Classical mechanics. Formation of schoolchildren's knowledge about the structure of physical theory Basic principles of 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 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

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 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

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 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.

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 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

From Wikipedia, the free encyclopedia

classical mechanics- a kind of mechanics (a section of physics that studies the laws of change in the positions of bodies in space over time and the causes that cause it), based on Newton's laws and Galileo's principle of relativity. Therefore, it is often called Newtonian mechanics».

Classical mechanics is subdivided into:

statics (which considers the equilibrium of bodies)

kinematics (which studies the geometric property of motion without considering its causes)

dynamics (which considers the movement of bodies).

Classical mechanics gives very accurate results if its application is limited to bodies whose speeds are much less than the speed of light, and whose dimensions are much larger than the dimensions of atoms and molecules. Relativistic mechanics is a generalization of classical mechanics for bodies moving at an arbitrary speed, and quantum mechanics for bodies whose dimensions are comparable to atomic ones. Quantum field theory considers quantum relativistic effects.

Nevertheless, classical mechanics retains its value because:

it is much easier to understand and use than other theories

in a wide range, it describes reality quite well.

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 insoluble contradictions. In particular, classical electrodynamics predicts that the speed of light is constant for all observers, which is inconsistent with classical mechanics. At the beginning of the 20th century, this led to the need to create a special theory of relativity. When considered together with thermodynamics, classical mechanics leads to the Gibbs paradox, in which it is impossible to accurately determine the amount of entropy, and the ultraviolet catastrophe, in which a completely black body must radiate an infinite amount of energy. Attempts to solve these problems led to the emergence and development of quantum mechanics.

10 ticket MECHANICAL PICTURE OF THE WORLD. THERMODYNAMICS

Thermodynamics(Greek θέρμη - “heat”, δύναμις - “force”) - a branch of physics that studies the relationships and transformations of heat and other forms of energy. Chemical thermodynamics, which studies physical and chemical transformations associated with the release or absorption of heat, as well as heat engineering, have separated into separate disciplines.

In thermodynamics, one does not deal with individual molecules, but with macroscopic bodies consisting of a huge number of particles. These bodies are called thermodynamic systems. In thermodynamics, thermal phenomena are described by macroscopic quantities - pressure, temperature, volume, ..., which are not applicable to individual molecules and atoms.

In theoretical physics, along with phenomenological thermodynamics, which studies the phenomenology of thermal processes, statistical thermodynamics is distinguished, which was created for the mechanical justification of thermodynamics and was one of the first sections of statistical physics.

Thermodynamics can be applied to a wide range of topics in science and technology, such as engines, phase transitions, chemical reactions, transport phenomena, and even black holes. Thermodynamics is important to other areas of physics and chemistry, chemical engineering, aerospace engineering, mechanical engineering, cell biology, biomedical engineering, materials science, and is useful in other areas such as economics [

11 ticket ELECTRODYNAMICS

Electrodynamics- a section of physics that studies the electromagnetic field in the most general case (that is, time-dependent variable fields are considered) and its interaction with bodies that have an electric charge (electromagnetic interaction). The subject of electrodynamics includes the relationship between electrical and magnetic phenomena, electromagnetic radiation (under different conditions, both free and in various cases of interaction with matter), electric current (generally speaking, alternating) and its interaction with an electromagnetic field (electric current can be considered under this as a set of moving charged particles). Any electrical and magnetic interaction between charged bodies is considered in modern physics as carried out through the electromagnetic field, and, therefore, is also the subject of electrodynamics.

Most often under the term electrodynamics the default is classical electrodynamics, which describes only the continuous properties of an electromagnetic field through a system of Maxwell's equations; to designate the modern quantum theory of the electromagnetic field and its interaction with charged particles, the stable term is usually used quantum electrodynamics.

