Semi-global symplectic invariants of the Euler top

George Papadopoulos; Holger R. Dullin

doi:10.3934/jgm.2013.5.215

Article Contents

2013, Volume 5, Issue 2: 215-232. Doi: 10.3934/jgm.2013.5.215

This issue Previous Article The supergeometry of Loday algebroids Next Article Twisted isotropic realisations of twisted Poisson structures

Semi-global symplectic invariants of the Euler top

George Papadopoulos^1, and
Holger R. Dullin^1,

1.
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006

Received: January 31, 2013

Revised: May 31, 2013

Published: May 2013

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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 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.

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Mathematics Subject Classification: Primary: 37J35; Secondary: 37J15, 70H06, 70E40.

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