# mechanics

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## mechanics,

branch of physics**physics,**

branch of science traditionally defined as the study of matter, energy, and the relation between them; it was called natural philosophy until the late 19th cent. and is still known by this name at a few universities.

**.....**Click the link for more information. concerned with motion

**motion,**

the change of position of one body with respect to another. The rate of change is the speed of the body. If the direction of motion is also given, then the velocity of the body is determined; velocity is a vector quantity, having both magnitude and direction, while speed

**.....**Click the link for more information. and the forces

**force,**

commonly, a "push" or "pull," more properly defined in physics as a quantity that changes the motion, size, or shape of a body. Force is a vector quantity, having both magnitude and direction.

**.....**Click the link for more information. that tend to cause it; it includes study of the mechanical properties of matter

**matter,**

anything that has mass and occupies space. Matter is sometimes called koinomatter (Gr.

*koinos*=common) to distinguish it from antimatter, or matter composed of antiparticles.

**.....**Click the link for more information. , such as density

**density,**

ratio of the mass of a substance to its volume, expressed, for example, in units of grams per cubic centimeter or pounds per cubic foot. The density of a pure substance varies little from sample to sample and is often considered a characteristic property of the

**.....**Click the link for more information. , elasticity

**elasticity,**

the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed. All solids are elastic for small enough deformations or strains, but if the stress exceeds a certain amount known as the elastic

**.....**Click the link for more information. , and viscosity

**viscosity,**

resistance of a fluid to flow. This resistance acts against the motion of any solid object through the fluid and also against motion of the fluid itself past stationary obstacles.

**.....**Click the link for more information. . Mechanics may be roughly divided into statics

**statics,**

branch of mechanics concerned with the maintenance of equilibrium in bodies by the interaction of forces upon them (see force). It incorporates the study of the center of gravity (see center of mass) and the moment of inertia.

**.....**Click the link for more information. and dynamics

**dynamics,**

branch of mechanics that deals with the motion of objects; it may be further divided into kinematics, the study of motion without regard to the forces producing it, and kinetics, the study of the forces that produce or change motion.

**.....**Click the link for more information. ; statics deals with bodies at rest and is concerned with such topics as buoyancy

**buoyancy**

, upward force exerted by a fluid on any body immersed in it. Buoyant force can be explained in terms of Archimedes' principle.

**.....**Click the link for more information. , equilibrium

**equilibrium,**

state of balance. When a body or a system is in equilibrium, there is no net tendency to change. In mechanics, equilibrium has to do with the forces acting on a body.

**.....**Click the link for more information. , and the principles of simple machines

**machine,**

arrangement of moving and stationary mechanical parts used to perform some useful work or to provide transportation. From a historical perspective, many of the first machines were the result of human efforts to improve war-making capabilities; the term

*engineer*

**.....**Click the link for more information. , while dynamics deals with bodies in motion and is sometimes further divided into kinematics (description of motion without regard to its cause) and kinetics (explanation of changes in motion as a result of forces). A recent subdiscipline of dynamics is nonlinear dynamics

**nonlinear dynamics,**

study of systems governed by equations in which a small change in one variable can induce a large systematic change; the discipline is more popularly known as chaos (see chaos theory).

**.....**Click the link for more information. , the study of systems in which small changes in a variable may have large effects. The science of mechanics may also be broken down, according to the state of matter being studied, into solid mechanics and fluid mechanics. The latter, the mechanics of liquids and gases, includes hydrostatics, hydrodynamics, pneumatics, aerodynamics, and other fields.

### Early Mechanics

Mechanics was studied by a number of ancient Greek scientists, most notably Aristotle, whose ideas dominated the subject until the late Middle Ages, and Archimedes, who made several contributions and whose approach was quite modern compared to other ancient scientists. In the Aristotelian view, ordinary motion required a material medium; a body was kept in motion by the medium rushing in behind it in order to prevent a vacuum, which, according to this philosophy, could not occur in nature. Celestial bodies, on the other hand, were kept in motion through the vacuum of space by various agents that, in the Christianized version of Aquinas and others, acquired an angelic character.

