Note on the Derivation of the Equation of Motion of a Charged Point-Particle from Hamilton's Principle

An alternative derivation of the equation of motion of a charged point particle from Hamilton’s principle is presented. The variational principle is restated as a Bolza problem of optimal control, the control variable u ⁱ , i = 0, …, 3, being the 4-velocity. The trajectory ${\overline{x}}^i$ (s) i and 4-velocity ū ⁱ (s) of the particle is an optimal pair, i.e., it furnishes an extremum to the action integral. The pair ( $\overline{x}$, ū) satisfies a set of necessary conditions known as the maximum principle. Because of the path dependence of proper time s, we are concerned with a control problem with a free end point in the space of coordinates (s, x ⁰ , …, x ³ ). To obtain the equation of motion, the transversality condition must be satisfied at the free end point.