The concept of the field. conservative forces

FORCE FIELD

A part of space (limited or unlimited), at each point of which the material placed there is affected by , the magnitude and direction of which depend either only on the coordinates x, y, z of this point, or on the coordinates and time t. In the first case, S., p. stationary, and in the second - non-stationary. If the force at all points of the S. p. has the same value, i.e., does not depend on the coordinates, then the S. p. is called. homogeneous.

S. p., in which the field forces acting on a material particle moving in it, depends only on the initial and final position of the particle and does not depend on the type of its trajectory, called. potential. This work can be expressed in terms of the potential energy of p-tsy P (x, y, z):

A=P(x1, y1, z1)-P(x2, y2, z2),

where x1, y1, z1 and x2, y2, z2 are the coordinates of the initial and final positions of the particle, respectively. When a particle moves in a potential S. p. under the action of only field forces, the law of conservation of mechanics takes place. energy, which makes it possible to establish a relationship between the speed of a particle and its position in the S. p.

Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .

FORCE FIELD

A part of space (limited or unlimited), at each point of which a material particle placed there is affected by a force determined in numerical value and direction, which depends only on the coordinates x, y, z this point. Such a S. p. stationary; if the strength of the field also depends on time, then the S. p. non-stationary; if the force at all points of the S. p. has the same value, i.e., does not depend on coordinates or time, S. p. homogeneous.

Stationary S. p. can be set by equations

where F x , F y , F z - projection of the field strength F.

If there is such a function U(x, y, z), called the force function, U(x, y, z), and the force F can be defined through this function by the equalities:

or . The condition for the existence of a force function for a given S. p. is that

or . When moving in a potential S. p. from a point M 1 (x 1 ,y 1 , z 1)exactly M 2 (x 2, y 2, z 2) the work of the field forces is determined by equality and does not depend on the type of trajectory along which the point of application of the force moves.

surfaces U(x, y, z) = const, on which the function preserves the post. Examples of potential S. p.: a homogeneous field of gravity, for which U=-mgz, where T - the mass of a particle moving in the field, g- acceleration of gravity (axis z directed vertically upwards). Newtonian gravitational field, for which U = km/r, where r = - distance from the center of attraction, k - coefficient constant for the given field. potential energy P associated with U addiction P(x,) = = - U(x, y, z). Study of particle motion in potentialpp. n. (in the absence of other forces) is greatly simplified, since in this case the law of conservation of mechanics takes place. energy, which makes it possible to establish a direct relationship between the velocity of a particle and its position in the SP. from. POWER LINES- a family of curves characterizing the spatial distribution of the vector field of forces; the direction of the field vector at each point coincides with the tangent to the S. l. Thus, ur-tion S. l. arbitrary vector field A (x, y, z) are written as:

Density S. l. characterizes the intensity (value) of the force field. The concept of S. l. introduced by M. Faraday in the study of magnetism, and then received further development in the works of J. K. Maxwell on electromagnetism. Maxwell tension tensor el.-mag. fields.

Along with the use of the concept of S. l. more often they simply talk about field lines: electric strength. fields E, magnetic induction. fields IN etc.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .

See what "POWER FIELD" is in other dictionaries:

Force field is an ambiguous term used in the following meanings: Force field (physics) vector field of forces in physics; Force field (science fiction) a kind of invisible barrier, the main function of which is the protection of some ... Wikipedia

A part of space, at each point of which a particle placed there is affected by a force of a certain magnitude and direction, depending on the coordinates of this point, and sometimes also on time. In the first case, the force field is called stationary, and in ... ... Big Encyclopedic Dictionary

force field- A region of space in which a force acts on a material point placed there, depending on the coordinates of this point in the reference frame under consideration and on time. [Collection of recommended terms. Issue 102. Theoretical Mechanics. Academy… … Technical Translator's Handbook

force field- jėgų laukas statusas T sritis Standartizacija ir metrologija apibrėžtis Vektorinis laukas, kurio bet kuriame taške esančią dalelę veikia tik nuo taško padėties priklausančios jėgos (nuostovusis jėgų lauėš nukas) arbaėš la pad… Penkiakalbis aiskinamasis metrologijos terminų žodynas

force field- jėgų laukas statusas T sritis fizika atitikmenys: angl. force field vok. Kraftfeld, n rus. force field, n; force field, n pranc. champ de forces, m … Fizikos terminų žodynas

