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### Wikipedia related words

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# Momentum

**Momentum****Kinematic momentum**(Or**Dynamic momentum**^{[1]}) Called. Also,**Angular momentum**^{[2]}In contrast to the amount of momentum that is different,**Linear momentum**^{[3]}It is also sometimes called.

## Overview

In daily life, the momentum of an object is perceived as a difficulty in stopping a moving object. In other words, a heavier and faster object has a larger momentum, and a larger object to stand still.ImpulseWill be required.

Isaac NewtonIs the relationship between temporal changes in momentum and forceSecond law of movementPresented as^{[4]}^{[5]}.

Analytical mechanicsNow, apart from the above definition, the momentum isGeneralized coordinatesとEuler-Lagrange equationGiven through. This momentum is the counterpart of the generalized velocity in the generalized coordinate system,**Generalized momentum**^{[6]}Called.

In particularHamiltonian formatIn the analytical mechanics ofCanonical equationGiven throughCanonical variableOne is called the coordinate and the other is**Momentum**Call^{[7]}.. Momentum in this sense is distinguished from the other,Canonical momentum^{[8]}Called. Also, canonical momentum is a pair of coordinates in the canonical equation,**Conjugate momentum**^{[9]}と 呼 ば れ る^{[10]}.. Momentum, in Hamiltonian mechanics, is more fundamental than velocity, and is the usual Hamiltonian description.Quantum mechanicsAlso plays an important role in.

As an example where the difference between conjugated momentum and normal kinematic momentum is outstanding,magnetic fieldExercise insideElectronicExamples of exercise (#Momentum in analytical mechanicsSee also).Electromagnetic fieldFor the electrons moving insideLorentz forceWorks, but generalized to correspond to this Lorentz forcePotential energySince there is an electron velocity term in, the conjugate momentum isLagrangianDepends on the potential term of^{[11]}.. At this time, the conjugate momentum and the kinematic momentum do not match. Also, in the classical Hamiltonian, which describes the motion of electrons in an electromagnetic field, all of the conjugate momentum is derived from the conjugate momentum.Vector potentialReplaced by subtracting the contribution of^{[11]}.

## Mathematical expression

Momentum isSecond law of movement, The rate of change over time isPowerIntroduced as an amount equal to.

That is, momentum **p** TheNewton's equation of motion,

Meet Power **F** TheVector quantityAnd momentum is also a vector quantity. Also, as is clear from the definition, momentum is the time t OffunctionIs the amount expressed as

Mass pointThe momentum of thespeedToProportionalTo do. The momentum of a mass is the velocity of the mass **v** Is expressed as m Then,

Given in.

Proportional coefficient introduced here m The**Inertial mass** (inertial mass) It is called and represents the difficulty of changing the velocity of the mass point.

The amount of change in the amount of exerciseImpulseHowever, if the inertial mass is constant during the motion, the change in velocity is the impulse divided by the inertial mass. Therefore, for the same magnitude of impulse, the larger the inertial mass, the smaller the change in velocity.

## Change over time

Times of Day *t*_{0} から *t*_{1} Change the momentum of the object between

And This object is the time t Power **F**(t) If you were exercising while receivingEquation of motionTo rate of change in momentum d**p**/dt Is power **F**(t) Is equal to Δ* p* The

Next power **F**(t) The time *t*_{0} から *t*_{1} UntilintegralEqual to what you did. The time integral of this force is**Impulse**(impulse) Is equal to the amount of change in momentum.

Time Δ*t*=*t*_{1}-*t*_{0} The force that an object receives ataverage The

Is defined by Force time average **F**_{bird} If you use

Becomes Especially time Δ*t* If is short enough and the force can be considered constant, then the impulse is simply the product of force and time.

Can be expressed as

That is, in order to apply a constant force to the object to increase the change in the momentum of the object, the time for which the force acts may be lengthened. Conversely, even if a large force is applied, if it is for a very short period of time, the impulse given to the object will be small.

## Mass movement

Momentum is an additive amount, and the total momentum of the system is represented by the sum of the partial momentums.

Total momentum of the mass system **P** Is the mass *i* = 1, 2, 3, ... Momentum of * p_{i}* =

*m*=

_{i}**v**_{i}*m*given that

_{i}d**r**_{i}/dtBecomes Where the total mass of the mass system M とCenter of mass **r**_{g} The

If introduced by

Becomes That is, the total momentum of the mass system is equal to the momentum when it is considered that the total mass is concentrated on the center of mass.

