The theory of classical mechanics. Formation of knowledge of schoolchildren about the structure of physical theory. The principle of operation in classical mechanics
Classic mechanics- the type of mechanics (a branch of physics that studies the laws of changes in the positions of bodies in space with time and the reasons that cause), based on Newton's laws and Galileo's principle of relativity. Therefore, it is often called “ Newtonian mechanics».
Classical mechanics are subdivided into:
- statics (which considers the balance of bodies)
- kinematics (which studies the geometric property of motion without considering its causes)
- dynamics (which deals with the movement of bodies).
There are several equivalent ways of formal mathematical description of classical mechanics:
- Lagrangian formalism
- Hamiltonian formalism
Classical mechanics gives very accurate results if its application is limited to bodies whose velocities are much less than the speed of light, and whose dimensions are much greater than the dimensions of atoms and molecules. The generalization of classical mechanics to bodies moving at an arbitrary speed is relativistic mechanics, and to bodies whose dimensions are comparable to atomic ones is quantum mechanics. Quantum field theory deals with quantum relativistic effects.
However, classical mechanics retain their significance because:
- it is much easier to understand and use than other theories
- in a wide range, it describes reality quite well.
Classical mechanics can be used to describe the movement of objects such as a top and a baseball, many astronomical objects (such as planets and galaxies), and sometimes even many microscopic objects such as molecules.
Classical mechanics is a self-consistent theory, that is, within its framework, there are no statements that contradict each other. However, its combination with other classical theories, such as classical electrodynamics and thermodynamics, leads to the appearance of insoluble contradictions. In particular, classical electrodynamics predicts that the speed of light is constant for all observers, which is incompatible with classical mechanics. At the beginning of the 20th century, this led to the need to create a special theory of relativity. When considered in conjunction with thermodynamics, classical mechanics leads to the Gibbs paradox, in which it is impossible to accurately determine the amount of entropy, and to an ultraviolet catastrophe, in which a black body must emit an infinite amount of energy. Attempts to solve these problems led to the emergence and development of quantum 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 velocity constant in absolute value and direction. These frames of reference are called inertial and move relative to each other uniformly and rectilinearly. In all inertial reference frames, the properties of space and time are the same, and all processes in mechanical systems ah obey the same laws. This principle can also be formulated as the absence of absolute frames of reference, that is, frames of reference that are somehow distinguished relative to others.
Newton's laws
The basis of classical mechanics is Newton's three laws.
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 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 complex movements of extended non-point objects. Euler's laws provide an extension of Newton's laws to this area. The concept of angular momentum is based on the same mathematical methods used to describe one-dimensional motion.
The equations of motion of a rocket extend the concept of velocity, where the momentum of an object changes over time, to account for an effect such 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 the momentum and kinetic energy valid only if there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for a current-carrying wire is violated if it does not include the contribution electromagnetic field into 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.
History
Ancient time
Classical mechanics originated in antiquity mainly in connection with problems that arose during construction. The first of the branches of mechanics to be developed was statics, the foundations of which were laid in the works of Archimedes in the 3rd century BC. NS. He formulated the rule of the lever, the theorem on the addition of parallel forces, introduced the concept of the center of gravity, laid the foundations of hydrostatics (the Archimedes force).
Middle Ages
New time
17th century
XVIII 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 oscillations, Routh, Zhukovsky and Lyapunov developed the theory of stability of mechanical systems. Coriolis developed the theory of relative motion by proving a theorem on the decomposition of acceleration into components. In the second half of the 19th century, kinematics was separated into a separate section of mechanics.
The advances in the field of continuum mechanics were especially significant in the 19th century. 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 ideal fluid hydrodynamics: Helmholtz's works on vortices, Kirchhoff, Zhukovsky and Reynolds on turbulence, Prandtl on boundary effects appear. Saint-Venant developed a mathematical model describing the plastic properties of metals.
Newest time
In the XX century, the interest of researchers switches 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. From the mechanics of a continuous medium, aerodynamics stands out, the foundations of which were developed by Zhukovsky. In the middle of the 20th century, a new direction in classical mechanics is actively developing - chaos theory. The issues of stability of complex dynamical systems also remain important.
