Force

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Decomposition of forces acting on a solid located on a tilted plane

In classical physics, the force (abbreviation F) is a phenomenon that modifies the movement of a body (accelerates it, slows it down, changes direction, etc.) or deforms it. Forces can be represented by vectors, since they have magnitude and direction. The concept of force should not be confused with effort or energy.

In the International System of Units, the unit of measure for force is the newton, which is represented by the symbol N, in recognition of Isaac Newton for his contribution to physics, especially classical mechanics. The newton is a unit derived from the International System of Units that is defined as the force necessary to provide an acceleration of 1m/s² to an object of mass 1kg.

Concepts related to force include: push, which increases the speed of an object; drag, which slows down an object; and motor torque, which produces changes in the speed of rotation of an object. In an extended body, each part usually applies forces on the adjacent parts; the distribution of these forces through the body is the internal mechanical stress. Such internal mechanical stresses do not cause any acceleration of that body, since the forces balance each other. Pressure, the distribution of many small forces applied over an area of a body, is a simple type of stress that, if unbalanced, can cause the body to accelerate. Stress usually causes deformation of solid materials, or flow in fluids.

Introduction

Force is a mathematical model of intensity of interactions, along with energy. Thus, for example, the gravitational force is the attraction between bodies that have mass, the weight is the attraction that the Earth exerts on objects in the vicinity of its surface, the elastic force is that exerted by a deformed spring (compressed or stretched). In physics, there are two types of equations of force: those "of causes", in which the origin of the attraction or repulsion is specified, as, for example, Newton's law of universal gravitation or Coulomb's law; and those "of effects", which is, fundamentally, Newton's second law.

Force is a physical quantity of a vector character capable of deforming a body (static effect), modifying its speed or overcoming its inertia and setting it in motion if they were stationary (dynamic effect). In this sense, force can be defined as any action or influence capable of modifying the state of movement or rest of a body (giving it an acceleration that modifies the magnitude or direction of its velocity).

Commonly we refer to the force applied on an object without taking into account the other object or objects with which it is interacting and that will, in turn, experience other forces. Currently, force can be defined as a mathematical physical entity, of a vectorial nature, associated with the interaction of the body with other bodies that constitute its environment. This concept is directly related to Newton's third law.

Development of the concept

Ancient philosophers used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes held to fundamental errors in their understanding of force. In part, this was due to an incomplete understanding of the sometimes non-obvious force of friction, and a consequently inadequate view of the nature of natural motion. A fundamental misconception was the belief that a force is required to maintain motion. motion, even at a constant speed. Most of the earlier misunderstandings about motion and force were eventually corrected by Galileo Galilei and Isaac Newton. With his mathematical acumen, Isaac Newton formulated the laws of motion that were not improved for almost three hundred years. At the beginning of the 20th century, Einstein developed a theory of relativity that correctly predicted the action of forces on objects with nearby increasing moments. at the speed of light, and also provided insight into the forces produced by gravitation and inertia.

With modern knowledge of quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe the forces between particles smaller than atoms. The Standard Model predicts that exchanged particles, called gauge bosons, are the fundamental medium by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational. High-energy particle physics Observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction.

History

Bust of Archimedes

The concept of force was originally described by Archimedes, although only in static terms. Archimedes and others believed that the "natural state" of material objects in the terrestrial sphere was rest and that bodies tended, by themselves, towards that state if they were not acted on in any way. According to Aristotle, the perseverance of movement always required an efficient cause (something that seems to agree with everyday experience, where frictional forces can go unnoticed).

Galileo Galilei (1564-1642) would be the first to give a dynamic definition of force, opposed to that of Archimedes, clearly establishing the law of inertia, stating that a body on which no force acts remains in motion unaltered. This law, which refutes Archimedes' thesis, is still not obvious today to most people without scientific training.

It is considered that Isaac Newton was the first to mathematically formulate the modern definition of force, although he also used the Latin term vis impressa ('impressed force') and vis motrix for other different concepts. Furthermore, Isaac Newton postulated that gravitational forces vary according to the inverse square law of distance.

