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If one body is motionless to begin with e. In this instance the initial velocities of the bodies would be non-zero, or the bodies would have to be massless.

One measure of the inelasticity of the collision is the coefficient of restitution C R , defined as the ratio of relative velocity of separation to relative velocity of approach.

In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula: [12].

The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation.

Rockets also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.

Real motion has both direction and velocity and must be represented by a vector. In a coordinate system with x , y , z axes, velocity has components v x in the x -direction, v y in the y -direction, v z in the z -direction.

The vector is represented by a boldface symbol: [14]. The equations in the previous sections, work in vector form if the scalars p and v are replaced by vectors p and v.

Each vector equation represents three scalar equations. For example,. The kinetic energy equations are exceptions to the above replacement rule.

The equations are still one-dimensional, but each scalar represents the magnitude of the vector , for example,.

Often coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results combined to produce a vector result.

A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision as in the figure.

The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a rocket ejecting fuel or a star accreting gas.

In analyzing such an object, one treats the object's mass as a function that varies with time: m t.

This equation does not correctly describe the motion of variable-mass objects. The correct equation is.

When considered together, the object and the mass dm constitute a closed system in which total momentum is conserved. Newtonian physics assumes that absolute time and space exist outside of any observer; this gives rise to Galilean invariance.

It also results in a prediction that the speed of light can vary from one reference frame to another.

This is contrary to observation. In the special theory of relativity , Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light c is invariant.

As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.

Consider, for example, one reference frame moving relative to another at velocity v in the x direction. The Galilean transformation gives the coordinates of the moving frame as.

Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the inertial mass m of an object a function of velocity:.

In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a fourth coordinate along with the three space coordinates.

These vectors are generally represented by capital letters, for example R for position. The expression for the four-momentum depends on how the coordinates are expressed.

Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length.

In all the coordinate systems, the contravariant relativistic four-velocity is defined by. Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

The magnitude of the momentum four-vector is equal to m 0 c :. In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle.

This is unlike the non-relativistic case where they travel at right angles. The four-momentum of a planar wave can be related to a wave four-vector [23].

Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by constraints.

For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot.

Many such constraints can be incorporated by changing the normal Cartesian coordinates to a set of generalized coordinates that may be fewer in number.

They introduce a generalized momentum , also known as the canonical or conjugate momentum , that extends the concepts of both linear momentum and angular momentum.

To distinguish it from generalized momentum, the product of mass and velocity is also referred to as mechanical , kinetic or kinematic momentum.

In Lagrangian mechanics , a Lagrangian is defined as the difference between the kinetic energy T and the potential energy V :.

If a coordinate q i is not a Cartesian coordinate, the associated generalized momentum component p i does not necessarily have the dimensions of linear momentum.

Even if q i is a Cartesian coordinate, p i will not be the same as the mechanical momentum if the potential depends on velocity.

In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as. Each component p j is said to be the conjugate momentum for the coordinate q j.

Now if a given coordinate q i does not appear in the Lagrangian although its time derivative might appear , then.

This is the generalization of the conservation of momentum. Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates.

An example is found in the section on electromagnetism. In Hamiltonian mechanics , the Lagrangian a function of generalized coordinates and their derivatives is replaced by a Hamiltonian that is a function of generalized coordinates and momentum.

The Hamiltonian is defined as. The Hamiltonian equations of motion are [29]. As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.

Conservation of momentum is a mathematical consequence of the homogeneity shift symmetry of space position in space is the canonical conjugate quantity to momentum.

That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem.

In Maxwell's equations , the forces between particles are mediated by electric and magnetic fields. The electromagnetic force Lorentz force on a particle with charge q due to a combination of electric field E and magnetic field B is.

These quantities form a four-vector, so the analogy is consistent; besides, the concept of potential momentum is important in explaining the so-called hidden-momentum of the electromagnetic fields [36].

In Newtonian mechanics, the law of conservation of momentum can be derived from the law of action and reaction , which states that every force has a reciprocating equal and opposite force.

Under some circumstances, moving charged particles can exert forces on each other in non-opposite directions.

The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.

The momentum density is proportional to the Poynting vector S which gives the directional rate of energy transfer per unit area: [38] [39].

