incorrectness of the statement that heavier bodies move faster than lighter ones, implying bodies of the same substance, i.e. such as those of which Aristotle speaks . In fact, tell me, Señor Simplicio, do you admit that every falling body has a certain speed by nature, which can be increased or decreased only by introducing a new force or obstacle?
Simplicio: I have no doubt that the same body in the same medium has a constant speed, determined by nature, which cannot increase except from the application of a new force, or decrease except from an obstacle that slows down the movement.
Salviati: Thus, if we have two falling bodies, the natural speeds of which are different, and we combine the faster one with the slower one, then it is clear that the motion of the body falling faster will be somewhat delayed, and the motion of the other will be somewhat accelerated. Do you object to this position?
Simplicio: I think that this is quite correct.
Salviati: But if this is so, and if at the same time it is true that a large stone moves, say, with a speed of eight cubits, while another, smaller one, with a speed of four cubits, then by joining them together, we should get a speed less than eight elbows; but two stones joined together make a body greater than the original, which had a speed of eight cubits; therefore, it turns out that a heavier body moves at a lower speed than a lighter one, and this is contrary to your assumption. You see now how, from the position that heavier bodies move faster than lighter ones, I could conclude that heavier bodies move less quickly.

The phenomena of a uniformly accelerated fall of a body on Earth were observed by numerous scientists before Galileo, but none of them could discover the true causes and correct laws that explain these everyday phenomena. Lagrange notes on this occasion that "an extraordinary genius was needed to discover the laws of nature in such phenomena that were always before our eyes, but the explanation of which, nevertheless, always eluded the research of philosophers."

So, Galileo was the founder of modern dynamics. Galileo clearly understood the laws of inertia and independent action of forces in their modern form.

Galileo was an outstanding observing astronomer and an ardent supporter of the heliocentric worldview. Radically improving the telescope, Galileo discovered the phases of Venus, the satellites of Jupiter, spots on the Sun. He waged a persistent, consistently materialistic struggle against the scholasticism of Aristotle, the dilapidated system of Ptolemy, and the anti-scientific canons of the Catholic Church. Galileo is one of the great men of science, "who knew how to break the old and create the new, in spite of any obstacles, in spite of everything."
The works of Galileo were continued and developed (1629-1695), who developed the theory of oscillations of a physical pendulum and installed laws of action of centrifugal forces. Huygens extended the theory of accelerated and decelerated motions of one point (translational motion of a body) to the case mechanical system points. This was a significant step forward, as it allowed the study of rotational movements. solid body. Huygens introduced the concept of moment of inertia of the body about the axis and defined the so-called swing center" physical pendulum. When determining the swing center of a physical pendulum, Huygens proceeded from the principle that "a system of weighty bodies moving under the influence of gravity cannot move in such a way that the common center of gravity of the bodies rises above its original position." Huygens also showed himself as an inventor. He created the design of pendulum clocks, invented the balancer-regulator of the pocket watch, built the best astronomical tubes of that time and was the first to clearly see the ring of the planet Saturn.

Collection output:

HISTORY OF FORMATIONANALYTICAL MECHANICS

Korolev Vladimir Stepanovich

Associate Professor, Cand. Phys.-Math. Sciences,

Saint Petersburg State University,
Russian Federation, St. Petersburg

HISTORY OF FORMATIONOF ANALYTICAL MECHANICS

Vladimir Korolev

candidate of Physical and Mathematical Sciences, assistant professor,

Saint-Petersburg State University,
Russia, Saint-Petersburg

annotation

The works of the classics of science in mechanics, which have been completed over the past years, are considered. An attempt was made to evaluate their contribution to the further development of science.

Abstract

Works of classics of science on mechanics which were performed for last years are considered. Attempt to estimate their contribution to further development of science is made.

Keywords: history of mechanics; development of science.

keywords: history of mechanics; development of science.

Introduction

Mechanics is the science of movement. The words theoretical or analytical show that the presentation does not use a constant reference to experiment, but is carried out by mathematical modeling on the basis of axiomatically accepted postulates and statements, the content of which is determined by the deep properties of the material world.

Theoretical mechanics is the fundamental basis of scientific knowledge. It is difficult to draw a clear line between theoretical mechanics and some branches of mathematics or physics. Many methods created in solving problems of mechanics, being formulated in the internal mathematical language, received an abstract continuation and led to the creation of new branches of mathematics and other sciences.

