Newton is the founder of classical mechanics. And although today, from the standpoint of modern science, Newton's mechanistic picture of the world seems rough and limited, it was it that gave impetus to the development of theoretical and applied sciences for the next almost 200 years. We owe Newton such concepts as absolute space, time, mass, force, speed, acceleration; he discovered the laws of motion of physical bodies, laying the foundation for the development of the science of physics. (However, none of this could have happened if Galileo, Copernicus and others had not been before him. No wonder he himself said: “I stood on the shoulders of giants.”) Let us dwell on the main achievement of Newton’s scientific research - a mechanistic picture of the world. It contains the following provisions:
- The statement that the whole world, the Universe is nothing but a collection of a huge number of indivisible and unchanging particles moving in space and time, interconnected by gravitational forces transmitted from body to body through the void. It follows that all events are rigidly predetermined and subject to the laws of classical mechanics, which makes it possible to predetermine and predict the course of events. The elementary unit of the world is an atom, and all bodies consist of absolutely solid, indivisible, unchanging corpuscles - atoms. When describing mechanical processes, he used the concepts of "body" and "corpuscle". The movement of atoms and bodies was presented as a simple movement of bodies in space and time. The properties of space and time, in turn, were presented as unchanging and independent of the bodies themselves. Nature was presented as a large mechanism (machine), in which each part had its own purpose and strictly obeyed certain laws. The essence of this picture of the world is the synthesis of natural science knowledge and the laws of mechanics, which reduced (reduced) the whole variety of phenomena and processes to mechanical ones.
| № | classical science | postclassical science |
| 1. | Taking the subject out of the object. | Recognition of the subjectivity of knowledge and cognition. |
| 2. | Installation on rationality. | Accounting for non-rational ways of knowing. |
| 3. | The dominance of dynamic laws. | Accounting for the role and significance of probabilistic-statistical regularities. |
| 4. | The object of study is the macrocosm. | The object of study is the micro-, macro- and mega-world. |
| 5. | The leading method of cognition is experiment. | Modeling (including mathematical). |
| 6. | Unconditional visibility. | Conditional visibility. |
| 7. | A clear line between the natural sciences and the humanities. | Erase this edge. |
| 8. | Responsible discipline. The predominance of differentiation of sciences. | Differentiation and integration (system theory, synergetics, structural method). |
- Variety of types of scientific knowledge. Empirical knowledge, its structure and features. Structure and specific features of theoretical knowledge. Foundations of science.
- as a problematic and unreliable form of knowledge; as a method of scientific knowledge.
- compliance with the laws established in science; consistency with the actual material; consistency from the point of view of formal logic (if we are talking about the contradiction of objective reality itself, then the hypothesis must contain contradictions); the absence of subjective, arbitrary assumptions (which does not cancel the activity of the subject himself); the possibility of its confirmation or refutation either in the course of direct observation, or indirectly - by deriving consequences from the hypothesis.
- The theory should not contradict the data of facts and experience and be verifiable on the available experimental material. It should not contradict the principles of formal logic, and at the same time be distinguished by logical simplicity, “naturalness”. A theory is "good" if it encompasses and links together a wide range of subjects into a coherent system of abstractions.
Mechanics- this is a part of physics that studies the laws of mechanical movement and the reasons that cause or change this movement.
Mechanics, in turn, is divided into kinematics, dynamics and statics.
mechanical movement- this is a change in the relative position of bodies or body parts over time.
Weight is a scalar physical quantity that quantitatively characterizes the inert and gravitational properties of matter.
inertia- this is the desire of the body to maintain a state of rest or uniform rectilinear motion.
inertial mass characterizes the ability of a body to resist a change in its state (rest or motion), for example, in Newton's second law
gravitational mass characterizes the body's ability to create a gravitational field, which is characterized by a vector quantity called tension. The intensity of the gravitational field of a point mass is equal to:
The gravitational mass characterizes the body's ability to interact with the gravitational field:
P equivalence principle gravitational and inertial masses: each mass is both inertial and gravitational at the same time.