If momentum is to be conserved over the volume V over a region Q , changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum.

If P mech is the momentum of all the particles in Q , and the particles are treated as a continuum, then Newton's second law gives.

The quantity T ij is called the Maxwell stress tensor , defined as. The above results are for the microscopic Maxwell equations, applicable to electromagnetic forces in a vacuum or on a very small scale in media.

It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary.

The definition of electromagnetic momentum density is modified to. The electromagnetic stress tensor depends on the properties of the media.

In quantum mechanics , momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once.

In quantum mechanics, position and momentum are conjugate variables. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms.

For example, in momentum space the momentum operator is represented as. Electromagnetic radiation including visible light , ultraviolet light, and radio waves is carried by photons.

Even though photons the particle aspect of light have no mass, they still carry momentum. This leads to applications such as the solar sail.

The calculation of the momentum of light within dielectric media is somewhat controversial see Abraham—Minkowski controversy.

In fields such as fluid dynamics and solid mechanics , it is not feasible to follow the motion of individual atoms or molecules.

Instead, the materials must be approximated by a continuum in which there is a particle or fluid parcel at each point that is assigned the average of the properties of atoms in a small region nearby.

Consider a column of water in hydrostatic equilibrium. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced.

The first is gravity, which acts directly on each atom and molecule inside. The second force is the sum of all the forces exerted on its surface by the surrounding water.

The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the pressure p.

The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is [44].

If the forces are not balanced, the droplet accelerates. Instead, the material derivative is needed: [45]. Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to advection as fluid is carried past the point.

This is equal to the net force on the droplet. Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above.

In addition, surface forces can deform the droplet. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another.

If the speed in the x direction varies with z , the tangential force in direction x per unit area normal to the z direction is. This is also a flux , or flow per unit area, of x-momentum through the surface.

Including the effect of viscosity, the momentum balance equations for the incompressible flow of a Newtonian fluid are.

These are known as the Navier—Stokes equations. The momentum balance equations can be extended to more general materials, including solids.

The local conservation of momentum is expressed by the Cauchy momentum equation :. The Cauchy momentum equation is broadly applicable to deformations of solids and liquids.

The relationship between the stresses and the strain rate depends on the properties of the material see Types of viscosity.

A disturbance in a medium gives rise to oscillations, or waves , that propagate away from their source.

In a fluid, small changes in pressure p can often be described by the acoustic wave equation :. In a solid, similar equations can be obtained for propagation of pressure P-waves and shear S-waves.

In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero.

However, nonlinear effects can give rise to a nonzero average. In about AD, working in Alexandria, Byzantine philosopher John Philoponus developed a concept of momentum in his commentary to Aristotle 's Physics.

Aristotle claimed that everything that is moving must be kept moving by something. For example, a thrown ball must be kept moving by motions of the air.

Most writers continued to accept Aristotle's theory until the time of Galileo, but a few were skeptical.

Philoponus pointed out the absurdity in Aristotle's claim that motion of an object is promoted by the same air that is resisting its passage.

He proposed instead that an impetus was imparted to the object in the act of throwing it. He agreed that an impetus is imparted to a projectile by the thrower; but unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, he viewed it as a persistent, requiring external forces such as air resistance to dissipate it.

Buridan, who in about was made rector of the University of Paris, referred to impetus being proportional to the weight times the speed.

Moreover, Buridan's theory was different from his predecessor's in that he did not consider impetus to be self-dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus.

This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and more important, he believed that it is speed rather than velocity that is conserved.

So for Descartes if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion.

Leibniz , in his " Discourse on Metaphysics ", gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances.

He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved.

Christiaan Huygens concluded quite early that Descartes's laws for the elastic collision of two bodies must be wrong, and he formulated the correct laws.

The war ended in , and Huygens announced his results to the Royal Society in The first correct statement of the law of conservation of momentum was by English mathematician John Wallis in his work, Mechanica sive De Motu, Tractatus Geometricus : "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result".

His Definition II defines quantitas motus , "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum.

From Wikipedia, the free encyclopedia. Redirected from Orders of magnitude momentum. This article is about linear momentum. It is not to be confused with angular momentum.

This article is about momentum in physics. For other uses, see Momentum disambiguation. Conserved physical quantity related to the motion of a body.