The subject of the study of theoretical mechanics are separate material bodies or selected systems of bodies in the process of their movement and interaction between themselves and the surrounding world when changing relative position in space and time. It is generally accepted that the objects around us are almost absolutely solid bodies. Deformable bodies, liquid and gaseous media are almost not considered or taken into account indirectly through their influence on the movement of selected mechanical systems. Theoretical mechanics deals with general patterns mechanical forms of motion and the construction of mathematical models to describe the possible behavior of mechanical systems. It is based on the laws established in experiments or special physical experiments and taken as axioms or truth that does not require proof, and also uses a large set of fundamental (common to many branches of science) and special concepts and definitions. They are only approximately correct and have been questioned, which has led to the emergence of new theories and directions for further research. We are not given an ideal fixed space or its metric, as well as processes uniform motion, which can be used to count absolutely accurate time intervals.

As a science, it originated in the 4th century BC in the works of ancient Greek scientists, as knowledge was accumulated along with physics and mathematics, it was actively developed by various philosophical schools until the first century and stood out as an independent direction. To date, there have been many scientific directions, trends, methods and research opportunities that create separate hypotheses or theories for description and modeling based on all accumulated knowledge. Many achievements in the natural sciences develop or supplement the basic concepts in the problems of mechanics. This space, which is determined by the dimension and structure, matter or a substance that fills the space, movement as a form of existence of matter, energy as one of the main characteristics of the movement.

The founders of classical mechanics

· archite Tarentum (428-365 BC), representative Pythagorean school philosophy, one of the first began to develop the problems of mechanics.

· Plato(427-347), a student of Socrates, developed and discussed many problems within the philosophical school, created the theory of the ideal world and the doctrine of the ideal state.

· Aristotle(384-322), student of Plato, formed general principles movements, created the theory of the movement of the celestial spheres, the principle of virtual speeds, considered the source of movements to be forces due to external influences.

Picture 1.

· Euclid(340-287), formulated many mathematical postulates and physical hypotheses, laid the foundations of geometry, which is used in classical mechanics.

· Archimedes(287-212), laid the foundations of mechanics and hydrostatics, the theory of simple machines, invented the Archimedes screw for water supply, the lever and many different lifting and military machines.

Figure 2.

· Hipparchus(180-125), created the theory of the motion of the Moon, explained the apparent motion of the Sun and planets, and introduced geographical coordinates.

· Heron Alexandrian (1st century BC), explored lifting mechanisms and devices, invented automatic doors, a steam turbine, was the first to create programmable devices, studied hydrostatics and optics.

· Ptolemy(100-178 AD), mechanic, optician, astronomer, proposed a geocentric system of the world, studied the apparent movement of the Sun, Moon and planets.

Figure 3

Science has been further developed in renaissance in the studies of many European scientists.

· Leonardo da Vinci(1452-1519), a universal creative person, did a lot of theoretical and practical mechanics, studied the mechanics of human movements and the flight of birds.

· Nicholas Copernicus(1473-1543), designed heliocentric system world and published in the work "On the Revolution of the Celestial Spheres".

· Tycho Brahe(1546-1601), left the most accurate observations of the movement celestial bodies, tried to combine the systems of Ptolemy and Copernicus, but in his model the Sun and Moon revolved around the Earth, and all other planets around the Sun.

Figure 4

· Galileo Galilei(1564-1642), conducted research on statics, dynamics and mechanics of materials, outlined the most important principles and laws that outlined the path to the creation of new dynamics, invented the telescope and discovered the satellites of Mars and Jupiter.

Figure 5

· Johannes Kepler(1571-1630), proposed the laws of planetary motion and laid the foundation for celestial mechanics. The discovery of the laws of planetary motion was made by the results of processing the tables of observations of the astronomer Tycho Brahe.

Figure 6

The founders of analytical mechanics

Analytical Mechanics was created by the labors of representatives of three generations almost closely following each other.

By 1687, the publication of Newton's "Principles of Mathematics of Natural Philosophy" dates back. In the year of his death, twenty-year-old Euler published his first work on the application of mathematical analysis in mechanics. For many years he lived in St. Petersburg, published hundreds of scientific papers and thus contributed to the formation of the Russian Academy of Sciences. Five years after Euler. Lagrange publishes Analytical Dynamics at the age of 52. Another 30 years will pass, and the works on analytic dynamics of three famous contemporaries will be published: Hamilton, Ostrogradsky and Jacobi. Mechanics received its main development in the studies of European scientists.