12 ticket CONCEPT OF SYMMETRY IN NATURAL SCIENCE

Emmy Noether's theorem asserts that each continuous symmetry of a physical system corresponds to a certain conservation law. Thus, the law of conservation of energy corresponds to the homogeneity of time, the law of conservation of momentum to the homogeneity of space, the law of conservation of momentum to the isotropy of space, the law of conservation of electric charge to gauge symmetry, etc.

The theorem is usually formulated for systems with an action functional and expresses the invariance of the Lagrangian with respect to some continuous group of transformations.

The theorem was established in the works of the scientists of the Göttingen school D. Gilbert, F. KleinaiE. Noether. The most common formulation was proved by Emmy Noether in 1918.

Symmetry types found in mathematics and natural sciences:

bilateral symmetry - symmetry with respect to mirror reflection. (Bilateral symmetry)

symmetry of the nth order - symmetry with respect to rotations through an angle of 360 ° / n around any axis. Described by the group Z n .

axial symmetry (radial symmetry, ray symmetry) - symmetry with respect to rotations through an arbitrary angle around an axis. Described by the SO(2) group.

spherical symmetry - symmetry with respect to rotations in three-dimensional space through arbitrary angles. Described by the SO(3) group. Local spherical symmetry of space or medium is also called isotropy.

rotational symmetry is a generalization of the previous two symmetries.

translational symmetry - symmetry with respect to shifts of space in any direction by a certain distance.

Lorentz invariance - symmetry with respect to arbitrary rotations in Minkowski's space-time.

gauge invariance is the independence of the type of equations of gauge theories in quantum field theory (in particular, Yang-Mills theories) under gauge transformations.

supersymmetry - the symmetry of the theory with respect to the replacement of bosons by fermions.

higher symmetry - symmetry in group analysis.

Kainosymmetry is a phenomenon of electronic configuration (the term was introduced by S. A. Shchukarev, who discovered it), which determines the secondary periodicity (discovered by E. V. Biron).

13 ticket service station

Special theory of relativity(HUNDRED; also private theory of relativity) is a theory that describes movement, laws of mechanics, space-time relations at arbitrary speeds of movement that are less than the speed of light in vacuum, including those close to the speed of light. Within the framework of special relativity, Newton's classical mechanics is an approximation of low velocities. The generalization of SRT for gravitational fields is called the general theory of relativity.

The deviations in the course of physical processes from the predictions of classical mechanics described by the special theory of relativity are called relativistic effects, and the rates at which such effects become significant are relativistic speeds.

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General theory of relativity(general relativity; it. allgemeine Relativitätstheorie) is a geometric theory of gravity that develops the special theory of relativity (SRT), published by Albert Einstein in 1915-1916. Within the framework of the general theory of relativity, as in other metric theories, it is postulated that gravitational effects are due to non-force interaction of bodies and fields located in space-time, but to the deformation of space-time itself, which is associated, in particular, with the presence of mass-energy. General relativity differs from other metric theories of gravity by using Einstein's equations to relate the curvature of space-time to the matter present in it.

General relativity is currently the most successful theory of gravity, well supported by observations. The first success of general relativity was to explain the anomalous precession of Mercury's perihelion. Then, in 1919, Arthur Eddington reported observing the deflection of light near the Sun at the time of a total eclipse, which qualitatively and quantitatively confirmed the predictions of general relativity. Since then, many other observations and experiments have confirmed a significant number of the theory's predictions, including gravitational time dilation, gravitational redshift, signal delay in a gravitational field, and, so far only indirectly, gravitational radiation. In addition, numerous observations are interpreted as confirmation of one of the most mysterious and exotic predictions of the general theory of relativity - the existence of black holes.

Despite the stunning success of the general theory of relativity, there is discomfort in the scientific community, connected, firstly, with the fact that it cannot be reformulated as the classical limit of quantum theory, and secondly, with the fact that the theory itself indicates the limits of its applicability, since it predicts the appearance of irremovable physical divergences when considering black holes and, in general, space-time singularities. To solve these problems, a number of alternative theories have been proposed, some of which are also quantum. Current experimental evidence, however, indicates that any type of deviation from general relativity should be very small, if it exists at all.