Momentum of a pool cue ball is transferred to the racked balls after collision. Second law of motion. History Timeline Textbooks.

Newton's laws of motion. Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton—Jacobi equation Appell's equation of motion Koopman—von Neumann mechanics.

Core topics. Circular motion Rotating reference frame Centripetal force Centrifugal force reactive Coriolis force Pendulum Tangential speed Rotational speed.

Main article: Conservation of momentum. Main article: Elastic collision. Main article: Inelastic collision. See also: Variable-mass system.

See also: Mass in special relativity and Tests of relativistic energy and momentum. Main article: Four-momentum. See also: Analytical mechanics.

Further information: Momentum operator. Main article: Cauchy momentum equation. This section needs attention from an expert in History of Science.

The specific problem is: Dispute over originator of conservation of momentum. See the talk page for details. WikiProject History of Science may be able to help recruit an expert.

November See also: Theory of impetus. Physics portal. Crystal momentum Galilean cannon Momentum transfer Newton's cradle Planck momentum Position and momentum space.

Archived from the original on Retrieved Engineering Mechanics, An Introduction to Dynamics 3rd ed.

PWS Publishing Company. Invitation to Contemporary Physics illustrated ed. World Scientific. Archived from the original on 18 August Retrieved 2 August Jewett, Jr Archived from the original on 22 August Archived from the original on 16 July Biomechanics of sport and exercise 2nd ed.

Champaign, IL [u. An Introduction to Mechanics. Boisvert July Fantasia Independent Online South Africa.

Marianopolis Matters. Marianopolis College: 8— Fall Archived from the original PDF on March 4, Rotten Tomatoes.

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In Newtonian mechanics , linear momentum , translational momentum , or simply momentum pl. It is a vector quantity, possessing a magnitude and a direction.

Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it.

Momentum depends on the frame of reference , but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change.

Momentum is also conserved in special relativity with a modified formula and, in a modified form, in electrodynamics , quantum mechanics , quantum field theory , and general relativity.

It is an expression of one of the fundamental symmetries of space and time: translational symmetry. Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics , allow one to choose coordinate systems that incorporate symmetries and constraints.

In these systems the conserved quantity is generalized momentum , and in general this is different from the kinetic momentum defined above.

The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function.

The momentum and position operators are related by the Heisenberg uncertainty principle. In continuous systems such as electromagnetic fields , fluid dynamics and deformable bodies , a momentum density can be defined, and a continuum version of the conservation of momentum leads to equations such as the Navier—Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids.

Momentum is a vector quantity : it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide.

Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations see multiple dimensions.

The momentum of a particle is conventionally represented by the letter p. It is the product of two quantities, the particle's mass represented by the letter m and its velocity v : [1].

The unit of momentum is the product of the units of mass and velocity. Being a vector, momentum has magnitude and direction.

The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m 1 and m 2 , and velocities v 1 and v 2 , the total momentum is.

A system of particles has a center of mass , a point determined by the weighted sum of their positions:.

If one or more of the particles is moving, the center of mass of the system will generally be moving as well unless the system is in pure rotation around it.

This is known as Euler's first law. In differential form, this is Newton's second law ; the rate of change of the momentum of a particle is equal to the instantaneous force F acting on it, [1].

If the net force experienced by a particle changes as a function of time, F t , the change in momentum or impulse J between times t 1 and t 2 is.

Under the assumption of constant mass m , it is equivalent to write. In a closed system one that does not exchange any matter with its surroundings and is not acted on by external forces the total momentum is constant.

This fact, known as the law of conservation of momentum , is implied by Newton's laws of motion.

Because of the third law, the forces between them are equal and opposite. If the velocities of the particles are u 1 and u 2 before the interaction, and afterwards they are v 1 and v 2 , then.

This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero.

This conservation law applies to all interactions, including collisions and separations caused by explosive forces. Momentum is a measurable quantity, and the measurement depends on the motion of the observer.

For example: if an apple is sitting in a glass elevator that is descending, an outside observer, looking into the elevator, sees the apple moving, so, to that observer, the apple has a non-zero momentum.

