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. Google Scholar [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. Google Scholar [3] A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology, Classification,'', Chapman & Hall/CRC, (2004). doi: 10.1201/9780203643426. Google Scholar [4] W. E. Boyce and R. C. DiPrima, "Elementary Differential Equations and Boundary Value Problems,'', $7^{th}$ edition, (2001). Google Scholar [5] C. H. Clemens, "A Scrapbook of Complex Curve Theory,'', The University Series in Mathematics, (1980). Google Scholar [6] R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems,'', Birkhäuser Verlag, (1997). doi: 10.1007/978-3-0348-8891-2. Google Scholar [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. Google Scholar [8] H. R. Dullin and S. Vũ Ngoc, Symplectic invariants near hyperbolic-hyperbolic points,, Regular and Chaotic Dynamics, 12 (2007), 689. doi: 10.1134/S1560354707060111. Google Scholar [9] H. R. Dullin, Semi-global symplectic invariants of the spherical pendulum,, Journal of Differential Equations, 254 (2013), 2942. doi: 10.1016/j.jde.2013.01.018. Google Scholar [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. doi: 10.1016/S0167-2789(01)00257-3. Google Scholar [11] D. D. Holm and J. E. Marsden, The rotor and the pendulum,, in, 99 (1991), 189. Google Scholar [12] E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point,'', Springer Tracts in Natural Philosophy, 7 (1965). doi: 10.1007/978-3-642-88412-2. Google Scholar [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 (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [14] K. Meyer, G. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'', Second edition, 90 (2009). doi: 10.1007/978-0-387-09724-4. Google Scholar [15] S. Vũ Ngoc, On semi-global invariants for focus-focus singularities,, Topology, 42 (2003), 365. doi: 10.1016/S0040-9383(01)00026-X. Google Scholar [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. doi: 10.1007/BF02355730. Google Scholar [17] George Papadopoulos, "Semi-Global Symplectic Invariants of the Euler Top,'', M.S. thesis, (2013). Google Scholar [18] Anne Toulet, "Classification of Integrable Systems on Two-Dimensional Symplectic Manifolds,'', Ph.D thesis, (1996). Google Scholar [19] Nguyen Tien Zung, Convergence versus integrability in Birkhoff normal form,, Annals of Mathematics (2), 161 (2005), 141. doi: 10.4007/annals.2005.161.141. Google Scholar

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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. Google Scholar [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. Google Scholar [3] A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology, Classification,'', Chapman & Hall/CRC, (2004). doi: 10.1201/9780203643426. Google Scholar [4] W. E. Boyce and R. C. DiPrima, "Elementary Differential Equations and Boundary Value Problems,'', $7^{th}$ edition, (2001). Google Scholar [5] C. H. Clemens, "A Scrapbook of Complex Curve Theory,'', The University Series in Mathematics, (1980). Google Scholar [6] R. H. Cushman and L. M. Bates, "Global Aspects of Classical Integrable Systems,'', Birkhäuser Verlag, (1997). doi: 10.1007/978-3-0348-8891-2. Google Scholar [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. Google Scholar [8] H. R. Dullin and S. Vũ Ngoc, Symplectic invariants near hyperbolic-hyperbolic points,, Regular and Chaotic Dynamics, 12 (2007), 689. doi: 10.1134/S1560354707060111. Google Scholar [9] H. R. Dullin, Semi-global symplectic invariants of the spherical pendulum,, Journal of Differential Equations, 254 (2013), 2942. doi: 10.1016/j.jde.2013.01.018. Google Scholar [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. doi: 10.1016/S0167-2789(01)00257-3. Google Scholar [11] D. D. Holm and J. E. Marsden, The rotor and the pendulum,, in, 99 (1991), 189. Google Scholar [12] E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point,'', Springer Tracts in Natural Philosophy, 7 (1965). doi: 10.1007/978-3-642-88412-2. Google Scholar [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 (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [14] K. Meyer, G. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'', Second edition, 90 (2009). doi: 10.1007/978-0-387-09724-4. Google Scholar [15] S. Vũ Ngoc, On semi-global invariants for focus-focus singularities,, Topology, 42 (2003), 365. doi: 10.1016/S0040-9383(01)00026-X. Google Scholar [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. doi: 10.1007/BF02355730. Google Scholar [17] George Papadopoulos, "Semi-Global Symplectic Invariants of the Euler Top,'', M.S. thesis, (2013). Google Scholar [18] Anne Toulet, "Classification of Integrable Systems on Two-Dimensional Symplectic Manifolds,'', Ph.D thesis, (1996). Google Scholar [19] Nguyen Tien Zung, Convergence versus integrability in Birkhoff normal form,, Annals of Mathematics (2), 161 (2005), 141. doi: 10.4007/annals.2005.161.141. Google Scholar
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