Formas normais e ações de grupos no estudo de sistemas Hamiltonianos

Abstract

The central goal of this work is the development of the normal forms theory for Hamiltonians defined in a finite-dimensional symplectic vector space. To achieve this, we develop the adjoint operator method and an alternative algebraic method which takes into account a given group S of matrices in terms of the linear part of the associated Hamiltonian system. In this process, we use tools from the representation theory and invariant theory of linear Lie groups, as well as from the symplectic geometry. We exemplify the methods by obtaining normal forms of specific Hamiltonians, even under the action of a linear Lie group. In this last case, the normal form can inherit the symmetries of the original Hamiltonian, implying that the symmetries of the associated Hamiltonian field are also preserved