This explanation was rejected in the 14th cent. by several philosophers, who revived the impetus theory proposed by John Philoponos in the 6th cent. A.D.; according to this theory a body acquired a quantity called impetus when it was set in motion, and it eventually came to rest as the impetus died out. The impetus school flourished in Paris and elsewhere during the 14th and 15th cent. and included William of Occam (Ockham), Jean Buridan, Albert of Saxony, Nicolas Oresme, and Nicolas of Cusa, although it was never successful in replacing the dominant Aristotelian mechanics.

### Modern Mechanics

Modern mechanics dates from the work of Galileo, Simon Stevin, and others in the late 16th and early 17th cent. By means of experiment and mathematical analysis, Galileo made a number of important studies, particularly of falling bodies and projectiles. He enunciated the principle of inertia**inertia**

, in physics, the resistance of a body to any alteration in its state of motion, i.e., the resistance of a body at rest to being set in motion or of a body in motion to any change of speed or change in direction of motion. Inertia is a property common to all matter.**.....** Click the link for more information. and used it to explain not only the mechanics of bodies on the earth but also that of celestial bodies (which, however, he believed moved in uniform circular orbits). The philosopher René Descartes advocated the application of the mathematical-mechanical approach to all fields and founded the mechanistic philosophy that was so important in science for the next two centuries or more.

The first system of modern mechanics to explain successfully all mechanical phenomena, both terrestrial and celestial, was that of Isaac Newton, who in his *Principia (Mathematical Principles of Natural Philosophy,* 1687) derived three laws of motion and showed how the principle of universal gravitation**gravitation,**

the attractive force existing between any two particles of matter. **The Law of Universal Gravitation**

Since the gravitational force is experienced by all matter in the universe, from the largest galaxies down to the smallest particles, it is often called**.....** Click the link for more information. can be used to explain both the behavior of falling bodies on the earth and the orbits of the planets in the heavens. Newton's system of mechanics was developed extensively over the next two centuries by many scientists, including Johann and Daniel Bernoulli, Leonhard Euler, J. le Rond d'Alembert, J. L. Lagrange, P. S. Laplace, S. D. Poisson, and W. R. Hamilton. It found application to the explanation of the behavior of gases and thermodynamics in the statistical mechanics of J. C. Maxwell, Ludwig Boltzmann, and J. W. Gibbs.

In 1905, Albert Einstein showed that Newton's mechanics was an approximation, valid for cases involving speeds much less than the speed of light; for very great speeds the relativistic mechanics of his theory of relativity**relativity,**

physical theory, introduced by Albert Einstein, that discards the concept of absolute motion and instead treats only relative motion between two systems or frames of reference.**.....** Click the link for more information. was required. Einstein showed further in his general theory of relativity (1916) that gravitation could be explained in terms of the effect of a massive body on the framework of space and time around it, this effect applying not only to the motions of other bodies possessing mass but also to light. In the quantum mechanics developed during the 1920s as part of the quantum theory**quantum theory,**

modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics.**.....** Click the link for more information. , the motions of very tiny particles, such as the electrons in an atom, were explained using the fact that both matter and energy have a dual nature—sometimes behaving like particles and other times behaving like waves. Two different but mathematically equivalent forms of quantum mechanics were elaborated, the wave mechanics of Erwin Schrödinger and the matrix mechanics of Werner Heisenberg.

### Bibliography

See I. B. Cohen, *Introduction to Newton's Principia* (1971); E. Mach, *Science of Mechanics* (6th ed. 1973); J. Gleick, *Chaos* (1987).

## Mechanics

In its original sense, mechanics refers to the study of the behavior of systems under the action of forces. Mechanics is subdivided according to the types of systems and phenomena involved.

