Fundamentals of classical mechanics
Mechanics- a branch of physics that studies the laws of mechanical motion of bodies.
Body– a tangible material object.
Mechanical movement- change provisions body or its parts in space over time.
Aristotle represented this type of movement as a direct change by a body of its place relative to other bodies, since in his physics the material world was inextricably linked with space and existed together with it. He considered time to be a measure of the movement of a body. Subsequent changes in views on the nature of movement led to the gradual separation of space and time from physical bodies. Finally, absolutization Newton's concept of space and time generally took them beyond the limits of possible experience.
However, this approach made it possible by the end of the 18th century to build a complete system mechanics, now called classical. Classicism is that it:
1) describes most mechanical phenomena in the macrocosm using a small number of initial definitions and axioms;
2) strictly mathematically justified;
3) is often used in more specific areas of science.
Experience shows that classical mechanics applies to the description of the motion of bodies with speeds v<< с ≈ 3·10 8 м/с. Ее основные разделы:
1) statics studies the conditions of equilibrium of bodies;
2) kinematics - the movement of bodies without taking into account its causes;
3) dynamics - the influence of the interaction of bodies on their movement.
Basic mechanics concepts:
1) A mechanical system is a mentally identified set of bodies that are essential in a given task.
2) A material point is a body whose shape and dimensions can be neglected within the framework of this problem. A body can be represented as a system of material points.
3) An absolutely rigid body is a body whose distance between any two points does not change under the conditions of a given problem.
4) The relativity of motion lies in the fact that a change in the position of a body in space can only be established in relation to some other bodies.
5) Reference body (RB) – an absolutely rigid body relative to which motion is considered in this problem.
6) Frame of reference (FR) = (TO + SC + clock). The origin of the coordinate system (OS) is combined with some TO point. Clocks measure periods of time.
Cartesian SK:
Figure 5
Position material point M is described radius vector of the point, – its projections on the coordinate axes.
If you set the initial time t 0 = 0, then the movement of point M will be described vector function or three scalar functions x(t),y(t), z(t).
Linear characteristics of the movement of a material point:
1) trajectory – line of motion of a material point (geometric curve),
2) path ( S) – the distance traveled along it in a period of time,
3) moving,
4) speed,
5) acceleration.
Any motion of a rigid body can be reduced to two main types - progressive And rotational around a fixed axis.
Forward movement- one in which the straight line connecting any two points of the body remains parallel to its original position. Then all points move equally, and the movement of the whole body can be described movement of one point.
Rotation around a fixed axis - a movement in which there is a straight line rigidly connected to the body, all points of which remain motionless in a given reference frame. The trajectories of the remaining points are circles with centers on this line. In this case it is convenient angular characteristics movements that are the same for all points of the body.
Angular characteristics of the movement of a material point:
1) angle of rotation (angular path), measured in radians [rad], where r– radius of the point’s trajectory,
2) angular displacement, the module of which is the angle of rotation over a short period of time dt,
3) angular velocity,
4) angular acceleration.
Figure 6
Relationship between angular and linear characteristics:
Dynamics uses concept of strength, measured in newtons (H), as a measure of the influence of one body on another. This impact is the cause of movement.
The principle of superposition of forces– the resulting effect of the influence of several bodies on a body is equal to the sum of the effects of the influence of each of these bodies separately. The quantity is called the resultant force and characterizes the equivalent effect on the body n tel.
Newton's laws generalize experimental facts of mechanics.
Newton's 1st law. There are reference systems relative to which a material point maintains a state of rest or uniform rectilinear motion in the absence of force acting on it, i.e. if , then .
Such motion is called motion by inertia or inertial motion, and therefore frames of reference in which Newton's 1st law is satisfied are called inertial(ISO).
Newton's 2nd law. , where is the momentum of the material point, m– its mass, i.e. if , then and, consequently, the movement will no longer be inertial.
Newton's 3rd law. When two material points interact, forces arise and are applied to both points, and .
Mechanics is a branch of physics that studies one of the simplest and most common forms of motion in nature, called mechanical motion.
Mechanical movement consists in changing the position of bodies or their parts relative to each other over time. Thus, mechanical motion is performed by planets revolving in closed orbits around the Sun; various bodies moving on the surface of the Earth; electrons moving under the influence of an electromagnetic field, etc. Mechanical motion is present in other more complex forms of matter as an integral, but not exhaustive part.
Depending on the nature of the objects being studied, mechanics is divided into the mechanics of a material point, the mechanics of a solid body and the mechanics of a continuous medium.
The principles of mechanics were first formulated by I. Newton (1687) on the basis of an experimental study of the motion of macrobodies with small velocities compared to the speed of light in a vacuum (3·10 8 m/s).
Macrobodies are called ordinary bodies that surround us, that is, bodies consisting of a huge number of molecules and atoms.
Mechanics, which studies the movement of macrobodies at speeds much lower than the speed of light in a vacuum, is called classical.
Classical mechanics is based on Newton’s following ideas about the properties of space and time.
Any physical process occurs in space and time. This is evident from the fact that in all areas of physical phenomena, each law explicitly or implicitly contains space-time quantities - distances and time intervals.
Space, which has three dimensions, obeys Euclidean geometry, that is, it is flat.
Distances are measured by scales, the main property of which is that two scales that once coincide in length always remain equal to each other, that is, they coincide with each subsequent overlap.
Time intervals are measured in hours, and the role of the latter can be performed by any system that performs a repeating process.