The mass of the body depends on the density of the substance ρ and the size of the body (body volume V):
The concept of mass is not identical to the concepts of weight and gravity. It does not depend on the fields of gravity and accelerations.
Moment of inertia is a tensor physical quantity that quantitatively characterizes the inertia of a solid body, which manifests itself in rotational motion.
When describing the rotational motion, it is not enough to specify the mass. The inertia of a body in rotational motion depends not only on the mass, but also on its distribution relative to the axis of rotation.
1. Moment of inertia of a material point
where m is the mass of a material point; r is the distance from the point to the axis of rotation.
2. Moment of inertia of the system of material points
3. Moment of inertia of a perfectly rigid body
Force- this is a vector physical quantity, which is a measure of the mechanical impact on the body from other bodies or fields, as a result of which the body acquires acceleration or deforms (changes its shape or size).
Mechanics uses various models to describe mechanical motion.
Material point(m.t.) is a body with a mass, the dimensions of which can be neglected in this problem.
Absolutely rigid body(a.t.t.) is a body that does not deform in the process of movement, that is, the distance between any two points in the process of movement remains unchanged.
§ 2. Laws of motion.
First Law n newton : any material point (body) retains a state of rest or uniform rectilinear motion until the impact from other bodies makes it change this state.
Newton's second law (the main law of the dynamics of translational motion): the rate of change in the momentum of a material point (body) is equal to the sum of the forces acting on it
Newton's third law : any action of material points (bodies) on each other has the character of interaction; the forces with which the material points act on each other are always equal in absolute value, oppositely directed and act along the straight line connecting these points
here is the force acting on the first material point from the second; - the force acting on the second material point from the side of the first. These forces are applied to different material points (bodies), always act in pairs and are forces of the same nature.
,
here is the gravitational constant. .
Conservation laws in classical mechanics.
The laws of conservation are fulfilled in closed systems of interacting bodies.
A system is called closed if no external forces act on the system.
Pulse - vector physical quantity that quantitatively characterizes the stock of translational motion:
Law of conservation of momentum systems of material points(m.t.): in closed systems, m.t. total momentum is conserved
where is the speed of the i-th material point before the interaction; is its speed after interaction.
angular momentum is a physical vector quantity that quantitatively characterizes the reserve of rotational motion.
is the momentum of the material point, is the radius vector of the material point.
Law of conservation of angular momentum
:
in a closed system, the total angular momentum is conserved:
The physical quantity that characterizes the ability of a body or system of bodies to do work is called energy.
Energy is a scalar physical quantity, which is the most general characteristic of the state of the system.
The state of the system is determined by its movement and configuration, i.e., by the mutual arrangement of its parts. The motion of the system is characterized by the kinetic energy K, and the configuration (being in the potential field of forces) is characterized by the potential energy U.
total energy defined as the sum:
E = K + U + E int,
where E ext is the internal energy of the body.
The kinetic and potential energies add up to mechanical energy .
Einstein formula(relationship of energy and mass):
In the reference frame associated with the center of mass of the m.t. system, m \u003d m 0 is the rest mass, and E \u003d E 0 \u003d m 0. c 2 - rest energy.
Internal energy is determined in the frame of reference associated with the body itself, that is, the internal energy is at the same time the rest energy.
Kinetic energy is the energy of the mechanical movement of a body or system of bodies. The relativistic kinetic energy is determined by the formula
At low speeds v
.
Potential energy is a scalar physical quantity that characterizes the interaction of bodies with other bodies or with fields.
Examples:
potential energy of elastic interaction
potential energy of gravitational interaction of point masses
Law of energy conservation : the total energy of a closed system of material points is conserved
In the absence of dissipation (scattering) of energy, both total and mechanical energies are conserved. In dissipative systems, total energy is conserved, while mechanical energy is not conserved.
§ 2. Basic concepts of classical electrodynamics.
The source of the electromagnetic field is an electric charge.