Mass point i Momentum of **p**_{i} The time change of i Force acting on **F**_{i} Equal to

Meet Mass here i The force acting on is divided into external force acting from the outside of the mass system and internal interaction with other masses included in the system. Mass point i External force acting on **f**_{i}, Mass point j From mass i Internal force acting on **f**_{ij} given that

Is expressed as However, the mass i From mass i The force acting on oneself * f_{ii}* = 0 And Considering the change over time in total exercise

Becomes hereSecond law of movementFrom the mass j From mass i Force acting on **f**_{ij} And mass i From mass j Force acting on **f**_{ji} Are equal in size and opposite in sign

And the sum of all internal forces 0 Becomes Therefore

And the time change of the total momentum of the mass system isSumIs equal to This means that under a simple external force such as gravity, the movement of the center of mass can be separated from the movement of the relative position.

## Conservation law

In the mass movement, especially when the external forces acting are balanced

Holds. In other words, in this system, the total momentum of the system does not change with time. this is**Law of conservation of momentum** (law of conservation of momentum) Called. The law of conservation of momentum is Newtonian mechanicsLaw of action and reaction, But the law of conservation of momentum itself is a law that generally holds better than the law of action-reaction.^{[12]}.. For example,ElectromagneticsSuch asField theoryThen.Proximity theoryTake the position ofRemote action theoryIs a general lawLaw of action and reactionIs not the basis for that. However, the law of conservation of momentum holds in electromagnetism, and the definition of momentum is expanded accordingly.^{[13]}.

## Moment

In physics,vectorOf the physical quantity represented byCross productTheMomentSay. Momentum of momentum is**Angular momentum** (angular momentum) And is defined as follows.

The classical angular momentum magnitude is the position vector **r** Size and momentum **p** Of **r** ToOrthogonalIt is expressed as the product of the sizes of the components. Two vectors * r*,

*Two vectors on the plane containing*

**p***,*

**r***The angle between θ Then, the magnitude of the angular momentum is expressed as follows.*

**p**In analytical mechanics, the angular momentum is角度Is obtained as a generalized momentum corresponding to.

The angular momentum isNewton's equation of motionEquation similar to,

Meet here * N* :=

*×*

**r***Acts on an objectMoment of force.*

**F**## Momentum in analytical mechanics

Analytical mechanicsAtGeneralized coordinates q_{i} Corresponding to**Generalized momentum** (generalized momentum) p_{i} Is thatsystem OfLagrangian *L*(* q*, ) Generalization speed of

*byPartial differentiationIs defined as*

_{i}Where Lagrangian *L*(* q*, ) It is,Physical energy K,potential U Is defined as the difference between.

Hamiltonian formatIn the dynamics of, the generalized momentum is used as a dynamic variable instead of the generalized velocity.Hamiltonian *H*(* q*,

*) Lagrangian*

**p***L*(

*, ) OfLegendre conversionIs defined as*

**q**^{[14]}.. Legendre conversion

^{[15]}

Maximize the right-hand side of Considering, the area to convert Legendre D Lagrangian inConvexAndSmooth enoughThen such Satisfies the following relation.

This is the Hamiltonian variable **p** Means equal to generalized momentum.

### Cartesian coordinate system

Three-dimensionalCartesian coordinate system * x* =

*x*+

*y*+

*z*Inpotential speed When not depending on

And then the generalized momentum **p** Is the product of mass and velocity. this isNewton formatEqual to the momentum of.

### Polar coordinate system

Two-dimensional as generalized coordinatesPolar coordinates * x* = (

*r*,

*θ*) Is selected, Lagrangian and

*r*,

*θ*Momentum conjugate to

*p*,

_{r}*p*Are each

_{θ}Becomes here,θ Momentum conjugate toAngular momentumHas become. Also r The conjugate momentum of represents the momentum in the radial direction.

### Generalized potential

Sometimes the potential depends on velocity. Then the generalized momentum in Cartesian coordinate system is different from that in Newtonian mechanics.