15 ticket EXPANSION OF THE UNIVERSE.HUBBLE LAW

Universe expansion- a phenomenon consisting in an almost uniform and isotropic expansion of outer space on the scale of the entire Universe. Experimentally, the expansion of the Universe is observed in the form of the implementation of the Hubble law. Science considers the so-called Big Bang to be the beginning of the expansion of the Universe. Theoretically, the phenomenon was predicted and substantiated by A. Friedman at an early stage of development of the general theory of relativity from general philosophical considerations about the homogeneity and isotropy of the Universe.

Hubble law(the law of the general recession of galaxies) is an empirical law that relates the redshift of the galaxy to the distance to them in a linear way:

where z- redshift of the galaxy D- distance to it H 0 is a proportionality factor, called the Hubble constant. With a small value z the approximate equality holds cz=V r, where V r is the speed of the galaxy along the observer's line of sight, c- the speed of light. In this case, the law takes the classical form:

This age is the characteristic time of the expansion of the Universe at the moment and, up to a factor of 2, corresponds to the age of the Universe calculated using the standard Friedman cosmological model.

16 ticket FRIEDMAN MODEL. SINGULARITY

Friedman's universe(Friedman-Lemaitre-Robertson-Walker metric) is one of the cosmological models that satisfy the field equations of the general theory of relativity, the first of the non-stationary models of the Universe. Received by Alexander Fridman in 1922. The Friedman model describes a homogeneous isotropic non-stationary A universe with matter that has a positive, zero, or negative constant curvature. This work of the scientist became the main theoretical development of general relativity after the work of Einstein in 1915-1917.

gravitational singularity- the region of space-time through which it is impossible to continue the geodetic line. Often in it the curvature of the space-time continuum turns to infinity, or the metric has other pathological properties that do not allow physical interpretation (for example, cosmological singularity- the state of the Universe at the initial moment of the Big Bang, characterized by an infinite density and temperature of matter);

17 ticket BIG BANG THEORY. RELICT RADIATION

Relic radiation(or cosmic microwave background radiation from English cosmic microwave background radiation) - cosmic electromagnetic radiation with a high degree of isotropy and with a spectrum characteristic of an absolutely black body with a temperature of 2.725 K.

The existence of the CMB was predicted theoretically within the framework of the Big Bang theory. Although many aspects of the original Big Bang theory have now been revised, the fundamentals that made it possible to predict the temperature of the CMB have not changed. It is believed that the relict radiation has been preserved from the initial stages of the existence of the Universe and evenly fills it. Its existence was experimentally confirmed in 1965. Along with the cosmological redshift, the cosmic microwave background radiation is considered as one of the main confirmations of the Big Bang theory.

Big Bang(English) big bang) is a cosmological model describing the early development of the Universe, namely, the beginning of the expansion of the Universe, before which the Universe was in a singular state.

Usually now automatically combine the theory of the Big Bang and the model of the hot Universe, but these concepts are independent and historically there was also a concept of a cold initial Universe near the Big Bang. It is the combination of the Big Bang theory with the theory of the hot Universe, supported by the existence of cosmic microwave background radiation, that is considered further.

18 ticket SPACE VACUUM

Vacuum(rel. vacuum- void) - space free from matter. In engineering and applied physics, vacuum is understood as a medium containing gas at pressures well below atmospheric pressure. Vacuum is characterized by the ratio between the mean free path of gas molecules λ and the characteristic size of the medium d. Under d the distance between the walls of the vacuum chamber, the diameter of the vacuum pipeline, etc. can be taken. Depending on the value of the ratio λ / d distinguish between low (), medium () and high () vacuum.

It is necessary to distinguish between concepts physical vacuum and technical vacuum.

