Darwin Lagrangian

The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order ${\textstyle {v^{2}}/{c^{2}}}$ between two charged particles in a vacuum where c  is the speed of light. It was derived before the advent of quantum mechanics and resulted from a more detailed investigation of the classical, electromagnetic interactions of the electrons in an atom. From the Bohr model it was known that they should be moving with velocities approaching the speed of light.^[1]

The full Lagrangian for two interacting particles is $${\displaystyle L=L_{\text{f}}+L_{\text{int}},}$$ where the free particle part is $${\displaystyle L_{\text{f}}={\frac {1}{2}}m_{1}v_{1}^{2}+{\frac {1}{8c^{2}}}m_{1}v_{1}^{4}+{\frac {1}{2}}m_{2}v_{2}^{2}+{\frac {1}{8c^{2}}}m_{2}v_{2}^{4},}$$ The interaction is described by $${\displaystyle L_{\text{int}}=L_{\text{C}}+L_{\text{D}},}$$ where the Coulomb interaction in Gaussian units is $${\displaystyle L_{\text{C}}=-{\frac {q_{1}q_{2}}{r}},}$$ while the Darwin interaction is $${\displaystyle L_{\text{D}}={\frac {q_{1}q_{2}}{r}}{\frac {1}{2c^{2}}}\mathbf {v} _{1}\cdot \left[\mathbf {1} +{\hat {\mathbf {r} }}{\hat {\mathbf {r} }}\right]\cdot \mathbf {v} _{2}.}$$ 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 ${\displaystyle {\hat {\mathbf {r} }}}$ is the unit vector in the direction of r.

The first part 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.^{[2]: 596–598}