Electric charge is the property of some elementary particles to enter into electromagnetic interaction.
Electric charge properties :
1. The electric charge can be positive and negative (it is generally accepted that the proton is positively charged, and the electron is negatively charged).
2. Electric charge is quantized. A quantum of electric charge is an elementary electric charge (е = 1.610 –19 C). In the free state, all charges are multiples of an integer number of elementary electric charges:
3. The law of conservation of charge: the total electric charge of a closed system is preserved in all processes involving charged particles:
q 1 + q 2 +...+ q N = q 1 * + q 2 * +...+ q N * .
4. relativistic invariance: the value of the total charge of the system does not depend on the motion of charge carriers (the charge of moving and resting particles is the same). In other words, in all ISOs, the charge of any particle or body is the same.
Description of the electromagnetic field.
The charges interact with each other (Fig. 1). The magnitude of the force with which charges of the same sign repel each other, and charges of opposite signs attract each other, is determined using the empirically established Coulomb's law:
Here, is the electric constant.
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Fig.1 |
And what is the mechanism of interaction of charged bodies? One can put forward the following hypothesis: bodies with an electric charge generate an electromagnetic field. In turn, the electromagnetic field acts on other charged bodies that are in this field. A new material object emerged – an electromagnetic field.
Experience shows that in any electromagnetic field, a force acts on a stationary charge, the magnitude of which depends only on the magnitude of the charge (the magnitude of the force is proportional to the magnitude of the charge) and its position in the field. It is possible to assign to each point of the field a certain vector , which is the coefficient of proportionality between the force acting on a fixed charge in the field and the charge . Then the force with which the field acts on a fixed charge can be determined by the formula:
The force acting from the side of the electromagnetic field on a fixed charge is called electric force. The vector value characterizing the state of the field that causes the action is called the electric strength of the electromagnetic field.
Further experiments with charges show that the vector does not completely characterize the electromagnetic field. If the charge begins to move, then some additional force appears, the magnitude and direction of which are in no way related to the magnitude and direction of the vector. The additional force that occurs when a charge moves in an electromagnetic field is called magnetic force. Experience shows that the magnetic force depends on the charge and on the magnitude and direction of the velocity vector. If we move a trial charge through any fixed point of the field with the same velocity, but in different directions, then the magnetic force will be different each time. However, always. Further analysis of the experimental facts made it possible to establish that for each point of the electromagnetic field there is a single direction MN (Fig. 2), which has the following properties:
Fig.2
If a certain vector is directed along the MN direction, which has the meaning of the coefficient of proportionality between the magnetic force and the product, then setting , and uniquely characterizes the state of the field that causes the appearance of . The vector was called the vector of electromagnetic induction. Since and , then
In an electromagnetic field, an electromagnetic Lorentz force acts on a charge moving at a speed q (Fig. 3):
.
The vectors and , that is, the six numbers , are equal components of a single electromagnetic field (components of the electromagnetic field tensor). In a particular case, it may turn out that all or all ; then the electromagnetic field is reduced to either electric or magnetic fields.
The experiment confirmed the correctness of the constructed two-vector model of the electromagnetic field. In this model, each point of the electromagnetic field is given a pair of vectors and . The model we have constructed is a model of a continuous field, since the functions and describing the field are continuous functions of the coordinates.
The theory of electromagnetic phenomena using the continuous field model is called classical.
In reality, the field, like matter, is discrete. But this begins to affect only at distances comparable to the sizes of elementary particles. The discreteness of the electromagnetic field is taken into account in quantum theory.
The principle of superposition.
Fields are usually depicted using lines of force.
force line is a line, the tangent to which at each point coincides with the field strength vector.
D
For point immobile charges, the pattern of force lines of the electrostatic field is shown in fig. 6.
The intensity vector of the electrostatic field created by a point charge is determined by the formula (Fig. 7 a and b) the magnetic field line is constructed so that at each point of the line of force the vector is directed tangentially to this line. The lines of force of the magnetic field are closed (Fig. 8). This suggests that the magnetic field is a vortex field.
