Modern classical mechanics is a branch of physics. The history of the formation of analytical mechanics
Mechanics is a branch of physics that studies the simplest form of motion of matter - mechanical movement, which consists in changing the position of bodies or their parts over time. The fact that mechanical phenomena occur in space and time is reflected in any law of mechanics that explicitly or implicitly contains space-time relationships - distances and time intervals.
Mechanics sets itself two main tasks:
the study of various movements and the generalization of the results obtained in the form of laws with the help of which the nature of the movement in each specific case can be predicted. The solution of this problem led to the establishment by I. Newton and A. Einstein of the so-called dynamic laws;
finding common properties inherent in any mechanical system in the process of its movement. As a result of solving this problem, the laws of conservation of such fundamental quantities as energy, momentum, and angular momentum were discovered.
Dynamic laws and the laws of conservation of energy, momentum and angular momentum are the basic laws of mechanics and constitute the content of this chapter.
§one. Mechanical movement: basic concepts
Classical mechanics consists of three main sections - statics, kinematics and dynamics. In statics, the laws of the addition of forces and the conditions for the equilibrium of bodies are considered. In kinematics, a mathematical description of all kinds of mechanical motion is given, regardless of the reasons that cause it. In dynamics, the influence of the interaction between bodies on their mechanical motion is studied.
In practice, everything physical problems are solved approximately: real complex movement considered as a set of simple movements, a real object replaced by an idealized model this object, etc. For example, when considering the motion of the Earth around the Sun, one can neglect the size of the Earth. In this case, the description of the movement is greatly simplified - the position of the Earth in space can be determined by one point. Among the models of mechanics, the determining ones are material point and absolutely solid.
Material point (or particle) is a body, the shape and dimensions of which can be neglected under the conditions of this problem. Any body can be mentally divided into very big number parts, arbitrarily small compared to the dimensions of the whole body. Each of these parts can be considered as a material point, and the body itself - as a system of material points.
If the deformations of the body during its interaction with other bodies are negligible, then it is described by the model absolutely rigid body.
Absolutely rigid body (or rigid body) is a body, the distance between any two points of which does not change in the process of motion. In other words, this is a body, the shape and dimensions of which do not change during its movement. An absolutely rigid body can be considered as a system of material points rigidly interconnected.
The position of a body in space can only be determined in relation to some other bodies. For example, it makes sense to talk about the position of a planet in relation to the Sun, an aircraft or a ship in relation to the Earth, but one cannot indicate their position in space without regard to any particular body. An absolutely rigid body, which serves to determine the position of an object of interest to us, is called a reference body. To describe the movement of an object, a reference body is associated with any coordinate system, for example, a rectangular Cartesian coordinate system. The coordinates of an object allow you to set its position in space. The smallest number of independent coordinates that must be set to fully determine the position of the body in space is called the number of degrees of freedom. For example, a material point freely moving in space has three degrees of freedom: a point can make three independent movements along the axes of a Cartesian rectangular coordinate system. An absolutely rigid body has six degrees of freedom: to determine its position in space, three degrees of freedom are needed to describe translational motion along the coordinate axes and three to describe rotation about the same axes. The coordinate system is equipped with a clock to keep time.
The set of the reference body, the coordinate system associated with it and the set of clocks synchronized with each other form the reference frame.
classical mechanics- a type of mechanics (a branch of physics that studies the laws of change in the positions of bodies in space over time and the causes that cause it), based on Newton's laws and Galileo's principle of relativity. Therefore, it is often called Newtonian mechanics».
Classical mechanics is subdivided into:
- statics (which considers the equilibrium of bodies)
- kinematics (which studies geometric property movement without considering its causes)
- dynamics (which considers body movement).
There are several equivalent ways to formally describe classical mechanics mathematically:
- Lagrangian formalism
- Hamiltonian formalism
Classical mechanics gives very accurate results if its application is limited to bodies whose speeds are much less than the speed of light, and whose dimensions are much larger than the sizes of atoms and molecules. A generalization of classical mechanics to bodies moving at an arbitrary speed is relativistic mechanics, and to bodies whose dimensions are comparable to atomic ones - quantum mechanics. Quantum field theory considers quantum relativistic effects.