FORCE FIELD- In physics, this term can be given a precise definition, in psychology it is used, as a rule, metaphorically and usually refers to any or all influences on behavior. It is usually used in a rather holistic way - a force field... ... Explanatory Dictionary of Psychology

A part of space (limited or unlimited), at each point of which a material particle placed there is affected by a force determined in magnitude and direction, depending either only on the coordinates x, y, z of this point, or on ... ... Great Soviet Encyclopedia

A part of space, at each point to which a particle placed there is affected by a force of a certain magnitude and direction, which depends on the coordinates of this point, and sometimes also on time. In the first case, S. p. stationary, and in the second ... ... Natural science. encyclopedic Dictionary

force field- A region of space in which a force acts on a material point placed there, depending on the coordinates of this point in the reference frame under consideration and on time ... Polytechnic terminological explanatory dictionary

force field called a physical space that satisfies the condition that forces acting on the points of a mechanical system located in this space depend on the position of these points or on the position of the points and time (but not on their velocities).

Force field, whose forces do not depend on time, is called stationary(examples of a force field are gravity field, electrostatic field, elastic force field).

Potential force field.

Stationary force field called potential, if the work of the field forces acting on the mechanical system does not depend on the shape of the trajectories of its points and is determined only by their initial and final positions. These forces are called potential forces or conservative forces.

Let us prove that the above condition is satisfied if there is a single-valued function of coordinates:

called the force function of the field, the partial derivatives of which with respect to the coordinates of any point M i (i=1, 2...n) are equal to the projections tions of the force applied to this point on the corresponding axes, i.e.

The elementary work of the force applied to each point can be determined by the formula:

The elementary work of forces applied to all points of the system is equal to:

Using the formulas we get:

As can be seen from this formula, the elementary work of the forces of the potential field is equal to the total differential of the force function. The work of the field forces on the final displacement of the mechanical system is equal to:

i.e., the work of forces acting on the points of a mechanical system in a potential field is equal to the difference between the values of the force function in the final and initial positions of the system and does not depend on the shape of the trajectories of the points of this system. The positions of the system and does not depend on the shape of the trajectories of the points of this system. It follows from this that the force field for which there is a force function is indeed potential.

And science fiction literature, as well as in literature of the fantasy genre, which denotes a kind of invisible (less often visible) barrier, the main function of which is to protect a certain area or goal from external or internal penetrations. This idea may be based on the concept of a vector field. In physics, this term also has several specific meanings (see Force field (physics)).

Force fields in literature

The concept of "force field" is quite common in fiction, movies and computer games. According to many works of art, force fields have the following properties and characteristics, and are also used for the following purposes.

Atmospheric energy barrier that allows you to work in rooms that are openly in contact with vacuum (for example, space). The force field keeps the atmosphere inside the room and does not allow it to go outside this room: at the same time, solid and liquid objects can freely pass in both directions
A barrier that protects against various enemy attacks, whether they are attacks with energy (including beam), kinetic or torpedo weapons.
To hold (not let go) the target within the space limited by the force field.
Blocks the teleportation of enemy (and sometimes friendly) troops to the ship, military base, etc.
A barrier that limits the spread of certain substances in the air, such as toxic gases and vapours. (Often this is a form of technology used to create a barrier between space and the interior of a ship/space station.
The means of extinguishing a fire, which limits the flow of air (and oxygen) into the fire area, - the fire, having consumed all available oxygen (or other strong oxidizing gas) in the area closed by the force field, completely goes out.
A shield to protect something from the effects of natural or man-made (including weapons) forces. For example, in Star Control, in some situations, the force field can be large enough to cover an entire planet.
The force field can be used to create a temporary living space in a place that is not initially habitable for sentient beings using it (for example, in space or under water).
As a security measure to guide someone or something in the right direction for capture.
Instead of doors and bars of cells in prisons.
In the fantasy series Star Trek: The Next Generation, sections of the spacecraft had internal force field generators that allowed the crew to turn on force fields to prevent any matter or energy from passing through them. They were also used as "windows" that separate the vacuum of space from the habitable atmosphere, to protect against depressurization due to damage or local destruction of the ship's main hull.
The force field can completely cover the surface of the human body to protect against external influences. In particular, Star Trek: The Animation Series, Federation astronauts use energy field suits instead of mechanical ones. And in the Stargate there are personal energy shields.

