The full Lagrangian for two interacting particles is L = L f + L int , where the free particle part is L f = 1 2 m 1 v 1 2 + 1 8 c 2 m 1 v 1 4 + 1 2 m 2 v 2 2 + 1 8 c 2 m 2 v 2 4 , The interaction is described by L int = L C + L D , where the Coulomb interaction in Gaussian units is L C = − − q 1 q 2 r , while the Darwin interaction is L D = q 1 q 2 r 1 2 c 2 v 1 ⋅ ⋅ [ 1 + r ^ ^ r ^ ^ ] ⋅ ⋅ v 2 . 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, v 1 and v 2 are the velocities of the particles, c is the speed of light, r is the vector between the two particles, and r ^ ^ is the unit vector in the direction of r.