Charles Coulomb was the first to prove that the interaction between point electric or electronic charges also varies according to the inverse square law of distance (1784).

In 1798, Henry Cavendish was able to experimentally measure the force of gravitational attraction between two small masses using a torsion balance. Thanks to which he was able to determine the value of the universal gravitation constant and, therefore, he was able to calculate the mass of the Earth.

With the development of quantum electrodynamics, in the mid-XX century, it was found that "force" was a purely macroscopic magnitude arising from the conservation of linear momentum or momentum for elementary particles. For this reason, the so-called fundamental forces are often called "fundamental interactions."

Pre-Newtonian Concepts

Aristotle famously described a force like everything that makes an object suffer a “non-natural movement”

Since ancient times the concept of force has been recognized as an integral part of the functioning of each of the simple machines. The mechanical advantage given by a simple machine allowed less force to be used in exchange for that force acting over a greater distance for the same amount of work. The analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids.

Aristotle provided a philosophical discussion of the concept of force as an integral part of Aristotelian Cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest in different "natural places" of it. Aristotle believed that the unmoving objects on Earth, those composed mostly of the elements earth and water, were in their natural place on the ground and would remain so if left alone. He distinguished between the innate tendency of objects to find their "natural place" (for example for heavy bodies to fall), which led to "natural motion," and unnatural or forced motion, which required the continuous application of a force. force. This theory, based on everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual problems in explaining the behavior of projectiles, such as the flight of arrows. The place where the archer moves the projectile was at the beginning of the flight, and while the projectile sailed through the air, no discernible efficient cause acted on it. Aristotle was aware of this problem and proposed that the air displaced through the path of the projectile carries it to its target. This explanation requires a continuum like air for change of place in general.

Aristotelian physics began to face criticism in medieval science, first from John Philopon in the sixth century.

The shortcomings of Aristotelian physics would not be fully corrected until the 17th century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were rolled down a slope to disprove Aristotle's theory of motion. He showed that bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their velocity unless acted upon by a force, such as friction.

In the early 17th century, before Newton's Principles, the term "force" (in Latin, vis) was applied to many physical and non-physical phenomena, for example, for the acceleration of a point. The product of a point mass times the square of its velocity was called vis viva (live force) by Leibniz. The modern concept of force corresponds to Newton's vis motrix. (force of acceleration).

See also: Aristotelian physics and Theory of the impetus.

Newton's Laws

First Law

Main article: Newton's First Law

Newton's first law of motion states that objects continue to move in a state of constant velocity unless acted on by a net external force. (resultant force). This law is an extension of Galileo's idea that constant velocity was associated with no net force (see a more detailed description of this below). Newton proposed that every object with mass has an innate inertia that functions as the "natural state" of fundamental equilibrium instead of the Aristotelian idea of the "natural state of rest." That is, Newton's first empirical law contradicts Aristotle's intuitive belief that a net force is required to keep an object moving with constant velocity. By making "rest" physically indistinguishable from "constant nonzero velocity," Newton's first law directly connects inertia to the concept of relative velocities. Specifically, in systems where objects move with different speeds, it is impossible to determine which object is "in motion" and which object is "at rest." The laws of physics are the same in all inertial reference frames, that is, in all frames related by a Galilean transformation.

For example, while traveling in a moving vehicle at a constant speed, the laws of physics do not change as a result of your motion. If a person inside the vehicle throws a ball upwards, that person will observe that it rises vertically and falls vertically and will not have to apply a force in the direction the vehicle is moving. Another person, watching the passing of the moving vehicle, would observe that the ball follows a curved parabolic trajectory in the same direction as the movement of the vehicle. It is the inertia of the ball, associated with its constant speed in the direction of movement of the vehicle, that keeps the ball moving even when it is thrown up and falls again. From the perspective of the person in the car, the vehicle and everything inside it is at rest: It is the outside world that is moving with a constant velocity in the opposite direction of the vehicle. Since there is no experiment that can distinguish whether it is the vehicle that is at rest or the outside world that is at rest, the two situations are considered to be physically indistinguishable. Therefore, inertia applies equally to constant velocity motion as it does to rest.

