Darwin Lagrangian

The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order formula_1 between two charged particles in a vacuum and is given by
where the free particle Lagrangian is
and the interaction Lagrangian is
where the Coulomb interaction is
and the Darwin interaction is
Here "q"1 and "q"2 are the charges on particles 1 and 2 respectively, "m"1 and "m"2 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 formula_7 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.
Derivation in vacuum.
The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is
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 (Gaussian units)
where the transverse current J"t" 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 motion.
The equation of motion for one of the particles is
where p1 is the momentum of the particle.
Free particle.
The equation of motion for a free particle neglecting interactions between the two particles is
Interacting particles.
For interacting particles, the equation of motion becomes
Hamiltonian for two particles in a vacuum.
The Darwin Hamiltonian for two particles in a vacuum is related to the Lagrangian by a Legendre transformation
The Hamiltonian becomes
Hamiltonian equations of motion.
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.