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]