Newton's law of motion is about a single free particle, and J Le R D'Alembert extended it to the motion of constrained particles. J.-L. Lagrange further studied the motion of constrained particles and summarized these results in his book Analytical Mechanics (first edition 1788), from which analytical mechanics was founded. Prior to this, L. Euler established the dynamic equation of rigid body (1758). So far, the classical mechanics centered on the motion law of particle system and rigid body has been perfect. In this development process, the theory of finite freedom motion and vibration is later than the theory of elastic chord, which is a rare inconsistency between historical order and logical order, because the study of elastic vibration is driven by acoustics. In 1787, Klanyi made experiments on the vibration modes of bars and plates. Lagrange's analytical mechanics in 1788 has fully discussed the micro-vibration with finite degrees of freedom. Later, к Wilstras pointed out its defects in 1858 and ои Somov in 1859.
Euler is the scholar who has made the greatest contribution to mechanics after Newton. In addition to listing the equations of motion and dynamics of rigid body motion and obtaining some solutions, he also made a pioneering study on elastic stability, opened up the theoretical analysis of fluid mechanics and laid the foundation of ideal fluid mechanics. He played a connecting role in the establishment of classical mechanics in this period and the growth of elastic mechanics and fluid mechanics into independent branches in the next period. D'Alembert also studied the motion of fluid, and reached the conclusion that the fluid resistance of moving objects is zero, that is, the D 'Alembert paradox. Newton's resistance formula (1723) and D 'Alembert's paradox (1752), as well as their differences and experimental results of fluid resistance, have promoted the research of fluid mechanics for a long time and promoted the emergence of branches of fluid mechanics in the next period. /kloc-the development of solid mechanics in the 0/9th century is mainly the establishment of mathematical elasticity besides the perfection of material mechanics and the gradual development of bar structure mechanics. Material mechanics and structural mechanics were closely related to civil construction technology, machinery manufacturing, transportation and so on at that time, while elastic mechanics had almost no direct application background at that time, mainly to explore natural laws.
In 1807, T. Yang put forward the concept of elastic modulus, and pointed out that shearing is an elastic deformation as well as stretching. Although the form of Young's modulus is different from the modern definition, and Yang doesn't know that shear and expansion should have different moduli, Yang's work has become a prelude to the establishment of elasticity theory. C.-L.-M.-H. Naville published his research results of 182 1. This report is based on the molecular structure theory (Boskovich model of 1763 assumes that matter is composed of many discrete molecules interacting with a central force). A.-L. Cauchy changed the discrete molecular model to the continuum model in 1823 (A. C. Clairaux first proposed the continuum model in 17 13), discussed the stress-strain theory in detail, and established the basic equations of equilibrium and motion of isotropic elastic materials, including two elastic constants. The elastic mechanics equation published by S.-D. Poisson in 1829 returns to the discrete particle model which gives the elastic constant equation, but it is pointed out that longitudinal stretching causes lateral contraction, and the strain ratio between them is constant, equal to one quarter. Whether the elastic constant of an isotropic elastic solid is 1 or 2, or whether it is 15 or 2 1 in a general elastic body, has aroused heated debates and promoted the development of elasticity theory. Finally, G Green gave a correct conclusion from the elastic potential and G Lame from the physical meaning of two constants: the elastic constants should be two, not one (the general elastic material is 2 1).
The theory of elastic vibration was developed on the basis of studying the vibration of strings and rods in the18th century. The representative work in this field is two volumes of Rayleigh's Acoustic Theory (1877 ~ 1878), which summarizes the achievements in this field at that time. The elastic wave theory developed on the basis of elastic dynamics and vibration theory points out that there are not only longitudinal waves and transverse waves (as Poisson pointed out in 1829), but also surface waves (Rayleigh, A.E.H Love, H. Lamb, etc. ), which has theoretical significance for explaining geophysical phenomena such as earthquakes. Interestingly, the earliest achievements of elastic waves were not obtained from mechanical research, but were put forward by A.-J. Fresnel in Optical Research 182 1. He pointed out that there were shear waves in elastic media, and at that time he thought that light propagated in elastic media (ether).
