# 💪 ｜ Gym trainer explains "basic easy training" that even beginners of the topic "HIIT" can do ### Gym trainer explains "basic easy training" that even beginners of the topic "HIIT" can do

If you write the contents roughly
Squats and burpees alone should give you a lot of exercise, so give it a try!

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### Wikipedia related words

If there is no explanation, there is no corresponding item on Wikipedia. # Momentum

Momentum(Undo,(British: momentum) Is basicallyobject Ofmotion OfStatusRepresentsPhysical quantityso,massspeed OfproductIs defined as The momentum in this sense is distinguished from the generalized momentum described later,Kinematic momentum(OrDynamic momentum) Called. Also,Angular momentumIn contrast to the amount of momentum that is different,Linear momentumIt 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.

Analytical mechanicsNow, apart from the above definition, the momentum isGeneralized coordinatesEuler-Lagrange equationGiven through. This momentum is the counterpart of the generalized velocity in the generalized coordinate system,Generalized momentumCalled.

In particularHamiltonian formatIn the analytical mechanics ofCanonical equationGiven throughCanonical variableOne is called the coordinate and the other isMomentumCall.. Momentum in this sense is distinguished from the other,Canonical momentumCalled. Also, canonical momentum is a pair of coordinates in the canonical equation,Conjugate momentumと 呼 ば れ る.. 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.. 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.

## 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 TheInertial 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 t0 から t1 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 dp/dt Is power F(t) Is equal to Δp The Next power F(t) The time t0 から t1 UntilintegralEqual to what you did. The time integral of this force isImpulse(impulse) Is equal to the amount of change in momentum.

Time Δt=t1-t0 The force that an object receives ataverage The Is defined by Force time average Fbird If you use Becomes Especially time Δt 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 pi = mivi = midri/dt given that Becomes Where the total mass of the mass system MCenter of mass rg 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 pi The time change of i Force acting on Fi 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 fi, Mass point j From mass i Internal force acting on fij given that Is expressed as However, the mass i From mass i The force acting on oneself fii = 0 And Considering the change over time in total exercise Becomes hereSecond law of movementFrom the mass j From mass i Force acting on fij And mass i From mass j Force acting on fji 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 isLaw 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... 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..

## Moment

In physics,vectorOf the physical quantity represented byCross productTheMomentSay. Momentum of momentum isAngular 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, p Two vectors on the plane containing r, p The angle between θ Then, the magnitude of the angular momentum is expressed as follows. 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 × F Acts on an objectMoment of force.

## Momentum in analytical mechanics

Analytical mechanicsAtGeneralized coordinates qi Corresponding toGeneralized momentum (generalized momentum) pi Is thatsystem OfLagrangian L(q, ) Generalization speed of i byPartial differentiationIs defined as 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, p) Lagrangian L(q, ) OfLegendre conversionIs defined as.. Legendre conversion 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 pr, pθ Are each   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 velocityUsualMomentum depends on the interaction with the electromagnetic field eA 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 - eA Has been replaced with.

## 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 = mc2 Has become the formula.

## Quantum theory

Light(OrElectromagnetic wave) IsHowever, 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 = 2πν 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 areCorrespondence principleCalledAnalytical 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.

## footnote

 [How to use footnotes]
1. ^ British: kinetic momentum,dynamical momentum
2. ^ British: angular momentum
3. ^ British: linear momentum,translational momentum
4. ^ Matsuda 1993, p. 21.
5. ^ Newtons 1729, Axioms, or Laws of Motion; Law II.
6. ^ British: generalized momentum
7. ^ Sudo 2008, pp. 42–43,48–51, §5 Hamiltonian format and canonical transformation.
8. ^ British: canonical momentum
9. ^ British: conjugate momentum
10. ^ Sudo 2008, pp. 42,51, §5 Hamiltonian format and canonical transformation.
11. ^ a b c Sudo 2008, pp. 202–204, Appendix A The classical theory of electromagnetic fields.
12. ^ Sunagawa 1987, p. 234, Chapter 5 §2 Electromagnetic field energy and momentum.
13. ^ Sunagawa 1987, pp. 156–160; 234–240, Chapter 3 §5 Forces acting during steady-state currents; Chapter 5 §2 Electromagnetic field energy and momentum.
14. ^ Sudo 2008, pp. 45–47, 5.2 Legendre transformation.
15. ^ Tasaki 2000, pp. 259–270; 270–278, Appendix G. Convex function; H. Legendre transformation.
16. ^

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