Although Sir Isaac Newton's most famous equation is

{displaystyle scriptstyle {{vec {F}}=m{vec {a}}}}

, actually wrote a different form for his second law of movement that did not use differential calculation.

Second Law

Main article: Newton's Second Law

A modern statement of Newton's second law is a vector equation:

{displaystyle {vec {F}}={frac {mathrm {d} {vec {p}}}{mathrm {d} t}},}

where ${displaystyle {vec {p}}}$ It's time for the system, and ${displaystyle {vec {F}}}$ is the net force "suma of vectors". If a body is in balance, the "net" force is null by definition (although there may be balanced forces). On the contrary, the second law states that if there is a force Balanced that acts on an object, the moment of the object will change over time.

By the definition of moment,

{displaystyle {vec {F}}={frac {mathrm {d} {vec {p}}}{mathrm {d} t}}={frac {mathrm {d} left(m{vec {v}}right)}{mathrm {d} t}},}

where m It's the mass and ${displaystyle {vec {v}}}$ It's speed.

If Newton's second law is applied to a system of constant mass, m can move outside the derivative operator. The equation then becomes

{displaystyle {vec {F}}=m{frac {mathrm {d} {vec {v}}}{mathrm {d} t}}.}

Substituting the definition of acceleration, we get the algebraic version of Newton's second law:

{displaystyle {vec {F}}=m{vec {a}}.}

Newton has never explicitly stated the formula in the reduced form above.

Newton's second law states the direct proportionality of acceleration to force and the inverse proportionality of acceleration to mass. Accelerations can be defined by kinematic measurements. However, while kinematics is well described through reference frame analysis in advanced physics, there are still deep questions that remain as to what is the proper definition of mass. General relativity offers an equivalence between spacetime and mass, but in the absence of a coherent theory of quantum gravity, it is unclear how or if this connection is relevant at microscales. With some justification, Newton's second law can be taken as a quantitative definition of "mass" by writing the law as an equality; the relative units of force and mass are then fixed.

The use of Newton's second law as a definition of force has been neglected in some of the more rigorous textbooks, An exception to this rule is: because it is essentially a mathematical truism. Notable physicists, philosophers, and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll.

Newton's second law can be used to measure the strength of forces. For example, knowing the masses of the planets together with the accelerations of their orbits allows scientists to calculate the gravitational forces on the planets.

Third Law

Main article: Newton's Third Law

Whenever one body exerts one force on another, it simultaneously exerts an equal and opposite force on the former. In vector form, yes ${displaystyle scriptstyle {vec {F}}_{1,2}}$ is the force of the body 1 on the body 2 and ${displaystyle scriptstyle {vec {F}}_{2,1}}$ body 2 in body 1, then

{displaystyle {vec {F}}_{1,2}=-{vec {F}}_{2,1}.}

This law is sometimes called Law of Action-Reaction, with ${displaystyle scriptstyle {vec {F}}_{1,2}}$ Call action and ${displaystyle scriptstyle -{vec {F}}_{2,1}}$ the reaction.

Newton's third law is the result of applying symmetry to situations in which forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies, and therefore there is no force that is unidirectional or that acts on a single body.

In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:

{displaystyle {vec {F}}_{1,2}+{vec {F}}_{mathrm {2,1} }=0.}

More generally, in a closed system of particles, all internal forces are balanced. The particles can accelerate each other, but the center of mass of the system is not accelerated. If an external force acts on the system, it will cause the center of mass to accelerate in proportion to the magnitude of the external force divided by the mass of the system.