19 ticket QUANTUM MECHANICS

Quantum mechanics- a section of theoretical physics that describes physical phenomena in which the action is comparable in magnitude to Planck's constant. The predictions of quantum mechanics can differ significantly from the predictions of classical mechanics. Because Planck's constant is extremely small compared to the action of everyday objects, quantum effects mostly only show up on microscopic scales. If the physical action of the system is much greater than Planck's constant, quantum mechanics goes organically into classical mechanics. In turn, quantum mechanics is a non-relativistic approximation (that is, an approximation of small energies compared to the rest energy of the massive particles of the system) of quantum field theory.

Classical mechanics, which well describes systems of macroscopic scales, is not capable of describing phenomena at the level of atoms, molecules, electrons and photons. Quantum mechanics adequately describes the basic properties and behavior of atoms, ions, molecules, condensed matter, and other systems with an electron-nuclear structure. Quantum mechanics is also capable of describing the behavior of electrons, photons, and other elementary particles, but a more accurate relativistically invariant description of the transformations of elementary particles is built within the framework of quantum field theory. Experiments confirm the results obtained with the help of quantum mechanics.

The basic concepts of quantum kinematics are the concepts of an observable and a state.

The basic equations of quantum dynamics are the Schrödinger equation, the von Neumann equation, the Lindblad equation, the Heisenberg equation, and the Pauli equation.

The equations of quantum mechanics are closely related to many branches of mathematics, including: operator theory, probability theory, functional analysis, operator algebras, group theory.

Completely black body- physical idealization used in thermodynamics, a body that absorbs all electromagnetic radiation incident on it in all ranges and reflects nothing. Despite the name, a black body itself can emit electromagnetic radiation of any frequency and visually have a color. The radiation spectrum of a black body is determined only by its temperature.

The importance of a black body in the question of the spectrum of thermal radiation of any (gray and colored) bodies in general, in addition to being the simplest non-trivial case, is also in the fact that the question of the spectrum of equilibrium thermal radiation of bodies of any color and reflection coefficient is reduced by the methods of classical thermodynamics to the question of radiation from an absolutely black body (and historically this was already done by the end of the 19th century, when the problem of radiation from an absolutely black body came to the fore).

The blackest real substances, for example, soot, absorb up to 99% of the incident radiation (that is, they have an albedo equal to 0.01) in the visible wavelength range, but they absorb infrared radiation much worse. Among the bodies of the Solar System, the Sun has the properties of an absolutely black body to the greatest extent.

The term was introduced by Gustav Kirchhoff in 1862.

20 ticket PRINCIPLES OF QUANTUM MECHANICS

All the problems of modern physics can be divided into two groups: the problems of classical physics and the problems of quantum physics. When studying the properties of ordinary macroscopic bodies, one almost never encounters quantum problems, because quantum properties become tangible only in the microcosm. Therefore, the physics of the 19th century, which studied only macroscopic bodies, was completely unaware of quantum processes. This is classical physics. It is typical for classical physics that it does not take into account the atomistic structure of matter. Now, however, the development of experimental technology has pushed the boundaries of our acquaintance with nature so widely that we now know, and, moreover, in great detail, the strictness of individual atoms and molecules. Modern physics studies the atomic structure of matter and, therefore, the principles of the old classical physics of the 19th century. had to change in accordance with the new facts, and change radically. This change in principles is the transition to quantum physics.

21 tickets CORPUSCULAR-WAVE DUALISM

Corpuscular-wave dualism- the principle that any object can exhibit both wave and particle properties. It was introduced during the development of quantum mechanics to interpret the phenomena observed in the microcosm from the point of view of classical concepts. A further development of the principle of wave-particle duality was the concept of quantized fields in quantum field theory.

As a classic example, light can be interpreted as a stream of corpuscles (photons), which in many physical effects exhibit the properties of electromagnetic waves. Light exhibits the properties of a wave in the phenomena of diffraction and interference at scales comparable to the wavelength of light. For example, even single photons passing through the double slit create an interference pattern on the screen, determined by Maxwell's equations.