The main feature of the ideas of classical mechanics about the sizes of bodies and time intervals is their absoluteness: the scale always has the same length, no matter how it moves relative to the observer; two clocks that have the same speed and are once brought into line with each other show the same time regardless of how they move.
Space and time have remarkable properties symmetry, imposing restrictions on the occurrence of certain processes in them. These properties have been established experimentally and seem at first glance so obvious that there seems to be no need to isolate them and deal with them. Meanwhile, without spatial and temporal symmetry, no physical science could have arisen or developed.
It turns out that space homogeneously And isotropically, and time - homogeneously.
The homogeneity of space consists in the fact that the same physical phenomena under the same conditions occur in the same way in different parts of space. All points in space are thus completely indistinguishable, equal in rights, and any of them can be taken as the origin of the coordinate system. The homogeneity of space is manifested in the law of conservation of momentum.
Space also has isotropy: the same properties in all directions. The isotropy of space is manifested in the law of conservation of angular momentum.
The homogeneity of time lies in the fact that all moments of time are also equal, equivalent, that is, the occurrence of identical phenomena in the same conditions is the same, regardless of the time of their implementation and observation.
The uniformity of time is manifested in the law of conservation of energy.
Without these properties of homogeneity, a physical law established in Minsk would be unfair in Moscow, and one discovered today in the same place could be unfair tomorrow.
Classical mechanics recognizes the validity of the Galileo-Newton law of inertia, according to which a body, not subject to the influence of other bodies, moves rectilinearly and uniformly. This law asserts the existence of inertial frames of reference in which Newton's laws (as well as Galileo's principle of relativity) are satisfied. Galileo's principle of relativity states that all inertial frames of reference are mechanically equivalent to each other, all the laws of mechanics are the same in these reference frames, or, in other words, are invariant under Galilean transformations expressing the spatio-temporal relationship of any event in different inertial reference frames. Galilean transformations show that the coordinates of any event are relative, that is, they have different values in different reference systems; the moments in time when the event occurred are the same in different systems. The latter means that time flows in the same way in different reference systems. This circumstance seemed so obvious that it was not even stated as a special postulate.
In classical mechanics, the principle of long-range action is observed: the interactions of bodies propagate instantly, that is, with an infinitely high speed.
Depending on the speeds at which bodies move and the dimensions of the bodies themselves, mechanics is divided into classical, relativistic, and quantum.
As already indicated, the laws classical mechanics applicable only to the movement of macrobodies, the mass of which is much greater than the mass of an atom, at low speeds compared to the speed of light in a vacuum.
Relativistic mechanics considers the movement of macrobodies at speeds close to the speed of light in a vacuum.
Quantum mechanics- mechanics of microparticles moving at speeds much lower than the speed of light in a vacuum.
Relativistic quantum mechanics - the mechanics of microparticles moving at speeds approaching the speed of light in a vacuum.
To determine whether a particle belongs to macroscopic ones and whether classical formulas are applicable to it, you need to use Heisenberg's uncertainty principle. According to quantum mechanics, real particles can be characterized in terms of position and momentum only with some accuracy. The limit of this accuracy is determined as follows
Where
ΔX - coordinate uncertainty;
ΔP x - uncertainty of projection onto the momentum axis;
h is Planck’s constant equal to 1.05·10 -34 J·s;
"≥" - greater than magnitude, order...
Replacing momentum with the product of mass and velocity, we can write
From the formula it is clear that the smaller the mass of the particle, the less certain its coordinates and speed become. For macroscopic bodies, the practical applicability of the classical method of describing motion is beyond doubt. Let us assume, for example, that we are talking about the movement of a ball with a mass of 1 g. Usually the position of the ball can practically be determined with an accuracy of a tenth or a hundredth of a millimeter. In any case, it hardly makes sense to talk about an error in determining the position of a ball that is smaller than the size of an atom. Let us therefore put ΔX=10 -10 m. Then from the uncertainty relation we find
The simultaneous smallness of the values of ΔX and ΔV x is proof of the practical applicability of the classical method of describing the motion of macrobodies.
Let's consider the movement of an electron in a hydrogen atom. The electron mass is 9.1·10 -31 kg. The error in the position of the electron ΔX in any case should not exceed the size of the atom, that is, ΔX<10 -10 м. Но тогда из соотношения неопределенностей получаем
This value is even greater than the speed of an electron in an atom, which is an order of magnitude equal to 10 6 m/s. In this situation, the classical picture of movement loses all meaning.
Mechanics are divided into kinematics, statics and dynamics. Kinematics describes the movement of bodies without being interested in the reasons that determined this movement; statics considers the conditions of equilibrium of bodies; dynamics studies the movement of bodies in connection with those reasons (interactions between bodies) that determine this or that nature of movement.
The real movements of bodies are so complex that when studying them, it is necessary to abstract from details that are unimportant for the movement under consideration (otherwise the problem would become so complicated that it would be practically impossible to solve it). For this purpose, concepts (abstractions, idealizations) are used, the applicability of which depends on the specific nature of the problem we are interested in, as well as on the degree of accuracy with which we want to obtain the result. Among these concepts, an important role is played by the concepts material point, system of material points, absolutely rigid body.
A material point is a physical concept with the help of which the translational motion of a body is described, if only its linear dimensions are small in comparison with the linear dimensions of other bodies within the given accuracy of determining the coordinates of the body, and the mass of the body is assigned to it.