Rice. eight
And if the field creates not one, but several point charges? Do the charges influence each other, or does each of the system's charges contribute to the resulting field independently of the others? Will the electromagnetic field created by the i-th charge in the absence of other charges be the same as the field created by the i-th charge in the presence of other charges?
Superposition principle : the electromagnetic field of an arbitrary system of charges is the result of the addition of fields that would be created by each of the elementary charges of this system in the absence of the others:
and .
Laws of the electromagnetic field
The laws of the electromagnetic field are formulated as a system of Maxwell's equations.
First
It follows from Maxwell's first equation that electrostatic field - potential (converging or diverging) and its source are motionless electric charges.
Second Maxwell's equation for a magnetostatic field:
It follows from Maxwell's second equation that the magnetostatic field is vortex non-potential and has no point sources.
Third Maxwell's equation for an electrostatic field:
It follows from Maxwell's third equation that the electrostatic field is not vortex.
In electrodynamics (for a variable electromagnetic field), Maxwell's third equation is:
i.e. the electric field is not potential (not Coulomb), but vortex and is created by a variable flux of the magnetic field induction vector.
Fourth Maxwell's equation for a magnetostatic field
It follows from the fourth Maxwell equation in magnetostatics that the magnetic field is vortex and is created by direct electric currents or moving charges. The direction of twisting of the magnetic field lines is determined by the right screw rule (Fig. 9).
R
Fig.9
In electrodynamics, Maxwell's fourth equation is:
The first term in this equation is the conduction current I associated with the movement of charges and creating a magnetic field.
The second term in this equation is the "displacement current in vacuum", i.e., the variable flux of the electric field strength vector.
The main provisions and conclusions of Maxwell's theory are as follows.
A change in time of the electric field leads to the appearance of a magnetic field and vice versa. Therefore, there are electromagnetic waves.
The transfer of electromagnetic energy occurs at a finite speed . The speed of transmission of electromagnetic waves is equal to the speed of light. From this followed the fundamental identity of electromagnetic and optical phenomena.
The pinnacle of I. Newton's scientific work is his immortal work "The Mathematical Principles of Natural Philosophy", first published in 1687. In it, he summarized the results obtained by his predecessors and his own research and created for the first time a single harmonious system of terrestrial and celestial mechanics, which formed the basis of all classical physics.
Here Newton gave definitions of the initial concepts - the amount of matter, equivalent to mass, density; amount of motion equivalent to momentum, and various types of force. Formulating the concept of quantity of matter, he proceeded from the idea that atoms consist of some single primary matter; Density was understood as the degree to which a unit volume of a body is filled with primary matter.
This work outlines Newton's doctrine of universal gravitation, on the basis of which he developed the theory of the motion of planets, satellites and comets that form the solar system. Based on this law, he explained the phenomenon of tides and the compression of Jupiter. Newton's concept was the basis for many technical advances over a long period of time. Many methods of scientific research in various fields of natural sciences were formed on its foundation.
The result of the development of classical mechanics was the creation of a unified mechanical picture of the world, within which the entire qualitative diversity of the world was explained by differences in the movement of bodies, subject to the laws of Newtonian mechanics.
Newton's mechanics, in contrast to previous mechanical concepts, made it possible to solve the problem of any stage of movement, both preceding and subsequent, and at any point in space with known facts that determine this movement, as well as the inverse problem of determining the magnitude and direction of these factors. at any point with known basic elements of motion. Because of this, Newtonian mechanics could be used as a method for the quantitative analysis of mechanical motion.
The law of universal gravitation.
The law of universal gravitation was discovered by I. Newton in 1682. According to his hypothesis, attractive forces act between all bodies of the Universe, directed along the line connecting the centers of mass. For a body in the form of a homogeneous ball, the center of mass coincides with the center of the ball.