Nevertheless, the experiment shows that a photon is not a short pulse of electromagnetic radiation, for example, it cannot be divided into several beams by optical beam splitters, which was clearly shown by an experiment conducted by French physicists Grangier, Roger and Aspe in 1986. The corpuscular properties of light are manifested in the photoelectric effect and in the Compton effect. A photon also behaves like a particle that is emitted or absorbed entirely by objects whose dimensions are much smaller than its wavelength (for example, atomic nuclei), or can generally be considered pointlike (for example, an electron).

At present, the concept of wave-particle duality is only of historical interest, since it served only as an interpretation, a way to describe the behavior of quantum objects, choosing analogies from classical physics for it. In fact, quantum objects are neither classical waves nor classical particles, acquiring the properties of the former or the latter only in some approximation. Methodologically more correct is the formulation of quantum theory in terms of path integrals (propagator), free from the use of classical concepts.

22 ticket THE CONCEPT OF THE STRUCTURE OF THE ATOM. MODELS OF THE ATOM

Thomson model of the atom(model "Pudding with raisins", eng. plum pudding model).J. J. Thomson proposed to consider the atom as some positively charged body with electrons enclosed inside it. It was finally refuted by Rutherford after his famous experiment on the scattering of alpha particles.

Nagaoka's early planetary model of the atom. In 1904, the Japanese physicist Hantaro Nagaoka proposed a model of the atom, built by analogy with the planet Saturn. In this model, electrons, united in rings, revolved around a small positive nucleus in orbits. The model turned out to be wrong.

Bohr-Rutherford planetary model of the atom. In 1911, Ernest Rutherford, having done a series of experiments, came to the conclusion that the atom is a kind of planetary system in which electrons move in orbits around a heavy positively charged nucleus located in the center of the atom ("Rutherford's model of the atom"). However, such a description of the atom came into conflict with classical electrodynamics. The fact is that, according to classical electrodynamics, an electron, when moving with centripetal acceleration, must radiate electromagnetic waves, and, consequently, lose energy. Calculations showed that the time it takes for an electron in such an atom to fall onto the nucleus is absolutely negligible. To explain the stability of atoms, Niels Bohr had to introduce postulates that boiled down to the fact that an electron in an atom, being in some special energy states, does not radiate energy (“the Bohr-Rutherford model of the atom”). Bohr's postulates showed that classical mechanics is not applicable to describe the atom. Further study of the radiation of the atom led to the creation of quantum mechanics, which made it possible to explain the overwhelming majority of the observed facts.

Atom(other Greek ἄτομος- indivisible) - the smallest chemically indivisible part of a chemical element, which is the carrier of its properties. An atom consists of an atomic nucleus and electrons. The nucleus of an atom is made up of positively charged protons and uncharged neutrons. If the number of protons in the nucleus coincides with the number of electrons, then the atom as a whole is electrically neutral. Otherwise, it has some positive or negative charge and is called an ion. Atoms are classified according to the number of protons and neutrons in the nucleus: the number of protons determines whether an atom belongs to a certain chemical element, and the number of neutrons determines the isotope of this element.

Atoms of different types in different quantities, connected by interatomic bonds, form molecules.

23 ticket FUNDAMENTAL INTERACTIONS

Fundamental interactions- qualitatively different types of interaction of elementary particles of bodies composed of them.

Today, the existence of four fundamental interactions is reliably known:

gravitational

electromagnetic

strong

weak

At the same time, electromagnetic and weak interactions are manifestations of a single electroweak interaction.

Searches are underway for other types of fundamental interactions, both in the phenomena of the microworld and on a cosmic scale, but so far no other type of fundamental interaction has been discovered.