In nature, material points do not exist. One and the same body, depending on the conditions, can be considered either as a material point or as a body of finite dimensions. Thus, the Earth moving around the Sun can be considered a material point. But when studying the rotation of the Earth around its axis, it can no longer be considered a material point, since the nature of this movement is significantly influenced by the shape and size of the Earth, and the path traversed by any point on the earth’s surface in a time equal to the period of its revolution around its axis is comparable with the linear dimensions of the globe. An airplane can be considered as a material point if we study the movement of its center of mass. But if it is necessary to take into account the influence of the environment or determine the forces in individual parts of the aircraft, then we must consider the aircraft as an absolutely rigid body.
An absolutely rigid body is a body whose deformations can be neglected under the conditions of a given problem.
A system of material points is a collection of bodies under consideration that represent material points.
The study of the motion of an arbitrary system of bodies comes down to the study of a system of interacting material points. It is natural, therefore, to begin the study of classical mechanics with the mechanics of one material point, and then move on to the study of a system of material points.
INTRODUCTION
Physics is a science of nature that studies the most general properties of the material world, the most general forms of motion of matter that underlie all natural phenomena. Physics establishes the laws that these phenomena obey.
Physics also studies the properties and structure of material bodies and indicates ways of practical use of physical laws in technology.
In accordance with the variety of forms of matter and its movement, physics is divided into a number of sections: mechanics, thermodynamics, electrodynamics, physics of vibrations and waves, optics, physics of the atom, nucleus and elementary particles.
At the intersection of physics and other natural sciences, new sciences arose: astrophysics, biophysics, geophysics, physical chemistry, etc.
Physics is the theoretical basis of technology. The development of physics served as the foundation for the creation of such new branches of technology as space technology, nuclear technology, quantum electronics, etc. In turn, the development of technical sciences contributes to the creation of completely new methods of physical research, which determine the progress of physics and related sciences.
PHYSICAL FOUNDATIONS OF CLASSICAL MECHANICS
I. Mechanics. General concepts
Mechanics is a branch of physics that examines the simplest form of motion of matter - mechanical motion.
Mechanical motion is understood as a change in the position of the body being studied in space over time relative to a certain goal or system of bodies conventionally considered motionless. Such a system of bodies together with a clock, for which any periodic process can be chosen, is called reference system(S.O.). S.O. often chosen for reasons of convenience.
For a mathematical description of movement with S.O. They associate a coordinate system, often rectangular.
The simplest body in mechanics is a material point. This is a body whose dimensions can be neglected in the conditions of the present problem.
Any body whose dimensions cannot be neglected is considered as a system of material points.
Mechanics are divided into kinematics, which deals with the geometric description of motion without studying its causes, dynamics, which studies the laws of motion of bodies under the influence of forces, and statics, which studies the conditions of equilibrium of bodies.
2. Kinematics of a point
Kinematics studies the spatiotemporal movement of bodies. It operates with such concepts as displacement, path, time t, speed, acceleration.
The line that a material point describes during its movement is called a trajectory. According to the shape of the movement trajectories, they are divided into rectilinear and curvilinear. Vector , connecting the initial I and final 2 points is called movement (Fig. I.I).
Each moment of time t has its own radius vector:
Thus, the movement of a point can be described by a vector function.
which we define vector way of specifying movement, or three scalar functions
x= x(t); y= y(t); z= z(t) , (1.2)
which are called kinematic equations. They determine the movement task coordinate way.
The movement of a point will also be determined if for each moment of time the position of the point on the trajectory is established, i.e. addiction
It determines the movement task natural way.
Each of these formulas represents law movement of the point.
3. Speed
If the moment of time t 1 corresponds to the radius vector , and , then during the interval the body will receive displacement . In this case average speedt is the quantity
which, in relation to the trajectory, represents a secant passing through points I and 2. Speed at time t is called a vector
From this definition it follows that the speed at each point of the trajectory is directed tangentially to it. From (1.5) it follows that the projections and magnitude of the velocity vector are determined by the expressions:
If the law of motion (1.3) is given, then the magnitude of the velocity vector will be determined as follows:
Thus, knowing the law of motion (I.I), (1.2), (1.3), you can calculate the vector and modulus of the doctor of speed and, conversely, knowing the speed from formulas (1.6), (1.7), you can calculate the coordinates and path.
4. Acceleration
During arbitrary movement, the velocity vector continuously changes. The quantity characterizing the rate of change of the velocity vector is called acceleration.
If in. moment of time t 1 is the speed of the point, and at t 2 - , then the speed increment will be (Fig. 1.2). The average acceleration in this case
and instantaneous
For the projection and acceleration module we have: , (1.10)
If a natural method of movement is given, then acceleration can be determined this way. The speed changes in magnitude and direction, the speed increment is divided into two quantities; - directed along (increase in speed in magnitude) and - directed perpendicularly (increment in speed in direction), i.e. = + (Fig. I.З). From (1.9) we obtain:
Tangential (tangential) acceleration characterizes the rate of change in magnitude (1.13)
normal (centripetal acceleration) characterizes the speed of change in direction. To calculate a n consider
OMN and MPQ under the condition of small movement of the point along the trajectory. From the similarity of these triangles we find PQ:MP=MN:OM:
The total acceleration in this case is determined as follows:
5. Examples
I. Equally variable rectilinear motion. This is motion with constant acceleration() . From (1.8) we find
or where v 0 - speed at time t 0 . Believing t 0 =0, we find , and the distance traveled S from formula (I.7):
Where S 0 is a constant determined from the initial conditions.
