Geodesic
Geodesics in the sub-Riemannian problem on the group $\mathrm{SO}(3)$
Sbornik. Mathematics, Tome 207 (2016) no. 7, pp. 915-941 Cet article a éte moissonné depuis la source Math-Net.Ru

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Geodesics of left-invariant sub-Riemannian structures are considered on the group $\mathrm{SO}(3)$. A complete description of periodic geodesics, their elementary properties, certain necessary conditions for minimality and estimates for the cut time and the diameter of the metric are presented. Bibliography: 32 titles.
Keywords: sub-Riemannian geometry, almost Riemannian geometry, optimal control, geodesic curves, cut time. 
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I. Yu. Beschastnyi; Yu. L. Sachkov. Geodesics in the sub-Riemannian problem on the group $\mathrm{SO}(3)$. Sbornik. Mathematics, Tome 207 (2016) no. 7, pp. 915-941. http://geodesic.mathdoc.fr/item/SM_2016_207_7_a1/

[1] U. Boscain, T. Chambrion, G. Charlot, “Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimal energy”, Discrete Contin. Dyn. Syst. Ser. B, 5:4 (2005), 957–990 | DOI | MR | Zbl

[2] I. Moiseev, Yu. L. Sachkov, “Maxwell strata in sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 16:2 (2010), 380–399 | DOI | MR | Zbl

[3] Yu. L. Sachkov, “Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 16:4 (2010), 1018–1039 | DOI | MR | Zbl

[4] Yu. L. Sachkov, “Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 17:2 (2011), 293–321 | DOI | MR | Zbl

[5] U. Boscain, R. Duits, F. Rossi, Y. Sachkov, “Curve cuspless reconstruction via sub-Riemannian geometry”, ESAIM Control Optim. Calc. Var., 20:3 (2014), 748–770 | DOI | MR | Zbl

[6] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr., 91, Amer. Math. Soc., Providence, RI, 2002, xx+259 pp. | DOI | MR | Zbl

[7] A. Agrachev, D. Barilari, “Sub-Riemanian structures on 3D Lie groups”, J. Dyn. Control Syst., 18:1 (2012), 21–44 | DOI | MR | Zbl

[8] A. M. Vershik, V. Ya. Gershkovich, “Nonholonomic dynamical systems, geometry of distributions and variational problems”, Dynamical systems. VII, Encyclopaedia Math. Sci., 16, Springer-Verlag, Berlin, 1994, 1–81 | MR | Zbl

[9] U. Boscain, F. Rossi, “Invariant Carnot–Carathéodory metrics on $S^3$, $SO(3)$, $SL(2)$, and lens spaces”, SIAM J. Control Optim., 47:4 (2008), 1851–1878 | DOI | MR | Zbl

[10] V. N. Berestovskii, I. A. Zubareva, “Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups”, Siberian Math. J., 42:4 (2001), 613–628 | DOI | MR | Zbl

[11] O. Calin, D.-C. Chang, I. Markina, “Sub-Riemannian geometry on the sphere $\mathbb S^3$”, Canad. J. Math., 61:4 (2009), 721–739 | DOI | MR | Zbl

[12] D.-C. Chang, I. Markina, A. Vasil'ev, “Sub-Riemannian geodesics on the 3-D sphere”, Complex Anal. Oper. Theory, 3:2 (2009), 361–377 | DOI | MR | Zbl

[13] V. N. Berestovskii, “Geodesics of a left-invariant nonholonomic Riemannian metric on the group of motions of the Euclidean plane”, Siberian Math. J., 35:6 (1994), 1083–1088 | DOI | MR | Zbl

[14] Y. A. Butt, Yu. L. Sachkov, A. I. Bhatti, “Extremal trajectories and Maxwell strata in sub-Riemannian problem on group of motions of pseudo-Euclidean plane”, J. Dyn. Control Syst., 20:3 (2014), 341–364 | DOI | MR | Zbl

