Kinetic momentum


Kinetic momentum

When a charged particle is interacting with an electromagnetic field, the kinetic momentum is a nonstandard term for the mass times velocity. It is distinguished from the canonical momentum, because the canonical momentum includes a contribution from the vector potential.

The nonrelativistic Hamiltonian for a particle in interaction with an electromagnetic field is:

:H = {(vec p -evec A(vec x))^2 over 2m } + ephi(vec x)

Where A is the vector potential and phi is the scalar potential.The Hamiltonian is an expression for the total energy as a sum of the kinetic energy and the potential energy. The quantity scriptstyle p-evec A is the kinetic momentum, which is equal to mass times velocity. The quantity scriptstyle vec p(t) is the canonical momentum, which is not equal to the kinetic momentum. Following this nonstandard terminology, the quantity scriptstyle evec A is the potential momentum.

; Relativistic Dynamics

In relativity, the Lagrangian for the particle interacting with the field is

:L = msqrt{1-dot{x}^2} + e A(x)dot x - e phi(x),

The action is the relativistic arclength of the path of the particle in spacetime, minus the potential energy contribution, plus an extra contribution whichquantum mechanically is an extra phase a charged particle gets when it is movingalong a vector potential.

The momentum conjugate to x, the canonical momentum, is defined from the variation of the lagrangian::p = {partial L over partial dot{x} } = {mv over sqrt{1-v^2 + eA,

and the kinetic momentum, the relativistic momentum of a particle moving with velocity v, is p-eA. The kinetic momentum is:p-eA = {mv over sqrt{1-v^2,

The Hamiltonian is the usual relativistic expression for the energy, but now in terms of the kinetic momentum::H= pdot{x} - L = {mover sqrt{1-dot{x}^2 + e phi = sqrt{(p -eA)^2 + m^2} + e phi,

The equations of motion derived by extremizing the action::{dp over dt} =-{partial L over partial x} = e {partial A_i over partial x} dot{x}^i - e {partial phi over partial x},

:p - eA = {mv over sqrt{1-v^2,

are the same as Hamilton's equations of motion:

:{dxover dt} = {partial over partial p}(sqrt{(p-eA)^2 +m^2} + ephi),:{dpover dt} = -{partial over partial x}(sqrt{(p-eA)^2 + m^2} + ephi) ,

And both are equivalent to the noncanonical form::{d over dt}({mv over sqrt{1-v^2) = e(E + v imes B),,

Which gives the rate at which the Lorentz force adds relativistic momentum to the particle.

External links

* [http://www.physics.nmt.edu/~raymond/classes/ph13xbook/node90.html Kinetic and Potential Momentum]
* [http://www.physics.nmt.edu/~raymond/classes/ph13xbook/node140.html Potential Momentum]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • kinetic momentum — judesio kiekis statusas T sritis Standartizacija ir metrologija apibrėžtis Dydis, išreiškiamas kūno masės ir jo judėjimo greičio sandauga. atitikmenys: angl. kinetic moment; kinetic momentum; linear momentum; quantity of motion vok.… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • kinetic momentum — judesio kiekis statusas T sritis fizika atitikmenys: angl. kinetic momentum; momentum; quantity of motion vok. Bewegungsgröße, f; Impuls, m rus. импульс, m; количество движения, n pranc. impulsion, f; quantité de mouvement, f …   Fizikos terminų žodynas

  • kinetic moment — judesio kiekis statusas T sritis Standartizacija ir metrologija apibrėžtis Dydis, išreiškiamas kūno masės ir jo judėjimo greičio sandauga. atitikmenys: angl. kinetic moment; kinetic momentum; linear momentum; quantity of motion vok.… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • momentum — judesio kiekis statusas T sritis fizika atitikmenys: angl. kinetic momentum; momentum; quantity of motion vok. Bewegungsgröße, f; Impuls, m rus. импульс, m; количество движения, n pranc. impulsion, f; quantité de mouvement, f …   Fizikos terminų žodynas

  • Kinetic energy — The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity . Having gained this energy during its acceleration …   Wikipedia

  • Momentum — This article is about momentum in physics. For other uses, see Momentum (disambiguation). Classical mechanics Newton s Second Law …   Wikipedia

  • Kinetic theory — [ temperature of an ideal monatomic gas is a measure related to the average kinetic energy of its atoms as they move. In this animation, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure.… …   Wikipedia

  • Angular momentum — For a generally accessible and less technical introduction to the topic, see Introduction to angular momentum. Classical mechanics Newton s Second Law …   Wikipedia

  • linear momentum — judesio kiekis statusas T sritis Standartizacija ir metrologija apibrėžtis Dydis, išreiškiamas kūno masės ir jo judėjimo greičio sandauga. atitikmenys: angl. kinetic moment; kinetic momentum; linear momentum; quantity of motion vok.… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • Crystal momentum — In solid state physics crystal momentum or quasimomentum[1] is a momentum like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors k of this lattice, according to (where is the reduced Planck s… …   Wikipedia