2. Uniform movement in a circle. In this case, the speed changes only in direction, that is, centripetal acceleration.
I. Basic concepts
The movement of bodies in space is the result of their mechanical interaction with each other, as a result of which a change in the movement of bodies or their deformation occurs. As a measure of mechanical interaction in dynamics, a quantity is introduced - force. For a given body, force is an external factor, and the nature of the movement depends on the properties of the body itself - compliance with external influences exerted on it or the degree of inertia of the body. The measure of inertia of a body is its mass T, depending on the amount of body matter.
Thus, the basic concepts of mechanics are: moving matter, space and time as forms of existence of moving matter, mass as a measure of inertia of bodies, force as a measure of mechanical interaction between bodies. The relationships between these concepts are determined by laws! movements that were formulated by Newton as a generalization and clarification of experimental facts.
2. Laws of mechanics
1st law. Every body maintains a state of rest or uniform rectilinear motion as long as external influences do not change this state. The first law contains the law of inertia, as well as the definition of force as a cause that violates the inertial state of the body. To express it mathematically, Newton introduced the concept of momentum or momentum of a body:
then if
2nd law. The change in momentum is proportional to the applied force and occurs in the direction of action of this force. Selecting units of measurement m and so that the proportionality coefficient is equal to unity, we get
If when moving m= const , That
In this case, the 2nd law is formulated as follows: force is equal to the product of the body’s mass and its acceleration. This law is the basic law of dynamics and allows us to find the law of motion of bodies based on given forces and initial conditions. 3rd law. The forces with which two bodies act on each other are equal and directed in opposite directions, i.e., (2.4)
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Classical mechanics (Newtonian mechanics)
The birth of physics as a science is associated with the discoveries of G. Galileo and I. Newton. Particularly significant is the contribution of I. Newton, who wrote down the laws of mechanics in the language of mathematics. I. Newton outlined his theory, which is often called classical mechanics, in his work “Mathematical Principles of Natural Philosophy” (1687).
The basis of classical mechanics is made up of three laws and two provisions regarding space and time.
Before considering I. Newton's laws, let us recall what a reference system and an inertial reference system are, since I. Newton's laws are not satisfied in all reference systems, but only in inertial reference systems.
A reference system is a coordinate system, for example, rectangular Cartesian coordinates, supplemented by a clock located at each point of a geometrically solid medium. A geometrically solid medium is an infinite set of points, the distances between which are fixed. In I. Newton's mechanics, it is assumed that time flows regardless of the position of the clock, i.e. The clocks are synchronized and therefore time flows the same in all reference frames.
In classical mechanics, space is considered Euclidean, and time is represented by the Euclidean straight line. In other words, I. Newton considered space absolute, i.e. it is the same everywhere. This means that non-deformable rods with divisions marked on them can be used to measure lengths. Among the reference systems, we can distinguish those systems that, due to taking into account a number of special dynamic properties, differ from the rest.
The reference system in relation to which the body moves uniformly and rectilinearly is called inertial or Galilean.
The fact of the existence of inertial reference systems cannot be verified experimentally, since in real conditions it is impossible to isolate a part of matter and isolate it from the rest of the world so that the movement of this part of matter is not affected by other material objects. To determine in each specific case whether the reference frame can be taken as inertial, it is checked whether the velocity of the body is conserved. The degree of this approximation determines the degree of idealization of the problem.
For example, in astronomy, when studying the motion of celestial bodies, the Cartesian ordinate system is often taken as an inertial reference system, the origin of which is at the center of mass of some “fixed” star, and the coordinate axes are directed to other “fixed” stars. In fact, stars move at high speeds relative to other celestial objects, so the concept of a “fixed” star is relative. But due to the large distances between the stars, the position we have given is sufficient for practical purposes.
For example, the best inertial reference system for the Solar System will be one whose origin coincides with the center of mass of the Solar System, which is practically located at the center of the Sun, since more than 99% of the mass of our planetary system is concentrated in the Sun. The coordinate axes of the reference system are directed to distant stars, which are considered stationary. Such a system is called heliocentric.
I. Newton formulated the statement about the existence of inertial reference systems in the form of the law of inertia, which is called Newton’s first law. This law states: Every body is in a state of rest or uniform rectilinear motion until the influence of other bodies forces it to change this state.
Newton's first law is by no means obvious. Before G. Galileo, it was believed that this effect does not determine the change in speed (acceleration), but the speed itself. This opinion was based on facts known from everyday life, such as the need to continuously push a cart moving along a horizontal, level road so that its movement does not slow down. We now know that by pushing a cart, we balance the force exerted on it by friction. But without knowing this, it is easy to come to the conclusion that the impact is necessary to maintain the movement unchanged.
Newton's second law states: rate of change of particle momentum equal to the force acting on the particle:
Where T- weight; t- time; A-acceleration; v- velocity vector; p = mv- impulse; F- force.
By force is called a vector quantity that characterizes the influence on a given body from other bodies. The modulus of this value determines the intensity of the impact, and the direction coincides with the direction of the acceleration imparted to the body by this impact.
Weight is a measure of the inertia of a body. Under inertia understand the intractability of the body to the action of force, i.e. the property of a body to resist a change in speed under the influence of a force. In order to express the mass of a certain body as a number, it is necessary to compare it with the mass of the reference body, taken as a unit.
