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

show all references

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

[1]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[2]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[3]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[4]

Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383

[5]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347

[6]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[7]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605

[8]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[9]

Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415

[10]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

[11]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[12]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[13]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025

[14]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[15]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[16]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[17]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[18]

Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370

[19]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[20]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]