M. Fabbrichesi , SISSA

Transformation of electromagnetic fields and relativistic Lagrangian

The first part of this lesson is dedicated to the transformations of electromagnetic fields [Jackson, Sect. 11.10]. As an example of transformation it is considered the one of an observer in a reference system K that sees a charge q moving by a straight-line path with a velocity **v**. The charge is at rest in the system K'. These transformations show that **E **and **B **fields have no independent existence. In the second part of the lecture, the Lagrangian of a charged particle in external fields is derived by writing down the action integral and by finding an invariant expression for the interaction part: -e/(γc)u_{α}A^{α}. Hence the conjugate moment *p*^{α }is not only the kinetic one but contains also the vector potential contribution. Finally Lagrange (that corresponds to Lorentz's force) and Hamilton equations are derived [Jackson, Sect 12.1].