Limitations of classical mechanics
Classical mechanics gives accurate results for systems that we meet in Everyday life... But its predictions become incorrect for systems whose speed approaches 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 the 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 opportunities for approximation and application for a very wide class of physical objects, from the usual ones, such as a top or a ball, to large astronomical objects (planets, galaxies) and very microscopic (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 more modern physics... In particular, the equations of classical electrodynamics are not invariant under the Galilean transformations. The speed of light enters into them as a constant, which means that classical electrodynamics and classical mechanics could be compatible only in one selected 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, within which the equations of mechanics were modified. The principles of classical mechanics are also incompatible with some statements of classical thermodynamics, which leads to the Gibbs paradox, according to which it is impossible to accurately determine the entropy, and to an ultraviolet catastrophe in which a black body must emit an infinite amount of energy. To overcome these incompatibilities, quantum mechanics was created.
Notes (edit)
Internet links
- Video lecture 1. Physics: Classical Mechanics (Fall 1999)// Massachusetts Institute of Technology Lectures: 8.01
Literature
- Arnold V.I. A. Ergodic problems of classical mechanics .. - RKhD, 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. - Edition 5, stereotyped. - Moscow: Fizmatlit, 2006. - T. I. Mechanics. - 560 p. - ISBN 5-9221-0715-1
- A. N. Matveev. Mechanics and Theory of Relativity. - 3rd ed. - M .: ONIX 21st century: Peace 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
(January 4, 1643, Woolsthorpe, near Grantham, Lincolnshire, England - March 31, 1727, London) - English mathematician, mechanic, astronomer and physicist, creator of classical mechanics, member (1672) and president (since 1703) of the Royal Society of London.
One of the founders of modern physics, formulated the basic laws of mechanics and was the actual creator of a unified physical program for describing all physical phenomena on the basis of mechanics; discovered the law of universal gravitation, explained the motion of planets around the Sun and the Moon around the Earth, as well as tides in the oceans, laid the foundations of continuum mechanics, acoustics and physical optics.
Childhood
Isaac Newton was born in a small village in the family of a small farmer who died three months before the birth of his son. The baby was premature; there is a legend that he was so small that he was placed in a sheepskin mitten lying on a bench, from which he once fell out and hit his head hard on the floor.
When the child was three years old, his mother remarried and left, leaving him in the care of his grandmother. Newton grew up sickly and uncommunicative, prone to daydreaming. He was attracted by poetry and painting, he, far from his peers, made kites, invented windmill, water clock, pedal carriage.
The beginning of school life was difficult for Newton. He studied poorly, was a weak boy, and one day his classmates beat him until he lost consciousness. To endure such a humiliating situation was unbearable for the proud Newton, and there was only one thing left: to stand out for his academic success. Through hard work, he achieved the first place in the class.
An interest in technology made Newton think about natural phenomena; he also studied mathematics in depth. Jean Baptiste Biot wrote about this later: “One of his uncles, finding him one day under a hedge with a book in his hands, immersed in deep thought, took the book from him and found that he was busy solving a mathematical problem. Struck by such a serious and active direction of such a young man, he persuaded his mother not to resist further the desire of her son and to send him to continue his studies. " After serious preparation, Newton entered Cambridge in 1660 as a Subsizzfr (this was the name of the poor students who were obliged to serve the college members, which could not but weigh Newton).
The beginning of creativity. Optics
For six years, Newton completed all college degrees and prepared all his further great discoveries. In 1665, Newton became a Master of Arts.
In the same year, when the plague was raging in England, he decided to temporarily settle in Woolsthorpe. It was there that he began to actively engage in optics; The search for ways to eliminate chromatic aberration in lens telescopes led Newton to investigate what is now called dispersion, that is, the dependence of the refractive index on frequency. Many of his experiments (and there are more than a thousand of them) have become classical and are being repeated today in schools and institutes.
