Classical Mechanics in the History of Mechanics

Galileo's main contributions to dynamics were his principle of inertia and his experiments on acceleration. He studied the motions of free fall, inclined motion, and projectiles on the ground, established the concept of acceleration and discovered the law of uniformly accelerated motion. Using a combination of scientific experiments and theoretical analysis, he pointed out the errors of the traditional Aristotelian view of motion and endeavored to promote the heliocentric theory. His Discourses and Mathematical Proofs Concerning Two New Sciences, published in 1638, was the first work on dynamics.C. Huygens proposed important concepts such as centripetal force, centrifugal force, moment of inertia, and center of oscillation of a compound pendulum in his study of dynamics. On the other hand, Kepler summarized the three laws of planetary motion on the basis of 30 years of astronomical observations in Tiguya (1609, 1619). I. Newton inherited and developed these results, and formulated the laws of motion of bodies and the law of universal gravitation. His achievements were collected in the Mathematical Principles of Natural Philosophy, published in 1687. The three laws of motion he gave in this book are: ① The first law: any body will remain in its state of rest or in uniform linear motion unless a force applied to it compels it to change this state. The second law: the change in the amount of motion of an object is proportional to the force applied and occurs in the direction of the line of action of the force. The third law: for any action there must be a counteraction of equal size and opposite direction. The first law has been described in Galileo's work, in 1644 R. Descartes made improvements in the form. The third law was derived by Newton by summarizing the results of C. Raine, J. Wallis and Huygens. Newton's law of gravitation was something he began to consider between 1665 and 1666, and was later inspired by R. Hooke's 1679 suggestion.

Newton's laws of motion were for a single free mass, and J.le R. D'Alembert extended them to the motions of constrained masses.J.-L. Lagrange further investigated the motions of constrained masses and summarized the results in his work Analytical Mechanics (first edition, 1788), from which analytical mechanics was founded. Before that, L. Euler established the dynamical equations of rigid bodies (1758). Thus classical mechanics, in which the laws of motion of a system of masses and rigid bodies are the main objects of study, was perfected. In the course of this development, the theory of motion and vibration of finite degrees of freedom came a little later than the theory of vibration of elastic strings and rods, which is a rare inconsistency in the historical and logical order, because the study of elastic vibration was facilitated by acoustics. 1787 Cluny made experiments on the modes of vibration of rods and plates, and the theory of finite-degree-of-freedom micro-vibrations was fully discussed in the Analytical Mechanics of Lagrange in 1788. Verstras in 1858 and О.I.Somov in 1859 pointed out the defects.

Euler was the scholar who contributed the most to mechanics after Newton. In addition to the motion of rigid bodies to list the equations of motion and kinetic equations and get some solutions, he made a pioneering study of elastic stability, and opened up the theoretical analysis of fluid mechanics, laid the foundation of ideal fluid mechanics, in this period of the creation of classical mechanics and the next period of elasticity mechanics, fluid mechanics, grew to become an independent branch of the next period of time between the role he played the role of a bridge between the beginning and the end. D'Alembert also studied the motion of fluids, and came to the conclusion that the fluid resistance to a moving object is zero, the D'Alembert feint. Newton's formula on resistance (1723), the D'Alembert's feint (1752), and the difference between them and the results of experiments on the resistance of fluids drove the study of fluid mechanics for a long period of time, and contributed to the emergence of the branch of fluid mechanics in the next period. The development of the mechanics of solids in the 19th century, in addition to the mechanics of materials more perfect and gradually developed into a rod system of structural mechanics, mainly the establishment of mathematical elastic mechanics. Mechanics of materials, structural mechanics and civil engineering and construction technology, machinery manufacturing, transportation and other closely related, and elastic mechanics at that time few direct application background, mainly for the exploration of the laws of nature and the basic research.

