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.
Classification : 49K15, 53C20, 70Q05, 81Q93
Mots clés : Euler−poinsot rigid body motion, conjugate locus on surfaces of revolution, Serret−Andoyer metric, spins dynamics
@article{COCV_2014__20_3_864_0, author = {Bonnard, Bernard and Cots, Olivier and Pomet, Jean-Baptiste and Shcherbakova, Nataliya}, title = {Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {864--893}, publisher = {EDP-Sciences}, volume = {20}, number = {3}, year = {2014}, doi = {10.1051/cocv/2013087}, zbl = {1293.49040}, mrnumber = {3264227}, language = {en}, url = {archive.numdam.org/item/COCV_2014__20_3_864_0/} }
Bonnard, Bernard; Cots, Olivier; Pomet, Jean-Baptiste; Shcherbakova, Nataliya. Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 864-893. doi : 10.1051/cocv/2013087. http://archive.numdam.org/item/COCV_2014__20_3_864_0/
[1] A Gauss-Bonnet-like Formula on Two-Dimensional Almost-Riemannian Manifolds. Discrete Contin. Dyn. Syst. A 20 (2008) 801-822. | MR 2379474 | Zbl 1198.49041
, and ,[2] 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). | MR 997295 | Zbl 0386.70001
,[3] The conjugate locus for the Euler top. I. The axisymmetric case. Int. Math. Forum 2 (2007) 2109-2139. | MR 2354391 | Zbl 1151.53348
and ,[4] Dynamical Systems, vol. IX. AMS Colloquium Publications (1927). | Zbl 0171.05402
,[5] Integrable Hamiltonian Systems. Geometry, Topology, Classification. Translated from the Russian original 1999. Chapman & Hall/CRC, Boca Raton, FL (2004) 730. | MR 2036760 | Zbl 1056.37075
and ,[6] 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. | Numdam | MR 2542715 | Zbl 1184.53036
, , and ,[7] Conjugate-cut loci and injectivity domains on two-spheres of revolution. ESAIM: COCV 19 (2013) 533-554. | Numdam | MR 3049722 | Zbl 1267.53042
, and ,[8] O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems. J. Math. Phys. 51 (2010) 092705, 44. | MR 2742815
,[9] Optimal Control in laser-induced population transfer for two and three-level quantum systems. J. Math. Phys. 43 (2002) 2107-2132. | MR 1893663 | Zbl 1059.81195
, , , and ,[10] Nonisotropic 3-level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy. Discrete Contin. Dyn. Systems B 5 (2005) 957-990. | MR 2170218 | Zbl 1084.81083
, and ,[11] Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2), and lens spaces. SIAM J. Control Optim. 47 (2008) 1851-1878. | MR 2421332 | Zbl 1170.53016
and ,[12] O. Cots and J. Gergaud, Differential continuation for regular optimal control problems. Optim. Methods Softw. 27 (2011) 177-196. | MR 2901956 | Zbl 1248.49025
,[13] Introduction to quantum control and dynamics. Appl. Nonlinear Sci. Ser. Chapman & Hall/CRC (2008). | MR 2357229 | Zbl 1139.81001
,[14] Introduction to nonlinear differential and integral equations. Dover Publications Inc., New York (1962). | MR 181773 | Zbl 0106.28904
,[15] The Serret−Andoyer formalism in rigid-body dynamics I. Symmetries and perturbations. Regul. Chaotic Dyn. 12 (2007) 389-425. | MR 2350331 | Zbl 1229.37112
, , and ,[16] The cut loci and the conjugate loci on ellipsoids. Manuscripta Math. 114 (2004) 247-264. | MR 2067796 | Zbl 1076.53042
and ,[17] Geometric Control Theory, vol. 52. Camb. Stud. Adv. Math. Cambridge University Press, Cambridge (1997). | MR 1425878 | Zbl 0940.93005
,[18] Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer. Phys. Rev. A 65 (2002) 032301. | MR 1891763
, and ,[19] Closed form Integration of the Hitzl-Breakwell problem in action-angle variables. IAA-AAS-DyCoSS1-01-02 (AAS 12-302), 27-39.
and ,[20] Elliptic Functions and Applications, vol. 80. Appl. Math. Sci. Springer-Verlag, New York (1989). | MR 1007595 | Zbl 0689.33001
,[21] Spin dynamics, basis of Nuclear Magnetic Resonance, 2nd edition. John Wiley and sons (2007).
,[22] Sur les lignes géodésiques des surfaces convexes. Trans. Amer. Math. Soc. 6 (1905) 237-274. | JFM 36.0669.01 | MR 1500710
,[23] The mathematical theory of optimal processes. Interscience Publishers John Wiley & Sons, Inc., New York-London (1962). | MR 166037 | Zbl 0117.31702
, , and ,[24] The Geometry of Total Curvature on Complete Open Surfaces, vol. 159. Camb. Tracts Math. Cambridge University Press, Cambridge (2003). | MR 2028047 | Zbl 1086.53056
, and ,[25] The cut locus of a two-sphere of revolution and Toponogov's comparison theorem. Tohoku Math. J. 59 (2007) 379-399. | MR 2365347 | Zbl 1158.53033
and ,[26] 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. | Zbl 0797.58007
and ,[27] Geometry, optimal control and quantum computing, Ph.D. Thesis. Harvard (2006). | MR 2708758
[28] Elliptic functions and efficient control of Ising spin chains with unequal coupling. Phys. Rev. A 77 (2008) 032340.
, and ,