Combining Newton's Second and Third Law, it is possible to demonstrate that the linear moment of a system is preserved. In a two-particle system, yes. ${displaystyle scriptstyle {vec {p}}_{1}}$ is the time of object 1 and ${displaystyle scriptstyle {vec {p}}_{2}}$ the moment of object 2, then

{displaystyle {frac {mathrm {d} {vec {p}}_{1}}{mathrm {d} t}}+{frac {mathrm {d} {vec {p}}_{2}}{mathrm {d} t}}={vec {F}}_{1,2}+{vec {F}}_{2,1}=0.}

Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost or gained.

Force in Newtonian Mechanics

Force can be defined from the time derivative of linear momentum:

${displaystyle mathbf {F} ={frac {dmathbf {p} }{dt}}={frac {d(mmathbf {v})}{dt}}}$

If the mass remains constant, we can write:

(♪) ${displaystyle mathbf {F} =m{frac {dmathbf {v} }{dt}}=mmathbf {a} }$

where m is the mass and a is the acceleration, which is the traditional expression of Newton's second law. In the case of statics, where there are no accelerations, the acting forces can be deduced from equilibrium considerations.

The equation (*) is useful above all to describe the motion of particles or bodies whose shape is not important for the problem. But even if it is a question of studying the mechanics of rigid solids, additional postulates are needed to define the angular velocity of the solid, or its angular acceleration as well as its relationship with the applied forces. For an arbitrary reference system the equation (*) must be replaced by:

${displaystyle mathbf {F} =m{frac {d^{2}mathbf {r} }{dt^{2}}}+2mathbf {A} _{t}{frac {dmathbf {r} }{dt}}+left({frac {dmathbf {A} _{t}}{dt}}-mathbf {A} _{t}^{2}right)mathbf {r} }$

Where:

${displaystyle mathbf {A} _{t}={begin{pmatrix}0&omega _{z}(t)&-omega _{y}(t)\-omega _{z}(t)&0&omega _{x}(t)\omega _{y}(t)&-omega _{x}(t)&0end{pmatrix}},qquad mathbf {A} _{t}mathbf {u} ={boldsymbol {omega }}(t)times mathbf {u} }$

Contact forces and distance forces

In a strict sense, all natural forces are forces produced at a distance as a product of the interaction between bodies; However, from the macroscopic point of view, it is customary to divide forces into two general types:

Contact forceswhich are given as a product of the interaction of the bodies in direct contact; that is, impacting their free surfaces (such as normal force).
Distance forces, such as gravitational force or coulombic between loads, due to the interaction between fields (gravitatory, electric, etc.) and that occur when the bodies are separated some distance from each other, for example: weight.

Internal and contact forces

F_N represents normal force exercised by the tilted plane on the object placed on it

In solids, the Pauli exclusion principle, together with the conservation of energy, leads to atoms having their electrons distributed in shells and having impenetrability despite being 99% empty. Impenetrability results from the fact that atoms are "extensive" and that electrons in the outer shells exert repulsive electrostatic forces that make matter macroscopically impenetrable.

The above means that two bodies placed in «contact» will experience normal (or approximately normal) resultant forces to the surface that will prevent the overlapping of the electronic clouds of both bodies.

The internal forces are similar to the contact forces between both bodies and although they have a more complicated form, since there is no macroscopic surface through which the surface occurs. The complication translates, for example, in that the internal forces need to be modeled by means of a stress tensor in which the force per unit area experienced by a point inside depends on the direction along which the forces are considered.

The above refers to solids, in fluids at rest the internal forces depend essentially on pressure, and in fluids in motion also viscosity can play an important role.

Friction

Main article: Friction

Friction in solids can occur between their free surfaces in contact. In the treatment of problems through Newtonian mechanics, the friction between solids is frequently modeled as a tangent force on any of the contact planes between their surfaces, with a value proportional to the normal force.

The friction between solid-liquid and inside a liquid or a gas depends essentially on whether the flow is considered laminar or turbulent and on its constitutive equation.

