Birkhoff’s normal form

In such a domain we may perform the construction of Birkhoff s normal form illustrated in Section 2.5. To this end we split again the Hamiltonian as a series... 

Let us now proceed in a different manner, which however is essentially equivalent to the procedure illustrated there. Instead of looking directly for first integrals, we go through the process of constructing a Birkhoff s normal form. This is what we have done in Section 2.5, at least formally we have actually seen that we can construct a normal form up to an arbitrary large order. Thus, we may give the Hamiltonian the form... 

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