[15] A. P. Mashtakov, Yu. L. Sachkov, “Superintegrability of left-invariant sub-Riemannian structures on unimodular three-dimensional Lie groups”, Diff. Equ., 51:11 (2015), 1476–1483 | DOI | Zbl

[16] B. Kruglikov, “Examples of integrable sub-Riemannian geodesic flows”, J. Dynam. Control Systems, 8:3 (2002), 323–340 | DOI | MR | Zbl

[17] D. Barilari, L. Rizzi, Comparison theorems for conjugate points in sub-Riemannian geometry, arXiv: 1401.3193v2

[18] U. Boscain, T. Chambrion, J.-P. Gauthier, “On the $K+P$ problem for a three-level quantum system: optimality implies resonance”, J. Dynam. Control Systems, 8:4 (2002), 547–572 | DOI | MR | Zbl

[19] B. Bonnard, O. Cots, J.-B. Pomet, N. Shcherbakova, “Riemannian metrics on 2D-manifolds related to the Euler–Poinsot rigid body motion”, ESAIM Control Optim. Calc. Var., 20:3 (2014), 864–893 | DOI | MR | Zbl

[20] B. Bonnard, M. Chyba, “Two applications of geometric optimal control to the dynamics of spin particle”, The impact of applications on mathematics, Math. Ind. (Tokyo), 1, Springer, Tokyo, 2014, 67–83 | DOI | MR | Zbl

[21] A. Agrachev, D. Barilari, U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, Preprint SISSA 09/2012/M, 2015, 341 pp. http://webusers.imj-prg.fr/~davide.barilari/Notes.php

[22] D. Yu. Burago, Yu. D. Burago, S. V. Ivanov, Kurs metricheskoi geometrii, In-t kompyuternykh issledovanii, M.–Izhevsk, 2004, 512 pp.

[23] A. A. Agrachev, Yu. L. Sachkov, Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005 | MR | Zbl

[24] D. F. Lawden, Elliptic functions and applications, Appl. Math. Sci., 80, Springer-Verlag, New York, 1989, xiv+334 pp. | DOI | MR | Zbl

[25] L. D. Landau, E. M. Lifschitz, Lehrbuch der theoretischen Physik, v. 1, Mechanik, H. Deutsch, Frankfurt am Main, 1997, 231 pp. | MR | Zbl | Zbl

[26] E. T. Whittaker, A treatise on the analytic dynamics of particles and rigid bodies, With an introduction to the problem of three bodies, Cambridge Math. Lib., Reprint of the 1937 ed., Cambridge Univ. Press, Cambridge, 1988, xx+456 pp. | DOI | MR | Zbl

[27] V. Jurdjevic, Geometric control theory, Cambridge Stud. Adv. Math., 52, Cambridge Univ. Press, Cambridge, 1997, xviii+492 pp. | MR | Zbl

[28] C. Schiller, Motion mountain. The adventure of physics, v. 5, Motion inside matter – pleasure, technology and stars, CreateSpace Independent Publ. Platform, 2013, 442 pp.

[29] A. Hatcher, Algebraic topology, Cambridge Univ. Press, Cambridge, 2002, xii+544 pp. | MR | Zbl

[30] A. A. Agrachev, G. Charlot, J. P. A. Gauthier, V. M. Zakalyukin, “On sub-Riemannian caustics and wave fronts for contact distributions in the three-space”, J. Dynam. Control Systems, 6:3 (2000), 365–395 | DOI | MR | Zbl

[31] J. H. Conway, D. A. Smith, On quaternions and octonions: their geometry, arithmetic and symmetry, A. K. Peters, Ltd., Natick, MA, 2003, xii+159 pp. | MR | Zbl

[32] P. F. Byrd, M. D. Friedman, “Hyperelliptic integrals”, Handbook of elliptic integrals for engineers and scientists, Grundlehren Math. Wiss., 67, Springer-Verlag, New York–Heidelberg, 1971, 252–271 | DOI | MR | Zbl