In 1807, T. Yang put forward the concept of elastic modulus, pointing out that shear is also a kind of elastic deformation like expansion and contraction. Although the form of Young's modulus is not the same as the modern definition, and Young was not clear that shear and expansion should have different moduli, Young's work became a prelude to the establishment of the theory of elasticity.C.-L.-M.-H. Navey published in 1827 the results of his research in 1821, "Report on the study of elasticity equilibrium and the laws of motion", which was based on the theory of the structure of the molecule (Boscovich model in 1763), which assumed that matter is formed by a central force. From the theory of molecular structure (the Boskovitch model of 1763, which assumes that matter consists of many discrete molecules interacting with each other with a central force), he established the equations of isotropic elastic solids, which have only one elastic constant. a.-l. Cauchy changed the discrete molecule model into the continuum model in 1823 (the continuum model was the first to be proposed by a.c. Clyde Clayroe in 1713), and explored the theories of stress and strain in detail, and established the basic equations of equilibrium and motion for isotropic elastic materials. The basic equations of equilibrium and motion of materials, which have two elastic constants.The equations of elastic mechanics published by S.-D. Poisson in 1829 returned to the discrete-particle model that gave an equation for an elastic constant, but it pointed out that longitudinal stretching caused transverse contraction, and the ratio of the two strains was a constant equal to one-fourth. Whether there are one or two constants of elasticity for isotropic elastic solids, or whether there are 15 or 21 in a general elastomer, had given rise to heated debates and contributed to the development of the theory of elasticity. Finally G. Green from the elastic potential, G. Ramey from the physical significance of the two constants gave the correct conclusion: the elastic constants should be two, not one (general elastic materials is 21).

The theory of elastic vibration in the 18th century strings, rods and other vibration research on the basis of the development of this area is represented by Riley's "Theory of Acoustics" in two volumes (1877 ~ 1878) summarizes the results of this area at the time. In the elastic dynamics and vibration theory developed on the basis of the theory of elastic waves pointed out that not only the existence of longitudinal and transverse waves (as Poisson pointed out in 1829), there is also the existence of surface waves (Riley, A.E.H. Lefebvre, H. Lamb, etc.), which explains the earthquakes and other geophysical phenomena have theoretical significance. It is interesting to note that the earliest results of elastic waves were not obtained from mechanical studies, but were proposed in 1821 by A.-J. Fresnel in his optical studies, when he pointed out the existence of transverse waves in elastic media, and it was then believed that light traveled in an elastic medium (Ether).

After the establishment of the basic equations of elastic mechanics A.J.C.B .de St. Venant embarked on the equations to solve, and obtained some valuable principle results, such as pointing out that the local equilibrium system of forces on a large scale of the elastic effect can be ignored. In the 19th century, solutions in specific cases were successively obtained, and these results were summarized in the two volumes of The Theory of Mathematical Elasticity (1892-1893) by Lefebvre. In the first half of the 20th century, there were more questions and answers from engineering and technology. In the 19th century, the strength and stiffness problems of solid mechanics, which arose in large numbers in construction and machinery, also had to rely on the mechanics of materials and structural mechanics for their calculations. Many scientists, including the physicist J.C. Maxwell, have successively studied practical solutions in structural mechanics, such as the graphical solution method. In addition, since most of the rods in the structure where the phenomenon of instability occurs do not belong to the slender rods considered by Euler, many scholars, such as Φ.C. Yashinsky, W.J.M. Rankin, etc., have given some semiempirical formulas on the basis of experiments. The results of research on the laws of plasticity and yielding of materials also began to appear, such as the publication of the Bauzinger effect in 1886 (before J. Bauzinger, Widmann had already observed this effect in experiments in 1858 and 1859), and the theory of plastic flow and shear stress yielding of Tresca in 1864. The development of mechanics in relation to fluids during this period was similar to that of solids, and a number of empirical or semi-empirical formulas were developed for hydraulics under the impetus of practice; on the other hand, the most important advance in mathematical theory was the establishment of the basic equations of motion for viscous fluids, i.e., the Navier-Stokes equations. Navi inherited Euler's work, published the incompressible viscous fluid equations of motion in 1821, its starting point is the discrete molecular model. 1831 Poisson changed to a viscous fluid model to explain and promote Navi's results, the first complete viscous fluid isomerism relationship. G.G. Stokes in 1845 will be discrete molecular averaging, using the continuum model, assuming that the stress is linearly dependent on the deformation velocity of six components. linearly dependent on the six components of the deformation velocity to obtain the fundamental equations of motion for viscous fluids, the right-angle component form of the Navier-Stokes equations in the modern literature. Prior to this, G. H. L. Hagen in 1839 and J.-L.-M. Posuyet in 1840-1841 published experimental results and derived formulas, respectively, on the flow of pipes, which served as illustrations of Stokes' equations. Stokes also considered the case where there is a general nonlinear functional relationship between stress and deformation velocity, but this study of non-Newtonian fluids, either theoretically or practically, only developed in the 1940s.