Gravitational force

Gravitational forces between two particles

In Newtonian mechanics, the force of attraction between two masses, whose centers of gravity are far apart compared to the dimensions of the body, is given by Newton's law of universal gravitation:

${displaystyle mathbf {F} _{21}=-G{frac {m_{1}m_{2}}{|mathbf {r} _{21}|^{2}}}mathbf {e} _{21}=-G{frac {m_{1}m_{2}}{|mathbf {r} _{21}|^{3}}}mathbf {r} _{21}}$

Where:

{displaystyle mathbf {F} _{21}}

is the force that acts on body 2, exercised by body 1.

G,

constant universal gravitation.

{displaystyle mathbf {r} _{21}=mathbf {r} _{2}-mathbf {r} _{1}}

relative position vector of the body 2 regarding the body 1.

{displaystyle mathbf {e} _{21}}

It's the unit vector led from 1 to 2.

{displaystyle m_{1},m_{2},}

bodies 1 and 2.

When the mass of one of the bodies is very large compared to that of the other (for example, if it has planetary dimensions), the previous expression becomes a simpler one:

${displaystyle mathbf {F} =-mleft(G{frac {M}{R_{0}^{2}}}right){hat {mathbf {u} }}_{r}=-mg{hat {mathbf {u} }}_{r}=mmathbf {g} }$

Where:

{mathbf {F}}

is the strength of a large body (like a planet or a star) over the small body.

{mathbf {u}}_{r}

is a unitary vector directed from the center of the body of large mass to the body of smaller mass.

{displaystyle R_{0},}

is the distance between the center of the large mass body and the lower mass.

Stationary Field Forces

Main article: Field (physics)

In Newtonian mechanics it is also possible to model some time-constant forces as force fields. For example, the force between two immobile electric charges can be adequately represented by Coulomb's law:

${displaystyle mathbf {F} _{12}=-kappa {frac {q_{1}q_{2}}{|mathbf {r} _{12}|^{3}}}mathbf {r} _{12}}$

Where:

{displaystyle mathbf {F} _{12}}

is the force exercised by load 1 on load 2.

kappa ,

a constant that will depend on the unit system for the load.

{displaystyle mathbf {r} _{12}}

load position vector 2 regarding load 1.

{displaystyle q_{1},q_{2},}

load value.

Static magnetic fields and those due to static charges with more complex distributions can also be summarized in two vector functions called electric field and magnetic field such that a particle in motion with respect to the static sources of said fields is given by the expression for Lorentz:

${displaystyle mathbf {F} =q(mathbf {E} +mathbf {v} times mathbf {B}),}$

Where:

{displaystyle mathbf {E} }

It's the electric field.

{displaystyle mathbf {B} }

It's the magnetic field.

{mathbf {v}}

It's the particle speed.

q,

is the total load of the particle.

Non-constant force fields however present a difficulty especially when they are created by fast moving particles, because in such cases relativistic lag effects can be important, and classical mechanics gives rise to a treatment of action a distance which may be inappropriate if the forces change rapidly over time.

Electric force

The electrical force is also an action at a distance, but sometimes the interaction between the bodies acts as an attractive force while, at other times, it has the opposite effect., that is, it can act as a repulsive force.

Units of force

In the International System of Units (SI) and in the Cegesimal (cgs), the fact of defining the force from the mass and acceleration (magnitude in which length and time intervene), leads to the fact that the force be a derived quantity. On the contrary, in the Technical System force is a Fundamental Unit and from it the unit of mass is defined in this system, the technical unit of mass, abbreviated u.t.m. (has no symbol). This fact meets the evidence that current physics has, expressed in the concept of fundamental forces, and is reflected in the International System of Units.

