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 and is given by^{: 596–598}

L=L_{\text{f}}+L_{\text{int}},

where the free particle Lagrangian is

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},

and the interaction Lagrangian is

L_{\text{int}}=L_{\text{C}}+L_{\text{D}},

where the Coulomb interaction is

L_{\text{C}}=-{\frac {q_{1}q_{2}}{r}},

and the Darwin interaction is

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 $q 1$ 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 ${\hat {\mathbf {r} }}$ 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.

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