Darwin Lagrangian

The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order ${\displaystyle {\frac {v^{2}}{c^{2}}}}$ 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 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.

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