International Unit System (IS)
- newton (N)
Technical Unit System
- kilogram-force (kg_f) or kilopondium (kp)
Cegesimal Unit System
- dyna (dyn)
Anglo-Saxon unit system
- Poundal
- Free strength (lb_f)
- KIP (= 1000 lb_f)

Equivalences: 1 newton = 100 000 dynasiums; 1 kilogram-force = 9,80665 newtons; 1 pound strength ≡ 4,448222 newtons

Force in relativistic mechanics

In special relativity, force must be defined only as the derivative of linear momentum, since in this case the force is not simply proportional to the acceleration:

${displaystyle mathbf {F} ={frac {d}{dt}}left({frac {mmathbf {v} }{sqrt {1-{frac {v^{2}}{c^{2}}}}}}right)={frac {mmathbf {v} }{left[1-{frac {v^{2}}{c^{2}}}right]^{3/2}}}left({frac {mathbf {v} }{c^{2}}}cdot mathbf {a} right)+{frac {mmathbf {a} }{sqrt {1-{frac {v^{2}}{c^{2}}}}}}}$

In fact, in general, the acceleration vector and the force vector will not even be parallel, only in the uniform circular movement and in any rectilinear movement will the force and acceleration vector be parallel, but in general the magnitude of the force will depend as much on velocity as acceleration.

Gravitational "Force"

In the theory of general relativity the gravitational field is not treated as a real force field, but as an effect of the curvature of space-time. A mass particle that does not suffer the effect of any other interaction, other than gravitational, will follow a geodesic trajectory of least curvature through space-time, and therefore its equation of motion will be:

${displaystyle {cfrac {d^{2}x^{mu }}{ds^{2}}}+sum _{sigmanu }Gamma _{sigma nu }^{mu }{cfrac {dx^{sigma }}{ds}}{cfrac {dx^{nu }}{ds}}=0}$

Where:

{displaystyle x^{mu },}

are the position coordinates of the particle.

s,

the arch parameter, which is proportional to the time proper to the particle.

{displaystyle Gamma _{sigma nu }^{mu },}

are Christoffel symbols corresponding to the time-space metric.

The apparent gravitational force comes from the term associated with the Christoffel symbols. An observer in "free fall" will form a moving reference frame in which these Christoffel symbols are zero, and therefore will not perceive any gravitational force, as sustained by the equivalence principle that helped Einstein formulate his ideas about the gravitational field.

Electromagnetic force

The effect of the electromagnetic field on a relativistic particle is given by the covariant expression of the Lorentz force:

${displaystyle f_{alpha }=sum _{beta }q F_{alpha beta } u^{beta },}$

Where:

{displaystyle f_{alpha },}

are the covariant components of the quadruple experienced by the particle.

{displaystyle F_{alpha beta },}

are the components of the electromagnetic field tensor.

{displaystyle u^{alpha },}

are the components of the particle quadrantility.

The equation of movement of a particle in a curved space-time and subjected to the action of the previous force is given by:

${displaystyle m{frac {Du^{mu }}{Dtau }}=mleft({cfrac {d^{2}x^{mu }}{dtau ^{2}}}+Gamma _{sigma nu }^{mu }{cfrac {dx^{sigma }}{dtau }}{cfrac {dx^{nu }}{dtau }}right)=f^{mu }}$

Where the previous expression has been applied the Einstein summation convention for repeated indices, the member on the right represents the quadriaceleration and the other magnitudes being:

{displaystyle f^{mu }=g^{mu alpha }f_{alpha },}

are the contravarianetes components of the electromagnetic quadruple on the particle.

m,

is the mass of the particle.

Force in quantum physics

Force in quantum mechanics

In quantum mechanics it is not easy to define for many systems a clear equivalent of force. This happens because in quantum mechanics a mechanical system is described by a wave function or state vector ${displaystyle scriptstyle |psi rangle }$ which in general represents the entire system as a whole and cannot be separated into parts. Only for systems where the state of the system can break down in a non-ambiguous way ${displaystyle scriptstyle |psi rangle =|psi _{A}rangle +|psi _{B}rangle }$ where each of these two parts represents a part of the system it is possible to define the concept of force. However, in most interesting systems this decomposition is not possible. For example if we consider the set of electrons of an atom, which is a set of identical particles it is not possible to determine a magnitude that represents the force between two specific electrons, because it is not possible to write a wave function that describes separately the two electrons.

