Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-20T17:09:24.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion∗

Published online by Cambridge University Press:  10 June 2014

Bernard Bonnard ,
Olivier Cots ,
Jean-Baptiste Pomet  and
Nataliya Shcherbakova
Bernard Bonnard
Affiliation:
Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France. bernard.bonnard@u-bourgogne.fr
Olivier Cots
Affiliation:
INRIA Sophia-Antipolis Méditerranée, 2004, route des Lucioles, 06902 Sophia Antipolis, France; olivier.cots@inria.fr
Jean-Baptiste Pomet
Affiliation:
INRIA Sophia-Antipolis Méditerranée, 2004, route des Lucioles, 06902 Sophia Antipolis, France; olivier.cots@inria.fr
Nataliya Shcherbakova
Affiliation:
Université de Toulouse, INPT, UPS, Laboratoire de Génie Chimique, 4 allée Emile Monso, 31432 Toulouse, France; nataliya.shcherbakova@inp-toulouse.fr
Get access

Abstract

The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S2 associated to the dynamics of spin particles with Ising coupling is analysed using both geometric techniques and numerical simulations.

Keywords

Euler−poinsot rigid body motionconjugate locus on surfaces of revolutionSerret−Andoyer metricspins dynamics
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 864 - 893
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrachev, A.A., Boscain, U. and Sigalotti, M., A Gauss–Bonnet-like Formula on Two-Dimensional Almost-Riemannian Manifolds. Discrete Contin. Dyn. Syst. A 20 (2008) 801-822. Google Scholar
V.I. Arnold, Mathematical Methods of Classical Mechanics, vol. 60. Translated from the Russian, edited by K. Vogtmann and A. Weinstein. 2nd edition. Grad. Texts Math. Springer-Verlag, New York (1989).
Bates, L. and Fassò, F., The conjugate locus for the Euler top. I. The axisymmetric case. Int. Math. Forum 2 (2007) 2109-2139. Google Scholar
G.D. Birkhoff, Dynamical Systems, vol. IX. AMS Colloquium Publications (1927).
A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification. Translated from the Russian original 1999. Chapman & Hall/CRC, Boca Raton, FL (2004) 730.
Bonnard, B., Caillau, J.-B., Sinclair, R. and Tanaka, M., Conjugate and cut loci of a two-sphere of revolution with application to optimal control. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009) 1081-1098. Google Scholar
Bonnard, B., Caillau, J.-B. and Janin, G., Conjugate-cut loci and injectivity domains on two-spheres of revolution. ESAIM: COCV 19 (2013) 533-554. Google Scholar
Bonnard, B., Cots, O., Shcherbakova, N. and Sugny, D., The energy minimization problem for two-level dissipative quantum systems. J. Math. Phys. 51 (2010) 092705, 44. Google Scholar
Boscain, U., Charlot, G., Gauthier, J.-P., Guérin, S. and Jauslin, H.-R., Optimal Control in laser-induced population transfer for two and three-level quantum systems. J. Math. Phys. 43 (2002) 2107-2132. Google Scholar
Boscain, U., Chambrion, T. and Charlot, G., Nonisotropic 3-level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy. Discrete Contin. Dyn. Systems B 5 (2005) 957-990. Google Scholar
Boscain, U. and Rossi, F., Invariant Carnot-Caratheodory metrics on S 3, SO(3), SL(2), and lens spaces. SIAM J. Control Optim. 47 (2008) 1851-1878. Google Scholar
Caillau, J.-B., Cots, O. and Gergaud, J., Differential continuation for regular optimal control problems. Optim. Methods Softw. 27 (2011) 177–196. Google Scholar
D. D’Alessandro, Introduction to quantum control and dynamics. Appl. Nonlinear Sci. Ser. Chapman & Hall/CRC (2008).
H.T. Davis, Introduction to nonlinear differential and integral equations. Dover Publications Inc., New York (1962).
Gurfil, P., Elipe, A., Tangren, W. and Efroimsky, M., The Serret−Andoyer formalism in rigid-body dynamics I. Symmetries and perturbations. Regul. Chaotic Dyn. 12 (2007) 389-425. Google Scholar
Itoh, J. and Kiyohara, K., The cut loci and the conjugate loci on ellipsoids. Manuscripta Math. 114 (2004) 247-264. Google Scholar
V. Jurdjevic, Geometric Control Theory, vol. 52. Camb. Stud. Adv. Math. Cambridge University Press, Cambridge (1997).
Khaneja, N., Brockett, R. and Glaser, S.J., Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer. Phys. Rev. A 65 (2002) 032301. Google Scholar
M. Lara and S. Ferrer, Closed form Integration of the Hitzl-Breakwell problem in action-angle variables. IAA-AAS-DyCoSS1-01-02 (AAS 12-302), 27-39.
D. Lawden, Elliptic Functions and Applications, vol. 80. Appl. Math. Sci. Springer-Verlag, New York (1989).
M.H. Levitt, Spin dynamics, basis of Nuclear Magnetic Resonance, 2nd edition. John Wiley and sons (2007).
Poincaré, H., Sur les lignes géodésiques des surfaces convexes. Trans. Amer. Math. Soc. 6 (1905) 237-274. Google Scholar
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. Interscience Publishers John Wiley & Sons, Inc., New York-London (1962).
K. Shiohama, T. Shioya and M. Tanaka, The Geometry of Total Curvature on Complete Open Surfaces, vol. 159. Camb. Tracts Math. Cambridge University Press, Cambridge (2003).
Sinclair, R. and Tanaka, M., The cut locus of a two-sphere of revolution and Toponogov’s comparison theorem. Tohoku Math. J. 59 (2007) 379-399. Google Scholar
A.M. Vershik and V.Ya. Gershkovich, Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems. in Dynamical Systems VII. In vol. 16 of Encyclopedia of Math. Sci. Springer Verlag (1991) 10-81.
H. Yuan Geometry, optimal control and quantum computing, Ph.D. Thesis. Harvard (2006).
Yuan, H., Zeier, R. and Khaneja, N., Elliptic functions and efficient control of Ising spin chains with unequal coupling. Phys. Rev. A 77 (2008) 032340. Google Scholar