An important distinction is based on the size of the system. Those systems that are large enough can be adequately described by the newtonian laws of classical mechanics; in this category, for example, are celestial mechanics and fluid mechanics. On the other hand, the behavior of microscopic systems such as molecules, atoms, and nuclei can be interpreted only by the concepts and mathematical methods of quantum mechanics.

Mechanics may also be classified as nonrelativistic or relativistic mechanics, the latter applying to systems with material velocities comparable to the velocity of light. This distinction pertains to both classical and quantum mechanics.

Finally, statistical mechanics uses the methods of statistics for both classical and quantum systems containing very large numbers of similar subsystems to obtain their large-scale properties. *See* Classical field theory, Classical mechanics, Dynamics, Fluid mechanics, Quantum mechanics, Statics, Statistical mechanics

## Mechanics

the science of the mechanical motion of material bodies and the interactions that take place between the bodies during the motion. The term “mechanical motion” is understood to mean a time variation in the relative position in space of bodies or the particles making up the bodies. Various motions are studied by the methods of mechanics. Examples of such motions in nature are the motions of celestial bodies, the vibrations of the earth’s crust, air currents, sea currents, and the thermal motion of molecules. Examples in technology are the motions of various aircraft and means of transport; the motions of the parts of all types of engines, machinery, and mechanisms; the deformations of the elements of various structures and buildings; and the motions of liquids and gases.

Mechanics considers those actions of bodies on each other that result in changes in the mechanical motion of these bodies. Some examples of these interactions are the attractions of bodies according to the law of universal gravitation, the relative pressures of contiguous bodies, and the effects of the particles of a liquid or gas on each other and on bodies moving within the liquid or gas. The term “mechanics” is usually understood to mean classical mechanics, which is based on Newton’s laws of motion and which deals with the study of the motion of all material bodies (except elementary particles) for velocities that are small in comparison with the velocity of light. The motion of bodies with velocities of the order of the velocity of light is considered in the theory of relativity, and subatomic phenomena and the motion of elementary particles are studied in quantum mechanics.

In studying the motion of material bodies, a number of abstract concepts that reflect various properties of real bodies are introduced in mechanics, including (1) a material point, (2) a perfectly rigid body, and (3) a continuous, variable medium. A material point is an object of negligibly small dimensions that has a mass; this concept is applicable if in the motion under study the dimensions of the body may be disregarded in comparison with the distances traversed by the points making up the body. A perfectly rigid body is a body for which the distance between any two points always remains constant; this concept is applicable when deformations of the body may be disregarded. The concept of a continuous, variable medium is applicable when in the study of the motion of a variable medium (a deformable body, a liquid, or a gas) the molecular structure of the medium can be disregarded.

Abstractions that under given conditions reflect the most significant properties of the corresponding real bodies are used in the study of continuous media. These abstractions include an ideally elastic body, a plastic body, an ideal fluid, a viscous fluid, and an ideal gas. Accordingly, mechanics is divided into (1) the mechanics of a material point, (2) the mechanics of a system of material points, (3) the mechanics of a perfectly rigid body, and (4) the mechanics of a continuous medium (continuum mechanics). The last, in turn, is subdivided into such areas as elasticity theory, plasticity theory, hydromechanics, aeromechanics, and gas dynamics. In each of these branches, we distinguish the following areas in accordance with the nature of the problems to be solved: statics—the study of the equilibrium of bodies under the action of forces; kinematics—the study of the geometrical properties of the motion of bodies; and dynamics—the study of the motion of bodies under the action of forces. Two fundamental problems are considered in dynamics: (1) the determination of the forces under the action of which a given motion of a body takes place and (2) the determination of the motion of a body when the forces acting on the body are known.

Various mathematical methods, many of which owe their very origin and development to mechanics, are widely used in solving the problems of mechanics. The study of the fundamental laws and principles that are obeyed by the mechanical motion of bodies and the study of the general theorems and equations that result from these laws and principles constitute the subject of general, or theoretical, mechanics. Branches of mechanics that are of great independent importance include the theory of vibrations, the theory of the stability of equilibrium and the stability of motion, gyroscope theory, the mechanics of bodies of variable mass, automatic control theory, and impact theory.