After the basic equation of elasticity is established, A.J.C.B De Saint-Venant set out to solve the equation, and obtained some valuable principled results, such as pointing out that the local equilibrium force system can be ignored for a large range of elastic effects. In the19th century, some concrete solutions were obtained one after another, and these results were summarized in two volumes (1892 ~ 1893) written by Love. In the first half of the 20th century, more questions were answered from engineering technology. /kloc-in the 0/9th century, a large number of solid mechanical strength and stiffness problems in architecture and machinery have to be calculated by material mechanics and structural mechanics. Many scientists, including physicist J.C. Maxwell, have studied practical solutions in structural mechanics, such as graphic solutions. In addition, because most of the unstable members in the structure do not belong to the slender members considered by Euler, many scholars, such as φ C Jasinski and W. J.M Rankin, have given some semi-empirical formulas on the basis of experiments. Research results on plasticity and yield law of materials have also begun to appear, such as the Bauschinger effect published in 1886 (this effect was observed in the experiments of 1858 and 1859 Videman before J. Bauschinger) and the plastic flow and shear stress yield theory published in 1864. During this period, the development of fluid mechanics was similar to that of solids, and many empirical or semi-empirical formulas were exhibited in hydraulics under the impetus of practice. On the other hand, the most important progress in mathematical theory is the establishment of the basic equation of viscous fluid motion, namely Naville-Stokes equation. Naville inherited Euler's work and published the motion equation of incompressible viscous fluid in 182 1, starting from the discrete molecular model. In 183 1, Poisson changed to viscous fluid model to explain and generalize Naville's results, and gave the constitutive relation of viscous fluid completely for the first time. 1845, G.G. Stokes averaged the discrete molecules, adopted the continuum model, and assumed that the six components of stress linearly depended on the six components of deformation velocity, and obtained the basic equation of viscous fluid motion, that is, the right-angle component form of Navier-Stokes equation in modern literature. Prior to this, G. H.L Hagen published the experimental results and formulas about pipeline flow at 1839 and J-L-M Marie Poi at1840 ~1,which became an example of Stokes equation. Stokes also thinks that there is a general nonlinear functional relationship between stress and deformation velocity, but the research on this kind of non-Newtonian fluid did not develop until the 1940s, both in theory and in practice.
In the mechanics of compressible fluid or gas, many basic laws are discovered according to experiments. 1839, Saint-Venant gave the calculation formula of gas passing through small holes. In acoustic theory, besides Rayleigh's elastic vibration theory, gas wave theory has also made great progress. For supersonic flow, E. Mach published the experimental results of projectile flying in air in 1887, and put forward the dimensionless number of the ratio of velocity to sound speed. Later, this parameter was called Mach number (1929), and its sine was called Mach angle (1907). Rankin and P.H. Xu Hongniu considered the discontinuous changes of pressure and density before and after one-dimensional shock wave (shock wave) in 1870 and 1887 respectively.
The basic work about the transition (or transition) from laminar flow to turbulent flow and flow instability is the pipeline experiment of O. Reynolds in 1883. He pointed out the dynamic similarity law of flow in the experiment, and a dimensionless number-Reynolds number played a key role in it. Renault also began the difficult research of theory of turbulence.
Lamb summed up the theoretical achievements of19th century fluid mechanics in his Mathematical Theory of Fluid Motion (first edition 1878, later renamed Fluid Mechanics). However, in practice, many problems in fluid mechanics have to rely on empirical or semi-empirical formulas in hydraulics, such as introducing some empirical coefficients into Bernoulli theorem, which represents mechanical energy, and adding correction coefficients considering non-uniformity into Hagen-Poiseuille flow formula, which is only suitable for uniform pipe flow. Many mechanical problems in hydraulic engineering and hydraulic machinery are solved by this method, such as the open channel flow formula of A. De Xie Cai and R. Manning, and many hydraulic studies made by L. A. Pelton, J. B. Francis and V. Kaplan to improve the efficiency of hydraulic machinery. Petrov's research on the flow between two eccentric cylinders in 1890 is related to the lubrication of bearings. The main achievement of analytical mechanics is the development from Lagrangian mechanics to Hamiltonian mechanics based on integral variational principle. The establishment of the variational principle of integral form is of great significance to the development of mechanics, both in modern times and in theory and application. The variational principle in integral form was put forward not only by W R Hamilton in 1834, but also by C F Gauss in 1829. Hamilton's other contribution is the canonical equation and its related canonical transformation, which provides a method to solve the mechanical motion equation. Jacobi further pointed out the relationship between regular equations and partial differential equations. The mechanical theories from Newton, Lagrange to Hamilton constitute the classical mechanical part of physics.
In addition, the research on nonholonomic systems began at the end of 19. For example, P.-Appel established the motion equation of nonholonomic system expressed by "acceleration energy".
1846, Neptune made a prediction through calculation, and then confirmed it through observation, which promoted the research of celestial mechanics based on Newton's law of motion and the law of universal gravitation. The French Academy of Sciences once offered a reward for the research results of the three-body system. Many research results of Poincare not only promote the development of motion stability and perturbation theory in mechanics, but also promote the development of topology and qualitative theory of differential equations in mathematics. On the other hand, engineering technology and other aspects of celestial mechanics have also raised many problems of motion stability. Other contributors include E.J. Laosi, не Zhukovsky, especially A.M. Liapunov, whose monograph General Problems of Motion Stability (1892) is still meaningful until the middle of the 20th century. /kloc-classical mechanical problems in the 0 th and 9 th centuries are rewarded for solving not only three-body but also fixed-point motions of heavy rigid bodies. In the application results, the fixed-point motion equation of heavy rigid body obtained by C B Kovalevskaya is the third integrable equation besides the two integrable equations obtained by Euler and Lagrange. 1906 V.F. Hess proved that there are only the above three integrable equations under general conditions.
In terms of application, the development of large machines put forward a large number of kinematics and dynamics problems related to machine transmission and solved them, gradually forming the current mechanical principles and other disciplines. The representative of applied mechanics is J.-V. Poncelet, who wrote "Practical Mechanics of Craftsmen and Workers" from 1827 to 1829.