Semi-global symplectic invariants of the Euler top

TL;DR

This paper computes semi-global symplectic invariants of the Euler top near hyperbolic equilibrium points using Birkhoff normal form and Picard-Fuchs equations, revealing non-equivalence to the pendulum near the separatrix.

Contribution

It introduces a method to compute semi-global symplectic invariants of the Euler top using Lie series and Picard-Fuchs equations, and analyzes their convergence and implications.

Findings

Birkhoff normal form computed via Lie series.

Actions near hyperbolic points derived from Picard-Fuchs equations.

The pendulum is not symplectically equivalent to any Euler top near the separatrix.

Abstract

We compute the semi-global symplectic invariants near the hyperbolic equilibrium points of the Euler top. The Birkhoff normal form at the hyperbolic point is computed using Lie series. The actions near the hyperbolic point are found using Frobenius expansion of its Picard-Fuchs equation. We show that the Birkhoff normal form can also be found by inverting the regular solution of the Picard-Fuchs equation. Composition of the singular action integral with the Birkhoff normal form gives the semi-global symplectic invariant. Finally, we discuss the convergence of these invariants and show that in a neighbourhood of the separatrix the pendulum is not symplectically equivalent to any Euler top.