Normalization of a Periodic Hamiltonian System

First, the normal form in the vicinity of the stationary solution of the autonomous Hamiltonian system is recalled. Next, linear periodic Hamiltonian systems are considered. For them, normal forms of the Hamiltonians in the complex and real cases are found. A feature of the real case in the situation of parametric resonance is discovered. Next, normal forms of Hamiltonians of nonlinear periodic systems are found. Using an additional canonical transformation, such a normal form can be always reduced to an autonomous Hamiltonian system that preserves all small parameters and symmetries of the original system. The local families of fixed points of this autonomous system are associated with families of periodic solutions to the original system. A similar theory is constructed in the neighborhood of the periodic solution to the autonomous system. All transformations are algorithmic and can be implemented in a computer algebra system.