Experimental studies conducted by means of various mechanical, optical, electrical, and other physical methods and instruments occupy an important place in mechanics, particularly in continuum mechanics.

Mechanics is closely related to many other branches of physics. Upon appropriate generalization, a number of concepts and methods of mechanics find application in such fields as optics, statistical physics, quantum mechanics, electrodynamics, and the theory of relativity. Moreover, the methods and equations of theoretical mechanics and of such fields as thermodynamics, molecular physics, and the theory of electricity are used together in solving a number of problems in such fields as gas dynamics, explosion theory, heat exchange in moving liquids and gases, the aerodynamics of rarefied gases, and magnetohydrodynamics. Mechanics is of great importance for many branches of astronomy, particularly celestial mechanics.

Some subdivisions of mechanics are directly connected with technology. These subdivisions include numerous general technical and special disciplines, such as hydraulics, the strength of materials, the kinematics of mechanisms, the dynamics of machines and mechanisms, the theory of gyroscopic devices, exterior ballistics, rocket dynamics, the theory of the regulation and control of the motion of various objects, structural mechanics, and the theory of motion of various types of land, water, and air transportation. All these disciplines make use of the equations and methods of theoretical mechanics. Thus, mechanics is one of the scientific foundations of many fields of modern technology.

** Fundamental concepts and methods of mechanics.** The main kinematic measures of motion in mechanics are for a point, its velocity and acceleration, and for a rigid body, the velocity and acceleration of translational motion and the angular velocity and angular acceleration of the rotational motion of the body. The kinematic state of a deformable solid is characterized by the relative elongations between, and displacements of, the particles of the solid; the set of these quantities defines the deformation tensor. For liquids and gases the kinematic state is characterized by the tensor of the deformation rates. Moreover, the concept of curl, which characterizes the rotation of a particle, is used in the study of the velocity field of a moving fluid.

The main measure of the mechanical interaction of material bodies in mechanics is the force. At the same time, the concept of the moment of a force about a point and about an axis is used extensively. In continuum mechanics, forces are defined by their surface or volume distributions, that is, by the ratio of the magnitude of the force to the surface area (for surface forces) or to the volume (for mass forces) on which the corresponding force acts. The internal stresses that arise in a continuous medium are characterized, at each point of the medium, by tangential and normal stresses, the set of which represents a quantity called the stress tensor. The arithmetic mean of the three normal stresses taken with opposite sign defines a quantity called the pressure at a given point of the medium.

In addition to effective forces, the motion of a body depends on the inertia of the body, that is, on how quickly the body changes its motion under applied forces. The measure of the inertia of a material point is a quantity called the mass of the point. The inertia of a material body depends not only on its total mass but also on the mass distribution in the body, which is characterized by the position of the center of mass and by quantities called the moments of inertia about axes and the products of inertia; the set of these quantities defines the inertia tensor. The inertia of a liquid or a gas is characterized by the density of the liquid or the gas.

Mechanics is based on Newton’s laws of motion. The first two laws are valid for an inertial frame of reference. The second law gives the basic equations for solving problems of particle dynamics and, together with the third law, for solving problems of the dynamics of a system of material points. In continuum mechanics, in addition to Newton’s laws, use is made of laws that reflect the properties of a given medium and that establish for the medium the relation between the stress tensor and the deformation tensor or deformation-rate tensor. Examples of such laws are Hooke’s law for a linearly elastic body and Newton’s law for a viscous fluid.

Concepts of dynamic measures of motion, such as momentum, angular momentum (or moment of momentum), and kinetic energy, and concepts of measures of the action of a force, such as impulse and work, are of great importance for solving the problems of mechanics. The relationship between the measures of motion and the measures of the action of a force is given by theorems on variation in momentum, in angular momentum, and in kinetic energy, which are called the general theorems of dynamics. These theorems, together with the laws of conservation of momentum, angular momentum, and mechanical energy that follow from these theorems, express the properties of the motion of any system of material points and a continuous medium.

