  A geometric formulation of the linear beam dynamics in accelerator physics is
presented. In particular, it is proved that the linear transverse and
longitudinal dynamics can be interpret geometrically as an approximation to the
Jacobi equation of an affine averaged Lorentz connection. We introduce a
specific notion reference trajectory as integral curves of the main velocity
vector field. A perturbation caused by the statistical nature of the bunch of
particles is considered.