Formula (3.1) is called the equation of particle motion. Expression (3.2) is the second formulation of Newton’s second law: the product of a particle's mass and its acceleration is equal to the force that acts on the particle.
Formula (3.2) is also valid for extended bodies if they move translationally. If several forces act on a body, then under the force F in formulas (3.1) and (3.2) their resultant is implied, i.e. sum of forces.
From (3.2) it follows that when F= 0 (i.e. the body is not affected by other bodies) acceleration A is equal to zero, so the body moves rectilinearly and uniformly. Thus, Newton's first law is, as it were, included in the second law as its special case. But Newton's first law is formed independently of the second, since it contains a statement about the existence of inertial reference systems in nature.
Equation (3.2) has such a simple form only with a consistent choice of units for measuring force, mass and acceleration. With an independent choice of units of measurement, Newton's second law is written as follows:
Where To - proportionality factor.
The influence of bodies on each other is always in the nature of interaction. In the event that the body A affects the body IN with force FBA then the body IN affects the body And with by force F AB .
Newton's third law states that the forces with which two bodies interact are equal in magnitude and opposite in direction, those.
Therefore, forces always arise in pairs. Note that the forces in formula (3.4) are applied to different bodies, and therefore they cannot balance each other.
Newton's third law, like the first two, is satisfied only in inertial frames of reference. In non-inertial reference systems it is not valid. In addition, deviations from Newton's third law will be observed in bodies that move at speeds close to the speed of light.
It should be noted that all three of Newton's laws appeared as a result of generalization of data from a large number of experiments and observations and are therefore empirical laws.
In Newtonian mechanics, not all reference systems are equal, since inertial and non-inertial reference systems differ from each other. This inequality indicates the lack of maturity of classical mechanics. On the other hand, all inertial frames of reference are equal and in each of them Newton's laws are the same.
G. Galileo in 1636 established that in an inertial frame of reference, no mechanical experiments can determine whether it is at rest or moving uniformly and rectilinearly.
Let us consider two inertial frames of reference N And N", and the system jV" moves relative to the system N along the axis X at constant speed v(Fig. 3.1).
Rice. 3.1.
We will start counting time from the moment when the origin of coordinates O and o" coincided. In this case, the coordinates X And X" arbitrarily taken point M will be related by the expression x = x" + vt. With our choice of coordinate axes y - y z~ Z- In Newtonian mechanics it is assumed that time flows the same in all reference systems, i.e. t = t". Consequently, we received a set of four equations:
Equations (3.5) are called Galilean transformations. They make it possible to move from the coordinates and time of one inertial reference system to the coordinates and time of another inertial reference system. Let us differentiate with respect to time / the first equation (3.5), keeping in mind that t = t therefore the derivative with respect to t will coincide with the derivative with respect to G. We get:
The derivative is the projection of the particle's velocity And in system N
per axis X of this system, and the derivative is the projection of the particle velocity O"in system N"on the axis X"of this system. Therefore we get
Where v = v x =v X "- projection of the vector onto the axis X coincides with the projection of the same vector onto the axis*".
Now we differentiate the second and third equations (3.5) and get:
Equations (3.6) and (3.7) can be replaced by one vector equation
Equation (3.8) can be considered either as a formula for converting the particle velocity from the system N" into the system N, or as the law of addition of speeds: the speed of a particle relative to the system Y is equal to the sum of the speed of the particle relative to the system N" and system speed N" relative to the system N. Let us differentiate equation (3.8) with respect to time and obtain:
therefore, particle accelerations relative to systems N and UU are the same. Force F, N, equal to force F", which acts on a particle in the system N", those.
Relationship (3.10) will be satisfied, since the force depends on the distances between a given particle and the particles interacting with it (as well as on the relative velocities of the particles), and these distances (and velocities) in classical mechanics are assumed to be the same in all inertial frames of reference. Mass also has the same numerical value in all inertial frames of reference.
From the above reasoning it follows that if the relation is satisfied ta = F, then the equality will be satisfied ta = F". Reference systems N And N" were taken arbitrarily, so the result means that the laws of classical mechanics are the same for all inertial frames of reference. This statement is called Galileo's principle of relativity. We can say it differently: Newton's laws of mechanics are invariant under Galileo's transformations.
Quantities that have the same numerical value in all reference systems are called invariant (from lat. invariantis- unchanging). Examples of such quantities are electric charge, mass, etc.
Equations whose form does not change during such a transition are also called invariant with respect to the transformation of coordinates and time when moving from one inertial reference system to another. The quantities that enter into these equations may change when moving from one reference system to another, but the formulas that express the relationship between these quantities remain unchanged. Examples of such equations are the laws of classical mechanics.
- By particle we mean a material point, i.e. a body whose dimensions can be neglected compared to the distance to other bodies.
The emergence of classical mechanics was the beginning of the transformation of physics into a strict science, that is, a system of knowledge that asserts the truth, objectivity, validity and verifiability of both its initial principles and its final conclusions. This emergence took place in the 16th-17th centuries and is associated with the names of Galileo Galilei, Rene Descartes and Isaac Newton. It was they who carried out the “mathematization” of nature and laid the foundations for an experimental-mathematical view of nature. They presented nature as a set of “material” points that have spatial-geometric (shape), quantitative-mathematical (number, magnitude) and mechanical (motion) properties and connected by cause-and-effect relationships that can be expressed in mathematical equations.