To someone inside the elevator, the apple does not move, so, it has zero momentum. The two observers each have a frame of reference , in which, they observe motions, and, if the elevator is descending steadily, they will see behavior that is consistent with those same physical laws.

Suppose a particle has position x in a stationary frame of reference. From the point of view of another frame of reference, moving at a uniform speed u , the position represented by a primed coordinate changes with time as.

This is called a Galilean transformation. Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law is unchanged.

Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion.

This independence of reference frame is called Newtonian relativity or Galilean invariance. A change of reference frame, can, often, simplify calculations of motion.

For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest.

Another, commonly used reference frame, is the center of mass frame — one that is moving with the center of mass.

In this frame, the total momentum is zero. By itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision.

Another property of the motion, kinetic energy , must be known. This is not necessarily conserved. If it is conserved, the collision is called an elastic collision ; if not, it is an inelastic collision.

An elastic collision is one in which no kinetic energy is absorbed in the collision. Perfectly elastic "collisions" can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps them apart.

A slingshot maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two pool balls is a good example of an almost totally elastic collision, due to their high rigidity , but when bodies come in contact there is always some dissipation.

A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies.

If the velocities are u 1 and u 2 before the collision and v 1 and v 2 after, the equations expressing conservation of momentum and kinetic energy are:.

A change of reference frame can simplify analysis of a collision. For example, suppose there are two bodies of equal mass m , one stationary and one approaching the other at a speed v as in the figure.

Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed v.

The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion.

Therefore, the final velocities are given by [4]. In general, when the initial velocities are known, the final velocities are given by [9].

If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change.

In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy such as heat or sound.

Examples include traffic collisions , [10] in which the effect of loss of kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms as in the Franck—Hertz experiment ; [11] and particle accelerators in which the kinetic energy is converted into mass in the form of new particles.

In a perfectly inelastic collision such as a bug hitting a windshield , both bodies have the same motion afterwards.

A head-on inelastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies.

If the velocities are u 1 and u 2 before the collision then in a perfectly inelastic collision both bodies will be travelling with velocity v after the collision.

The equation expressing conservation of momentum is:. If one body is motionless to begin with e.

In this instance the initial velocities of the bodies would be non-zero, or the bodies would have to be massless.

One measure of the inelasticity of the collision is the coefficient of restitution C R , defined as the ratio of relative velocity of separation to relative velocity of approach.

In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula: [12].

The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation.

Rockets also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.

Real motion has both direction and velocity and must be represented by a vector. In a coordinate system with x , y , z axes, velocity has components v x in the x -direction, v y in the y -direction, v z in the z -direction.

The vector is represented by a boldface symbol: [14]. The equations in the previous sections, work in vector form if the scalars p and v are replaced by vectors p and v.

Each vector equation represents three scalar equations. For example,. The kinetic energy equations are exceptions to the above replacement rule.

The equations are still one-dimensional, but each scalar represents the magnitude of the vector , for example,. Often coordinates can be chosen so that only two components are needed, as in the figure.

Each component can be obtained separately and the results combined to produce a vector result. A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision as in the figure.

The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a rocket ejecting fuel or a star accreting gas.

In analyzing such an object, one treats the object's mass as a function that varies with time: m t.

This equation does not correctly describe the motion of variable-mass objects. The correct equation is. When considered together, the object and the mass dm constitute a closed system in which total momentum is conserved.

Newtonian physics assumes that absolute time and space exist outside of any observer; this gives rise to Galilean invariance.

It also results in a prediction that the speed of light can vary from one reference frame to another. This is contrary to observation. In the special theory of relativity , Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light c is invariant.

As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.

Consider, for example, one reference frame moving relative to another at velocity v in the x direction. The Galilean transformation gives the coordinates of the moving frame as.

Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the inertial mass m of an object a function of velocity:.

In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a fourth coordinate along with the three space coordinates.

These vectors are generally represented by capital letters, for example R for position. The expression for the four-momentum depends on how the coordinates are expressed.

Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length.

In all the coordinate systems, the contravariant relativistic four-velocity is defined by. Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

The magnitude of the momentum four-vector is equal to m 0 c :. In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle.

This is unlike the non-relativistic case where they travel at right angles. The four-momentum of a planar wave can be related to a wave four-vector [23].

Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by constraints. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot.

Many such constraints can be incorporated by changing the normal Cartesian coordinates to a set of generalized coordinates that may be fewer in number.

They introduce a generalized momentum , also known as the canonical or conjugate momentum , that extends the concepts of both linear momentum and angular momentum.

To distinguish it from generalized momentum, the product of mass and velocity is also referred to as mechanical , kinetic or kinematic momentum.

In Lagrangian mechanics , a Lagrangian is defined as the difference between the kinetic energy T and the potential energy V :. If a coordinate q i is not a Cartesian coordinate, the associated generalized momentum component p i does not necessarily have the dimensions of linear momentum.

Even if q i is a Cartesian coordinate, p i will not be the same as the mechanical momentum if the potential depends on velocity.

In this mathematical framework, a generalized momentum is associated with the generalized coordinates.

Its components are defined as. Each component p j is said to be the conjugate momentum for the coordinate q j. Now if a given coordinate q i does not appear in the Lagrangian although its time derivative might appear , then.

This is the generalization of the conservation of momentum. Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates.

An example is found in the section on electromagnetism. In Hamiltonian mechanics , the Lagrangian a function of generalized coordinates and their derivatives is replaced by a Hamiltonian that is a function of generalized coordinates and momentum.

The Hamiltonian is defined as. The Hamiltonian equations of motion are [29]. As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.

Conservation of momentum is a mathematical consequence of the homogeneity shift symmetry of space position in space is the canonical conjugate quantity to momentum.

That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem.

In Maxwell's equations , the forces between particles are mediated by electric and magnetic fields. The electromagnetic force Lorentz force on a particle with charge q due to a combination of electric field E and magnetic field B is.

These quantities form a four-vector, so the analogy is consistent; besides, the concept of potential momentum is important in explaining the so-called hidden-momentum of the electromagnetic fields [36].

In Newtonian mechanics, the law of conservation of momentum can be derived from the law of action and reaction , which states that every force has a reciprocating equal and opposite force.

Under some circumstances, moving charged particles can exert forces on each other in non-opposite directions. The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.

The momentum density is proportional to the Poynting vector S which gives the directional rate of energy transfer per unit area: [38] [39].

If momentum is to be conserved over the volume V over a region Q , changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum.

If P mech is the momentum of all the particles in Q , and the particles are treated as a continuum, then Newton's second law gives.

The quantity T ij is called the Maxwell stress tensor , defined as. The above results are for the microscopic Maxwell equations, applicable to electromagnetic forces in a vacuum or on a very small scale in media.

It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary.

The definition of electromagnetic momentum density is modified to. The electromagnetic stress tensor depends on the properties of the media.

In quantum mechanics , momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once.

In quantum mechanics, position and momentum are conjugate variables. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms.

For example, in momentum space the momentum operator is represented as. Electromagnetic radiation including visible light , ultraviolet light, and radio waves is carried by photons.

Even though photons the particle aspect of light have no mass, they still carry momentum. This leads to applications such as the solar sail.

The calculation of the momentum of light within dielectric media is somewhat controversial see Abraham—Minkowski controversy. In fields such as fluid dynamics and solid mechanics , it is not feasible to follow the motion of individual atoms or molecules.

Instead, the materials must be approximated by a continuum in which there is a particle or fluid parcel at each point that is assigned the average of the properties of atoms in a small region nearby.

Consider a column of water in hydrostatic equilibrium. All the forces on the water are in balance and the water is motionless.

On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The second force is the sum of all the forces exerted on its surface by the surrounding water.

The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the pressure p.

The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is [44].

If the forces are not balanced, the droplet accelerates. Instead, the material derivative is needed: [45]. Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to advection as fluid is carried past the point.

This is equal to the net force on the droplet. Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above.

In addition, surface forces can deform the droplet. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another.

If the speed in the x direction varies with z , the tangential force in direction x per unit area normal to the z direction is. Momentum received generally negative reviews.

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July 23, Retrieved July 23, The Numbers. Boisvert July Fantasia Independent Online South Africa. Marianopolis Matters.

Marianopolis College: 8— Fall Archived from the original PDF on March 4, Rotten Tomatoes. Retrieved November 1,

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