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9. Let U (T ) = Dn T n be the neighborhood of the Liouville torus we x a certain non-trivial cycle on each of the Liouville tori which depends continuously on the torus. Consider the corresponding action function I s (f1 : : : fn ) = 21 where the integral is taken over the cycle lying on the torus that corresponds to given values f1 : : : fn of the rst integrals. Then all the trajectories of the Hamiltonian vector eld sgrad s are closed with the same period 2 and are homologous to the cycle . Proof.

Consider the Hamiltonian H as a function of s1 s2 . Then 1 = @H=@s @H=@s : 2 Proof. Since the form ! takes the canonical form in action-angle variables, we get @H @=@' + @H @=@' : sgrad H = @s 1 @s 2 1 2 Now the desired formula follows directly from the de nition of . In fact, the notion of a rotation number is topological. It can be seen, for example, from the following proposition. If T is resonant, then all integral trajectories of v are closed and homologous to each other (and even isotopic on the torus).

It is convenient to consider x and y as coordinates on the 2-plane that covers the torus T . Let x(t) and y(t) be the coordinates of a point of an arbitrary integral trajectory of v (after its lifting to the covering plane). 12. 1 y(t) : Proof. By choosing new periodic coordinates on the torus, we only change the shape of the fundamental domain (Fig. 3). Moreover, we may suppose that the vertices of the lattice remain the same. 1 y (t) c2 as was to be proved. Copyright 2004 by CRC Press LL The above formula can be taken as the de nition of the rotation number.

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