Nevertheless, classical mechanics retains its value because:
- it is much easier to understand and use than other theories
- in a wide range, it describes reality quite well.
Classical mechanics can be used to describe the motion of objects such as tops and baseballs, many astronomical objects (such as planets and galaxies), and sometimes even many microscopic objects such as molecules.
Classical mechanics is a self-consistent theory, that is, within its framework there are no statements that contradict each other. However, its combination with other classical theories, such as classical electrodynamics and thermodynamics, leads to insoluble contradictions. In particular, classical electrodynamics predicts that the speed of light is constant for all observers, which is inconsistent with classical mechanics. At the beginning of the 20th century, this led to the need to create a special theory of relativity. When considered together with thermodynamics, classical mechanics leads to the Gibbs paradox, in which it is impossible to accurately determine the amount of entropy, and to the ultraviolet catastrophe, in which a blackbody must radiate an infinite amount of energy. Attempts to solve these problems led to the emergence and development of quantum mechanics.
Basic concepts
Classical mechanics operates with several basic concepts and models. Among them should be highlighted:
Basic Laws
Galileo's principle of relativity
The basic principle on which classical mechanics is based is the principle of relativity, formulated on the basis of empirical observations by G. Galileo. According to this principle, there are infinitely many frames of reference in which a free body is at rest or moves with a constant speed in absolute value and direction. These frames of reference are called inertial and move relative to each other uniformly and rectilinearly. In all inertial frames of reference, the properties of space and time are the same, and all processes in mechanical systems obey the same laws. This principle can also be formulated as the absence of absolute reference systems, that is, reference systems that are somehow distinguished relative to others.
Newton's laws
Newton's three laws are the basis of classical mechanics.
Newton's second law is not enough to describe the motion of a particle. Additionally, a description of the force is required, obtained from consideration of the essence of the physical interaction in which the body participates.
Law of energy conservation
The law of conservation of energy is a consequence of Newton's laws for closed conservative systems, that is, systems in which only conservative forces act. From a more fundamental point of view, there is a relationship between the law of conservation of energy and the homogeneity of time, expressed by Noether's theorem.
Beyond the applicability of Newton's laws
Classical mechanics also includes descriptions of the complex motions of extended non-point objects. Euler's laws provide an extension of Newton's laws to this area. The concept of angular momentum is based on the same mathematical methods used to describe one-dimensional motion.
The equations of rocket motion expand the concept of velocity when an object's momentum changes over time to account for such effects as mass loss. There are two important alternative formulations of classical mechanics: Lagrange mechanics and Hamiltonian mechanics. These and other modern formulations, as a rule, bypass the concept of "power", and emphasize other physical quantities, such as energy or action, to describe mechanical systems.
The above expressions for momentum and kinetic energy are valid only in the absence of a significant electromagnetic contribution. In electromagnetism, Newton's second law for a wire carrying current is violated if it does not include the contribution electromagnetic field into the momentum of the system expressed in terms of the Poynting vector divided by c 2 , where c is the speed of light in free space.
Story
ancient time
Classical mechanics originated in antiquity mainly in connection with the problems that arose during construction. The first of the sections of mechanics to be developed was statics, the foundations of which were laid in the works of Archimedes in the 3rd century BC. e. He formulated the rule of the lever, the theorem on the addition of parallel forces, introduced the concept of center of gravity, laid the foundations of hydrostatics (Archimedes force).
Middle Ages
new time
17th century
18th century
19th century
In the 19th century, the development of analytical mechanics takes place in the works of Ostrogradsky, Hamilton, Jacobi, Hertz, and others. In the theory of vibrations, Routh, Zhukovsky, and Lyapunov developed a theory of the stability of mechanical systems. Coriolis developed the theory of relative motion by proving the acceleration theorem. In the second half of the 19th century, kinematics was separated into a separate section of mechanics.
