# American Institute of Mathematical Sciences

January  2014, 13(1): 435-452. doi: 10.3934/cpaa.2014.13.435

## Geometric conditions for the existence of a rolling without twisting or slipping

 1 Mathematisches Institut, Georg-August-Universität, Bunsen-str. 3-5, D-37073 Göttingen, Germany 2 Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020

Received  June 2012 Revised  May 2013 Published  July 2013

We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that, up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each curve coincide. By using the anti-developments of the curves, which we claim can be seen as a generalization of the geodesic curvatures, we are able to extend the result to arbitrary absolutely continuous curves. For a manifold of constant sectional curvature rolling on itself, two such curves can only differ by an isometry. In the case of surfaces, we give conditions for when loops in the manifolds lift to loops in the configuration space of the rolling.
Citation: Mauricio Godoy Molina, Erlend Grong. Geometric conditions for the existence of a rolling without twisting or slipping. Communications on Pure & Applied Analysis, 2014, 13 (1) : 435-452. doi: 10.3934/cpaa.2014.13.435
##### References:
 [1] A. Agrachev, Rolling balls and octonions,, Proc. Steklov Inst. Math., 258 (2007), 13. doi: 10.1134/S0081543807030030. Google Scholar [2] A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,'', Springer, (2004). doi: 10.1007/978-3-662-06404-7. Google Scholar [3] A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics,, Notices Amer. Math. Soc., 52 (2005), 324. Google Scholar [4] G. Bor and R. Montgomery, $G_2$ and the rolling distribution,, L'Ens. Math., 55 (2009), 157. Google Scholar [5] É. Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre,, Ann. Sci. \'Ecole Norm. Sup., 27 (1910), 109. Google Scholar [6] S. A. Chaplygin, On some generalization of the area theorem, with applications to the problem of rolling balls, (Russian), Mat. Sbornik, XX (1897), 1. doi: 10.1134/S1560354712020086. Google Scholar [7] S. A. Chaplygin, On a ball's rolling on a horizontal plane, (Russian), Mat. Sbornik, XXIV (1903), 139. doi: 10.1070/RD2002v007n02ABEH000200. Google Scholar [8] Y. Chitour, A. Marigo and B. Piccoli, Quantization of the rolling-body problem with applications to motion planning,, Systems Control Lett., 54 (2005), 999. doi: 10.1016/j.sysconle.2005.02.012. Google Scholar [9] Y. Chitour, M. Godoy Molina and P. Kokkonen, Symmetries of the rolling model, preprint,, \arXiv{1301.2579}., (). Google Scholar [10] Y. Chitour and P. Kokkonen, Rolling manifolds: Intrinsic formulation and controllability, preprint,, \arXiv{1011.2925}., (). Google Scholar [11] Y. Chitour and P. Kokkonen, Rolling manifolds on space forms,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 29 (2012), 927. doi: 10.1016/j.anihpc.2012.05.005. Google Scholar [12] B. E. J. Dahlberg, The converse of the four vertex theorem,, Proc. Amer. Math. Soc., 133 (2005), 2131. doi: 10.1090/S0002-9939-05-07788-9. Google Scholar [13] M. Godoy Molina, E. Grong, I. Markina and F. Silva Leite, An intrinsic formulation of the problem on rolling manifolds,, J. Dyn. Control Syst., 18 (2012), 181. doi: 10.1007/s10883-012-9139-2. Google Scholar [14] E. Grong, Controllability of rolling without twisting or slipping in higher dimensions,, SIAM J. Control Optim., 50 (2012), 2462. doi: 10.1137/110829581. Google Scholar [15] E. Hsu, "Stochastic Analysis on Manifolds,'' Graduate Studies in Mathematics 38,, American Mathematical Society, (2002). Google Scholar [16] K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Grassmann manifolds,, J. Dyn. Control Syst., 13 (2007), 467. doi: 10.1007/s10883-007-9027-3. Google Scholar [17] B. D. Johnson, The nonholonomy of the rolling sphere,, Amer. Math. Monthly, 114 (2007), 500. Google Scholar [18] V. Jurdjevic and J. A. Zimmerman, Rolling sphere problems on spaces of constant curvature,, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729. doi: 10.1017/S0305004108001084. Google Scholar [19] M. Levi, Geometric phases in the motion of rigid bodies,, Arch. Rational Mech. Anal., 122 (1993), 213. doi: 10.1007/BF00380255. Google Scholar [20] K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact,, T\^ohoku Math. J., 30 (1978), 623. doi: 10.2748/tmj/1178229921. Google Scholar [21] J. W. Robbin and D. A. Salamon, "Introduction to Differential Geometry,", Available from: \url{http://www.math.ethz.ch/\, (). Google Scholar [22] R. W. Sharpe, "Differential Geometry,'', GTM 166, (1997). Google Scholar [23] M. Spivak, "A Comprehensive Introduction to Differential Geometry'',, Volume IV, (1999). Google Scholar [24] I. Zelenko, On variational approach to differential invariants of rank two distributions,, Differential Geom. Appl., 24 (2006), 235. doi: 10.1016/j.difgeo.2005.09.004. Google Scholar [25] I. Zelenko, Fundamental form and the Cartan tensor of $(2,5)$-distributions coincide,, J. Dyn. Control Syst., 12 (2006), 247. doi: 10.1007/s10450-006-0383-1. Google Scholar [26] J. A. Zimmerman, Optimal control of the sphere $S^n$ rolling on $E^n$,, Math. Control Signals Systems, 17 (2005), 14. doi: 10.1007/s00498-004-0143-2. Google Scholar