Force fields in scientific interpretation

Notes

Literature

Andrews, Dana G.(2004-07-13). "Things to do While Coasting Through Interstellar Space" (PDF) in 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit..AIAA 2004-3706. Retrieved 2008-12-13.
Martin, A.R. (1978). “Bombardment by Interstellar Material and Its Effects on the Vehicle, Project Daedalus Final Report.”

A field of forces is a region of space, at each point of which a particle placed there is affected by a force that naturally changes from point to point, for example, the Earth's gravity field or the field of resistance forces in a fluid (gas) flow. If the force at each point of the force field does not depend on time, then such a field is called stationary. It is clear that a force field that is stationary in one frame of reference may turn out to be non-stationary in another frame. In a stationary force field, the force depends only on the position of the particle.

The work done by the field forces when moving a particle from a point 1 exactly 2 , generally speaking, depends on the path. However, among the stationary force fields there are those in which this work does not depend on the path between points 1 And 2 . This class of fields, having a number of important properties, occupies a special place in mechanics. We now turn to the study of these properties.

Let us explain what has been said on the example of the following force. On fig. 5.4 shows the body ABCD, at the point ABOUT which force is applied , permanently connected with the body.

Let's move the body from position I into position II two ways. Let us first choose a point as a pole ABOUT(Fig. 5.4a)) and turn the body around the pole at an angle π / 2 opposite to the direction of clockwise rotation. The body will take a position A"B"C"D". Let us now inform the body of translational displacement in the vertical direction by the value OO". The body will take a position II (A"B"C"D"). The work of the force on the perfect displacement of the body from the position I into position II equals zero. The vector of movement of the pole is represented by a segment OO".

In the second method, we choose a point as a pole K rice. 5.4b) and turn the body around the pole by an angle π/2 counterclockwise. The body will take a position A"B"C"D"(Fig. 5.4b). Now let's move the body vertically upwards with the pole displacement vector KK", after which we give the body a horizontal displacement to the left by the amount K"K". As a result, the body will take a position II, the same as in the position, Fig.5.4 but) of Figure 5.4. However, now the vector of displacement of the pole will be different than in the first method, and the work of the force in the second method of moving the body from the position I into position II is equal to A \u003d F K "K", i.e., it is different from zero.

Definition: a stationary force field in which the work of the field force on the path between any two points does not depend on the shape of the path, but depends only on the position of these points, is called potential, and the forces themselves - conservative.

Potential such forces ( potential energy) is the work done by them on moving the body from the final position to the initial one, and the initial position can be chosen arbitrarily. This means that the potential energy is determined up to a constant.

If this condition is not met, then the force field is not potential, and the field forces are called non-conservative.

In real mechanical systems, there are always forces whose work is negative during the actual movement of the system (for example, friction forces). Such forces are called dissipative. They are a special kind of non-conservative forces.

Conservative forces have a number of remarkable properties, to reveal which we introduce the concept of a force field. The force field is space(or part of it), in which a certain force acts on a material point placed at each point of this field.

Let us show that in a potential field the work of the field forces on any closed path is equal to zero. Indeed, any closed path (Fig. 5.5) can be divided arbitrarily into two parts, 1a2 And 2b1. Since the field is potential, then, by condition, . On the other hand, it is obvious that . That's why

Q.E.D.

Conversely, if the work of the field forces on any closed path is zero, then the work of these forces on the path between arbitrary points 1 And 2 does not depend on the form of the path, i.e., the field is potential. To prove this, we take two arbitrary paths 1a2 And 1b2(see figure 5.5). Let's make a closed path 1a2b1. Work on this closed path is equal to zero by condition, i.e. . From here. But, therefore

Thus, the zero work of the field forces on any closed path is a necessary and sufficient condition for the independence of work from the shape of the path, and can be considered a hallmark of any potential field of forces.

