Dynamics near the p:−qp:−q resonance

We study the dynamics near the truncated p:±qp:±q resonant Hamiltonian equilibrium for pp, qq coprime. The critical values of the momentum map of the Liouville integrable system are found. The three basic objects reduced period, rotation number, and non-trivial action for the leading order dynamics are computed in terms of complete hyperelliptic integrals. A relation between the three functions that can be interpreted as a decomposition of the rotation number into geometric and dynamic phase is found. Using this relation we show that the p:−qp:−q resonance has fractional monodromy. Finally we prove that near the origin of the 1:−q1:−q resonance the twist vanishes.