  The goal of this contribution is to introduce the Hamiltonian formalism of
theoretical mechanics for analysing motion in generic linear and non-linear
dynamical systems, including particle accelerators. This framework allows the
derivation and integration of equations of motion, in order to describe the
particle trajectory evolution with respect to time. First, basic concepts are
re-visited for describing particle propagation through the resolution of
differential equations, applied to linear and non-linear motion. These
equations of motion can be obtained by the Lagrangian of the system, which is
the natural step leading to Hamiltonian Formalism and its properties. The
accelerator ring Hamiltonian is derived, starting with the relativistic
Hamiltonian of particles in the influence of E/M fields and a series of
canonical (or symplectic) transformations and approximations. Thereby,
introductory concepts of beam dynamics such as invariants and transport
matrices are revisited and extended towards generic concepts such as
action-angle variables and symplectic maps. To this end, the ground is prepared
for the advanced methods and tools used for studying non-linear motion in
particle accelerators.
