Analytic extension of Birkhoff normal forms for Hamiltonian systems of one degree of freedom : Simple pendulum and free rigid body dynamics (Exponential Analysis of Differential Equations and Related Topics)

Abstract

Birkhoff normal form is a power series expansion associated with the local behavior of a Hamiltonian system near the critical point. It is known that one can take convergent canonical transformation which puts the Hamiltonian into Birkhoff normal form for integrable systems under some non-degeneracy conditions. By means of an expression of the inverse of Birkhoff normal form by a period integral, analytic continuation of the Birkhoff normal forms is considered for two examples of Hamiltonian systems of one degree of freedom, the simple pendulum dynamics and the free rigid body dynamics on SO(3). It is shown that the analytic continuation of the inverse derivative for the Birkhoff normal forms has monodromy structure, which is explicitly calculated, and that in the free rigid body case the monodromy coincides with that of an elliptic fibration which naturally arises from the dynamics.