# Kinetic momentum

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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:

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

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 $scriptstyle vec p\left(t\right)$ 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

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

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::$p = \left\{partial L over partial dot\left\{x\right\} \right\} = \left\{mv over sqrt\left\{1-v^2 + eA,$

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

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

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

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

are the same as Hamilton's equations of motion:

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

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

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

* [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]

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