In subsequent years, Newton tried to find a physical explanation for the laws of planetary motion discovered by I. Kepler at the beginning of the 17th century, and to give a quantitative expression for gravitational forces. So, knowing how the planets move, Newton wanted to determine what forces act on them. This path is called the inverse problem of mechanics.
If the main task of mechanics is to determine the coordinates of a body of known mass and its speed at any moment of time from the known forces acting on the body, then when solving the inverse problem, it is necessary to determine the forces acting on the body if it is known how it moves.
The solution of this problem led Newton to the discovery of the law of universal gravitation: "All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them."
There are several important remarks to be made about this law.
1, its action explicitly extends to all physical material bodies in the Universe without exception.
2 the force of gravity of the Earth at its surface equally affects all material bodies located anywhere on the globe. Right now, the force of gravity is acting on us, and we really feel it as our own weight. If we drop something, it, under the influence of the same force, will rush to the ground with uniform acceleration.
Many phenomena are explained by the action of forces of universal gravitation in nature: the movement of the planets in the solar system, artificial satellites of the Earth - all of them are explained on the basis of the law of universal gravitation and the laws of dynamics.
Newton was the first to suggest that gravitational forces determine not only the movement of the planets of the solar system; they act between any bodies of the Universe. One of the manifestations of the force of universal gravitation is the force of gravity - this is how it is customary to call the force of attraction of bodies to the Earth near its surface.
The force of gravity is directed towards the center of the earth. In the absence of other forces, the body falls freely to the Earth with free fall acceleration.
Three principles of mechanics.
Newton's laws of mechanics, the three laws underlying the so-called. classical mechanics. Formulated by I. Newton (1687).
First law: "Every body continues to be held in its state of rest or uniform and rectilinear motion, until and insofar as it is forced by applied forces to change this state."
The second law: "The change in momentum is proportional to the applied driving force and occurs in the direction of the straight line along which this force acts."
The third law: "There is always an equal and opposite reaction to an action, otherwise, the interactions of two bodies against each other are equal and directed in opposite directions." N. h. m. appeared as a result of the generalization of numerous observations, experiments and theoretical studies of G. Galileo, H. Huygens, Newton himself, and others.
According to modern ideas and terminology, in the first and second laws, a body should be understood as a material point, and under movement - movement relative to an inertial frame of reference. The mathematical expression of the second law in classical mechanics has the form or mw = F, where m is the mass of the point, u is its speed, a w is the acceleration, F is the acting force.
N. h. m cease to be valid for the movement of objects of very small sizes (elementary particles) and for movements with speeds close to the speed of light
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LECTURE 1
INTRODUCTION TO CLASSICAL MECHANICS
classical mechanics studies the mechanical motion of macroscopic objects that move at speeds much less than the speed of light (=3 10 8 m/s). Macroscopic objects are understood as objects whose dimensions are m (on the right is the size of a typical molecule).
Physical theories that study systems of bodies whose motion occurs at velocities much lower than the speed of light are among the nonrelativistic theories. If the velocities of the particles of the system are comparable with the speed of light, then such systems are related to relativistic systems, and they must be described on the basis of relativistic theories. The basis of all relativistic theories is the special theory of relativity (SRT). If the dimensions of the physical objects under study are small, then such systems are quantum systems, and their theories are quantum theories.
Thus, classical mechanics should be considered as a non-relativistic non-quantum theory of particle motion.
1.1 Frames of reference and principles of invariance
mechanical movement- this is a change in the position of a body relative to other bodies over time in space.
The space in classical mechanics is considered to be three-dimensional (to determine the position of a particle in space, you must specify three coordinates), obeying Euclid's geometry (the Pythagorean theorem is valid in space) and absolute. Time is one-dimensional, unidirectional (changing from past to future) and absolute. The absoluteness of space and time means that their properties do not depend on the distribution and movement of matter. In classical mechanics, the following statement is accepted as true: space and time are not related to each other and can be considered independently of each other.
