The proper choice of the Lagrangian for a relativistic particle in external fields

Abstract

It is an unsatisfactory feature of the educational literature that renowned authors propose apparently conflicting prescriptions for the construction of an appropriate Lagrangian to describe the interaction of a relativistic point particle with external fields. The point where opinions sharply differ is whether-if the independent parameter is to be the proper time of the particle-the equations of motion should be derived from Hamilton's principle by constrained or unconstrained variation. Therefore, it is shown that there exists a unifying point of view, which allows a consistent derivation of all these seemingly unrelated or even incompatible Lagrangians, completely clarifies their status and interrelations, and, moreover, provides a sound basis for discussing coherently a multitude of related issues that have emerged in the literature in an often obscure and even misconstrued manner.