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 where the free particle Lagrangian is and the interaction Lagrangian is where the Coulomb interaction is and the Darwin interaction is Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and is the unit vector in the direction of . 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 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 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 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 . 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 . The inverse Fourier transform of the vector potential is where (see ). The Darwin interaction term in the Lagrangian is then where again we kept only the lowest order term in . Lagrangian equations of motion The equation of motion for one of the particles is where is the momentum of the particle. Free particle The equation of motion for a free particle neglecting interactions between the two