The beginning of the transformation of physics into a strict science was laid by G. Galileo. Galileo formulated a number of fundamental principles and laws of mechanics. Namely:
- principle of inertia, according to which when a body moves along a horizontal plane without encountering any resistance to movement, then its movement is uniform and would continue constantly if the plane extended in space without end;
- principle of relativity, according to which in inertial systems all the laws of mechanics are the same and there is no way, being inside, to determine whether it moves rectilinearly and uniformly or is at rest;
- principle of conservation of speeds and preservation of spatial and time intervals during the transition from one inertial system to another. This is famous Galilean transformation.
Mechanics received a holistic view of a logically and mathematically organized system of basic concepts, principles and laws in the works of Isaac Newton. First of all, in the work “Mathematical Principles of Natural Philosophy” In this work, Newton introduces the concepts: weight, or amount of matter, inertia, or the property of a body to resist changes in its state of rest or movement, weight, as a measure of mass, force, or an action performed on a body to change its condition.
Newton distinguished between absolute (true, mathematical) space and time, which do not depend on the bodies in them and are always equal to themselves, and relative space and time - moving parts of space and measurable durations of time.
A special place in Newton's concept is occupied by the doctrine of gravity or gravity, in which he combines the movement of “celestial” and terrestrial bodies. This teaching includes the statements:
The gravity of a body is proportional to the amount of matter or mass contained in it;
Gravity is proportional to mass;
Gravity or gravity and is that force which acts between the Earth and the Moon in inverse proportion to the square of the distance between them;
This gravitational force acts between all material bodies at a distance.
Regarding the nature of gravity, Newton said: “I invent no hypotheses.”
Galileo-Newton mechanics, developed in the works of D. Alembert, Lagrange, Laplace, Hamilton... eventually received a harmonious form that determined the physical picture of the world of that time. This picture was based on the principles of self-identity of the physical body; its independence from space and time; determinacy, that is, a strict unambiguous cause-and-effect relationship between specific states of physical bodies; reversibility of all physical processes.
Thermodynamics.
Studies of the process of converting heat into work and back, carried out in the 19th century by S. Kalno, R. Mayer, D. Joule, G. Hemholtz, R. Clausius, W. Thomson (Lord Kelvin), led to the conclusions about which R. Mayer wrote: “Motion, heat..., electricity are phenomena that are measured by each other and transform into each other according to certain laws.” Hemholtz generalizes this statement of Mayer into the conclusion: “The sum of the tense and living forces existing in nature is constant.” William Thomson clarified the concepts of “intense and living forces” to the concepts of potential and kinetic energy, defining energy as the ability to do work. R. Clausius summarized these ideas in the formulation: “The energy of the world is constant.” Thus, through the joint efforts of the physics community, a fundamental principle for all physical knowledge of the law of conservation and transformation of energy.
Research into the processes of conservation and transformation of energy led to the discovery of another law - law of increasing entropy. “The transition of heat from a colder body to a warmer one,” wrote Clausius, “cannot take place without compensation.” Clausius called the measure of the ability of heat to transform entropy. The essence of entropy is expressed in the fact that in any isolated system processes must proceed in the direction of converting all types of energy into heat while simultaneously equalizing the temperature differences existing in the system. This means that real physical processes proceed irreversibly. The principle that asserts the tendency of entropy to a maximum is called the second law of thermodynamics. The first principle is the law of conservation and transformation of energy.
The principle of increasing entropy posed a number of problems to physical thought: the relationship between the reversibility and irreversibility of physical processes, the formality of conservation of energy, which is not capable of doing work when the temperature of bodies is homogeneous. All this required a deeper justification of the principles of thermodynamics. First of all, the nature of heat.
An attempt at such a substantiation was made by Ludwig Boltzmann, who, based on the molecular-atomic idea of the nature of heat, came to the conclusion that statistical the nature of the second law of thermodynamics, since due to the huge number of molecules that make up macroscopic bodies and the extreme speed and randomness of their movement, we observe only average values. Determining average values is a task in probability theory. At maximum temperature equilibrium, the chaos of molecular motion is also maximum, in which all order disappears. The question arises: can and, if so, how, can order emerge again from chaos? Physics will be able to answer this only in a hundred years, introducing the principle of symmetry and the principle of synergy.
Electrodynamics.
By the middle of the 19th century, the physics of electrical and magnetic phenomena had reached a certain completion. A number of the most important laws of Coulomb, Ampere's law, the law of electromagnetic induction, the laws of direct current, etc. were discovered. All these laws were based on long-range principle. The exception was the views of Faraday, who believed that electrical action is transmitted through a continuous medium, that is, based on short range principle. Based on Faraday's ideas, the English physicist J. Maxwell introduces the concept electromagnetic field and describes the state of matter “discovered” by him in his equations. “... The electromagnetic field,” writes Maxwell, “is that part of space that contains and surrounds bodies that are in an electric or magnetic state.” By combining the electromagnetic field equations, Maxwell obtains the wave equation, from which the existence of electromagnetic waves, the speed of propagation of which in the air is equal to the speed of light. The existence of such electromagnetic waves was experimentally confirmed by the German physicist Heinrich Hertz in 1888.
In order to explain the interaction of electromagnetic waves with matter, the German physicist Hendrik Anton Lorenz hypothesized the existence electron, that is, a small electrically charged particle, which is present in huge quantities in all weighty bodies. This hypothesis explained the phenomenon of splitting of spectral lines in a magnetic field, discovered in 1896 by the German physicist Zeeman. In 1897, Thomson experimentally confirmed the existence of the smallest negatively charged particle or electron.