Particularly significant in the 19th century were advances in continuum mechanics. Navier and Cauchy formulated the equations of elasticity theory in a general form. In the works of Navier and Stokes, differential equations of hydrodynamics were obtained taking into account the viscosity of the liquid. Along with this, there is a deepening of knowledge in the field of hydrodynamics of an ideal fluid: the works of Helmholtz on vortices, Kirchhoff, Zhukovsky and Reynolds on turbulence, and Prandtl on boundary effects appear. Saint-Venant developed a mathematical model describing the plastic properties of metals.
Newest time
In the 20th century, the interest of researchers switched to nonlinear effects in the field of classical mechanics. Lyapunov and Henri Poincaré laid the foundations for the theory of nonlinear oscillations. Meshchersky and Tsiolkovsky analyzed the dynamics of bodies of variable mass. Aerodynamics stands out from continuum mechanics, the foundations of which were developed by Zhukovsky. In the middle of the 20th century, a new direction in classical mechanics is actively developing - the theory of chaos. The issues of stability of complex dynamical systems also remain important.
Limitations of classical mechanics
Classical mechanics gives exact results for the systems we encounter in Everyday life. But her predictions become incorrect for systems approaching the speed of light, where it is replaced by relativistic mechanics, or for very small systems where the laws of quantum mechanics apply. For systems that combine both of these properties, relativistic mechanics is used instead of classical mechanics. quantum theory fields. For systems with a very large number of components, or degrees of freedom, classical mechanics also cannot be adequate, but methods of statistical mechanics are used.
Classical mechanics is widely used because, firstly, it is much simpler and easier to apply than the theories listed above, and, secondly, it has great possibilities for approximation and application for a very wide class of physical objects, starting from the usual, such as a spinning top or a ball, to large astronomical objects (planets, galaxies) and very microscopic ones (organic molecules).
Although classical mechanics is generally compatible with other "classical" theories such as classical electrodynamics and thermodynamics, there are some inconsistencies between these theories that were found in the late 19th century. They can be solved by methods more modern physics. In particular, the equations of classical electrodynamics are not invariant under Galilean transformations. The speed of light enters them as a constant, which means that classical electrodynamics and classical mechanics could only be compatible in one chosen frame of reference associated with the ether. However, experimental verification did not reveal the existence of the ether, which led to the creation of a special theory of relativity, in which the equations of mechanics were modified. The principles of classical mechanics are also inconsistent with some of the claims of classical thermodynamics, leading to the Gibbs paradox, according to which it is impossible to accurately determine entropy, and to the ultraviolet catastrophe, in which a black body must radiate an infinite amount of energy. To overcome these incompatibilities, quantum mechanics was created.
Notes
Internet links
- Video lecture 1. Physics: Classical mechanics (autumn 1999)// Massachusetts Lectures Institute of Technology: 8.01
Literature
- Arnold V.I. Avets A. Ergodic problems of classical mechanics. - RHD, 1999. - 284 p.
- B. M. Yavorsky, A. A. Detlaf. Physics for high school students and those entering universities. - M .: Academy, 2008. - 720 p. -( Higher education). - 34,000 copies. - ISBN 5-7695-1040-4
- Sivukhin D.V. General course physics. - 5th edition, stereotypical. - M .: Fizmatlit, 2006. - T. I. Mechanics. - 560 p. - ISBN 5-9221-0715-1
- A. N. MATVEEV Mechanics and the Theory of Relativity. - 3rd ed. - M .: ONYX 21st century: World and Education, 2003. - 432 p. - 5000 copies. - ISBN 5-329-00742-9
- C. Kittel, W. Knight, M. Ruderman Mechanics. Berkeley Physics Course. - M .: Lan, 2005. - 480 p. - (Textbooks for universities). - 2000 copies. - ISBN 5-8114-0644-4
Thus, the subject of study of classical mechanics is the laws and causes mechanical movement, understood as the interaction of macroscopic (consisting of a huge number of particles) physical bodies and their constituent parts, and the change in their position in space generated by this interaction, which occurs at sublight (nonrelativistic) speeds.