Motion is relative and, therefore, to describe it, you must choose reference body, i.e. the body relative to which the movement is considered. Since the movement occurs in space and time, one or another coordinate system and clock should be chosen to describe it (to arithmetize space and time). Due to the three-dimensionality of space, each of its points is associated with three numbers (coordinates). The choice of one or another coordinate system is usually dictated by the condition and symmetry of the task. In theoretical reasoning, we will usually use a rectangular Cartesian coordinate system (Figure 1.1).
In classical mechanics, to measure time intervals, due to the absoluteness of time, it is sufficient to have one clock placed at the origin of the coordinate system (this issue will be considered in detail in the theory of relativity). The body of reference and the hours and scales associated with this body (coordinate system) form reference system.
Let us introduce the concept of a closed physical system. closed physical system such a system of material objects is called, in which all objects of the system interact with each other, but do not interact with objects that are not included in the system.
As experiments show, the following principles of invariance turn out to be valid with respect to a number of reference systems.
The principle of invariance under spatial shifts(space is homogeneous): the course of processes inside a closed physical system is not affected by its position relative to the reference body.
The principle of invariance under spatial rotations(space is isotropic): the course of processes inside a closed physical system is not affected by its orientation relative to the reference body.
The principle of invariance with respect to time shifts(time is homogeneous): the time of the beginning of the processes does not affect the flow of processes inside a closed physical system.
The principle of invariance under mirror reflections(the space is mirror-symmetric): the processes occurring in closed mirror-symmetric physical systems are themselves mirror-symmetric.
Those frames of reference with respect to which space is homogeneous, isotropic and mirror-symmetric and time is uniformly called inertial reference systems(ISO).
Newton's first law claims that ISOs exist.
There is not one, but an infinite number of ISOs. That frame of reference, which moves relative to the ISO in a straight line and uniformly, will itself be the ISO.
The principle of relativity claims that the flow of processes in a closed physical system is not affected by its rectilinear uniform motion relative to the reference frame; the laws describing the processes are the same in different ISOs; the processes themselves will be the same if the initial conditions are the same.
1.2 Basic models and sections of classical mechanics
In classical mechanics, when describing real physical systems, a number of abstract concepts are introduced that correspond to real physical objects. Such basic concepts include: a closed physical system, a material point (particle), an absolutely rigid body, a continuous medium, and a number of others.
Material point (particle)- a body whose dimensions and internal structure can be neglected when describing its movement. In addition, each particle is characterized by its specific set of parameters - mass, electric charge. The model of a material point does not consider the structural internal characteristics of particles: moment of inertia, dipole moment, intrinsic moment (spin), etc. The position of a particle in space is characterized by three numbers (coordinates) or a radius vector (Fig. 1.1).
Absolutely rigid body
A system of material points, the distances between which do not change during their movement;
A body whose deformations can be neglected.
A real physical process is considered as a continuous sequence of elementary events.
elementary event is a phenomenon with zero spatial extent and zero duration (for example, a bullet hitting a target). The event is characterized by four numbers - coordinates; three spatial coordinates (or radius - vector) and one time coordinate: . In this case, the motion of a particle is represented as a continuous sequence of the following elementary events: the passage of a particle through a given point in space at a given time.
The law of motion of a particle is considered given if the dependence of the radius-vector of the particle (or its three coordinates) on time is known:
Depending on the type of objects being studied, classical mechanics is subdivided into the mechanics of particles and systems of particles, the mechanics of an absolutely rigid body, and the mechanics of continuous media (mechanics of elastic bodies, hydromechanics, aeromechanics).
According to the nature of the tasks to be solved, classical mechanics is divided into kinematics, dynamics and statics. Kinematics studies the mechanical movement of particles without taking into account the causes that cause a change in the nature of the movement of particles (forces). The law of motion of the particles of the system is considered given. According to this law, velocities, accelerations, trajectories of the particles of the system are determined in kinematics. Dynamics considers the mechanical movement of particles, taking into account the causes that cause a change in the nature of the movement of particles. The forces acting between the particles of the system and on the particles of the system from bodies not included in the system are considered to be known. The nature of forces in classical mechanics is not discussed. Statics can be considered as a special case of dynamics, where the conditions of mechanical equilibrium of the particles of the system are studied.
