# American Institute of Mathematical Sciences

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] A. V. Bolsinov and H. R. Dullin, On the Euler case in rigid body dynamics and the Jacobi problem, (Russian) Regul. Khaoticheskaya Din., 2 (1997), 13-25. [2] A. V. Bolsinov and A. T. Fomenko, The geodesic flow of an ellipsoid is orbitally equivalent to the integrable Euler case in the dynamics of a rigid body, Dokl. Akad. Nauk, 339 (1994), 253-296. [3] A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology, Classification,'' Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426. [4] W. E. Boyce and R. C. DiPrima, "Elementary Differential Equations and Boundary Value Problems,'' $7^{th}$ edition, John Wiley & Sons, Inc., New York-London-Sydney, 2001. [5] C. H. Clemens, "A Scrapbook of Complex Curve Theory,'' The University Series in Mathematics, Plenum Press, New York-London, 1980. [6] R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems,'' Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2. [7] J.-P. Dufour, P. Molino and A. Toulet, Classification des systèmes intégrables en dimension $2$ et invariants des modèles de Fomenko, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 949-952. [8] H. R. Dullin and S. Vũ Ngoc, Symplectic invariants near hyperbolic-hyperbolic points, Regular and Chaotic Dynamics, 12 (2007), 689-716. doi: 10.1134/S1560354707060111. [9] H. R. Dullin, Semi-global symplectic invariants of the spherical pendulum, Journal of Differential Equations, 254 (2013), 2942-2963. doi: 10.1016/j.jde.2013.01.018. [10] H. R. Dullin, P. H. Richter, A. P. Veselov and H. Waalkens, Actions of the Neumann systems via Picard-Fuchs equations, Physica D, 155 (2001), 159-183. doi: 10.1016/S0167-2789(01)00257-3. [11] D. D. Holm and J. E. Marsden, The rotor and the pendulum, in "Symplectic Geometry and Mathematical Physics" (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, (1991), 189-203. [12] E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point,'' Springer Tracts in Natural Philosophy, 7, Springer-Verlag, Berlin-Heidelberg, 1965. doi: 10.1007/978-3-642-88412-2. [13] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems,'' Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [14] K. Meyer, G. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'' Second edition, Applied Mathematical Sciences, 90, Springer, New York, 2009. doi: 10.1007/978-0-387-09724-4. [15] S. Vũ Ngoc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X. [16] O. E. Orël, On the nonconjugacy of the Euler case in the dynamics of a rigid body and on the Jacobi problem of geodesics on an ellipsoid, Mat. Zametki, 61 (1997), 252-258. doi: 10.1007/BF02355730. [17] George Papadopoulos, "Semi-Global Symplectic Invariants of the Euler Top,'' M.S. thesis, The University of Sydney, 2013. [18] Anne Toulet, "Classification of Integrable Systems on Two-Dimensional Symplectic Manifolds,'' Ph.D thesis, Montpellier II University, 1996. [19] Nguyen Tien Zung, Convergence versus integrability in Birkhoff normal form, Annals of Mathematics (2), 161 (2005), 141-156. doi: 10.4007/annals.2005.161.141.

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##### References:
 [1] A. V. Bolsinov and H. R. Dullin, On the Euler case in rigid body dynamics and the Jacobi problem, (Russian) Regul. Khaoticheskaya Din., 2 (1997), 13-25. [2] A. V. Bolsinov and A. T. Fomenko, The geodesic flow of an ellipsoid is orbitally equivalent to the integrable Euler case in the dynamics of a rigid body, Dokl. Akad. Nauk, 339 (1994), 253-296. [3] A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology, Classification,'' Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426. [4] W. E. Boyce and R. C. DiPrima, "Elementary Differential Equations and Boundary Value Problems,'' $7^{th}$ edition, John Wiley & Sons, Inc., New York-London-Sydney, 2001. [5] C. H. Clemens, "A Scrapbook of Complex Curve Theory,'' The University Series in Mathematics, Plenum Press, New York-London, 1980. [6] R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems,'' Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2. [7] J.-P. Dufour, P. Molino and A. Toulet, Classification des systèmes intégrables en dimension $2$ et invariants des modèles de Fomenko, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 949-952. [8] H. R. Dullin and S. Vũ Ngoc, Symplectic invariants near hyperbolic-hyperbolic points, Regular and Chaotic Dynamics, 12 (2007), 689-716. doi: 10.1134/S1560354707060111. [9] H. R. Dullin, Semi-global symplectic invariants of the spherical pendulum, Journal of Differential Equations, 254 (2013), 2942-2963. doi: 10.1016/j.jde.2013.01.018. [10] H. R. Dullin, P. H. Richter, A. P. Veselov and H. Waalkens, Actions of the Neumann systems via Picard-Fuchs equations, Physica D, 155 (2001), 159-183. doi: 10.1016/S0167-2789(01)00257-3. [11] D. D. Holm and J. E. Marsden, The rotor and the pendulum, in "Symplectic Geometry and Mathematical Physics" (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, (1991), 189-203. [12] E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point,'' Springer Tracts in Natural Philosophy, 7, Springer-Verlag, Berlin-Heidelberg, 1965. doi: 10.1007/978-3-642-88412-2. [13] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems,'' Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [14] K. Meyer, G. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'' Second edition, Applied Mathematical Sciences, 90, Springer, New York, 2009. doi: 10.1007/978-0-387-09724-4. [15] S. Vũ Ngoc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X. [16] O. E. Orël, On the nonconjugacy of the Euler case in the dynamics of a rigid body and on the Jacobi problem of geodesics on an ellipsoid, Mat. Zametki, 61 (1997), 252-258. doi: 10.1007/BF02355730. [17] George Papadopoulos, "Semi-Global Symplectic Invariants of the Euler Top,'' M.S. thesis, The University of Sydney, 2013. [18] Anne Toulet, "Classification of Integrable Systems on Two-Dimensional Symplectic Manifolds,'' Ph.D thesis, Montpellier II University, 1996. [19] Nguyen Tien Zung, Convergence versus integrability in Birkhoff normal form, Annals of Mathematics (2), 161 (2005), 141-156. doi: 10.4007/annals.2005.161.141.
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