Kinetic momentum

In physics, in particular electromagnetism, the kinetic momentum is a nonstandard term for the momentum of a charged particle due to its inertia. When a charged particle interacts with an electromagnetic field (often abbreviated as EM field), there are two momenta: due to its inertia and due to the field. It is distinguished from the canonical momentum, because the kinetic momentum includes a contribution from the vector potential.

Definition and terminology

See also: Momentum in electromagnetism (section)

The momenta associated with a charged particle in an electromagnetic field is:

$\vec{\Pi} = m\vec{v} = \vec{P}- e\vec{A}\,\!$

or in terms of vector components:

$\Pi_j = m v_j = P_j - e A_j \,\!$

where:

• $\vec{P} \,\!$ is the canonical momentum,
• $\vec{\Pi} = m\vec{v}$ is the kinetic momentum,
• $e\vec{A}$ is the potential momentum (no standard symbol),

furthermore:

• e = charge of particle,
• m = mass of particle,
• $\vec{v} = \mathrm{d}\vec{q}/\mathrm{d}t$ is the velocity of the charged particle (q are generalized coordinates),
• $\vec{A}= \vec{A}(\vec{r})$ is the magnetic vector potential; aka the A-field,

The motivations for these names are as follows. An EM field possesses energy and momentum, a particle has charge. Charged particles interact with electromagnetic fields due to the electromagnetic interaction. So the momentum in the field is gained by the particle when it interacts with it. This contribution due to the field presents itself in terms of the particle's charge and the A-field. The particle also has mass and is moving with velocity, which is the familiar form of momentum as a quantity of motion, hence the term kinetic momentum. The total momentum possessed by the particle is the vector sum of these momenta: the canonical momentum.

The kinetic momentum $\vec{\Pi}$ satsifies the commutation relation:

$\left [ \Pi_j , \Pi_k \right ] = \frac{i\hbar e}{c} \epsilon_{jk\ell } B_\ell$

where:

• j, k, l are indices labelling vector components,
• B_l is a component of the magnetic field,
• ε_kjl is the permutation tensor, here in 3-dimensions.

Non-relativistic dynamics

The non-relativistic Hamiltonian for a particle in interaction with an electromagnetic field is:

$H = {\left | \vec p -e\vec A \right |^2 \over 2m } + e\phi,$

where $\phi = \phi (\vec{r})$ is the scalar potential.

The Hamiltonian H is an expression for the total energy as a sum of the kinetic energy T and the potential energy V:

H = T + V.

The kinetic energy always corresponds to the kinetic momentum:

$T = \frac{m|\vec{v}_\mathrm{kin}|^2}{2}=\frac{|\vec{p}_\mathrm{kin}|^2}{2m}$

which is a familiar relation from classical mechanics. For electrodynamics,

$\vec{p}_\mathrm{kin} = \vec{p}-e\vec{A}$

substituting to get the kinetic energy of the charged particle:

$T = \frac{|\vec{p}-e\vec{A}|^2}{2m} .$

The potential energy is simply the potential times the charge (since electric potential is defined as potential energy per unit charge):

V = eϕ.

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, that is the canonical momentum p, is defined from the variation of the lagrangian:

$p = \frac{\partial L}{\partial \dot{x} } = \frac{mv}{\sqrt{1-v^2}} + eA \,\!$

The kinetic momentum is the relativistic momentum of a particle moving with velocity v, still (p – eA), so we have:

$p-eA = \frac{mv}{\sqrt{1-v^2}} \,$

The Hamiltonian equals total energy (kinetic plus potential), and is the usual relativistic expression for the energy. So 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:

$\frac{\mathrm{d}p}{\mathrm{d}t} =\frac{\partial L}{\partial x} = e {\partial A_i \over \partial x} \dot{x}^i - e {\partial \phi \over \partial x} \,\!$

$p - eA = \frac{mv}{\sqrt{1-v^2}}\,$

are the same as Hamilton's equations of motion:

$\frac{\mathrm{d}x}{\mathrm{d}t} = \frac{\partial}{\partial p}\left ( \sqrt{(p-eA)^2 +m^2} + e\phi \right ) \,\!$

$\frac{\mathrm{d}p}{\mathrm{d}t} = -{\partial \over \partial x}\left ( \sqrt{(p-eA)^2 + m^2} + e\phi \right ) \,\!$

both are equivalent to the noncanonical form:

$\frac{\mathrm{d}}{\mathrm{d}t}\left ( {mv \over \sqrt{1-v^2}} \right ) = e\left ( \vec{E} + \vec{v} \times \vec{B} \right ) . \,\!$

This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.

See also

• Momentum
• Canonical commutation relation
• Magnetic vector potential
• Electromagnetic field
• Hamiltonian
• Lagrangian
• Generalized coordinates

Sources

• Classical Mechanics (2nd Edition), T.W.B. Kibble, European Physics Series, Mc Graw Hill (UK), 1973, ISBN 07-084018-0. Although concentrates on undergraduate-level classical Newtonian and Lagrangian mechanics, also contains a chapter on potential theory, includes electrodynamic fields: the ϕ and A fields and canonical momentum.
• Quantum mechanics, E. Abers, 2003, Benjamin-Cummings Publishers, ISBN(10) 0-1314-6100-1. Focuses on graduate-level quantum mechanics, but also contains simalar coverage to the above.
• Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8. Again concentrates on under/post-graduate level quantum mechanics, but does provide some exposure to lagrangian field theory and application to the EM field.