· Christian Huygens(1629-1695), invented the pendulum clock, the law of propagation of oscillations, developed the wave theory of light.

· Robert Hooke(1635-1703), studied the theory of planetary motions, expressed the idea of the law of universal gravitation in his letter to Newton, studied air pressure, surface tension of a liquid, discovered the law of deformation of elastic bodies.

Figure 7. Robert Hooke

· Isaac Newton(1643-1727), created the foundations of modern theoretical mechanics, in his main work "Mathematical Principles of Natural Philosophy" summarized the results of his predecessors, gave definitions of basic concepts and formulated the basic laws, carried out the justification and received common decision in the two-body problem. Translation from Latin into Russian was made by Academician A.N. Krylov.

Figure 8

· Gottfried Leibniz(1646-1716), introduced the concept of manpower, formulated the principle of least action, investigated the theory of resistance of materials.

· Johann Bernoulli(1667-1748), solved the problem of the brachistochrone, developed the theory of impacts, studied the motion of bodies in a resisting medium.

· Leonhard Euler(1707-1783), laid the foundations of analytical dynamics in the book "Mechanics or the science of motion in an analytical presentation", analyzed the case of the motion of a heavy rigid body fixed in the center of gravity, is the founder of hydrodynamics, developed the theory of projectile flight, introduced the concept of inertia force.

Figure 9

· jean Leron d'Alembert(1717-1783), received general rules drawing up equations of motion material systems, studied the motion of the planets, established the basic principles of dynamics in the book "Treatise on Dynamics".

· joseph Louis Lagrange(1736-1813), in his work "Analytical Dynamics" proposed the principle of possible displacements, introduced generalized coordinates and gave the equations of motion new form, discovered a new case of solvability of the equations rotary motion solid body.

The works of these scientists completed the construction of the foundations of modern classical mechanics, laid the foundation for the analysis of infinitesimals. A course in mechanics was developed, which was presented in a strictly analytical way on the basis of a general mathematical principle. This course was called "analytical mechanics". The advances in mechanics were so great that they influenced the philosophy of the time, which manifested itself in the creation of "mechanism".

The development of mechanics was also promoted by the interest of astronomers, mathematicians, and physicists in the problems of determining the motion of visible celestial bodies (the Moon, planets, and comets). The discoveries and works of Copernicus, Galileo and Kepler, the theory of the motion of the Moon by d’Alembert and Poisson, the five-volume Celestial Mechanics by Laplace and other classics made it possible to create a fairly complete theory of motion in a gravitational field, making it possible to apply analytical and numerical methods to the study of other problems of mechanics. The further development of mechanics is connected with the works of outstanding scientists of their time.

· Pierre Laplace(1749-1827), completed the creation of celestial mechanics based on the law of universal gravitation, proved the stability of the solar system, developed the theory of ebbs and flows, investigated the motion of the moon and determined the compression of the earth's spheroid, substantiated the hypothesis of the emergence of the solar system.

Figure 10.

· Jean Baptiste Fourier(1768-1830), created the theory of partial differential equations, developed the doctrine of the representation of functions in the form of trigonometric series, explored the principle of virtual work.

· Charles Gauss(1777-1855), a great mathematician and mechanic, published the theory of the motion of celestial bodies, established the position of the planet Ceres, studied the theory of potentials and optics.

· Louis Poinsot(1777-1859), proposed a general solution for the problem of body motion, introduced the concept of an ellipsoid of inertia, studied many problems of statics and kinematics.

· Simeon Poisson(1781-1840), was engaged in solving problems in gravitation and electrostatics, generalized the theory of elasticity and the construction of equations of motion based on the principle of living forces.

· Mikhail Vasilievich Ostrogradsky(1801-1862), a great mathematician and mechanic, his works relate to analytical mechanics, elasticity theory, celestial mechanics, hydromechanics, studied the general equations of dynamics.

· Carl Gustav Jacobi(1804-1851), proposed new solutions to the equations of dynamics, developed a general theory of integration of the equations of motion, used the canonical equations of mechanics and partial differential equations.