The variational principles of mechanics, in particular, the virtual work principle, the principle of least action, and the d’Alembert principle, provide effective methods for studying the equilibrium and motion of a constrained system of material points, that is, a system for which predetermined constraints, called mechanical constraints, are imposed on the motion of the system. In solving problems in mechanics, we make extensive use of the differential equations of motion of a material point, a rigid body, and a system of material points that follow from the laws and principles of mechanics; in particular, we use the Lagrange equations, Hamilton’s canonical equations of motion, and the Hamilton-Jacobi equation. In continuum mechanics we make use of the corresponding equations of the equilibrium or motion of the continuum, the continuity equation for the continuum, and the energy equation.

** History.** Mechanics is one of the oldest sciences, Its origin and development are inseparably linked with the development of the productive forces of society and with practical needs. Statics developed before the other branches of mechanics, mainly in response to the requirements of civil engineering. It may be assumed that elementary data on statics (the properties of the simplest machines) were known several thousand years before the Common Era. The remains of ancient Babylonian and Egyptian structures indirectly attest to this, but no direct evidence has been found. The works of Aristotle on natural philosophy (fourth century B.C.), which introduced the term “mechanics” into science, are among the earliest extant treatises on mechanics from ancient Greece. It is evident from these works that the laws of the addition and balancing of forces that are applied at the same point and that act along the same straight line were known at that time, as were the properties of the simplest machines and the law of the equilibrium of a lever. The scientific foundations of statics were developed by Archimedes (third century B.C.). His works contain a rigorous theory of the lever, the concept of static moment, the rule for the addition of parallel forces, a study of the equilibrium of suspended bodies, a study of the center of gravity, and the fundamentals of hydrostatics. Subsequent significant contributions to the study of statics that led to the establishment of the parallelogram law of forces and also led to the development of the concept of the moment of a force were made by J. Nemorarius (c. 13th century), by Leonardo da Vinci (15th century), by the Dutch scientist Stevin (16th century), and, especially, by the French scientist P. Varignon (17th century), who culminated these studies by constructing statics on the basis of the rules governing the addition and resolution of ferees and the theorem of the resultant moment. The development of the theory of couples of forces by the French scientist L. Poinsot and the construction of statics on this basis (1804) was the final stage in the development of geometrical statics. Another approach in statics that was based on the virtual work principle developed in close association with the study of motion.

The problem of the study of motion also arose in remote antiquity. Solutions of the simplest kinematic problems of the addition of motions can be found as early as in the works of Aristotle and in the astronomical theories of the ancient Greeks, especially in the theory of epicycles, which was completed by Ptolemy (second century A.D.). However, Aristotle’s dynamic doctrine, which prevailed until almost the 17th century, proceeded from erroneous concepts, including (1) that a moving body is always acted upon by some force (for a projectile, for example, this is the pushing force of air, which tends to fill the place vacated by the body; the possibility of the existence of a vacuum was excluded here), and (2) that the velocity of a falling body is proportional to its weight.