The leitmotif of all research was the desire to understand the physical nature of light. At first, Newton was inclined to think that light is waves in the all-pervading ether, but later he abandoned this idea, deciding that resistance from the ether should have noticeably slowed down the movement of celestial bodies. These arguments led Newton to the idea that light is a stream of special particles, corpuscles, escaping from a source and moving in a straight line until they meet obstacles. The corpuscular model explained not only the straightness of the propagation of light, but also the law of reflection (elastic reflection), and - true, not without additional assumptions - and the law of refraction. This assumption consisted in the fact that light corpuscles, flying up to the surface of the water, for example, should be attracted by it and therefore experience acceleration. According to this theory, the speed of light in water should be greater than in air (which contradicted later experimental data).
The laws of mechanics
The formation of corpuscular ideas about light was clearly influenced by the fact that at this time, basically, the work was completed, which was destined to become the main great result of Newton's works - the creation of a unified physical picture of the World based on the laws of mechanics formulated by him.
This picture was based on the idea of material points - physically infinitely small particles of matter and the laws governing their motion. It was precisely the precise formulation of these laws that gave Newton's mechanics completeness and completeness. The first of these laws was, in fact, the definition of inertial reference systems: it is in such systems that material points that do not experience any influences move uniformly and rectilinearly. The second law of mechanics plays a central role. It says that the change in quantity, motion (product of mass and speed) per unit of time is equal to the force acting on a material point. The mass of each of these points is constant; in general, all these points "do not wear out", according to Newton, each of them is eternal, that is, it can neither arise nor be destroyed. Material points interact, and force is a quantitative measure of impact on each of them. The task of figuring out what these forces are is the root problem of mechanics.
Finally, the third law - the law of "equality of action and reaction" explained why the total impulse of any body that does not experience external influences remains unchanged, no matter how its constituent parts interact with each other.
The law of universal gravitation
Having posed the problem of studying various forces, Newton himself gave the first brilliant example of its solution, formulating the law of universal gravitation: the force of gravitational attraction between bodies whose dimensions are much less than the distance between them is directly proportional to their masses, inversely proportional to the square of the distance between them and is directed along the connecting their straight. The law of universal gravitation allowed Newton to give a quantitative explanation of the motion of the planets around the Sun and the Moon around the Earth, to understand the nature of sea tides. This could not but make a huge impression on the minds of researchers. The program of a unified mechanical description of all natural phenomena - both "terrestrial" and "celestial" for many years has become firmly established in physics. Moreover, for two centuries the very question of the limits of applicability of Newton's laws seemed unjustified to many physicists.
In 1668 Newton returned to Cambridge and soon received the Lucas Department of Mathematics. This department before him was occupied by his teacher I. Barrow, who yielded the department to his beloved student in order to provide him financially. By that time, Newton was already the author of the binomial and the creator (simultaneously with Leibniz, but independently of him) of the fluxia method - what is now called differential and integral calculus. In general, that was the most fruitful period in Newton's work: in seven years, from 1660 to 1667, his main ideas were formed, including the idea of the law of universal gravitation. Not limiting himself to theoretical studies alone, he designed and began to create a reflector telescope (reflective) in the same years. This work led to the discovery of what later became known as "lines of equal thickness" interference. (Newton, realizing that "light extinguishing by light" is manifested here, which did not fit into the corpuscular model, tried to overcome the difficulties that arose here, introducing the assumption that corpuscles in the light move in waves - "tides"). The second of the telescopes made (improved) was the reason for the introduction of Newton to the members of the Royal Society of London. When Newton withdrew from membership, citing lack of funds to pay membership fees, it was considered possible, given his scientific merits, to make an exception for him, exempting him from paying them.
Being by nature a very cautious (not to say timid) person, Newton, against his will, sometimes found himself drawn into painful discussions and conflicts for him. Thus, his theory of light and colors, set forth in 1675, caused such attacks that Newton decided not to publish anything on optics while he was alive. Hooke, his most fierce opponent. Newton had to take part in political events. From 1688 to 1694 he was a member of parliament. By that time, in 1687, his main work "Mathematical Principles of Natural Philosophy" was published - the basis of the mechanics of all physical phenomena, from the movement of celestial bodies to the propagation of sound. For several centuries ahead, this program determined the development of physics, and its significance has not been exhausted to this day.
