Kinetic momentum

From Wikipedia, the free encyclopedia

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 -e\vec A(\vec x))^2 \over 2m } + e\phi(\vec x)

Where A is the vector potential and φ 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-e\vec 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 e\vec A is the potential momentum.

Relativistic Dynamics

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

L = m\sqrt{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 space time, minus the potential energy contribution, plus an extra contribution which quantum mechanically is an extra phase a charged particle gets when it is moving along 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= p\dot{x} - L = {m\over \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:

{dx\over dt} = {\partial \over \partial p}(\sqrt{(p-eA)^2 +m^2} + e\phi) \,
{dp\over dt} = -{\partial \over \partial x}(\sqrt{(p-eA)^2 + m^2} + e\phi) \,

And both are equivalent to the noncanonical form:

{d \over dt}({mv \over \sqrt{1-v^2}}) = e(E + v \times B) \,\,

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

[edit] External links

This relativity-related article is a stub. You can help Wikipedia by expanding it.