The field of central forces. Any force field is caused by the action of certain bodies. Force acting on a particle BUT in such a field is due to the interaction of this particle with these bodies. Forces that depend only on the distance between interacting particles and directed along a straight line connecting these particles are called central. An example of the latter are gravitational, Coulomb and elastic forces.

The central force acting on the particle BUT from the side of the particle IN, can be represented in general form:

where f(r) is a function that, for a given nature of interaction, depends only on r- distances between particles; - unit vector that specifies the direction of the radius-vector of the particle BUT relative to the particle IN(Fig. 5.6).

Let's prove that any stationary field of central forces is potentially.

To do this, we first consider the work of the central forces in the case when the force field is caused by the presence of one motionless particle IN. The elementary work of force (5.8) on displacement is . Since is the projection of the vector onto the vector , or onto the corresponding radius vector (Fig. 5.6), then . The work of this force along an arbitrary path from a point 1 to the point 2

The resulting expression depends only on the type of function f(r), i.e., on the nature of the interaction, and on the values r1 And r2 initial and final distances between particles BUT And IN. It has nothing to do with the shape of the path. And this means that this force field is potential.

Let us generalize the result obtained to the stationary force field caused by the presence of a set of immobile particles acting on the particle BUT with forces each of which is central. In this case, the work of the resulting force when moving the particle BUT from one point to another is equal to the algebraic sum of the work of individual forces. And since the work of each of these forces does not depend on the shape of the path, the work of the resulting force does not depend on it either.

Thus, indeed, any stationary field of central forces is potential.

Potential energy of a particle. The fact that the work of the forces of the potential field depends only on the initial and final positions of the particle makes it possible to introduce the extremely important concept of potential energy.

Imagine that we move a particle in a potential field of forces from different points P i to a fixed point ABOUT. Since the work of the field forces does not depend on the shape of the path, it remains dependent only on the position of the point R(at a fixed point ABOUT). And this means that this work will be some function of the radius vector of the point R. Denoting this function , we write

The function is called the potential energy of a particle in a given field.

Now let's find the work of the field forces when moving a particle from a point 1 exactly 2 (Fig. 5.7). Since the work does not depend on the path, we take the path passing through the point 0. Then the work on the path 1 02 can be presented in the form

or taking into account (5.9)

The expression on the right is the loss* of potential energy, i.e., the difference between the values of the particle's potential energy at the starting and ending points of the path.

_________________

* Change any value X can be characterized either by its increase or decrease. Increment X is called the difference of the final ( x2) and initial ( X 1) values of this quantity:

increment Δ X = X 2 - X 1.

Decline in size X is called the difference of its initial ( X 1) and final ( X 2) values:

decline X 1 - X 2 \u003d -Δ X,

i.e. decrease in value X is equal to its increment, taken with the opposite sign.

Increment and loss are algebraic quantities: if X 2 > x1, then the increase is positive and the decrease is negative, and vice versa.

Thus, the work of the field forces on the way 1 - 2 is equal to the decrease in the potential energy of the particle.

Obviously, a particle located at point 0 of the field can always be assigned any preselected value of potential energy. This corresponds to the circumstance that only the difference of potential energies at two points of the field can be determined by measuring the work, but not its absolute value. However, once the value is fixed

potential energy at any point, its values at all other points of the field are uniquely determined by formula (5.10).

Formula (5.10) makes it possible to find an expression for any potential force field. To do this, it is enough to calculate the work done by the field forces on any path between two points, and present it as a loss of some function, which is potential energy.

This is exactly what was done when calculating the work in the fields of elastic and gravitational (Coulomb) forces, as well as in a uniform gravitational field [see Fig. formulas (5.3) - (5.5)]. It is immediately clear from these formulas that the potential energy of a particle in these force fields has the following form:

1) in the field of elastic force

2) in the field of a point mass (charge)

3) in a uniform gravity field

We emphasize once again that the potential energy U is a function that is defined up to the addition of some arbitrary constant. This circumstance, however, is completely unimportant, because all formulas include only the difference in values U in two positions of the particle. Therefore, an arbitrary constant, the same for all points of the field, drops out. In this regard, it is usually omitted, which is done in the three previous expressions.