Newton's disease
Constant enormous nervous and mental stress led to the fact that in 1692 Newton fell ill with a mental disorder. The immediate impetus for this was the fire, in which all the manuscripts he had prepared perished. Only by 1694 he, according to the testimony Huygens, "... is already beginning to understand his book" Beginnings "".
The constant oppressive feeling of material insecurity was undoubtedly one of the causes of Newton's illness. Therefore, the position of superintendent of the Mint with the preservation of the professorship at Cambridge was of great importance to him. Eagerly getting down to work and quickly achieving noticeable success, he was appointed director in 1699. It was impossible to combine this with teaching, and Newton moved to London. At the end of 1703 he was elected president of the Royal Society. By that time, Newton had reached the pinnacle of fame. In 1705, he was elevated to the dignity of knighthood, but, having a large apartment, six servants and a rich exit, he remains lonely as before. The time for active creativity is over, and Newton is limited to preparing the publication of "Optics", the reprint of "Principles" and the interpretation of Holy Scripture (he owns the interpretation of the Apocalypse, a composition about the prophet Daniel).
Newton was buried at Westminster Abbey. The inscription on his grave ends with the words: "Let mortals rejoice that such a decoration of the human race lived in their midst."
Thus, the subject of study of classical mechanics is the laws and causes of mechanical motion, understood as the interaction of macroscopic (consisting of a huge number of particles) physical bodies and their constituent parts, and the change in their position in space generated by this interaction, which occurs with subluminal (nonrelativistic) velocities.
The place of classical mechanics in the system of physical sciences and the limits of its applicability are shown in Figure 1.
Figure 1. The area of applicability of classical mechanics
Classical mechanics is divided into statics (which considers the balance of bodies), kinematics (which studies the geometric property of motion without considering its causes) and dynamics (which considers the motion of bodies taking into account the causes that cause it).
There are several equivalent ways of formal mathematical description of classical mechanics: Newton's laws, Lagrangian formalism, Hamiltonian formalism, Hamilton - Jacobi formalism.
When classical mechanics is applied to bodies whose speeds are much less than the speed of light, and whose dimensions are much larger than the sizes of atoms and molecules, and at distances or conditions when the speed of propagation of gravity can be considered infinite, it gives extremely accurate results. Therefore, even today, classical mechanics retains its significance, since it is much easier to understand and use than other theories, and describes everyday reality quite well. Classical mechanics can be used to describe the motion of a very wide class of physical objects: ordinary objects of the macrocosm (such as a top and a baseball), and objects of astronomical size (such as planets and stars), and many microscopic objects.
Classical mechanics is the oldest of the physical sciences. Even in pre-antique times, people not only experienced the laws of mechanics, but also applied them in practice, constructing the simplest mechanisms. Already in the Neolithic and Bronze Age, a wheel appeared, a little later a lever and an inclined plane were used. V antique period the accumulated practical knowledge began to be generalized, the first attempts were made to define the basic concepts of mechanics, such as force, resistance, displacement, speed, and to formulate some of its laws. It was during the development of classical mechanics that the foundations were laid scientific method knowledge, suggesting some general rules scientific reasoning about empirically observed phenomena, making assumptions (hypotheses) that explain these phenomena, building models that simplify the studied phenomena while maintaining their essential properties, forming systems of ideas or principles (theories) and their mathematical interpretation.
However, the qualitative formulation of the laws of mechanics began only in the 17th century A.D. e., when Galileo Galilei discovered the kinematic law of addition of velocities and established the laws of free fall of bodies. Several decades after Galileo, Isaac Newton formulated the basic laws of dynamics. In Newtonian mechanics, the motion of bodies is considered at speeds much less than the speed of light in a vacuum. It is called classical or Newtonian mechanics, in contrast to relativistic mechanics, created at the beginning of the 20th century, mainly due to the work of Albert Einstein.
Modern classical mechanics, as a method for studying natural phenomena, uses their description using a system of basic concepts and building on their basis ideal models of real phenomena and processes.
