A Unified Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems

We prove that in all but one case the normal form of a real or complex Hamiltonian matrix which is irreducible and appropriately normalized can be computed by Lie series methods in formally the same manner as one computes the normal form of a nonlinear Hamiltonian function. Calculations are emphasized; the methods are illustrated with detailed examples, and for the sake of completeness the exceptional case is also reviewed and illustrated. Alternate methods are also discussed, along with detailed examples