And there is one more important circumstance that should not be forgotten. Potential energy, strictly speaking, should be attributed not to a particle, but to a system of particles and bodies interacting with each other, causing a force field. With a given character of interaction, the potential energy of the interaction of a particle with given bodies depends only on the position of the particle relative to these bodies.

Relationship between potential energy and force. According to (5.10), the work of the potential field force is equal to the decrease in the potential energy of the particle, i.e. BUT 12 = U 1 - U 2 = - (U 2 - U one). With an elementary displacement, the last expression has the form dA = - dU, or

F l dl= - dU. (5.14)

i.e., the projection of the field strength at a given point on the direction of displacement is equal with the opposite sign to the partial derivative of the potential energy in this direction.

, then with the help of formula (5.16) we have the possibility to restore the field of forces .

The locus of points in space at which the potential energy U has the same value, defines an equipotential surface. It is clear that for every value U corresponds to its equipotential surface.

It follows from formula (5.15) that the projection of the vector onto any direction tangent to the equipotential surface at a given point is equal to zero. This means that the vector is normal to the equipotential surface at the given point. In addition, the minus sign in (5.15) means that the vector is directed towards decreasing potential energy. This is explained in Fig. 5.8, referring to the two-dimensional case; here is a system of equipotentials, and U 1 < U 2 < U 3 < … .

In addition to contact interactions that occur between bodies in contact, there are also interactions between bodies that are distant from each other.

In addition to contact interactions that occur between bodies in contact, there are also interactions between bodies that are distant from each other. For example, the interaction between the Sun and the Earth, the Earth and the Moon, the Earth and a body raised above its surface, the interaction between electrified bodies. These interactions are carried out through physical fields, which are a special form of matter. Each body creates a special state in the space surrounding it, called power field. This field manifests itself in the action of forces on other bodies. For example, the Earth creates a gravitational field. In it, a force - mg acts on each body of mass m at each point near the surface of the Earth.

Forces whose work does not depend on the path along which the particle moved, but is determined only by the initial and final position of the particle, are called conservative.

Let us show that the work of conservative forces on any closed path is equal to zero.

Consider an arbitrary closed path. Let us divide it by arbitrarily chosen points 1 and 2 into two sections: I and II. The work done on a closed path is:

(18 .1 )

Fig.18.1. Work of conservative forces on a closed path

A change in the direction of movement along section II to the opposite is accompanied by the replacement of all elementary displacements dr by (-dr), which causes it to reverse its sign. Then:

(18 .2 )

Now, substituting (18.2.) into (18.1.), we get that A=0, i.e. the above assertion has been proved by us. Another definition of conservative forces can be formulated as follows: conservative forces are forces whose work on any closed path is zero.

All forces that are not conservative are called non-conservative. Non-conservative forces include friction and resistance forces.

If the forces acting on the particle are the same in magnitude and direction at all points of the field, then the field is called homogeneous.

A field that does not change with time is called stationary. In the case of a uniform stationary field: F=const.

Statement: the forces acting on a particle in a uniform stationary field are conservative.

Let's prove this statement. Since the field is uniform and stationary, then F=const. Let's take two arbitrary points 1 and 2 in this field (Fig. 18.2.) and calculate the work done on the particle when it moves from point 1 to point 2.

18.2. The work of forces in a uniform stationary field on the way from point 1 to point 2

The work of forces acting on a particle in a uniform stationary field is:

where r F is the projection of the displacement vector r 12 onto the direction of the force, r F is determined only by the positions of points 1 and 2, and does not depend on the shape of the trajectory. Then, the work of the force in this field does not depend on the shape of the path, but is determined only by the positions of the initial and final points of displacement, i.e. the forces of a uniform stationary field are conservative.

Near the Earth's surface, the gravity field is a uniform stationary field and the work done by the force mg is:

(18 .4 )

where (h 1 -h 2) is the projection of the displacement r 12 on the direction of the force, the force mg is directed vertically downwards, the force of gravity is conservative.

Forces that depend only on the distance between interacting particles and directed along a straight line passing through these particles are called central. Examples of central forces are: Coulomb, gravitational, elastic.