· William Rowan Hamilton(1805-1865), brought the equations of motion of an arbitrary mechanical system to a canonical form, introduced the concept of quaternions and vectors, established the general integral variational principle of mechanics.

Figure 11.

· Hermann Helmholtz(1821-1894), gave a mathematical interpretation of the law of conservation of energy, laid the foundation for the widespread application of the principle of least action to electromagnetic and optical phenomena.

· Nikolai Vladimirovich Maievsky(1823-1892), founder of the Russian scientific school of ballistics, created the theory of the rotational motion of a projectile, was the first to take into account air resistance.

· Pafnuty Lvovich Chebyshev(1821-1894), studied the theory of machines and mechanisms, created a steam engine, a centrifugal regulator, walking and rowing mechanisms.

Figure 12.

· Gustav Kirchhoff(1824-1887), studied the deformation, motion and balance of elastic bodies, worked on the logical construction of mechanics.

· Sofia Vasilievna Kovalevskaya(1850-1891), was engaged in the theory of the rotational motion of a body around a fixed point, discovered the third classical case of solving the problem, studied the Laplace problem on the equilibrium of Saturn's rings.

Figure 13.

· Henry Hertz(1857-1894), the main works are devoted to electrodynamics and general theorems of mechanics based on a single principle.

Modern development of mechanics

In the twentieth century, they were and are still engaged in solving many new problems in mechanics. This was especially active after the advent of modern computing tools. First of all, these are new complex problems of controlled motion, space dynamics, robotics, biomechanics, quantum mechanics. It is possible to note the work of outstanding scientists, many scientific schools of universities and research teams in Russia.

· Nikolay Egorovich Zhukovsky(1847-1921), the founder of aerodynamics, studied the motion of a rigid body with a fixed point and the problem of stability of motion, derived a formula for determining the lift force of a wing, and studied the theory of impact.

Figure 14.

· Alexander Mikhailovich Lyapunov(1857-1918), the main works are devoted to the theory of the stability of equilibrium and the movement of mechanical systems, the founder modern theory stability .

· Konstantin Eduardovich Tsiolkovsky(1857-1935), the founder of modern astronautics, aerodynamics and rocket dynamics, created the theory of the hovercraft and the theory of the movement of single-stage and multi-stage rockets.

· Ivan Vsevolodovich Meshchersky(1859-1935), studied the movement of bodies of variable mass, compiled a collection of problems in mechanics, which is still used today.

Figure 15.

· Alexey Nikolaevich Krylov(1863-1945), the main studies are related to structural mechanics and shipbuilding, the unsinkability of the ship and its stability, hydromechanics, ballistics, celestial mechanics, the theory of jet propulsion, the theory of gyroscopes and numerical methods, translated into Russian the works of many classics of science.

· Sergey Alekseevich Chaplygin(1869-1942), the main works relate to nonholonomic mechanics, hydrodynamics, the theory of aviation and aerodynamics, gave complete solution problems of the impact of an air flow on a streamlined body.

· Albert Einstein(1879-1955), formulated the special and general theory of relativity, created new system space-time relations and showed that gravity is an expression of the inhomogeneity of space and time, which is produced by the presence of matter.

· Alexander Alexandrovich Fridman(1888-1925), created a model of a non-stationary universe, where he predicted the possibility of the expansion of the universe.

· Nikolai Gurevich Chetaev(1902-1959) studied the properties of perturbed motions of mechanical systems, issues of motion stability, proved the basic theorems on the instability of equilibrium.

Figure 16.

· Lev Semenovich Pontryagin(1908-1988) explored the theory of oscillations, calculus of variations, control theory, creator mathematical theory optimal processes.

Figure 17.

It is possible that even in ancient times and subsequent periods there were centers of knowledge, scientific schools and directions of research into the science and culture of peoples or civilizations: Arab, Chinese or Indian in Asia, the Mayan people in America, where achievements appeared, but European philosophical and scientific schools developed in a special way, not always paying attention to the discoveries or theories of other researchers. IN different times the languages used for communication were Latin, German, French, English... Accurate translations of available texts and common notations in formulas were needed. This made it difficult, but did not stop development.

Modern science tries to study single complex of all that exists, which manifests itself so diversely in the world around us. To date, many scientific directions, trends, methods and research opportunities have been formed. When studying classical mechanics, kinematics, statics and dynamics are traditionally distinguished as the main sections. An independent section or science formed celestial mechanics, as part of theoretical astronomy, as well as quantum mechanics.

