Symmetry and integrability in Hamiltonian normal forms

In studies in the natural sciences assumptions of symmetries abound spherical symmetry cylindrical symmetry reection symmetry etc This is partly because such an assumption simplies the mathematical analysis but it is also based on the observation that equilibrium states as the outcome of a transient evolutionary process usually display symmetries In the sequel we will study the consequences of symmetry assumptions for nitedimensional Hamiltonian systems by normal form methods It should be noted that such techniques are approximation techniques they are related to in fact can be viewed as equivalent to averaging and multiple timescale methods Now and then we shall refer to the relation with averaging results We shall not refer to multiple timescale methods these are generally speaking useful techniques but they are not very ecient in the case of Hamiltonian system