Analytic Extension of the Birkhoff Normal Forms for the Free Rigid Body Dynamics on SO(3)

TL;DR

This paper explores the analytic continuation of Birkhoff normal forms for the free rigid body on SO(3), linking local Hamiltonian behavior to global properties and monodromy via period integrals.

Contribution

It introduces a method to analytically continue Birkhoff normal forms using period integrals, revealing their global structure and relation to monodromy in rigid body dynamics.

Findings

Analytic continuation of Birkhoff normal forms is achieved for the free rigid body.

The global behavior of these forms is connected to monodromy of an elliptic fibration.

The approach clarifies the relationship between local Hamiltonian analysis and global geometric properties.

Abstract

Birkhoff normal form is a power series expansion associated with the local behavior of the Hamiltonian systems near a critical point. It is known to be convergent 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 the free rigid body dynamics on SO(3). The global behavior of the Birkhoff normal forms is clarified in relation to the monodromy of an elliptic fibration which naturally arises from the free rigid body dynamics.