Basic tasks of dynamics consist in determining the motion of a system of bodies according to known active forces taken into account or in determining forces according to a known law of motion. Control in the problems of dynamics assumes that there is a possibility of changing for the conditions for the implementation of the process of movement according to our own choice of parameters or functions that determine the process or are included in the equations of motion, in accordance with the given requirements, wishes or criteria.

Analytical, Theoretical, Classical, Applied,

Rational, Managed, Celestial, Quantum…

It's all Mechanics in different presentations!

Bibliography:

Aleshkov Yu.Z. Excellent work in applied mathematics. SPb.: Ed. St. Petersburg State University, 2004. - 309 p.
Bogomolov A.N. Mathematics of mechanics. Biographical guide. Kyiv: Ed. Naukova Dumka, 1983. - 639 p.
Vavilov S.I. Isaac Newton. 4th ed., add. M.: Nauka, 1989. - 271 p.
Krylov A.N. Isaac Newton: Mathematical principles of natural philosophy. Translation from Latin with notes and explanations of the fleet by Lieutenant General A.N. Krylov. // Proceedings of the Nikolaev Marine Academy (Issue 4), Petrograd. Book 1. 1915. 276 p., Book 2. 1916. (Issue 5). 344 p. or in the book: A.N. Krylov. Collection of Works. M.-L. Publishing House of the Academy of Sciences of the USSR. T. 7. 1936. 696 p. or in the Classics of Science series: I. Newton. Mathematical principles of natural philosophy. Translation from lat. and comments by A.N. Krylov. M.: Science. 1989. - 687 p.
People of Russian science // Essays on outstanding figures of natural science and technology. (Mathematics. Mechanics. Astronomy. Physics. Chemistry). Collection of articles, ed. I.V. Kuznetsova. M.: Fizmatlit, 1961. 600 p.
Novoselov V.S., Korolev V.S. Analytical mechanics of a controlled system. SPb.: Ed. St. Petersburg State University, 2005. 298 p.
Novoselov V.S. quantum mechanics and statistical physics. SPb.: Ed. VVM, 2012. 182 p.
Polyakhova E.N. Classical celestial mechanics in the works of the Petersburg School of Mathematics and Mechanics in the 19th century. SPb.: Ed. Nestor-History, 2012. 140 p.
Polyakhova E.N., Korolev V.S., Kholshevnikov K.V. Translations of the works of the classics of science by Academician A.N. Krylov. "Natural and mathematical sciences in modern world» No. 2(26). Novosibirsk: Ed. SibAK, 2015. S. 108-128.
Poincare A. About science. Per. from fr. ed. L.S. Pontryagin. M.: Nauka, 1990. 736 p.
Tyulina I.A., Chinenova V.N. The history of mechanics through the prism of the development of ideas, principles and hypotheses. M.: URSS (Librocom), 2012. 252 p.

Basic concepts

Classical mechanics operates with several basic concepts and models. Among them should be highlighted:

Basic Laws

Galileo's principle of relativity

Newton's laws

Newton's three laws are the basis of classical mechanics.

Law of energy conservation

Beyond the applicability of Newton's laws

The above expressions for momentum and kinetic energy are valid only in the absence of a significant electromagnetic contribution. In electromagnetism, Newton's second law for a wire carrying current is violated if it does not include the contribution of the electromagnetic field to the momentum of the system expressed in terms of the Poynting vector divided by c 2 , where c is the speed of light in free space.

Story

ancient time

Middle Ages

new time

17th century

18th century

19th century

Particularly significant in the 19th century were advances in continuum mechanics. Navier and Cauchy formulated the equations of elasticity theory in a general form. In the works of Navier and Stokes, differential equations of hydrodynamics were obtained taking into account the viscosity of the liquid. Along with this, there is a deepening of knowledge in the field of hydrodynamics of an ideal fluid: the works of Helmholtz on vortices, Kirchhoff, Zhukovsky and Reynolds on turbulence, and Prandtl on boundary effects appear. Saint-Venant developed a mathematical model describing the plastic properties of metals.

