A Gauge-invariant Hamiltonian Description of the Motion of Charged Test   Particles
  New, gauge-independent, second-order Lagrangian for the motion of classical, charged test particles is used to derive the corresponding Hamiltonian formulation. For this purpose a Hamiltonian description of the theories derived from the second-order Lagrangian is presented. Unlike in the standard approach, the canonical momenta arising here are explicitely gauge-invariant and have a clear physical intepretation. The reduced symplectic form is equaivalent to the Souriau's form. This approach illustrates a new method of deriving equations of motion from field equations. 
D. Chruscinski, J. Kijowski