According to the method of describing systems, mechanics is divided into Newtonian and analytical mechanics.
1.3 Event coordinate transformations
Let us consider how the coordinates of events are transformed during the transition from one IFR to another.
1. Spatial shift. In this case, the transformations look like this:
Where is the spatial shift vector, which does not depend on the event number (index a).
2. Time shift:
Where is the time shift.
3. Spatial rotation:
Where is the infinitesimal rotation vector (Fig. 1.2).
4. Time inversion (time reversal):
5. Spatial inversion (reflection at a point):
6. Galilean transformations. We consider the transformation of the coordinates of events during the transition from one IFR to another, which moves relative to the first one in a straight line and uniformly with a speed (Fig. 1.3):
Where is the second ratio postulated(!) and expresses the absoluteness of time.
Differentiating with respect to time the right and left parts of the transformation of spatial coordinates, taking into account the absolute character of time, using the definition speed, as a derivative of the radius-vector with respect to time, the condition that =const, we obtain the classical law of addition of velocities
Here we should pay special attention to the fact that when deriving the last relation necessary take into account the postulate of the absolute character of time.
Rice. 1.2 Fig. 1.3
Differentiating with respect to time again using the definition acceleration, as a derivative of the speed with respect to time, we get that the acceleration is the same with respect to different ISOs (invariant with respect to the Galilean transformations). This statement mathematically expresses the principle of relativity in classical mechanics.
From a mathematical point of view, transformations 1-6 form a group. Indeed, this group contains a single transformation - an identical transformation corresponding to the absence of a transition from one system to another; for each of transformations 1-6 there is an inverse transformation that takes the system to its original state. The operation of multiplication (composition) is introduced as a successive application of the corresponding transformations. It should be especially noted that the group of rotation transformations does not obey the commutative (permutation) law, i.e. is non-abelian. The complete transformation group 1-6 is called the Galilean transformation group.
1.4 Vectors and scalars
Vector a physical quantity is called, which is transformed as the radius vector of a particle and is characterized by its numerical value and direction in space. With respect to the spatial inversion operation, vectors are divided into true(polar) and pseudovectors(axial). With spatial inversion, the true vector changes its sign, the pseudovector does not change.
Scalars characterized only by their numerical value. With respect to the spatial inversion operation, scalars are divided into true and pseudoscalars. With spatial inversion, the true scalar does not change, the pseudoscalar changes its sign.
Examples. Radius vector, velocity, particle acceleration are true vectors. The vectors of the angle of rotation, angular velocity, angular acceleration are pseudovectors. The vector product of two true vectors is a pseudovector, the vector product of a true vector and a pseudovector is a true vector. The scalar product of two true vectors is a true scalar, a true vector times a pseudovector is a pseudoscalar.
It should be noted that in a vector or scalar equality on the right and on the left there must be terms of the same nature with respect to the spatial inversion operation: true scalars or pseudoscalars, true vectors or pseudovectors.
Mechanics is a branch of physics that studies one of the simplest and most general 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. So mechanical movement is made by planets circulating 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 subdivided into the mechanics of a material point, the mechanics of a solid body, and the mechanics of a continuum.
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 vacuum (3·10 8 m/s).
macrobodies called ordinary bodies that surround us, that is, bodies consisting of a huge number of molecules and atoms.
The mechanics that studies the motion of macrobodies with velocities much lower than the speed of light in vacuum is called classical.
Classical mechanics is based on the following Newton's ideas about the properties of space and time.
Any physical process takes place in space and time. This can be seen at least from the fact that in all areas of physical phenomena, each law explicitly or implicitly contains space-time quantities - distances and time intervals.
A space that 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 coincided in length always remain equal to each other, that is, they coincide with each subsequent overlay.