Thus, within the framework of classical physics, a fairly harmonious and complete picture of the world arose, describing and explaining motion, gravity, heat, electricity and magnetism, and light. This gave rise to Lord Kelvin (Thomson) to say that the edifice of physics was almost complete, only a few details were missing...
Firstly, it turned out that Maxwell's equations are non-invariant under Galilean transformations. Secondly, the theory of the ether as an absolute coordinate system to which Maxwell’s equations are “tied” has not found experimental confirmation. The Michelson-Morley experiment showed that there is no dependence of the speed of light on the direction in a moving coordinate system No. A supporter of the preservation of Maxwell's equations, Hendrik Lorentz, “tied” these equations to the ether as an absolute frame of reference, sacrificed Galileo's principle of relativity, its transformations and formulated his own transformations. From G. Lorentz's transformations it followed that spatial and time intervals are non-invariant when moving from one inertial reference system to another. Everything would be fine, but the existence of an absolute medium - the ether - was not confirmed, as noted, experimentally. This is a crisis.
Non-classical physics. Special theory of relativity.
Describing the logic of the creation of the special theory of relativity, Albert Einstein in a joint book with L. Infeld writes: “Let us now collect together those facts that have been sufficiently verified by experience, without worrying any more about the problem of the ether:
1. The speed of light in empty space is always constant, regardless of the movement of the source or receiver of light.
2. In two coordinate systems moving rectilinearly and uniformly relative to each other, all the laws of nature are strictly the same, and there is no means of detecting absolute rectilinear and uniform motion...
The first position expresses the constancy of the speed of light, the second generalizes Galileo's principle of relativity, formulated for mechanical phenomena, to everything that happens in nature." Einstein notes that the acceptance of these two principles and the rejection of the principle of the Galilean transformation, since it contradicts the constancy of the speed of light, laid the foundation the beginning of the special theory of relativity. To the accepted two principles: the constancy of the speed of light and the equivalence of all inertial frames of reference, Einstein adds the principle of invariance of all laws of nature with respect to the transformations of G. Lorentz. Therefore, the same laws are valid in all inertial frames, and the transition from one system to another is given by Lorentz transformations.This means that the rhythm of a moving clock and the length of the moving rods depend on the speed: the rod will shrink to zero if its speed reaches the speed of light, and the rhythm of the moving clock will slow down, the clock would completely stop if it could move at the speed of light.
Thus, Newtonian absolute time, space, motion, which were, as it were, independent of moving bodies and their state, were eliminated from physics.
General theory of relativity.
In the book already cited, Einstein asks: “Can we formulate physical laws in such a way that they are valid for all coordinate systems, not only for systems moving rectilinearly and uniformly, but also for systems moving completely arbitrarily in relation to each other?” . And he answers: “It turns out to be possible.”
Having lost their “independence” from moving bodies and from each other in the special theory of relativity, space and time seemed to “find” each other in a single space-time four-dimensional continuum. The author of the continuum, mathematician Hermann Minkowski, published in 1908 the work “Foundations of the Theory of Electromagnetic Processes,” in which he argued that from now on, space itself and time itself should be relegated to the role of shadows, and only some kind of connection of both should continue to be preserved independence. A. Einstein’s idea was to represent all physical laws as properties of this continuum, as it is metric. From this new position, Einstein considered Newton's law of gravitation. Instead of gravity he began to operate gravitational field. Gravitational fields were included in the space-time continuum as its “curvature.” The continuum metric became a non-Euclidean, “Riemannian” metric. The "curvature" of the continuum began to be considered as a result of the distribution of masses moving in it. The new theory explained the trajectory of Mercury's rotation around the Sun, which is not consistent with Newton's law of gravity, as well as the deflection of a ray of starlight passing near the Sun.
Thus, the concept of an “inertial coordinate system” was eliminated from physics and the statement of a generalized principle of relativity: any coordinate system is equally suitable for describing natural phenomena.
Quantum mechanics.
The second, according to Lord Kelvin (Thomson), the missing element to complete the edifice of physics at the turn of the 19th and 20th centuries was a serious discrepancy between theory and experiment in the study of the laws of thermal radiation of an absolutely black body. According to the prevailing theory, it should be continuous, continual. However, this led to paradoxical conclusions, such as the fact that the total energy emitted by a black body at a given temperature is equal to infinity (Rayleigh-Jean formula). To solve the problem, the German physicist Max Planck put forward the hypothesis in 1900 that matter cannot emit or absorb energy except in finite portions (quanta) proportional to the emitted (or absorbed) frequency. The energy of one portion (quantum) E=hn, where n is the frequency of radiation, and h is a universal constant. Planck's hypothesis was used by Einstein to explain the photoelectric effect. Einstein introduced the concept of a quantum of light or photon. He also suggested that light, in accordance with Planck's formula, has both wave and quantum properties. The physics community started talking about wave-particle duality, especially since in 1923 another phenomenon was discovered confirming the existence of photons - the Compton effect.
In 1924, Louis de Broglie extended the idea of the dual corpuscular-wave nature of light to all particles of matter, introducing the idea of waves of matter. From here we can talk about the wave properties of the electron, for example, about electron diffraction, which were established experimentally. However, R. Feynman’s experiments with “shelling” electrons on a shield with two holes showed that it is impossible, on the one hand, to say through which hole the electron is flying, that is, to accurately determine its coordinate, and on the other hand, not to distort the distribution pattern of the detected electrons, without disturbing the nature of the interference. This means that we can know either the electron's coordinates or its momentum, but not both.
