Based on the generalized theory of wide electron beams focusing set up by one of the authors

the present paper takes a further step to investigate the electron motion around a curved optical axis in the electrostatic and magnetic fields described by electrostatic potential and scalar or vector magnetic potential

using the contravariant and covariant forms of Newton’s equation and Lorentz force

as well as the generalized variational principle

in which the Frenet-Serret local coordinate systems both fixed and twisted along the principal trajectory are used. In this paper the different forms of equations for electron motion and electron trajectory in most general case are given.