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The original expression of classical mechanics was given by Newton, and geometric methods and vectors are widely used as research tools, so it is also called vector mechanics (sometimes called Newtonian mechanics). Lagrange, Hamilton, jacoby and others established a set of mechanical expressions equivalent to vector mechanics by using generalized coordinates and variational methods. Compared with vector mechanics, analytical mechanics is more universal. Many complex problems in vector mechanics can be solved simply by analytical mechanics. The method of analytical mechanics can be extended to quantum mechanical systems and complex dynamic systems, and has important applications in both quantum mechanics and nonlinear dynamics. Analytical mechanics is a branch of theoretical mechanics and a highly mathematical expression of classical mechanics.
Different systems follow different differential equations of motion; Statistical mechanics is needed to study the system of a large number of particles; The process of quantum effect that cannot be ignored needs to be studied by quantum mechanics. However, the knowledge of analytical mechanics still plays an important role in statistical mechanics and quantum mechanics.
Analytical mechanics is a branch of general mechanics. It takes generalized coordinates as variables to describe particle systems, and based on Newton's laws of motion, it uses mathematical analysis methods to study mechanical problems in macroscopic phenomena. 1788 The book Analytical Mechanics published by J.-L. Lagrange laid the foundation of this discipline.
Development history
Since18th century, another mechanical system, namely analytical mechanics, has appeared in the history of mechanical development, which keeps pace with vector mechanics. The characteristic of this system is that the analysis of energy and work replaces the analysis of force and moment. In order to avoid the appearance of unknown ideal binding force, one method of analytical mechanics is to establish a direct relationship between ideal binding force and constraint equation, and derive the first kind of Lagrange equation, which is more obvious than the general method of vector mechanics. Another method of analytical mechanics is to use pure mathematical analysis method to express the dynamic equation described by independent coordinates with unified principles and formulas, which overcomes the shortcomings of relying on skills to establish this equation in vector dynamics. This unified equation is Lagrange's second kind equation. All the above work was done by J.L. Lagrange in 1788. Analytical mechanics based on Lagrangian equation is called Lagrangian mechanics. 1834, Hamiltonian transformed the Lagrangian equation of the second kind into a regular form, and summarized the basic principles of dynamics into Hamiltonian principles of variational forms, thus establishing Hamiltonian mechanics. For a dynamic system, although the establishment of Lagrangian second equation or Hamiltonian canonical equation of the system does not depend on skills, its mathematical derivation process is quite complicated, so it is quite difficult and easy to make mistakes when it is used to establish the dynamic equation of a multi-degree-of-freedom system. Using Lagrange equation of the first kind to solve the dynamic problem of the system, like the general method of vector dynamics, although it is easy to establish the equation, its solution scale is very large. It is for this reason that the Lagrange equation of the first kind is not superior to the general method of vector dynamics in the history of mechanical development and has been put aside.
With the development of modern computing technology, solving mathematical problems with stylized characteristics can be solved regardless of the scale. Therefore, the Lagrange equation of the first kind for solving dynamic problems has attracted extensive attention. It can be said that Lagrange's first equation and acceleration constraint equation are used as the dynamic model of the system in the computer-aided analysis software that successfully solves complex dynamic problems.
Analytical Mechanics published by Lagrange 1788 is the earliest work on analytical mechanics in the world. The basis of analytical mechanics is virtual work principle and D'Alembert principle. By combining the two, we can get the general dynamic equations, and then we can derive the dynamic equations of various systems of analytical mechanics.
During the period of 1760 ~ 176 1 year, Lagrange combined these two principles with ideal constraints and obtained the general equations of dynamics. Almost all dynamic equations of analytical mechanics are directly or indirectly derived from this equation.
1834, Hamilton deduced the dynamic equation expressed by generalized coordinates and generalized momentum, which is called the canonical equation. In the multidimensional space of Hamiltonian system, the path integral variational principle representing a point of the system can be used to study the mechanical problems of the complete system.
The rolling equation of a sphere on the horizontal plane was deduced from 186 1, and Appel equation was put forward by 1899 Appel in rational mechanics, which basically completed the linear nonholonomic constraint theory.
In the 20th century, analytical mechanics made further research on nonlinear, unsteady, variable mass and other mechanical systems, and made extensive research on the stability of motion.
classify
Analytical mechanics is divided into Lagrangian mechanics or Hamiltonian mechanics. The former describes the mechanical system with Lagrangian quantity, and the motion equation is called Lagrangian equation, while the latter describes the mechanical system with Hamiltonian quantity, and the motion equation is Hamiltonian canonical equation.