The place of classical mechanics in the system of physical sciences and the limits of its applicability are shown in Figure 1.
Figure 1. Scope of applicability of classical mechanics
Classical mechanics is subdivided into statics (which considers the equilibrium of bodies), kinematics (which studies the geometric property of motion without considering its causes), and dynamics (which considers the movement of bodies taking into account the causes that cause it).
There are several equivalent ways of formal mathematical description of classical mechanics: Newton's laws, Lagrange formalism, Hamiltonian formalism, Hamilton-Jacobi formalism.
When classical mechanics is applied to bodies much less than the speed of light and much larger than atoms and molecules, and at distances or conditions where the speed of gravity can be considered infinite, it gives exceptionally accurate results. Therefore, today, classical mechanics retains its significance, since it is much easier to understand and use than other theories, and describes everyday reality quite well. Classical mechanics can be used to describe the motion of a very wide class of physical objects: both ordinary objects in the macrocosm (such as a spinning top and a baseball), and objects of astronomical dimensions (such as planets and stars), and many microscopic objects.
Classical mechanics is the oldest of the physical sciences. Even in pre-antique times, people not only experienced the laws of mechanics, but also applied them in practice, designing the simplest mechanisms. Already in the Neolithic and Bronze Ages, a wheel appeared, a little later, a lever and an inclined plane were used. In the ancient period, the accumulated practical knowledge began to be generalized, the first attempts were made to define the basic concepts of mechanics, such as force, resistance, displacement, speed, and to formulate some of its laws. It was during the development of classical mechanics that the foundations were laid scientific method knowledge, which implies certain general rules for scientific reasoning about empirically observed phenomena, making assumptions (hypotheses) that explain these phenomena, building models that simplify the phenomena under study while maintaining their essential properties, forming systems of ideas or principles (theories) and their mathematical interpretation.
However, the qualitative formulation of the laws of mechanics began only in the 17th century AD. e., when Galileo Galilei discovered the kinematic law of addition of velocities and established the laws of free fall of bodies. A few decades after Galileo, Isaac Newton formulated the basic laws of dynamics. In Newtonian mechanics, the motion of bodies is considered at speeds much less than the speed of light in a vacuum. It is called classical or Newtonian mechanics, in contrast to relativistic mechanics, created at the beginning of the 20th century, mainly due to the work of Albert Einstein.
Modern classical mechanics as a research method natural phenomena uses their description with the help of a system of basic concepts and the construction on their basis of ideal models of real phenomena and processes.
Basic concepts of classical mechanics
Basic principles of classical mechanics
The issue of including methodological knowledge in the course of physics high school devoted to the works of famous domestic scientists, such as V.F. Efimenko, G.M. Golin, A.A. Bukh, V.G. Razumovsky, B.I. Spassky, V.V. , N.S. Purysheva and others. G.M. Golin singled out the following system of methodological knowledge:
- Scientific experiment and methods of experimental (empirical) knowledge.
- Physical theory and methods of theoretical knowledge.
- Core methodological ideas of physics.
- Basic regularities in the development of physics.
One of the elements of this system is the physical theory and methods of theoretical knowledge. Physical theory is an integral system of physical knowledge that fully describes a certain range of phenomena and is one of the structural elements of the physical picture of the world (see Table 1).
Table 1. The structure of the physical picture of the world
The school physics course is structured around four fundamental physical theories: classical mechanics, molecular kinetic theory, electrodynamics, quantum theory. The theoretical core of the school physics course embodies the four specified fundamental theories, specially adapted for the school course. “This allows us to single out general directions in the course of physics in the form of educational and methodological lines and then form all the material around these lines. Such a generalization educational material allows students to form adequate ideas about the structure of modern physics, as well as the implementation of the theoretical method of teaching…” . The generalization of educational material is aimed at ensuring the qualitative assimilation of the knowledge system, which is the scientific basis of general polytechnic education, at ensuring the effectiveness educational process and deep and integral perception of a certain field of knowledge; on the formation and development of a creative, scientific and theoretical way of thinking.