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##### References:
 [1] A. Agrachev, Rolling balls and octonions,, Proc. Steklov Inst. Math., 258 (2007), 13. doi: 10.1134/S0081543807030030. Google Scholar [2] A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,'', Springer, (2004). doi: 10.1007/978-3-662-06404-7. Google Scholar [3] A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics,, Notices Amer. Math. Soc., 52 (2005), 324. Google Scholar [4] G. Bor and R. Montgomery, $G_2$ and the rolling distribution,, L'Ens. Math., 55 (2009), 157. Google Scholar [5] É. Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre,, Ann. Sci. \'Ecole Norm. Sup., 27 (1910), 109. Google Scholar [6] S. A. Chaplygin, On some generalization of the area theorem, with applications to the problem of rolling balls, (Russian), Mat. Sbornik, XX (1897), 1. doi: 10.1134/S1560354712020086. Google Scholar [7] S. A. Chaplygin, On a ball's rolling on a horizontal plane, (Russian), Mat. Sbornik, XXIV (1903), 139. doi: 10.1070/RD2002v007n02ABEH000200. Google Scholar [8] Y. Chitour, A. Marigo and B. Piccoli, Quantization of the rolling-body problem with applications to motion planning,, Systems Control Lett., 54 (2005), 999. doi: 10.1016/j.sysconle.2005.02.012. Google Scholar [9] Y. Chitour, M. Godoy Molina and P. Kokkonen, Symmetries of the rolling model, preprint,, \arXiv{1301.2579}., (). Google Scholar [10] Y. Chitour and P. Kokkonen, Rolling manifolds: Intrinsic formulation and controllability, preprint,, \arXiv{1011.2925}., (). Google Scholar [11] Y. Chitour and P. Kokkonen, Rolling manifolds on space forms,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 29 (2012), 927. doi: 10.1016/j.anihpc.2012.05.005. Google Scholar [12] B. E. J. Dahlberg, The converse of the four vertex theorem,, Proc. Amer. Math. Soc., 133 (2005), 2131. doi: 10.1090/S0002-9939-05-07788-9. Google Scholar [13] M. Godoy Molina, E. Grong, I. Markina and F. Silva Leite, An intrinsic formulation of the problem on rolling manifolds,, J. Dyn. Control Syst., 18 (2012), 181. doi: 10.1007/s10883-012-9139-2. Google Scholar [14] E. Grong, Controllability of rolling without twisting or slipping in higher dimensions,, SIAM J. Control Optim., 50 (2012), 2462. doi: 10.1137/110829581. Google Scholar [15] E. Hsu, "Stochastic Analysis on Manifolds,'' Graduate Studies in Mathematics 38,, American Mathematical Society, (2002). Google Scholar [16] K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Grassmann manifolds,, J. Dyn. Control Syst., 13 (2007), 467. doi: 10.1007/s10883-007-9027-3. Google Scholar [17] B. D. Johnson, The nonholonomy of the rolling sphere,, Amer. Math. Monthly, 114 (2007), 500. Google Scholar [18] V. Jurdjevic and J. A. Zimmerman, Rolling sphere problems on spaces of constant curvature,, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729. doi: 10.1017/S0305004108001084. Google Scholar [19] M. