  Considering a Hamiltonian Dynamical System describing the motion of charged
particle in a Tokamak or a Stellarator, we build a change of coordinates to
reduce its dimension. This change of coordinates is in fact an intricate
succession of mappings that are built using Hyperbolic Partial Differential
Equations, Differential Geometry, Hamiltonian Dynamical System Theory and
Symplectic Geometry, Lie Transforms and a new tool which is here introduced :
Partial Lie Sums.