Analytical mechanics is a mechanical system suitable for studying macroscopic phenomena, and its research object is particle system. The particle system can be regarded as an ideal model of a mechanical system composed of macroscopic objects such as rigid body, elastic body, fluid and their synthesis, and the number of particles can range from one to infinite. For another example, the solar system can be regarded as a free particle system, and the interaction between stars is gravity. The research on celestial mechanics of planets of the solar system and satellites is closely related to analytical mechanics and promotes each other in methods. Most mechanical problems in engineering are constrained particle systems, and different mechanical systems are formed due to different types of constraint equations. Such as complete system, nonholonomic system, steady system, unsteady system, etc.
The main contents of analytical mechanics research are as follows: (1) Deriving the dynamic equations of various mechanical systems, such as Lagrange equation, canonical equation and Appel equation of nonholonomic systems; Explore the general principles of mechanics, such as Hamilton principle, principle of least action, etc. Discuss the characteristics of mechanical system; Study the methods of solving differential equations of motion, such as studying canonical transformation to solve canonical equations; The stability of the system is judged by studying the trajectory of representative points in phase space.
Analytical mechanics method is different from classical Newton mechanics method. Newton's method is to separate the object system into components in vitro, attach constraints to the reaction force according to the reaction law, and then list the equations of motion.
Variational principles (such as Hamilton principle) can also be used to derive differential equations of motion in analytical mechanics. Its advantage is that it can be extended to new fields (such as electrodynamics), and the approximation method in variational theory can be used to solve problems. Since the 1960s, in order to design complex spacecraft and robots, a multi-rigid-body system has been developed, and the dynamic equations have been established, instead of using the traditional method of derivation of dynamic functions. The established equation is easy to be calculated by electronic computer.
Before the establishment of quantum mechanics, physicists used analytical mechanics to study the mechanical problems of microscopic phenomena. Since 1923, quantum mechanics has been established and gradually improved, replacing analytical mechanics in the field of microscopic phenomena research. But mastering some basic knowledge of analytical mechanics is helpful to learn quantum mechanics well. For example, the Hamiltonian function is obtained by using the knowledge of analytical mechanics, and then transformed into Hamiltonian operator, and then transformed from Hamiltonian-Jacobian equation into Schrodinger equation, the basic equation of wave mechanics.
When Einstein put forward the theory of relativity, he also applied some methods of analytical mechanics to the study of relativistic mechanics whose speed is close to the speed of light.
fundamental principle
The basic principles of analytical mechanics are mainly virtual work principle and D'Alembert principle, while the former is the basis of analytical statics. By combining the former with the latter, we can get the general equations of dynamics, and then derive the dynamic equations of various systems of analytical mechanics. The research object is particle system. The particle system can be regarded as an ideal model of a mechanical system composed of all macroscopic objects. For example, rigid body, elastic body, fluid and their combinations can all be regarded as particle systems, and the number of particles can range from 1 to infinity. For another example, the solar system can be regarded as a free particle system. Celestial mechanics studies the motion of planets and satellites in the solar system, which is closely related to analytical mechanics and promotes each other in methods. Analytical mechanics is superior to solving the constrained particle system, because with the constrained equation, the degree of freedom of the system can be reduced, and the order of the differential equation of motion will be reduced, making it easier to solve.
differentiate
Analytical mechanics and theoretical mechanics are the same in principle, but in general textbooks, theoretical mechanics will first talk about some knowledge of general mechanics, and then the last chapter will talk about analytical mechanics, which means that theoretical mechanics is simple and easy to learn, and it is a relatively primary analytical mechanics.
Analytical mechanics takes generalized coordinates as variables to describe particle systems, and based on virtual displacement principle and D'Alembert principle, it studies mechanical problems in macroscopic phenomena by mathematical analysis methods. Analytical Mechanics published by J.-L. Lagrange in 1788 laid the foundation of this subject. 1834 and 1843, W.R. Hamilton established Hamilton principle and canonical equation, which further promoted the development of analytical mechanics. 1894, H.R. Hertz put forward that constraints and systems can be divided into complete and incomplete categories, and from then on, the research on analytical mechanics of nonholonomic systems began. The basic content of analytical mechanics is to explain the general principles of mechanics, from which the basic differential equations of motion of particle systems are derived, and to study these equations themselves and their integration methods. In recent 20 years, the principles and methods of studying analytical mechanics from the perspective of modern differential geometry have been developed. Analytical mechanics is one of the foundations of classical physics and the whole mechanics. It is widely used in structural analysis, machine dynamics and vibration, aerospace mechanics, multi-rigid-body system and robot dynamics and various engineering and technical fields, and can also be extended to continuum mechanics and relativistic mechanics.