Basic concepts of classical mechanics
Basic principles of classical mechanics
Mechanics- this is a branch of physics in which the simplest form of motion of matter is studied - mechanical movement, which consists in changing over time the position of bodies or their parts. The fact that mechanical phenomena take place in space and time is reflected in any law of mechanics that contains explicitly or implicitly space-time relationships - distances and time intervals.
Mechanics sets itself two main tasks:
the study of various movements and the generalization of the results obtained in the form of laws, with the help of which the nature of the movement can be predicted in each specific case. The solution of this problem led to the establishment by I. Newton and A. Einstein of the so-called dynamic laws;
finding general properties inherent in any mechanical system in the course of its movement. As a result of solving this problem, the laws of conservation of such fundamental quantities as energy, momentum and angular momentum were discovered.
Dynamic laws and laws of conservation of energy, momentum and angular momentum are the basic laws of mechanics and constitute the content of this chapter.
§1. Mechanical movement: basic concepts
Classical mechanics has three main sections - statics, kinematics and dynamics... In statics, the laws of the addition of forces and the conditions of equilibrium of bodies are considered. In kinematics, a mathematical description of all kinds of mechanical movement is given, regardless of the reasons that cause it. In dynamics, the influence of interaction between bodies on their mechanical movement is investigated.
In practice, all physical problems are solved approximately: real complex movement considered as a set of simplest movements, a real object replaced by an idealized model this object, etc. For example, when considering the motion of the Earth around the Sun, the size of the Earth can be neglected. In this case, the description of the motion is greatly simplified - the position of the Earth in space can be determined by one point. Among the models of mechanics, the determining ones are material point and absolutely rigid body.
Material point (or particle) is a body, the shape and size of which can be neglected under the conditions of this problem. Any body can be mentally divided into a very large number of parts, arbitrarily small in comparison with the size of the whole body. Each of these parts can be considered as a material point, and the body itself - as a system of material points.
If the deformations of a body during its interaction with other bodies are negligible, then it is described by the model absolutely solid.
Absolutely solid (or solid) is a body, the distances between any two points of which do not change during movement. In other words, it is a body, the shape and dimensions of which do not change during its movement. An absolutely rigid body can be viewed as a system of material points rigidly connected to each other.
The position of a body in space can be determined only in relation to some other bodies. For example, it makes sense to talk about the position of the planet in relation to the Sun, an airplane or ship in relation to the Earth, but it is impossible to indicate their position in space without regard to any particular body. An absolutely rigid body, which serves to determine the position of an object of interest to us, is called a reference body. To describe the movement of an object, a coordinate system is associated with a reference body, for example, a rectangular Cartesian coordinate system. The coordinates of an object allow you to establish its position in space. The smallest number of independent coordinates that must be set to completely determine the position of a body in space is called the number of degrees of freedom. So, for example, a material point freely moving in space has three degrees of freedom: a point can perform three independent movements along the axes of a Cartesian rectangular coordinate system. An absolutely rigid body has six degrees of freedom: to determine its position in space, three degrees of freedom are needed to describe translational motion along the coordinate axes and three - to describe rotation about the same axes. The coordinate system is equipped with a clock for counting time.
The set of the reference body, the associated coordinate system and the set of clocks synchronized with each other form the reference system.
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HISTORY OF FORMATIONANALYTICAL MECHANICS
Korolev Vladimir Stepanovich
Associate Professor, Cand. phys.-mat. sciences,
Saint Petersburg State University,
RF, 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 carried out over the past years, are considered. An attempt is made to assess 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 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 fundamental scientific knowledge... It is difficult to draw a clear line between theoretical mechanics and some areas 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 theoretical mechanics research is individual material bodies or selected systems of bodies in the process of their movement and interaction with each other and the surrounding world when the relative position in space and time changes. It is generally accepted that the objects around us are almost absolutely rigid bodies. Deformable bodies, liquid and gaseous media are hardly considered or are taken into account indirectly through their influence on the motion of the selected mechanical systems. Theoretical mechanics deals with the general laws of mechanical forms of motion and the construction of mathematical models to describe the possible behavior of mechanical systems. It relies on the laws established in experiments or special physical experiments and taken as axioms or truth, which 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 were questioned, which led to the emergence of new theories and directions for further research. We are not given an ideal stationary space or its metric, as well as processes of uniform motion, along which absolutely precise time intervals can be counted.
