Papadopoulos, George; Dullin, Holger R. Semi-global symplectic invariants of the Euler top. (English) Zbl 1334.37055 J. Geom. Mech. 5, No. 2, 215-232 (2013). Summary: 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 inverse of 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. Cited in 4 Documents MSC: 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) 70E15 Free motion of a rigid body 70E40 Integrable cases of motion in rigid body dynamics 70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 34M45 Ordinary differential equations on complex manifolds Keywords:Euler top; Picard-Fuchs equation; semiglobal symplectic invariants; Birkhoff normal form; Liouville integrable PDFBibTeX XMLCite \textit{G. Papadopoulos} and \textit{H. R. Dullin}, J. Geom. Mech. 5, No. 2, 215--232 (2013; Zbl 1334.37055) Full Text: DOI arXiv