However, in the case of an isolated particle subjected to the action of a conservative force, it is possible to describe the force by means of an external potential and introduce the notion of force. This situation is the one that occurs, for example, in the Schrödinger atomic model for a hydrogen atom where the electron and the nucleus are discernible from each other. In this and other cases of an isolated particle in a potential, Ehrenfest's theorem leads to a generalization of Newton's second law in the form:

${displaystyle {frac {d}{dt}}langle prangle =int Phi ^{*}V(mathbf {x}t)nabla Phi ~d^{3}mathbf {x} -int Phi ^{*}(nabla V(mathbf {x}t))Phi ~d^{3}mathbf {x} -int Phi ^{*}V(mathbf {x}t)nabla Phi ~d^{3}mathbf {x} }$ ${displaystyle =0-int Phi ^{*}(nabla V(mathbf {x}t))Phi ~d^{3}mathbf {x} -0=langle -nabla V(mathbf {x}t)rangle =langle Frangle}$

Where:

<math alttext="{displaystyle ,}" xmlns="http://www.w3.org/1998/Math/MathML">.p▪{displaystyle θp,}<img alt="{displaystyle ,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10501c7bff65669013790f43e054190239b7b076" style="vertical-align: -0.671ex; width:6.463ex; height:2.176ex;"/> is the expected value of the linear moment of the particle.

{displaystyle Phi (mathbf {x}),Phi ^{*}(mathbf {x})}

is the wave function of the particle and its complex conjugate.

{displaystyle V(mathbf {x}t),}

is the potential of which to derive “forces”.

nabla ,

denotes the operator.

In other cases, such as experiments involving the collision or dispersion of elementary particles with positive energy that are fired against other target particles, such as typical experiments carried out in particle accelerators, it is sometimes possible to define a potential that is related to the typical force that a colliding particle will experience, but even so in many cases we cannot speak of force in the classical sense of the word.

Fundamental forces in quantum field theory

Main article: Fundamental interactions

Explanatory table of the 4 fundamental forces

In quantum field theory, the term "force" has a slightly different meaning than it does in classical mechanics because of the specific difficulty noted in the previous section of defining a quantum equivalent of classical forces. For that reason the term "fundamental force" in quantum field theory refers to the mode of interaction between particles or quantum fields, rather than to a concrete measure of the interaction of two particles or fields.

Quantum field theory tries to give a description of the forms of interaction existing between the different forms of matter or quantum fields existing in the Universe. Thus the term "fundamental forces" now refers to the clearly differentiated modes of interaction that we know of. Each fundamental force will be described by a different theory and will postulate different interaction Lagrangians that describe what that peculiar mode of interaction is like.

When the idea of the fundamental force was formulated, it was considered that there were four "fundamental forces": gravitational, electromagnetic, strong nuclear, and weak nuclear. The description of the traditional "fundamental forces" is as follows:

Gravitational is the force of attraction that one mass exerts on another, and affects all bodies. Gravity is a very weak and one-way force, but of infinite reach.
The electromagnetic force affects the electrically charged bodies, and is the force involved in the physical and chemical transformations of atoms and molecules. It is much more intense than gravitational force, it can have two senses (attractive and repulsive) and its reach is infinite.
The strong nuclear force or interaction is the one that keeps the components of the atomic nuclei together, and acts indistinctly between any two nucleons, protons or neutrons. Its scope is of the order of nuclear dimensions, but it is more intense than the electromagnetic force.
The weak nuclear force or interaction is responsible for the beta disintegration of neutrons; neutrinos are sensitive only to this type of electromagnetic interaction (apart from gravitational) and its scope is even lower than that of strong nuclear interaction.

However, it should be noted that the number of fundamental forces in the above sense depends on our state of knowledge, so until the late 1960s the weak interaction and the electromagnetic interaction were considered different fundamental forces, but theoretical advances They allowed us to establish that in reality both types of interaction were phenomenologically different manifestations of the same "fundamental force", the electroweak interaction. One suspects that ultimately all "fundamental forces" are phenomenological manifestations of a single "force" that would be described by some kind of unified theory or theory of everything.

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