As a science, it originated in the IV century BC in the works of ancient Greek scientists as knowledge accumulated along with physics and mathematics, was actively developed by various schools of thought until the first century and emerged as an independent direction. By now, many scientific directions, trends, methods and research opportunities have been formed that create separate hypotheses or theories for description and modeling based on all the accumulated knowledge. Many achievements natural sciences develop or supplement the basic concepts in the problems of mechanics. space, which is determined by the dimension and structure, matter or a substance that fills a space traffic as a form of existence of matter, energy as one of the main characteristics of movement.
The founders of classical mechanics
· Archyt Tarentum (428-365 BC), a representative of the Pythagorean school of philosophy, was one of the first to develop the problems of mechanics.
· Plato(427-347), a student of Socrates, developed and discussed many problems within the framework of the philosophical school, created the theory of the ideal world and the doctrine of the ideal state.
· Aristotle(384-322), disciple of Plato, formed general principles motion, created the theory of motion of celestial spheres, the principle of virtual velocities, considered the source of motion to forces caused by 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, a lever and many different lifting and military vehicles.
Figure 2.
· Hipparchus(180-125), created the theory of the motion of the moon, explained the apparent motion of the sun and planets, introduced geographic coordinates.
· Heron Alexandria (1st century BC), researched lifting mechanisms and devices, invented automatic doors, a steam turbine, was the first to create programmable devices, was engaged in hydrostatics and optics.
· Ptolemy(100-178 AD), mechanic, optician, astronomer, proposed a geocentric system of the world, investigated the apparent motion of the Sun, Moon and planets.
Figure 3.
Science received further development in renaissance in the studies of many European scientists.
· Leonardo da Vinci(1452-1519), universal creative person, did a lot of theoretical and practical mechanics, investigated the mechanics of human movements and the flight of birds.
· Nikolay Copernicus(1473-1543), developed the heliocentric system of the world and published in the work "On the circulation of the celestial spheres."
· Tycho Brahe(1546-1601), left the most accurate observations of the movement of celestial bodies, tried to unite 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 way to create new dynamics, invented the telescope and discovered the moons 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 motion of the planets was made according to the results of processing the tables of observations of the astronomer Tycho Brahe.
Figure 6.
Founders of Analytical Mechanics
Analytical Mechanics was created by the works of representatives of almost closely following one after another three generations.
The publication of Newton's "Mathematical Principles of Natural Philosophy" dates back to 1687. In the year of his death, twenty-year-old Euler publishes his first work on the application of mathematical analysis to mechanics. For many years he lived in St. Petersburg, published hundreds of scientific works and thus contributed to the formation of the Academy of Sciences of Russia. Five years after Euler. Lagrange, 52, publishes Analytical Dynamics. Another 30 years will pass, and the works on the analytical dynamics of three famous contemporaries: Hamilton, Ostrogradsky and Jacobi will be published. The main development of mechanics was in the research of European scientists.
· Christian Huygens(1629-1695), invented the pendulum clock, the law on the 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" he generalized the results of his predecessors, gave definitions of the basic concepts and formulated the basic laws, completed the rationale and received common decision in the two-body problem. The 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 strength of materials.
· Johann Bernoulli(1667-1748), solved the problem of brachistochrone, developed the theory of impacts, investigated the movement of bodies in a resisting medium.
· Leonard 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 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 for drawing up the equations of motion of material systems, studied the motion of planets, established the basic principles of dynamics in the book "A 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 a new form, discovered a new case of solvability of the equations of rotational motion of a rigid body.
The work of these scientists completed the construction of the foundations of modern classical mechanics, laid the foundation for the analysis of the infinitesimal. A mechanics course was developed, which was stated strictly analytical method based on a common mathematical beginning. This course is called "analytical mechanics". The successes of mechanics were so great that they influenced the philosophy of that time, which manifested itself in the creation of "mechanism".