The scientific foundations of dynamics, and with it, of all of mechanics, were created in the 17th century. Bourgeois relations, which led to the significant development of the trades, commercial navigation, and military science (the improvement of firearms) first developed in the 15th and 16th centuries in the countries of Western and Central Europe. This raised a number of important scientific problems, such as the study of the flight of projectiles, the impact of a body, the strength of large ships, and the oscillations of a pendulum (in connection with the development of clocks). But the solution of these problems, which required the development of dynamics, could be found only by discarding the erroneous propositions of Aristotle’s doctrine, which prevailed at that time. The first important step in this direction was taken by N. Copernicus (16th century), whose theory greatly influenced the development of all natural science and gave to mechanics the concepts of the relativity of motion and the necessity of choosing a frame of reference in the study of motion. The next step was J. Kepler’s experimental discovery of the kinematic laws of planetary motion in the early 17th century. The erroneous propositions of Aristotelian dynamics were conclusively disproved by Galileo, who laid the scientific foundations of modern mechanics. He gave the first correct solution for the problem of the motion of a body under the action of a force by determining experimentally the law of the uniformly accelerated free fall of bodies in a vacuum. Galileo established two fundamental propositions of mechanics—the Galilean principle of relativity in classical mechanics and the law of inertia. Although he expressed the latter only for the case of motion in a horizontal plane, he did apply it in his studies in a completely general manner. By using the idea of the addition of motions—a horizontal motion (from inertia) and a vertical motion (accelerated motion)—he was the first to note that in a vacuum the trajectory of a projectile thrown at an angle to the horizontal is a parabola. He initiated the theory of vibrations after his discovery of the isochronism of small oscillations of a pendulum. In investigating the equilibrium conditions of simple machines and solving certain problems of hydrostatics, Galileo used the so-called golden rule of statics, which he stated in general form; this was the initial form of the principle of virtual displacements. He was also the first to study the strength of beams, thus giving rise to the science of the strength of materials. One of Galileo’s most important achievements was the systematic introduction of the scientific experiment into mechanics.

R. Descartes, a contemporary of Galileo, formulated the law of inertia in general form and expressed the law of conservation of momentum (but not in vector form); he based his investigations in mechanics on these laws. He also introduced the concept of impulse. The Dutch scientist C. Huygens made the next major contribution to the development of mechanics. He solved a number of problems in dynamics that were most important at that time—the study of the circular motion of a point, the oscillations of a compound pendulum, and the laws of the elastic impact of a body. He was the first to introduce the concepts of centrifugal force and centripetal force and the concept of the moment of inertia (L. Euler introduced the term itself). Huygens also made use of a principle essentially equivalent to the law of conservation of mechanical energy, a general mathematical expression of which was subsequently given by H. Helmholtz.

I. Newton is credited with completing (1687) the formulation of the fundamental laws of mechanics. After studying the works of his predecessors, Newton generalized the concept of feree and introduced into mechanics the concept of mass. The fundamental (second) law of mechanics, which Newton formulated, enabled him to solve a large number of problems pertaining primarily to celestial mechanics, which was based on his law of universal gravitation. He also formulated the third fundamental law of mechanics—the law of equal action and reaction—which is the basis of the mechanics of a system of material points. Newton’s investigations completed the creation of the foundations of classical mechanics. The two initial propositions of continuum mechanics were also established during this period. Newton, who studied the resistance of a fluid to bodies moving in it, discovered the fundamental law of internal friction (viscosity) in liquids and gases, and the English scientist R. Hooke experimentally established a law that expresses the relation between stresses and deformations in an elastic body.

The general analytic methods of solving problems in the mechanics of a material point, a system of points, and a rigid body and problems in celestial mechanics, which were based on the use of the infinitesimal calculus discovered by Newton and G. W. von Leibniz, were intensively developed in the 18th century. Euler was primarily responsible for applying the infinitesimal calculus to the solution of problems in mechanics. He worked out analytic methods for solving problems in the dynamics of a material point, developed the theory of moments of inertia, and laid the foundations of the mechanics of rigid bodies. He also conducted the first studies in the theory of marine architecture, the theory of the stability of elastic rods, and the theory of turbines, and he solved a number of applied problems in kinematics. A contribution to the development of applied mechanics was the establishment of the experimental laws of friction by the French scientists G. Amontons and C. Coulomb.