As an example of such a system,Electromagnetic fieldExercise insidechargehaveparticle OfNon-relativisticExercising. The Lagrangian of this system is

Is. here e Has an objectcharge,φ TheScalar potential,**A** TheVector potentialIs. At this time, the conjugate momentum is

Becomes The conjugate momentum at this time is the product of mass and velocity*Usual*Momentum depends on the interaction with the electromagnetic field e**A** Is added. At this time, Hamiltonian transforms Legendre

Than,

Becomes Compared to a system without vector potential, formally conjugate momentum **p** The kinematic momentum * p* -

*e*Has been replaced with

**A**^{[11]}.

## Theory of relativity

Momentum in the theory of relativityEnergy TheMinkowski spaceInQuaternion vectorTo do

Is (m Is the mass,τ TheCharacteristic time). The spatial component of this is

Becomes Non-relativistic limit *v*/*c* → 0 At, it agrees with the above-mentioned momentum (product of mass and velocity).

Momentum and energy

Meet the relationship. Momentum 0 Famous for *E* = *mc*^{2} Has become the formula.

## Quantum theory

Light(OrElectromagnetic wave) Is波However, it is also considered by experiments to be a particle with energy and momentum. Its energy and momentum

Is. (here h ThePlanck's constant,ν TheFrequency,*ω* = 2π*ν* TheAngular frequency,c TheSpeed of light in vacuum,λ Thewavelength,k TheWave numberIs)

Putting this relation in the above relational expression of energy and momentum,*ω* = *ck* So the mass of this particle is 0 It turns out that This mass 0 Particles ofPhotonThat.

In quantum mechanics, the above classical momentum Is the wave function Against

To sayoperatorIs considered to be. here, TheImaginary unit, TheNabla.

Or if it is expressed as a quaternion vector together with energy,

Is. They are**Correspondence principle**CalledAnalytical mechanicsInAction integral OfFunctional differentiation

It was inferred from that.

In addition,Canonical quantizationAccording to the method, position and momentum areCanonical exchange relationship

MeetPhysical quantityAsQuantizationIs done.

## Relationship with symmetry

Momentum corresponds to spatial uniformity (translational symmetry)Storage amountIs. Corresponds to time uniformityEnergyCorresponding to the isotropicity of spaceAngular momentumTogether with the basic physical quantity^{[16]}.

## footnote

**^**British: kinetic momentum,dynamical momentum**^**British: angular momentum**^**British: linear momentum,translational momentum**^**Matsuda 1993, p. 21.**^**Newtons 1729, Axioms, or Laws of Motion; Law II.**^**British: generalized momentum**^**Sudo 2008, pp. 42–43,48–51, §5 Hamiltonian format and canonical transformation.**^**British: canonical momentum**^**British: conjugate momentum**^**Sudo 2008, pp. 42,51, §5 Hamiltonian format and canonical transformation.- ^
^{a}^{b}^{c}Sudo 2008, pp. 202–204, Appendix A The classical theory of electromagnetic fields. **^**Sunagawa 1987, p. 234, Chapter 5 §2 Electromagnetic field energy and momentum.**^**Sunagawa 1987, pp. 156–160; 234–240, Chapter 3 §5 Forces acting during steady-state currents; Chapter 5 §2 Electromagnetic field energy and momentum.**^**Sudo 2008, pp. 45–47, 5.2 Legendre transformation.**^**Tasaki 2000, pp. 259–270; 270–278, Appendix G. Convex function; H. Legendre transformation.**^**Landau & Lifshits 2008.

## References

- Newton, Isaac (1729).
*The Mathematical Principles of Natural Philosophy*. translated by Andrew Motte (English ed.).- Wiki sauce. - Matsuda, Satoshi "Mechanics" Maruzen <Parity Physics Course>, 1993.
- Landau, LD,Lifshits, EM"Dynamics and Field Theory: Landau-Lifshitz Physics Lesson"Iwao Mito-Toshihiko Tsuneto-Toru HiroshigeTranslation,Chikuma Shobo〈Chikuma art library>,2008.ISBN 978-4-480-09111-6.
- Sudo, Yasushi "Analytical Mechanics/Quantum Theory", University of Tokyo Press, 2008.ISBN 978-4-13-062610-1.
- Sunagawa, Shigenobu "Electromagnetics" Iwanami Shoten 〈Physical text series 4〉, 1987, new edition.ISBN 4-00-007744-9.
- Tasaki, Haruaki "Thermodynamics from a modern perspective", Baifukan, April 2000, 4, first edition.ISBN 978-4-563-02432-1.

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