June  2013, 5(2): 215-232. doi: 10.3934/jgm.2013.5.215

Semi-global symplectic invariants of the Euler top

1. 

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia, Australia

Received  February 2013 Revised  June 2013 Published  July 2013

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.
Citation: George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215
References:
[1]

(Russian) Regul. Khaoticheskaya Din., 2 (1997), 13-25.  Google Scholar

[2]

Dokl. Akad. Nauk, 339 (1994), 253-296.  Google Scholar

[3]

Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426.  Google Scholar

[4]

$7^{th}$ edition, John Wiley & Sons, Inc., New York-London-Sydney, 2001.  Google Scholar

[5]

The University Series in Mathematics, Plenum Press, New York-London, 1980.  Google Scholar

[6]

Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.  Google Scholar

[7]

C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 949-952.  Google Scholar

[8]

Regular and Chaotic Dynamics, 12 (2007), 689-716. doi: 10.1134/S1560354707060111.  Google Scholar

[9]

Journal of Differential Equations, 254 (2013), 2942-2963. doi: 10.1016/j.jde.2013.01.018.  Google Scholar

[10]

Physica D, 155 (2001), 159-183. doi: 10.1016/S0167-2789(01)00257-3.  Google Scholar

[11]

in "Symplectic Geometry and Mathematical Physics" (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, (1991), 189-203.  Google Scholar

[12]

Springer Tracts in Natural Philosophy, 7, Springer-Verlag, Berlin-Heidelberg, 1965. doi: 10.1007/978-3-642-88412-2.  Google Scholar

[13]

Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[14]

Second edition, Applied Mathematical Sciences, 90, Springer, New York, 2009. doi: 10.1007/978-0-387-09724-4.  Google Scholar

[15]

Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X.  Google Scholar

[16]

Mat. Zametki, 61 (1997), 252-258. doi: 10.1007/BF02355730.  Google Scholar

[17]

M.S. thesis, The University of Sydney, 2013. Google Scholar

[18]

Ph.D thesis, Montpellier II University, 1996. Google Scholar

[19]

Annals of Mathematics (2), 161 (2005), 141-156. doi: 10.4007/annals.2005.161.141.  Google Scholar

show all references

References:
[1]

(Russian) Regul. Khaoticheskaya Din., 2 (1997), 13-25.  Google Scholar

[2]

Dokl. Akad. Nauk, 339 (1994), 253-296.  Google Scholar

[3]

Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426.  Google Scholar

[4]

$7^{th}$ edition, John Wiley & Sons, Inc., New York-London-Sydney, 2001.  Google Scholar

[5]

The University Series in Mathematics, Plenum Press, New York-London, 1980.  Google Scholar

[6]

Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.  Google Scholar

[7]

C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 949-952.  Google Scholar

[8]

Regular and Chaotic Dynamics, 12 (2007), 689-716. doi: 10.1134/S1560354707060111.  Google Scholar

[9]

Journal of Differential Equations, 254 (2013), 2942-2963. doi: 10.1016/j.jde.2013.01.018.  Google Scholar

[10]

Physica D, 155 (2001), 159-183. doi: 10.1016/S0167-2789(01)00257-3.  Google Scholar

[11]

in "Symplectic Geometry and Mathematical Physics" (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, (1991), 189-203.  Google Scholar

[12]

Springer Tracts in Natural Philosophy, 7, Springer-Verlag, Berlin-Heidelberg, 1965. doi: 10.1007/978-3-642-88412-2.  Google Scholar

[13]

Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[14]

Second edition, Applied Mathematical Sciences, 90, Springer, New York, 2009. doi: 10.1007/978-0-387-09724-4.  Google Scholar

[15]

Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X.  Google Scholar

[16]

Mat. Zametki, 61 (1997), 252-258. doi: 10.1007/BF02355730.  Google Scholar

[17]

M.S. thesis, The University of Sydney, 2013. Google Scholar

[18]

Ph.D thesis, Montpellier II University, 1996. Google Scholar

[19]

Annals of Mathematics (2), 161 (2005), 141-156. doi: 10.4007/annals.2005.161.141.  Google Scholar

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