Chapter 4

Liouville Equivalence of Integrable

Systems with Two Degrees of Freedom

Now we make more precise the denition of the Liouville equivalence for integrable

Hamiltonian systems. From now on, we shall assume that two Liouville foliations

are Liouville equivalent if and only if there exists a dieomorphism that sends

the leaves of the rst foliation to those of the second one and satises two conditions

related to the orientation. Namely, it preserves the orientation on 3-manifolds

Q and Q

, and moreover, it also preserves the orientation on the critical circles

given by the Hamiltonian ows.