In physics, mechanical energy is divided into two types - potential and kinetic energy. The reason for the change in the movement of bodies (changes in kinetic energy) is the force (potential energy) (see Newton's second law). Exploring the world around us, we can notice a wide variety of forces: gravity, thread tension, spring compression force, collision force of bodies , friction force, air resistance force, explosion force, etc. However, when the atomic structure of matter was clarified, it became clear that all the variety of these forces is the result of the interaction of atoms with each other. Since the main type of interatomic interaction is electromagnetic, it turned out that most of these forces are just various manifestations of electromagnetic interaction. One of the exceptions is, for example, the force of gravity, which is caused by the gravitational interaction between bodies that have mass.

24 ticket ELEMENTARY PARTICLES AND THEIR PROPERTIES

Elementary particle- a collective term referring to micro-objects on a sub-nuclear scale that cannot be broken down into their component parts.

It should be borne in mind that some elementary particles (electron, photon, quarks, etc.) are currently considered structureless and are considered as primary fundamental particles. Other elementary particles (so-called constituent particles-proton, neutron, etc.) have a complex internal structure, but, nevertheless, according to modern concepts, it is impossible to separate them into parts (see Confinement).

The structure and behavior of elementary particles is studied by elementary particle physics.

Main article:Quarks

Quarks and antiquarks have never been found in a free state - this is explained by the phenomenon of confinement. Based on the symmetry between leptons and quarks, which is manifested in electromagnetic interaction, hypotheses are put forward that these particles consist of more fundamental particles - preons.

25 ticket CONCEPT OF BIFURCATION. BIFURCATION POINT

Bifurcation is the acquisition of a new quality in the movements of a dynamic system with a small change in its parameters.

The central concept of the bifurcation theory is the concept of a (non)rough system (see below). Any dynamical system is taken and such a (multi)parametric family of dynamical systems is considered that the original system is obtained as a special case - for any one value of the parameter (parameters). If the qualitative picture of the partition of the phase space into trajectories is preserved for the value of the parameters sufficiently close to the given one, then such a system is called rough. Otherwise, if such a neighborhood does not exist, then the system is called rough.

Thus, regions of rough systems appear in the parameter space, which are separated by surfaces consisting of non-rough systems. The theory of bifurcations studies the dependence of a qualitative picture when a parameter changes continuously along a certain curve. The scheme by which the qualitative picture changes is called bifurcation diagram.

The main methods of bifurcation theory are the methods of perturbation theory. In particular, it applies small parameter method(Pontryagin).

bifurcation point- change of the established operating mode of the system. A term from non-equilibrium thermodynamics and synergetics.

bifurcation point- the critical state of the system, in which the system becomes unstable relative to fluctuations and uncertainty arises: will the state of the system become chaotic or will it move to a new, more differentiated and high level of order. A term from the theory of self-organization.

26 ticket SYNERGETICS - THE SCIENCE OF OPEN SELF-ORGANIZING SYSTEMS

Synergetics(Other Greek συν - prefix with the meaning of compatibility and ἔργον - "activity") - an interdisciplinary area of scientific research, the task of which is to study natural phenomena and processes based on the principles of self-organization of systems (consisting of subsystems). "... A science that studies the processes of self-organization and the emergence, maintenance, stability and decay of structures of the most diverse nature ...".

Synergetics was originally declared as an interdisciplinary approach, since the principles governing the processes of self-organization seem to be the same (regardless of the nature of the systems), and a common mathematical apparatus should be suitable for their description.

From an ideological point of view, synergetics is sometimes positioned as “global evolutionism” or “universal theory of evolution”, which provides a single basis for describing the mechanisms for the emergence of any innovations, just as cybernetics was once defined as “universal control theory”, equally suitable for describing any regulation and optimization operations. : in nature, in technology, in society, etc., etc. However, time has shown that the general cybernetic approach has far from justified all the hopes placed on it. Similarly, the broad interpretation of the applicability of synergetic methods is also criticized.