The formulation of the dynamics of constrained mechanical systems proved to be an important stage in the development of mechanics. The solution of this problem proceeded from (1) the virtual work principle, which expresses the general condition for the equilibrium of a mechanical system and which was developed and generalized in the 18th century by Johann Bernoulli, L. Carnot, J. Fourier, and J. L. Lagrange, and (2) a principle that was expressed in most general form by d’Alembert and bears his name. Using these two principles, Lagrange completed the development of analytic methods for solving problems in the dynamics of free and constrained mechanical systems and obtained the equations of motion for a system in generalized coordinates— equations which bear his name. He also worked out the principles of the modern theory of vibrations. Another approach in the solution of problems in mechanics proceeded from the principle of least action in a form that was expressed by P. Maupertuis and developed by Euler for a single point. This principle was generalized by Lagrange to the case of a mechanical system. Celestial mechanics was developed considerably by Euler, by d’Alembert, by Lagrange, and, especially, by P. Laplace.

The application of analytic methods to continuum mechanics led to the development of the theoretical foundations of the hydrodynamics of an ideal fluid. The fundamental works in this field were those of Euler, D. Bernoulli, Lagrange, and d’Alembert. The law of conservation of mass, which was discovered by M. V. Lomonosov, was of great significance in continuum mechanics.

The intensive development of all branches of mechanics continued in the 19th century. In the dynamics of a rigid body, the classical results of Euler and Lagrange, and later of S. V. Kovalevskaia, which were further developed by other scientists, served as the basis for the theory of the gyroscope, which acquired particularly great practical significance in the 20th century. The fundamental works of M. V. Ostrogradskii, W. Hamilton, K. Jacobi, and H. Hertz were devoted to the further development of the principles of mechanics.

Lagrange, the British scientist E. Routh, and N. E. Zhukovskii obtained a number of important results in solving the fundamental problem of mechanics and of all natural science—the stability of equilibrium and motion. A. M. Liapunov rigorously formulated the problem of the stability of motion and developed the most general methods of solving the problem. Further studies on the theory of vibrations and the problem of controlling the operation of machines were conducted in connection with the requirements of machine technology. I. A. Vyshnegradskii developed the principles of the modern theory of automatic control.

Kinematics, which acquired increasingly greater independent significance, also developed in parallel with dynamics in the 19th century. The French scientist G. Coriolis proved a theorem concerning the components of acceleration, which is basic in the mechanics of relative motion. The purely kinematic term “acceleration” (J. Poncelet and H.-A. Resal) appeared, replacing various terms, for example, “accelerating force.” Poinsot gave a number of clear geometrical interpretations for the motion of a rigid body. The importance of applied studies in the kinematics of mechanisms increased, and P. L. Chebyshev made important contributions in this field. In the second half of the 19th century, kinematics became an independent branch of mechanics.

Continuum mechanics also developed significantly in the 19th century. The general equations of elasticity theory were established by L. Navier (C. L. M. H. Navier) and A. Cauchy. Subsequent fundamental results in this field were obtained by G. Green, S. Poisson, B. de Saint-Venant, M. V. Ostrogradskii, G. Lamé, W. Thomson (Lord Kelvin), and G. Kirchhoff. The studies of Navier and G. Stokes led to the establishment of the differential equations of motion for a viscous fluid. Among those who made significant contributions to the further development of the dynamics of ideal and viscous fluids were Helmholtz (the study of vortices), Kirchhoff and Zhukovskii (the detached flow past bodies), O. Reynolds (first studies of turbulent flows), and L. Prandtl (the theory of the boundary layer). N. P. Petrov constructed the hydrodynamic theory of friction in a lubricant, which was further developed by others, including Zhukovskii (with S. A. Chaplygin) and Reynolds. Saint-Venant proposed the first mathematical theory of the plastic flow of a metal.

A number of new branches of mechanics developed in the 20th century. Problems raised in electrical and radio engineering, in automatic control, and in other areas gave rise to a new field of science—the theory of nonlinear vibrations—whose foundations were laid by Liapunov and J. H. Poincare. Another branch of mechanics, which formed the basis of the theory of jet propulsion, was the dynamics of bodies of variable mass. Its principles were developed in the late 19th century by I. V. Meshcherskii. The first studies of the theory of rocket motion were conducted by K. E. Tsiolkovskii.

