Normal Forms for Hamiltonian Systems in Some Nilpotent Cases

We study Hamiltonian systems with two degrees of freedom near an equilibrium point, when the linearized system is not semisimple. The invariants of the adjoint linear system determine the normal form of the full Hamiltonian system. For work on stability or bifurcation the problem is typically reduced to a semisimple (diagonalizable) case. Here we study the nilpotent cases directly by looking at the Poisson algebra generated by the polynomials of the linear system and its adjoint.