The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order between two charged particles in a vacuum and is given by[1]

where the free particle Lagrangian is

and the interaction Lagrangian is

where the Coulomb interaction is

and the Darwin interaction is

Here q1 and q2 are the charges on particles 1 and 2 respectively, m1 and m2 are the masses of the particles, v1 and v2 are the velocities of the particles, c is the speed of light, r is the vector between the two particles, and is the unit vector in the direction of r.

The free Lagrangian is the Taylor expansion of free Lagrangian of two relativistic particles to second order in v. The Darwin interaction term is due to one particle reacting to the magnetic field generated by the other particle. If higher-order terms in v/c are retained, then the field degrees of freedom must be taken into account, and the interaction can no longer be taken to be instantaneous between the particles. In that case retardation effects must be accounted for.

Contents

Derivation of the Darwin interaction in a vacuumEdit

The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is[2]

 

where u is the relativistic velocity of the particle. The first term on the right generates the Coulomb interaction. The second term generates the Darwin interaction.

The vector potential in the Coulomb gauge is described by[3] (Gaussian units)

 

where the transverse current Jt is the solenoidal current (see Helmholtz decomposition) generated by a second particle. The divergence of the transverse current is zero.

The current generated by the second particle is

 

which has a Fourier transform

 

The transverse component of the current is

 

It is easily verified that

 

which must be true if the divergence of the transverse current is zero. We see that

 

is the component of the Fourier transformed current perpendicular to k.

From the equation for the vector potential, the Fourier transform of the vector potential is

 

where we have kept only the lowest order term in v/c.

The inverse Fourier transform of the vector potential is

 

where

 

(see Common integrals in quantum field theory).

The Darwin interaction term in the Lagrangian is then

 

where again we kept only the lowest order term in v/c.

Lagrangian equations of motionEdit

The equation of motion for one of the particles is

 
 

where p1 is the momentum of the particle.

Free particleEdit

The equation of motion for a free particle neglecting interactions between the two particles is

 
 

Interacting particlesEdit

For interacting particles, the equation of motion becomes

 

 

 
 
 

Darwin Hamiltonian for two particles in a vacuumEdit

The Darwin Hamiltonian for two particles in a vacuum is related to the Lagrangian by a Legendre transformation

 

The Hamiltonian becomes

 

Hamiltonian equations of motionEdit

The Hamiltonian equations of motion are

 

and

 

which yield

 

and

 

Note that the quantum mechanical Breit equation originally used the Darwin Lagrangian with the Darwin Hamiltonian as its classical starting point though the Breit equation would be better vindicated by the Wheeler–Feynman absorber theory and better yet quantum electrodynamics.

See alsoEdit

ReferencesEdit

  1. ^ Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X. pp. 596-598
  2. ^ Jackson, pp. 580-581.
  3. ^ Jackson, p. 242.