Table 2. Structure of the physical theory
Based on the work of V.F. Efimenko, V.V. Multanovsky singled out the following structural elements physical theory: foundation, core, consequences and interpretations (see Table 2). Within the framework of a school course in physics, the structure of classical mechanics (see Table 3) and molecular-kinetic theory can be most fully considered. It is not possible to fully reveal the structure of such a fundamental theory as classical electrodynamics (in particular, due to the insufficient mathematical apparatus of a schoolchild). However, in this case, the formation of students' knowledge about the structure of physical theory can be carried out on the example of a particular theory - the Drude-Lorentz theory (see Table 4).
CLASSICAL MECHANICS |
|||
Base |
Consequences |
Interpretation |
|
observation of phenomena (movement of bodies, free fall, pendulum swing...)
mat. point, absolute solid body
|
Newton's laws, abs. tv. body, law gravity
ZSE, ZSI, ZSMI
Long-range actions, independence of action of forces, Galilean relativity
Homogeneity and isotropy of space, homogeneity of time.
gravit. constant |
Discovery of the planets Neptune and Pluto |
Limits of applicability of the theory: macroscopic bodies |
Table 3. Structure of classical mechanics
CLASSICAL ELECTRON DRUDE-LOrentz theory |
|||
Base |
Consequences |
Interpretation |
|
1) Rikke's experience (1901); 2) Experience of Mandelstam and Papaleksi (1913); 3) Experience of Tolman and Stewart (1916). |
The main provisions of the theory: 1) The motion of electrons obeys the laws of classical mechanics. 2) Electrons do not interact with each other. 3) Electrons interact only with ions of the crystal lattice, this interaction is reduced to collision. 4) In the intervals between collisions, the electrons move freely. 5) Conduction electrons form an electron gas, like an ideal gas, "electron gas" obeys the laws of an ideal gas. |
|
Limits of applicability and shortcomings of the theory: the classical theory cannot explain the Dulong and Petit law, the temperature dependence of the resistivity of metals, and superconductivity. |
Table 4. The structure of the classical electronic theory of Drude-Lorentz
The structure of the physical theory presented in Table 4 can be used to structure the content of the general lesson on the topic “ Electricity in metals”, which is the first lesson in the study of the topic “Electric current in various media” in grade 10. Generalization and systematization of knowledge at the level of physical theory contributes to students' awareness of methodological knowledge, understanding of the logic of the cognition process. In this case, it is very important that the process of cognition appears before the students in dynamics. It is in this case that the methodological nature of knowledge can be most fully reflected. In accordance with this, it is advisable to deploy the educational material according to the stages of the cognition cycle: experimental facts > hypothesis (model) > theoretical consequences > experiment (see Table 5). In this case, the reference summary in the students' notebooks can be presented in the form of table 4.
Table 5. Generalization of educational material when studying the topic “Electric current in metals”
Consideration of the limits of applicability of the Drude-Lorentz theory will protect students from dogmatism in the study of physics. It is very important that the studied material is not considered by students as a complete scheme, devoid of contradictions. It is necessary that schoolchildren understand that absolute truth is not achievable, and the process of cognition is a constant striving for absolute truth through a series of relative truths replacing each other. Thus, the teacher brings them to an understanding of the essence of the methodological principle of correspondence. (Subsequently, we can touch on the content of another methodological principle - the principle of complementarity, pointing out that the Maxwell theory and the Drude-Lorentz theory describe the phenomenon of electrical conductivity from different points of view and thus complement each other.)
AT<annex 1 > a detailed plan-summary of a lesson-generalization on the topic “Electric current in metals” is presented, in<annex 2 > - a generalized plan for studying the section “Electric current in various media” and a generalized plan for studying physical theory, in<annex 3 > - computer presentation on the topic.
