Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 083, 17 pages      arXiv:1907.07819      https://doi.org/10.3842/SIGMA.2019.083

Collective Heavy Top Dynamics

Tomoki Ohsawa
Department of Mathematical Sciences, The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080-3021, USA

Received July 20, 2019, in final form October 22, 2019; Published online October 30, 2019

Abstract
We construct a Poisson map $\mathbf{M}\colon T^{*}\mathbb{C}^{2} \to \mathfrak{se}(3)^{*}$ with respect to the canonical Poisson bracket on $T^{*}\mathbb{C}^{2} \cong T^{*}\mathbb{R}^{4}$ and the $(-)$-Lie-Poisson bracket on the dual $\mathfrak{se}(3)^{*}$ of the Lie algebra of the special Euclidean group $\mathsf{SE}(3)$. The essential part of this map is the momentum map associated with the cotangent lift of the natural right action of the semidirect product Lie group $\mathsf{SU}(2) \ltimes \mathbb{C}^{2}$ on $\mathbb{C}^{2}$. This Poisson map gives rise to a canonical Hamiltonian system on $T^{*}\mathbb{C}^{2}$ whose solutions are mapped by $\mathbf{M}$ to solutions of the heavy top equations. We show that the Casimirs of the heavy top dynamics and the additional conserved quantity of the Lagrange top correspond to the Noether conserved quantities associated with certain symmetries of the canonical Hamiltonian system. We also construct a Lie-Poisson integrator for the heavy top dynamics by combining the Poisson map $\mathbf{M}$ with a simple symplectic integrator, and demonstrate that the integrator exhibits either exact or near conservation of the conserved quantities of the Kovalevskaya top.

Key words: heavy top dynamics; collectivization; momentum maps; Lie-Poisson integrator.

pdf (932 kb)   tex (485 kb)  

References

  1. Audin M., Spinning tops. A course on integrable systems, Cambridge Studies in Advanced Mathematics, Vol. 51, Cambridge University Press, Cambridge, 1996.
  2. Austin M.A., Krishnaprasad P.S., Wang L.S., Almost Poisson integration of rigid body systems, J. Comput. Phys. 107 (1993), 105-117.
  3. Bogfjellmo G., Collective symplectic integrators on ${S}_2^n \times {T}^*\mathbb{R}^m$, arXiv:1809.06231.
  4. Guillemin V., Sternberg S., The moment map and collective motion, Ann. Physics 127 (1980), 220-253.
  5. Guillemin V., Sternberg S., Symplectic techniques in physics, 2nd ed., Cambridge University Press, Cambridge, 1990.
  6. Hairer E., Lubich C., Wanner G., Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd ed., Springer Series in Computational Mathematics, Vol. 31, Springer-Verlag, Berlin, 2006.
  7. Holm D.D., Marsden J.E., Ratiu T.S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), 1-81, arXiv:chao-dyn/9801015.
  8. Holmes P.J., Marsden J.E., Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Indiana Univ. Math. J. 32 (1983), 273-309.
  9. Kowalevski S., Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math. 12 (1889), 177-232.
  10. Leimkuhler B., Reich S., Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics, Vol. 14, Cambridge University Press, Cambridge, 2004.
  11. Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, 2nd ed., Texts in Applied Mathematics, Vol. 17, Springer-Verlag, New York, 1999.
  12. Marsden J.E., Ratiu T.S., Weinstein A., Reduction and Hamiltonian structures on duals of semidirect product Lie algebras, in Fluids and Plasmas: Geometry and Dynamics (Boulder, Colo., 1983), Contemp. Math., Vol. 28, Amer. Math. Soc., Providence, RI, 1984, 55-100.
  13. Marsden J.E., Ratiu T.S., Weinstein A., Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984), 147-177.
  14. McLachlan R.I., Modin K., Verdier O., Collective symplectic integrators, Nonlinearity 27 (2014), 1525-1542, arXiv:1308.6620.
  15. McLachlan R.I., Modin K., Verdier O., Collective Lie-Poisson integrators on $\mathbb{R}^3$, IMA J. Numer. Anal. 35 (2015), 546-560, arXiv:1307.2387.
  16. McLachlan R.I., Modin K., Verdier O., Geometry of discrete-time spin systems, J. Nonlinear Sci. 26 (2016), 1507-1523, arXiv:1505.04035.
  17. McLachlan R.I., Modin K., Verdier O., A minimal-variable symplectic integrator on spheres, Math. Comp. 86 (2017), 2325-2344, arXiv:1402.3334.

Previous article  Next article  Contents of Volume 15 (2019)