Newest time

Limitations of classical mechanics

Classical mechanics gives accurate results for the systems we encounter in everyday life. But her predictions become incorrect for systems approaching the speed of light, where it is replaced by relativistic mechanics, or for very small systems where the laws of quantum mechanics apply. For systems that combine both of these properties, relativistic quantum field theory is used instead of classical mechanics. For systems with a very large number of components, or degrees of freedom, classical mechanics also cannot be adequate, but methods of statistical mechanics are used.

Although classical mechanics is generally compatible with other "classical" theories such as classical electrodynamics and thermodynamics, there are some inconsistencies between these theories that were found in the late 19th century. They can be solved by methods of more modern physics. In particular, the equations of classical electrodynamics are not invariant under Galilean transformations. The speed of light enters them as a constant, which means that classical electrodynamics and classical mechanics could only be compatible in one chosen frame of reference associated with the ether. However, experimental verification did not reveal the existence of the ether, which led to the creation of a special theory of relativity, in which the equations of mechanics were modified. The principles of classical mechanics are also inconsistent with some of the claims of classical thermodynamics, leading to the Gibbs paradox, according to which it is impossible to accurately determine entropy, and to the ultraviolet catastrophe, in which a black body must radiate an infinite amount of energy. To overcome these incompatibilities, quantum mechanics was created.

Notes

Internet links

Video lecture 1. Physics: Classical mechanics (autumn 1999)// MIT Lectures: 8.01

Literature

Arnold V.I. Avets A. Ergodic problems of classical mechanics. - RHD, 1999. - 284 p.
B. M. Yavorsky, A. A. Detlaf. Physics for high school students and those entering universities. - M .: Academy, 2008. - 720 p. - (Higher education). - 34,000 copies. - ISBN 5-7695-1040-4
Sivukhin D.V. General course of physics. - 5th edition, stereotypical. - M .: Fizmatlit, 2006. - T. I. Mechanics. - 560 p. - ISBN 5-9221-0715-1
A. N. MATVEEV Mechanics and the Theory of Relativity. - 3rd ed. - M .: ONYX 21st century: World and Education, 2003. - 432 p. - 5000 copies. - ISBN 5-329-00742-9
C. Kittel, W. Knight, M. Ruderman Mechanics. Berkeley Physics Course. - M .: Lan, 2005. - 480 p. - (Textbooks for universities). - 2000 copies. - ISBN 5-8114-0644-4

Definition 1

Classical mechanics is a subsection of physics that studies the movement of physical bodies based on Newton's laws.

The basic concepts of classical mechanics are:

mass - is defined as the main measure of inertia, or the ability of a substance to maintain a state of rest in the absence of the influence of external factors on it;
force - acts on the body and changes the state of its movement, causing acceleration;
internal energy - determines the current state of the element under study.

Others no less important concepts This section of physics are: temperature, momentum, angular momentum and volume of matter. The energy of a mechanical system mainly consists of its kinetic energy of motion and potential force, which depends on the position of the elements acting in a particular system. With respect to the physical quantities the fundamental conservation laws of classical mechanics function.

Founders of classical mechanics

Remark 1

The foundations of classical mechanics were successfully laid by the thinker Galileo, as well as Kepler and Copernicus, when considering the patterns of rapid motion of celestial bodies.

Figure 1. Principles of classical mechanics. Author24 - online exchange of student papers

Interestingly, for a long period of time, physics and mechanics were studied in the context of astronomical events. In his scientific works, Copernicus argued that the correct calculation of the patterns of interaction of celestial bodies can be simplified if we deviate from the existing principles that were previously laid down by Aristotle and consider it the starting point for the transition from the geocentric to the heliocentric concept.

The ideas of the scientist were further formalized by his colleague Kepler in the three laws of motion of material bodies. In particular, the second law stated that absolutely all planets solar system carry out uniform movement in elliptical orbits, having the main focus of the Sun.

The next significant contribution to the development of classical mechanics was made by the inventor Galileo, who, studying the fundamental postulates of the mechanical motion of celestial bodies, in particular under the influence of the forces of gravity, presented the public with five universal laws of the physical motion of substances at once.

But still, contemporaries attribute the laurels of the key founder of classical mechanics to Isaac Newton, who in his famous scientific work"Mathematical Expression of Natural Philosophy" described the synthesis of those definitions of the physics of motion, which were previously presented by his predecessors.