Literature
- Golin G.M. Questions of the methodology of physics in the course of high school. - M. Enlightenment, 1987.
- Manshinyan A.A. Theoretical basis creation and application of learning technologies. - M.: Prometheus, 1999. - 136 p.
- Efimenko V.F. Methodological issues of the school course in physics. - M .: Pedagogy, 1976. - 224 p.
- Multanovsky V.V. Physical interactions and the picture of the world in school course- M .: Education, 1977. - 168 p.
- Theory and methods of teaching physics at school: General issues: Proc. allowance for students. higher ped. textbook institutions / S.E. Kamenetsky, N.S. Purysheva, N.E. Vazheevskaya and others; Ed. S.E. Kamenetsky, N.S. Purysheva. - M.: Publishing Center "Academy", 2000. - 368 p.
Sir ISAAC NEWTON (January 4, 1643 - March 31, 1727) - an outstanding English scientist who laid the foundations of modern natural science, the creator of classical physics, a member of the Royal Society of London and its president (since 1703). Born in Woolsthorpe. Graduated from Cambridge University in 1665. In March-June 1666, Newton visited Cambridge. However, in the summer, a new wave of plague forced him to leave home again. Finally, in early 1667, the epidemic subsided, and in April Newton returned to Cambridge. On October 1, he was elected a Fellow of Trinity College, and in 1668 became a master. He was given a spacious private room to live in, a salary of £2 a year, and a group of students with whom he conscientiously studied standard subjects for several hours a week. However, neither then nor later did Newton become famous as a teacher, his lectures were poorly attended. one
Having consolidated his position, Newton traveled to London, where shortly before, in 1660, the Royal Society of London was established - an authoritative organization of prominent scientists, one of the first Academies of Sciences. The printed organ of the Royal Society was the journal Philosophical Transactions.
In 1669, mathematical works began to appear in Europe using expansions into infinite series. Although the depth of these discoveries was no match for Newton's, Barrow insisted that his student fix his priority in this matter. 2 ______________________________
1. https://ru.wikipedia.org/
2. Akroyd P. “Isaac Newton. Biography". - M.: Hummingbird, Azbuka-Atticus, 2011
Newton wrote a brief but fairly complete summary of this part of his discoveries, which he called "Analysis using equations with an infinite number members." Barrow sent this treatise to London. Newton asked Barrow not to reveal the name of the author of the work (but he still let it slip). "Analysis" spread among specialists and gained some notoriety in England and beyond.
In the same year, Barrow accepted the invitation of the king to become a court chaplain and left teaching. On October 29, 1669, the 26-year-old Newton was elected as his successor, professor of mathematics and optics at Trinity College, with a high salary of £100 a year. Barrow left Newton an extensive alchemical laboratory; during this period, Newton became seriously interested in alchemy, conducted a lot of chemical experiments. Newton formulated the basic laws of classical mechanics, discovered the law of universal gravitation, the dispersion of light, developed the corpuscular theory of light, and developed differential and integral calculus. Summarizing the results of the research of his predecessors in the field of mechanics and his own, Newton created a huge work "Mathematical Principles of Natural Philosophy" ("Beginnings"), published in 1687. "Beginnings" contained the basic concepts of classical mechanics, in particular the concepts: mass, momentum, force, acceleration, centripetal force and three laws of motion. In the same work, his law of universal gravitation is given, on the basis of which Newton explained the motion of celestial bodies and created the theory of gravitation. 1 The discovery of this law finally confirmed the victory of the teachings of Copernicus. He showed that Kepler's three laws follow from the law of universal gravitation; explained the features of the movement of the moon, the phenomenon of the procession; developed the theory of the figure of the Earth, noting that it should be compressed at the poles, _____________________________
1. Akroyd P. “Isaac Newton. Biography". - M.: Hummingbird, Azbuka-Atticus, 2011
the theory of ebbs and flows; considered the problem of creating artificial satellite Lands, etc. Newton developed the law of resistance and the basic law of internal friction in liquids and gases, gave a formula for the speed of wave propagation.