  We study the motion of a charged particle in a tokamak magnetic field and
discuss its chaotic nature. Contrary to most of recent studies, we do not make
any assumption on any constant of the motion and solve numerically the
cyclotron gyration using Hamiltonian formalism. We take advantage of a
symplectic integrator allowing us to make long-time simulations. First
considering an idealized magnetic configuration, we add a non generic
perturbation corresponding to a magnetic ripple, breaking one of the invariant
of the motion. Chaotic motion is then observed and opens questions about the
link between chaos of magnetic field lines and chaos of particle trajectories.
Second, we return to a axi-symmetric configuration and tune the safety factor
(magnetic configuration) in order to recover chaotic motion. In this last
setting with two constants of the motion, the presence of chaos implies that no
third global constant exists, we highlight this fact by looking at variations
of the first order of the magnetic moment in this chaotic setting. We are
facing a mixed phase space with both regular and chaotic regions and point out
the difficulties in performing a global reduction such as gyrokinetics.