The basic concept of synergetics is the definition of structure as states, arising as a result of the multivariant and ambiguous behavior of such multi-element structures or multi-factor media that do not degrade to the thermodynamic averaging standard for closed systems, but develop due to openness, energy inflow from the outside, nonlinearity of internal processes, the appearance of special regimes with sharpening and the presence of more than one stable state. In the indicated systems, neither the second law of thermodynamics nor Prigogine's theorem on the minimum rate of entropy production is applicable, which can lead to the formation of new structures and systems, including those more complex than the original ones.

This phenomenon is interpreted by synergetics as a general mechanism of the direction of evolution observed everywhere in nature: from elementary and primitive to complex and more perfect.

In some cases, the formation of new structures has a regular, wave character, and then they are called autowave processes (by analogy with self-oscillations).

27 ticket THE CONCEPT OF LIFE. THE PROBLEM OF THE ORIGIN OF LIFE

A life- the active form of the existence of a substance, in a sense, the highest in comparison with its physical and chemical forms of existence; a set of physical and chemical processes occurring in the cell, allowing the exchange of matter and its division. The main attribute of living matter is the genetic information used for replication. More or less accurately define the concept of "life" can only enumerate the qualities that distinguish it from non-life. Life does not exist outside the cell, viruses exhibit the properties of living matter only after the transfer of genetic material into the cell [ source not specified 268 days] . Adapting to the environment, a living cell forms the whole variety of living organisms.

Also, the word "life" is understood as the period of existence of a single organism from the moment of occurrence to its death (ontogeny).

In 1860, the French chemist Louis Pasteur took up the problem of the origin of life. Through his experiments, he proved that bacteria are ubiquitous, and that non-living materials can easily be contaminated by living things if they are not properly sterilized. The scientist boiled various media in water in which microorganisms could form. Additional boiling killed the microorganisms and their spores. Pasteur attached a sealed flask with a free end to the S-shaped tube. Spores of microorganisms settled on a curved tube and could not penetrate into the nutrient medium. A well-boiled nutrient medium remained sterile; no life was found in it, despite the fact that air access was provided.

As a result of a series of experiments, Pasteur proved the validity of the theory of biogenesis and finally refuted the theory of spontaneous generation.

28 ticket THE CONCEPT OF THE ORIGIN OF OPARIN'S LIFE

Sir ISAAC NEWTON (January 4, 1643 - March 31, 1727) - an outstanding English scientist who laid the foundations of modern natural science, the creator of classical physics, a member of the Royal Society of London and its president (since 1703). Born in Woolsthorpe. Graduated from Cambridge University in 1665. In March-June 1666, Newton visited Cambridge. However, in the summer, a new wave of plague forced him to leave home again. Finally, in early 1667, the epidemic subsided, and in April Newton returned to Cambridge. On October 1, he was elected a Fellow of Trinity College, and in 1668 became a master. He was given a spacious private room to live in, a salary of £2 a year, and a group of students with whom he conscientiously studied standard subjects for several hours a week. However, neither then nor later did Newton become famous as a teacher, his lectures were poorly attended. one

Having consolidated his position, Newton traveled to London, where shortly before, in 1660, the Royal Society of London was established - an authoritative organization of prominent scientists, one of the first Academies of Sciences. The printed organ of the Royal Society was the journal Philosophical Transactions.

In 1669, mathematical works began to appear in Europe using expansions into infinite series. Although the depth of these discoveries did not go to any comparison with Newton's, Barrow insisted that his student fix his priority in this matter. 2 ______________________________

1. https://ru.wikipedia.org/

2. Akroyd P. “Isaac Newton. Biography". - M.: Hummingbird, Azbuka-Atticus, 2011

Newton wrote a brief but fairly complete summary of this part of his discoveries, which he called "Analysis using equations with an infinite number of terms." Barrow sent this treatise to London. Newton asked Barrow not to reveal the name of the author of the work (but he still let it slip). "Analysis" spread among specialists and gained some notoriety in England and beyond.