This experiment called into question the very concept of a particle in the classical sense of precise localization in space and time.
The explanation of the "non-classical" behavior of microparticles was first given by the German physicist Werner Heisenberg. The latter formulated the law of motion of a microparticle, according to which knowledge of the exact coordinate of a particle leads to complete uncertainty of its momentum, and vice versa, exact knowledge of the momentum of a particle leads to complete uncertainty of its coordinates. W. Heisenberg established the relationship between the uncertainties of the coordinates and momentum of a microparticle:
Dx * DP x ³ h, where Dx is the uncertainty in the coordinate value; DP x - uncertainty in the value of the impulse; h is Planck's constant. This law and the uncertainty relation are called uncertainty principle Heisenberg.
Analyzing the uncertainty principle, the Danish physicist Niels Bohr showed that, depending on the setup of the experiment, a microparticle reveals either its corpuscular nature or its wave nature, but not both at once. Consequently, these two natures of microparticles are mutually exclusive, and at the same time should be considered as complementary to each other, and their description based on two classes of experimental situations (corpuscular and wave) should be a holistic description of the microparticle. There is not a particle “in itself”, but a system “particle - device”. These conclusions of N. Bohr are called principle of complementarity.
Within the framework of this approach, uncertainty and additionality turn out to be not a measure of our ignorance, but objective properties of microparticles, microworld as a whole. It follows from this that statistical, probabilistic laws lie in the depths of physical reality, and the dynamic laws of unambiguous cause-and-effect dependence are only some particular and idealized case of expressing statistical laws.
Relativistic quantum mechanics.
In 1927, the English physicist Paul Dirac drew attention to the fact that to describe the movement of microparticles discovered by that time: electron, proton and photon, since they move at speeds close to the speed of light, the application of the special theory of relativity is required. Dirac composed an equation that described the motion of an electron taking into account the laws of both quantum mechanics and Einstein's theory of relativity. There were two solutions to this equation: one solution gave a known electron with positive energy, the other gave an unknown twin electron but with negative energy. This is how the idea of particles and antiparticles symmetrical to them arose. This raised the question: is a vacuum empty? After Einstein's "expulsion" of the ether, it seemed undoubtedly empty.
Modern, well-proven ideas say that the vacuum is “empty” only on average. A huge number of virtual particles and antiparticles are constantly being born and disappearing in it. This does not contradict the uncertainty principle, which also has the expression DE * Dt ³ h. Vacuum in quantum field theory is defined as the lowest energy state of a quantum field, the energy of which is zero only on average. So the vacuum is “something” called “nothing”.
On the way to constructing a unified field theory.
In 1918, Emmy Noether proved that if a certain system is invariant under some global transformation, then there is a certain conservation value for it. It follows from this that the law of conservation (of energy) is a consequence symmetries, existing in real space-time.
Symmetry as a philosophical concept means the process of existence and formation of identical moments between different and opposite states of the phenomena of the world. This means that when studying the symmetry of any systems, it is necessary to consider their behavior under various transformations and identify in the entire set of transformations those that leave unchangeable, invariant some functions corresponding to the systems under consideration.
In modern physics the concept is used gauge symmetry. By calibration, railway workers mean the transition from a narrow to a wide gauge. In physics, calibration was originally also understood as a change in level or scale. In special relativity, the laws of physics do not change with respect to translation or shift when calibrating distance. In gauge symmetry, the requirement of invariance gives rise to a certain specific type of interaction. Consequently, gauge invariance allows us to answer the question: “Why and why do such interactions exist in nature?” Currently, physics defines the existence of four types of physical interactions: gravitational, strong, electromagnetic and weak. All of them have a gauge nature and are described by gauge symmetries, which are different representations of Lie groups. This suggests the existence of a primary supersymmetric field, in which there is still no distinction between types of interactions. The differences and types of interaction are the result of a spontaneous, spontaneous violation of the symmetry of the original vacuum. The evolution of the Universe appears then as synergetic self-organizing process: During the process of expansion from a vacuum supersymmetric state, the Universe heated up to the “big bang”. The further course of its history ran through critical points - bifurcation points, at which spontaneous violations of the symmetry of the original vacuum occurred. Statement self-organization of systems through spontaneous violation of the original type of symmetry at bifurcation points and there is principle of synergy.
The choice of the direction of self-organization at bifurcation points, that is, at points of spontaneous violation of the original symmetry, is not accidental. It is defined as if it were already present at the level of vacuum supersymmetry by the “project” of a person, that is, the “project” of a being asking why the world is like this. This anthropic principle, which was formulated in physics in 1962 by D. Dicke.
The principles of relativity, uncertainty, complementarity, symmetry, synergy, the anthropic principle, as well as the affirmation of the deep-basic nature of probabilistic cause-and-effect dependencies in relation to dynamic, unambiguous cause-and-effect dependencies constitute the categorical-conceptual structure of modern gestalt, the image of physical reality.
Literature
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3. Bohr N. Causality and complementarity // Bohr N. Selected scientific works in 2 volumes. T.2. M., 1971.
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6. Heisenberg V. Physics and Philosophy. Part and whole. M. 1989.
8. Einstein A., Infeld L. Evolution of physics. M., 1965.