In this seminar we present the normal form theory as a tool for the local study of the qualitative behavior of hamiltonian vector fields under the action of a linear Lie group Γ. We introduce an algebraic method which takes into account a given group S of matrices in terms of the linear part of the associated hamiltonian system. The normal form can inherit the symmetries of the original hamiltonian function if the changes of coordinates are equivariant under the action Γ. For this study, we use tools from the representation theory and invariant theory of linear Lie groups.