What Is Kinetic Theory?

Dynamics is a branch of theoretical mechanics, which mainly studies the relationship between the force acting on an object and its motion. The research object of dynamics is macro-objects whose motion speed is much lower than the speed of light. Dynamics is the basis of physics and astronomy, as well as the basis of many engineering disciplines. Many mathematical advances are often related to solving dynamics problems, so mathematicians have a strong interest in dynamics.

Chinese name: dynamics
Foreign name: Dynamics
Belongs to: Theoretical mechanics
main research: The relationship between the force acting on an object and its motion

Research object: Macro objects that move much faster than the speed of light
Basic law: Newton's law of motion proposed by Isaac Newton
main effect: Basic courses for mechanical and aeronautical engineering
Mainly involved: Physics, mechanics, physics, theoretical mechanics

Overview of Dynamics

Dynamics is a branch of theoretical mechanics that studies the relationship between the force acting on an object and its motion. The research object of dynamics is macro-objects whose motion speed is much lower than the speed of light. The study of the dynamics of atoms and subatomic particles belongs to quantum mechanics, and the study of high-speed motion comparable to the speed of light belongs to relativity mechanics. Dynamics is the basis of physics and astronomy, as well as the basis of many engineering disciplines. Many mathematical advances are often related to solving dynamics problems, so mathematicians have a strong interest in dynamics.

The study of dynamics is based on Newton's laws of motion; the establishment of Newton's laws of motion is based on experiments. Dynamics is part of Newtonian mechanics or classical mechanics, but since the 20th century, dynamics has often been understood as a branch of mechanics that focuses on the application of engineering technology.

The basic contents of dynamics include particle dynamics, particle system dynamics, rigid body dynamics, Dalbert principle, and so on. Applied disciplines developed on the basis of dynamics include celestial mechanics, vibration theory, kinematic stability theory, gyro mechanics, external ballistics, variable mass mechanics, and dynamics of multi-rigid body systems in development, etc. Sex, variable mass body motion, multiple rigid body systems).

There are two basic types of particle dynamics: one is the motion of a known point, find the force acting on the particle, and the other is the force of the known point, and find the motion of the particle. The second-order derivative of the motion equation of the particle is used to obtain the acceleration of the particle, which can be obtained by substituting into Newton's second law; when solving the second type of problem, the differential equation of motion of the particle or the integral is required. The so-called differential equation of particle motion is the equation that writes the second law of motion as the derivative of the coordinate of the particle with respect to time.

The general theorem of dynamics is the basic theorem of particle system dynamics. It includes momentum theorem, momentum moment theorem, kinetic energy theorem, and some other theorems derived from these three basic theorems. Momentum, momentum moment, and kinetic energy (see energy) are the basic physical quantities describing particles, particle systems, and rigid body motion. The relationship between the forces or moments acting on the mechanical model and these physical quantities constitutes the general theorem of dynamics. The two-body problem and the three-body problem are classic problems in the dynamics of particle systems.

Rigid bodies are distinguished from other particle systems by the invariance of the distance between their points. The classic method for deriving rigid body attitudes is to use three independent Euler angles. Euler's kinetic equation is the basic equation of rigid body dynamics, and rigid body fixed-point rotation dynamics is a classic theory in dynamics. The formation of gyro mechanics shows that the application of rigid body dynamics in engineering technology is of great significance. Multi-rigid-body system dynamics is a new branch formed by the development of new technologies since the 1960s, and its research methods have been different from those of classical theory.

A brief history of dynamics

Dynamic universe view

Copernicus and Kepler's view of the universe

The development of mechanics has gone from the elaboration of the simplest laws of object balance to the establishment of general laws of motion, which has taken about 20 centuries. The large amount of mechanical knowledge accumulated by the predecessors played an important role in the research of dynamics, especially the astronomers Copernicus and Kepler's universe view.

17 Dynamics began in the 17th century

In the early 17th century, Galileo, an Italian physicist and astronomer, used experiments to reveal the principle of inertia of matter, and used acceleration and sliding experiments of objects on a smooth bevel to reveal the law of constant acceleration motion, and realized that the value of gravity acceleration near the ground does not change. The mass of an object varies, it approximates a constant, and the general laws of projectile motion and particle motion are studied. Galileo's research pioneered the academic methods that were commonly used by future generations, starting from experiments and using experiments to verify theoretical results.

In the 17th century, the calculus established by British great scientist Newton and German mathematician Leibniz brought dynamics research into a new era. Newton's great work "Mathematical Principles of Natural Philosophy" published in 1687 explicitly proposed the laws of inertia, the law of particle motion, the laws of action and reaction, and the laws of independent action of forces. In searching for the causes of falling and celestial movements, he discovered the law of universal gravitation and derived Kepler's law based on it, verifying the relationship between the centripetal acceleration of the moon's rotation around the earth and the acceleration of gravity, explaining the tidal phenomenon on the earth , Established a very strict and perfect system of mechanical laws.

Dynamics is centered on Newton's second law, which states the relationship between force, acceleration, and mass. Newton first introduced the concept of mass, and distinguished it from the gravity of the object, indicating that the gravity of the object is only the gravity of the object on the object. After the establishment of the law of action and reaction, people have carried out research on particle dynamics.

Vehicle dynamics Newton's mechanical work and calculus work are inseparable. Since then, dynamics has become a rigorous science based on experiments, observations, and mathematical analysis, thus laying the foundation for modern mechanics.

In the 17th century, the Dutch scientist Huygens observed the pendulum's gravitational acceleration and established the equation of motion of the pendulum. Huygens also established the concept of centrifugal force while studying the pendulum; in addition, he also proposed the concept of rotational inertia.