In the mechanics of compressible fluids or gases, a number of fundamental laws have been discovered on the basis of experiment. St. Venant in 1839 gave the formula for the passage of gases through small holes. In the theory of acoustics, in addition to Riley's theory of elastic vibrations mentioned above, there has been a great development in the fluctuation theory of gases. For supersonic flow, E. Mach's experimental results on the flight of projectiles in air, published beginning in 1887, proposed the dimensionless number of the ratio of the flow velocity to the speed of sound. Later this parameter became known as the Mach number (1929) and its inverse sine was called the Mach angle (1907). The discontinuous law of change of pressure and density before and after a one-dimensional shock wave (excitation) was considered by Rankin and P.H. Hügonnü in 1870 and 1887, respectively.

The seminal work on the turning (or transition) from laminar to turbulent flow, as well as on flow destabilization, was O. Reynolds's pipe experiment in 1883. In his experiments, he pointed out the dynamical similarity law of the flow, and the key role in it is a dimensionless number, that is, the Reynolds number. Reynolds also began the difficult study of the theory of turbulence.

Lamb summarized the theoretical achievements of 19th-century fluid mechanics in his Mathematical Theory of the Motion of Fluids, first published in 1878 and later renamed Hydrodynamics. But many practical problems in fluid mechanics, but also have to rely on empirical formulas or semi-empirical formulas in hydrodynamics, such as the characterization of mechanical energy Bernoulli's theorem in the introduction of a number of empirical coefficients in order to calculate the effect of resistance, in the Hagen-Poissuet flow formula that applies only to the uniform pipe flow, plus the addition of non-uniformity into the consideration of the correction coefficient, and so on. Many mechanical problems in hydraulic engineering and hydromechanics were solved by this approach, such as the open channel flow formulae of A. de Shetai and R. Manning, and many hydromechanical researches to improve the efficiency of hydromechanics by L.A. Pelton, J.B. Francis, V. Kaplan, etc. The researches of Н.П. Petrov in 1890 on the flow between two eccentric cylinders were connected to the problem of lubrication of the bearings. The main achievements in analytical mechanics were made by N.P. Petrov. The main achievement in analytical mechanics was the development of Lagrangian mechanics into Hamiltonian mechanics based on the principle of integral forms of differentiation. Integral form of the principle of variation of the establishment of the development of mechanics, whether in modern times or modern, whether in theory or application, are of great significance. In addition to the variational principle of integral form proposed by W.R. Hamilton in 1834, there is also the principle of minimum constraint proposed by C.F. Gauss in 1829. Another contribution of Hamilton was the regular equations and the associated regular transformations, which provide the means for solving the equations of motion in mechanics.C.G.J. Jacobi further pointed out the relationship between the regular equations and a partial differential equation. Theories of mechanics from Newton, Lagrange to Hamilton constitute the classical mechanics part of physics.

In addition, the study of incomplete systems began at the end of the 19th century, as P.-┵. Appel established the equations of motion for incomplete systems expressed in terms of "acceleration energy".

In 1846, Neptune was predicted by calculation and then confirmed by observation, which promoted the study of celestial mechanics based on Newton's laws of motion and the law of gravity. The French Academy of Sciences had offered a reward for the research results of the three-body problem, and H. Poincaré's many research results for this purpose not only promoted the development of the theory of stability of motion and the theory of regeneration in mechanics, but also promoted the development of topology and the qualitative theory of differential equations in two branches of mathematics. On the other hand, other aspects of engineering and celestial mechanics have also raised a number of motion stability problems. Contributing to this were E.J. Rauth, N.E. Jukovsky, and especially A.M. Lyapunov, whose monograph General Problems of the Stability of Motion (1892) was still relevant until the middle of the 20th century. 19th century bounty solving of the classical problems of mechanics, in addition to the three-body problem, was also the motion of the fixed point of a heavy rigid body. The equations of motion of the fixed point are the third equations of integrable form in addition to the two already obtained by Euler and Lagrange, and in 1906 V.F. Hess proved that there are only three equations of integrable form above under general conditions.

In terms of application, the development of large machines raised a large number of kinematics and dynamics problems related to machine transmissions and solved them, gradually forming the present principles of mechanics and other subjects. The representative of applied mechanics is worth mentioning is J.-V. Ponselle, who specialized in writing Practical Mechanics for Artisans and Workers between 1827 and 1829.