The development of mechanics was also facilitated 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 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 associated 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 stability Solar System, developed the theory of ebb and flow, investigated the motion of the moon and determined the compression of the earth's spheroid, substantiated the hypothesis of the origin of the solar system.
Figure 10.
· Jean Baptiste Fourier(1768-1830), created the theory of partial differential equations, developed the theory of representing functions in the form of trigonometric series, investigated 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 Poinseau(1777-1859), proposed a general solution for the problem of body motion, introduced the concept of an ellipsoid of inertia, investigated many problems of statics and kinematics.
· Simeon Poisson(1781-1840), was engaged in solving problems in gravity 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, the theory of elasticity, celestial mechanics, hydromechanics, investigated 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 the 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.
· Nikolay Vladimirovich Mayevsky(1823-1892), the 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.
· Pafnutiy 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 deformation, motion and balance of elastic bodies, worked on the logical construction of mechanics.
· Sofya Vasilievna Kovalevskaya(1850-1891), studied the theory of the rotational motion of a body around a fixed point, discovered the third classical case of solving the problem, investigated 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 on the basis of a single principle.
Modern development of mechanics
In the twentieth century, they were engaged and continue to be engaged in the solution of many new problems in mechanics. This was especially active after the advent of modern computing facilities. First of all, these are new complex problems of controlled motion, space dynamics, robotics, biomechanics, and quantum mechanics. One can note the work of outstanding scientists, many scientific schools of universities and research teams of Russia.
· Nikolay Egorovich Zhukovsky(1847-1921), the founder of aerodynamics, investigated the motion of a rigid body with a fixed point and the problem of stability of movements, derived a formula for determining the lift of a wing, was engaged in the theory of impact.
Figure 14.
· Alexander Mikhailovich Lyapunov(1857-1918), the main works are devoted to the theory of stability, equilibrium and motion of mechanical systems, the founder of the modern theory of stability.
· Konstantin Eduardovich Tsiolkovsky(1857-1935), the founder of modern astronautics, aerodynamics and rocket dynamics, created the theory of the hovercraft train and the theory of motion of single-stage and multistage rockets.
· Ivan Vsevolodovich Meshchersky(1859-1935), investigated the motion 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 research relates to structural mechanics and shipbuilding, the unsinkability of a ship and its stability, hydromechanics, ballistics, celestial mechanics, the theory of jet propulsion, to 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 a complete solution to the problem of the effect of air flow on a streamlined body.
· Albert Einstein(1879-1955), formulated the special and general theory of relativity, created a new system of 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.
· Nikolay Guryevich Chetaev(1902-1959) investigated the properties of disturbed motions of mechanical systems, issues of motion stability, proved the basic theorems on the instability of equilibrium.
Figure 16.
· Lev Semyonovich Pontryagin(1908-1988) researched the theory of oscillations, calculus of variations, control theory, the creator of the mathematical theory of optimal processes.
Figure 17.
It is possible that even in ancient times and subsequent periods there were centers of knowledge, scientific schools and areas of study of the science and culture of peoples or civilizations: Arab, Chinese or Indian in Asia, the Maya 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. At different times, the languages used for communication were Latin, German, French, English ... Accurate translations of the available texts and general designations in formulas were needed. This made it difficult, but did not stop development.
Modern science is trying to study single complex of all that exists, which manifests itself so diversely in the world around us. By now, many scientific directions, trends, methods and research opportunities have been formed. In the study of classical mechanics, kinematics, statics and dynamics are traditionally singled out as the main sections. Celestial mechanics, as a part of theoretical astronomy, as well as quantum mechanics were formed as an independent branch or science.
The main tasks of dynamics consist in determining the motion of a system of bodies according to known taken into account acting forces or in determining forces according to a known law of motion. Control in problems of dynamics assumes that there is a possibility of changing for the conditions of the realization of the process of motion 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, Controlled, Heavenly, Quantum ...
It's all Mechanics in different ways!
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- Bogomolov A.N. Mechanics mathematicians. Biographical reference book. Kiev: Publishing house. Naukova Dumka, 1983 .-- 639 p.
- Vavilov S.I. Isaac Newton. 4th ed., Add. Moscow: Nauka, 1989 .-- 271 p.
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