Time intervals are measured by hours, and the role of the latter can be played by any system that performs a repeating process.
The main feature of the ideas of classical mechanics about the size 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 having the same rate and once brought into line with each other show the same time, no matter how they move.
Space and time have remarkable properties symmetry that impose restrictions on the flow of certain processes in them. These properties have been established by experience and seem so obvious at first glance that there seems to be no need to single them out and deal with them. Meanwhile, if there were no spatial and temporal symmetry, no physical science could arise or develop.
It turns out that the space uniformly and isotropically, and the time is uniformly.
The homogeneity of space lies in the fact that the same physical phenomena under the same conditions occur in the same way in different parts of space. All points of space, therefore, are 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 course of identical phenomena in the same conditions is the same, regardless of the time of their implementation and observation.
The homogeneity of time is manifested in the law of conservation of energy.
Without these homogeneity properties, the physical law established in Minsk would be unfair in Moscow, and the law discovered today in the same place could be unfair tomorrow.
In classical mechanics, the validity of the Galileo-Newton law of inertia is recognized, according to which a body that is not subject to action from other bodies moves in a straight line 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) hold. 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 frames of reference, or, in other words, they are invariant with respect to the Galilean transformations expressing the space-time connection of any event in different inertial frames of reference. Galilean transformations show that the coordinates of any event are relative, that is, they have different values in different reference systems; the instants of time when the event occurred are the same in different systems. The latter means that time flows in the same way in different frames of reference. This circumstance seemed so obvious that it was not even mentioned as a special postulate.
In classical mechanics, the principle of long-range action is observed: the interactions of bodies propagate instantly, that is, at an infinitely high speed.
Depending on the speed with which bodies move and what are the sizes of the bodies themselves, mechanics is divided into classical, relativistic, and quantum.
As already mentioned, laws classical mechanics are applicable only to the motion 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 vacuum.
Relativistic mechanics considers the motion of macrobodies with velocities close to the speed of light in vacuum.
Quantum mechanics- mechanics of microparticles moving at speeds much lower than the speed of light in 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, whether classical formulas are applicable to it, one must use Heisenberg's uncertainty principle. According to quantum mechanics, real particles can only be characterized in terms of position and momentum with some accuracy. The limit of this accuracy is defined as follows
where
ΔX - coordinate uncertainty;
ΔP x - uncertainty of the projection on the momentum axis;
h - Planck's constant, equal to 1.05·10 -34 J·s;
"≥" - more than a value, of the order of ...
Replacing momentum with the product of mass times velocity, we can write
It can be seen from the formula that the smaller the mass of a 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. Suppose, 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 the ball, which is smaller than the dimensions of the atom. Let us therefore ΔX=10 -10 m. Then from the uncertainty relation we find
The simultaneous smallness of the values ΔX and ΔV x is the proof of the practical applicability of the classical method of describing the motion of macrobodies.
Consider the motion of an electron in a hydrogen atom. The mass of an electron is 9.1 10 -31 kg. The error in the position of the electron ΔX in any case should not exceed the dimensions 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 equal in order of magnitude 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 causes that caused this movement; statics considers the conditions for the equilibrium of bodies; dynamics studies the movement of bodies in connection with those causes (interactions between bodies) that determine one or another character of movement.
The real movements of bodies are so complex that, when studying them, it is necessary to abstract from details that are not essential 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 of interest to us, as well as on the degree of accuracy with which we want to obtain the result. Among these concepts, the most important are the concepts material point, system of material points, absolutely rigid body.
A material point is a physical concept that describes the translational motion of a body, if only its linear dimensions are small in comparison with the linear dimensions of other bodies within the given accuracy of determining the body coordinate, moreover, the body mass is attributed 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 traveled by any point on the earth's surface in a time equal to the period of its revolution around its axis, we compare with the linear dimensions of the globe. An aircraft 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.
The system of material points is a set of bodies under consideration, which are material points.
The study of the motion of an arbitrary system of bodies is reduced 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 proceed to the study of a system of material points.