In the same year, Barrow accepted the invitation of the king to become a court chaplain and left teaching. On October 29, 1669, the 26-year-old Newton was elected as his successor, professor of mathematics and optics at Trinity College, with a high salary of £100 a year. Barrow left Newton an extensive alchemical laboratory; during this period, Newton became seriously interested in alchemy, conducted a lot of chemical experiments. Newton formulated the basic laws of classical mechanics, discovered the law of universal gravitation, the dispersion of light, developed the corpuscular theory of light, and developed differential and integral calculus. Summarizing the results of the research of his predecessors in the field of mechanics and his own, Newton created a huge work "Mathematical Principles of Natural Philosophy" ("Beginnings"), published in 1687. "Beginnings" contained the basic concepts of classical mechanics, in particular the concepts: mass, momentum, force, acceleration, centripetal force and three laws of motion. In the same work, his law of universal gravitation is given, on the basis of which Newton explained the motion of celestial bodies and created the theory of gravitation. 1 The discovery of this law finally confirmed the victory of the teachings of Copernicus. He showed that Kepler's three laws follow from the law of universal gravitation; explained the features of the movement of the moon, the phenomenon of the procession; developed the theory of the figure of the Earth, noting that it should be compressed at the poles, _____________________________

1. Akroyd P. “Isaac Newton. Biography". - M.: Hummingbird, Azbuka-Atticus, 2011

the theory of ebbs and flows; considered the problem of creating an artificial satellite of the Earth, etc. Newton developed the law of resistance and the basic law of internal friction in liquids and gases, gave a formula for the speed of wave propagation.

Collection output:

HISTORY OF FORMATIONANALYTICAL MECHANICS

Korolev Vladimir Stepanovich

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

St. 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 the relative position in space and time changes. 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 the general laws of 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 immobile space or its metric, as well as processes of 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 up to the first century and stood out as an independent direction. To date, many scientific directions, trends, methods and research opportunities have been formed 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, motion as a form of existence of matter, energy as one of the main characteristics of the movement.

The founders of classical mechanics

· archite Tarentsky (428-365 BC), a representative of the Pythagorean school of philosophy, was one of the first to develop problems in 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), a student of Plato, formed the general principles of motion, created the theory of motion of the celestial spheres, the principle of virtual speeds, considered the source of motion 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, was engaged in 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), developed the heliocentric system of the world and published it in On the Revolution of the Celestial Spheres.

· Tycho Brahe(1546-1601), left the most accurate observations of the movement of 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 the 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 paper on the application of mathematical analysis to 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 the basic concepts and formulated the basic laws, carried out the justification and received a general solution to the problem of two bodies. 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 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.

· 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 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 the general rules for compiling the equations of motion of 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 a new form, discovered a new case of solvability of the equations of rotational motion of a rigid 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 stability of 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 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 researches 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 related to nonholonomic mechanics, hydrodynamics, the theory of aviation and aerodynamics, gave a complete solution to the problem of the effect of an 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.

· 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 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 Mayan people in America, where achievements appeared, but European philosophical and scientific schools developed in a special way, without always paying attention to the discoveries or theories of other researchers. At different times, Latin, German, French, English were used for communication... Accurate translations of available texts and common notation in formulas were needed. This made it difficult, but did not stop development.

Modern science tries to study single complex of everything that exists, which manifests itself in such a diverse way 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 motion process 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 the 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.

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.

Other equally important concepts of 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 these physical quantities, the fundamental laws of conservation of classical mechanics operate.

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 the planets of the solar system carry out uniform movement in elliptical orbits, with 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 to the public at once five universal laws of the physical motion of substances.

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 in the physics of motion that 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 of free falling bodies. 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 its initial locations and orientation relative to the material body of reference do not affect the course of any processes within a closed system.
The principle of relativity: the flow of physical processes in an isolated system is not affected by its rectilinear motion relative to 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 major variational proposition in classical mechanics is the principle of the straightest path, where any free system is in a state of calm or uniform motion along specific lines compared to 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, the correct choice of 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.