One hundred years after Newton's Law was published, Lagrange, a French mathematician, established the Lagrange equation that can be applied to complete systems. This set of equations is different from the form of force and acceleration of Newton's second law, but is expressed by Lagrange function using generalized coordinates as independent variables. Lagrangian system is more convenient to study certain types of problems (such as small oscillation theory and rigid body dynamics) than Newton's law.

18 18th Century Newton's Second Law

The concept of a rigid body was introduced by Euler. In the 18th century, the Swiss scholar Euler extended Newton's second law to rigid bodies. He applied three Euler angles to represent the angular displacement of a rigid body around a fixed point, defined the moment of inertia, and derived the differential equation of motion for a fixed point of rigid body rotation. In this way, a universal equation of motion for a rigid body with six degrees of freedom is completely established. For rigid bodies, the sum of the work done by the internal force is zero. Therefore, rigid body dynamics has become an approximate theory for studying general solid motion.

In 1755 Euler established the dynamic equation of the ideal fluid; in 1758 Bernoulli obtained the energy integral along the streamline (called Bernoulli

Molecular reaction dynamics Equation); In 1822, Navier obtained the dynamic equation of incompressible fluid; in 1855, Higonneau, France studied shock waves in continuous media. In this way, kinetics penetrates into the realm of various forms of matter. For example, in elastic mechanics, due to the need to study the problems of collision, vibration, and elastic wave propagation, elastic dynamics is established, which can be used to study the transmission of seismic waves.

19 19th Century Hamilton Regular Equation

In the 19th century, the British mathematician Hamilton used the variational principle to derive the Hamilton regular equation. This equation is a system of first-order equations with generalized coordinates and generalized momentum as variables and represented by the Hamilton function. Its form is symmetrical. The system formed by motion using regular equations is called the Hamilton system or Hamilton dynamics. It is the basis of classical statistical mechanics and an example of quantum mechanics. The Hamilton system is applicable to perturbation theory, such as the perturbation problem of celestial mechanics, and plays an important role in understanding the general properties of the motion of complex mechanical systems.

Lagrangian and Hamiltonian dynamics are based on the same principles as Newton's, and they are equivalent in the category of classical mechanics, but their approaches or methods are different. Mechanical systems that directly apply Newton's equations are sometimes called vector mechanics; the dynamics of Lagrange and Hamilton are called analytical mechanics.

Kinetic content

The basic contents of dynamics include particle dynamics, particle system dynamics, rigid body dynamics, D'Alembert's principle, and so on. Applied disciplines developed based on dynamics include celestial mechanics, vibration theory, kinematic stability theory, gyro mechanics, external ballistics, variable mass mechanics, and multi-rigid-body system dynamics and crystal dynamics that are being developed. ^[1]

Rotor dynamics

Two abstract models of dynamics

Particles and particle systems. A particle is an object with a certain mass and negligible geometry and size.

Two basic types of dynamics

There are two basic types of particle dynamics: one is the motion of a known particle and finds the force acting on the particle; the other is the force of a known particle and finds the movement of the particle. When solving the first type of problem, as long as you take the second derivative of the motion equation of the particle, get the acceleration of the particle, and substitute it into Newton's second law, you can find the force; when solving the second type of problem, you need to solve the differential equation of particle motion or calculate the integral. ^[2]

General theorem of dynamics

The general theorem of dynamics is the basic theorem of particle system dynamics. It includes momentum theorem, momentum moment theorem, kinetic energy theorem, and some other theorems derived from these three basic theorems. Momentum, momentum moment, and kinetic energy are the basic physical quantities that describe the motion of particles, particle systems, and rigid bodies. The relationship between the force or moment acting on the mechanical model and these physical quantities constitutes the general theorem of dynamics.

Dynamic rigid body

A rigid body is characterized by the invariance of the distance between its particles. Euler's kinetic equation is the basic equation of rigid body dynamics, and rigid body fixed-point rotation dynamics is a classic theory in dynamics. The formation of gyro mechanics shows that the application of rigid body dynamics in engineering technology is of great significance. Multi-rigid body system dynamics is a new branch formed by the development of new technologies since the 1960s, and its research methods differ from those of classical theory.

D'Alembert's Principle

The D'Alembert principle is a universal and effective method for studying the dynamics of non-free particle systems. This method introduces the concept of inertial force on the basis of Newton's laws of motion, and thus uses the method of studying equilibrium in statics to study the problem of imbalance in dynamics. ^[3]

Application of Kinetics

The study of dynamics has enabled people to grasp the laws of motion of objects and can provide better services to humans. For example, Newton discovered the law of gravity, explained Kepler's law, and opened the way for modern interstellar navigation, launching aircraft to inspect the moon, Mars, Venus, and so on.

Since the introduction of relativity in the early 20th century, the concept of space-time in Newtonian mechanics and other basic concepts of mechanical quantities have changed significantly. experiment

Celestial geodynamics The results also show that when the speed of the object is close to the speed of light, classical dynamics are completely unsuitable. However, in practical problems such as engineering, the moving speed of the macro objects in contact with them is far less than the speed of light. The research using Newtonian mechanics is not only accurate enough, but also much simpler than relativity calculations. Therefore, classical dynamics is still the basis for solving practical engineering problems.

In the currently studied mechanical systems, factors that need to be considered are gradually increasing, such as variable mass, non-integer, non-linear, non-conservative, plus feedback control, random factors, etc., which make the motion differential equations more and more complex and can be correct. There are fewer and fewer problems to solve, and many dynamic problems need to be solved approximately by numerical calculations. The application of micro, high-speed, and large-capacity electronic computers solves the difficulty of complex calculations.

At present, the research field of dynamic systems is still expanding. For example, increasing the heat and electricity to become system dynamics; increasing the activities of living systems to become biological dynamics, etc., which makes dynamics have further depth and breadth development of. ^[4]