Figure 2. Variational principles of classical mechanics. Author24 - online exchange of student papers

Newton clearly formulated the three basic laws of motion, which were named after him, as well as the theory of universal gravitation, which drew a line under Galileo's research and explained the phenomenon free fall tel. Thus, a new, more improved picture of the world was developed.

Basic and variational principles of classical mechanics

Classical mechanics provides researchers with accurate results for systems that are often encountered in everyday life. But they eventually become incorrect for other concepts, the speed of which is almost equal to the speed of light. Then it is necessary to use the laws of relativistic and quantum mechanics in experiments. For systems that combine several properties at once, instead of classical mechanics, the theory of the field of quanta is used. For concepts with many components, or levels of freedom, the direction of study in physics is also adequate when using the methods of statistical mechanics.

Today, the following main principles of classical mechanics are distinguished:

The principle of invariance with respect to spatial and temporal displacements (rotations, shifts, symmetries): space is always homogeneous, and the course of any processes within a closed system is not affected by its initial locations and orientation relative to the material reference body.
The principle of relativity: the course of physical processes in an isolated system is not affected by its rectilinear motion regarding the very concept of reference; the laws that describe such phenomena are the same in different branches of physics; the processes themselves will be the same if the initial conditions were identical.

Definition 2

Variational principles are the initial, basic provisions of analytical mechanics, mathematically expressed in the form of unique variational relations, from which differential formulas of motion follow as a logical consequence, as well as all kinds of provisions and laws of classical mechanics.

In most cases, the main feature by which the real motion can be distinguished from the considered class of kinematic motions is the stationarity condition, which ensures the invariance of the further description.

Figure 4. The principle of long-range action. Author24 - online exchange of student papers

The first of the variational rules of classical mechanics is the principle of possible or virtual displacements, which allows you to find the correct equilibrium positions for a system of material points. Therefore, this pattern helps to solve complex problems of statics.

The next principle is called the least constraint. This postulate presupposes a certain movement of a system of material points, directly interconnected in a chaotic way and subject to any influences from the environment.

Another main variational position in classical mechanics, this is the principle of the most direct path, where any free system is in a calm state or uniform motion along specific lines in comparison with any other arcs allowed by relationships and having a common starting point and tangent in concept.

Operating principle in classical mechanics

Newton's equations of mechanical motion can be formulated in many ways. One is through the Lagrange formalism, also called Lagrangian mechanics. Although this principle is quite equivalent to Newton's laws in classical physics, but the interpretation of action is better suited for generalizations of all concepts and plays an important role in modern science. Indeed, this principle is a complex generalization in physics.

In particular, this is fully understood within the framework of quantum mechanics. The interpretation of quantum mechanics by Richard Feynman through the use of path integrals is based on the principle of constant interaction.

Many problems in physics can be solved by applying the principle of operation, which is able to find the fastest and easiest way to solve the problems.

For example, light can find its way out through an optical system, and the trajectory of a material body in a gravitational field can be detected using the same operating principle.

Symmetries in any situation can be better understood by applying this concept, together with the Euler-Lagrange equations. In classical mechanics right choice further action can be experimentally proved from Newton's laws of motion. And, conversely, from the principle of action, Newtonian equations are implemented in practice, with a competent choice of action.

Thus, in classical mechanics, the principle of action is considered ideally equivalent to Newton's equations of motion. The application of this method greatly simplifies the solution of equations in physics, since it is a scalar theory, with applications and derivatives that apply elementary calculus.

The history of the formation of analytical mechanics. The principles of classical mechanics Which was the main classical mechanics

Basic concepts

Basic Laws

Galileo's principle of relativity

Newton's laws

Law of energy conservation

Beyond the applicability of Newton's laws

Story

ancient time

Middle Ages

new time

17th century

18th century

19th century

Newest time

Limitations of classical mechanics

Notes

Internet links

Literature

Basic concepts

Basic Laws

Galileo's principle of relativity

Newton's laws

Law of energy conservation

Beyond the applicability of Newton's laws

Story

ancient time

Middle Ages

new time

17th century

18th century

19th century

Newest time

Limitations of classical mechanics

Notes

Internet links

Literature

Founders of classical mechanics

Basic and variational principles of classical mechanics

Operating principle in classical mechanics

Collection of ideal social studies essays

Anti-Soviet rebellion in Hungary (1956)