Particle in Electromagnetic Fields

The prefactor of the interaction term has been suitably chosen in order to yield the correct equation of motion. We emphasize that the 4-potential AH = AH(x(t)) represents an external field only and therefore the Lagrangian does not dynamically depend on AH. Straightforward evaluation of the two terms of the Euler-Lagrange equations as given by Eq. (3.147) yields 

In analogy to our argumentation for the free particle above, we require the three-dimensional formulation of the Lagrangian L = Li r,v,t) to yield exactly the same action S as the explicitly covariant Lagrangian L2 = L2 x, u), i.e., the interaction term has to be rewritten as 

This finally gives rise to the relativistic Hamiltonian for a particle subject to an external electromagnetic field, 

---