Two important new branches arose in the study of continuum mechanics—aerodynamics, whose foundations, like those of all aviation science, were established by Zhukovskii, and gas dynamics, whose foundations were laid by Chaplygin. The works of Zhukovskii and Chaplygin proved to be of great importance for the development of all modern aerohydrodynamics.

** Current problems in mechanics.** Among the most important problems in modern mechanics are the problems of the theory of vibrations (especially nonlinear vibrations), the dynamics of a rigid body, the theory of the stability of motion (all of which were discussed above), the mechanics of bodies of variable mass, and the dynamics of space flight. Problems in which “probabilistic” quantities, that is, quantities for which only the probability that they can have various values is known, must be considered instead of “determinate” quantities, that is, a priori quantities (for example, effective forces or laws of motion of individual objects), are acquiring increasing importance in all fields of mechanics. The study of the behavior of macroparticles during a change in shape is an extremely timely problem in continuum mechanics. This problem is connected with the development of a more rigorous theory of the turbulent flows of fluids, the solution of the problems of plasticity and creep, and the creation of a well-grounded theory of the strength and fracture of solids.

A large number of problems in mechanics involve the study of the motion of a plasma in a magnetic field (magnetohydrodynamics), for example, the solution of one of the most urgent problems in modern physics—the realization of a controlled thermonuclear fusion reaction. In hydrodynamics a number of the most important problems are connected with problems of high velocities in aviation, ballistics, turbine construction, and engine construction. Many new problems are arising on the border between mechanics and other scientific fields. Among these are, for example, the problems of hydrothermochemistry, (that is, the study of mechanical processes in liquids and gases that enter into chemical reactions) and the study of the forces that cause cell division and the mechanism by which a muscle produces a force.

Various types of computers are widely used to solve many problems in mechanics. The development of methods for the computer solution of new problems in mechanics (especially in continuum mechanics) is also an extremely pressing problem.

Research in various fields of mechanics is being conducted at universities and technical higher educational institutions in the USSR, at the Institute of Problems of Mechanics of the Academy of Sciences of the USSR, and in many other scientific research institutes both in the USSR and abroad.

The results of research in various fields of mechanics are published in numerous periodicals: *Doklady Akademii Nauk SSSR* (series *Matematika. Fizika*), since 1965; *Izvestiia Akademii Nauk SSSR* (series *Mekhanika tverdogo tela* [Mechanics of a rigid body] and *Mekhanika zhidkosti i gaza* [Mechanics of liquids and gases]), since 1966; *Prikladnaia matematika i mekhanika*, since 1933; *Zhurnalprikladnoi mekhaniki i tekhnicheskoi fiziki*, published since 1960 by the Siberian Division of the Academy of Sciences of the USSR; *Prikladnaia mekhanika*, published since 1955 by the Academy of Sciences of the Ukrainian SSR; and *Mekhanika polimerov*, published since 1965 by the Academy of Sciences of the Latvian SSR. Research results are also published in other journals, such as the *Vestnik* and *Trudy* of various higher educational institutions.

International congresses on theoretical and applied mechanics and conferences devoted to individual fields of mechanics are held periodically to coordinate scientific research in mechanics. They are organized by the International Union of Theoretical and Applied Mechanics (IUTAM), where the USSR is represented by the National Committee for Theoretical and Applied Mechanics of the USSR. The committee, in conjunction with other scientific institutions, periodically holds All-Union congresses and conferences devoted to research in various fields of mechanics.

### REFERENCES

Galilei, G.*Soch*., vol. 1. Moscow-Leningrad, 1934.

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*Sobr. trudov*, vol. 7. Moscow-Leningrad, 1936.

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*Istoriia fiziki*. Moscow, 1970. (Translated from Italian.)

S. M. TARG

## mechanics

[mi′kan·iks]## mechanics

**1.**the branch of science, divided into statics, dynamics, and kinematics, concerned with the equilibrium or motion of bodies in a particular frame of reference

**2.**the science of designing, constructing, and operating machines

**3.**the working parts of a machine