Equations of Motion from Field Equations and a Gauge-invariant   Variational Principle for the Motion of Charged Particles
  New, gauge-independent, second-order Lagrangian for the motion of classical, charged test particles is proposed. It differs from the standard, gauge-dependent, first order Lagrangian by boundary terms only. A new method of deriving equations of motion from field equations is developed. When applied to classical electrodynamics, this method enables us to obtain unambigously the above, second order Lagrangian from the general energy-momentum conservation principle. 
D. Chruscinski, J. Kijowski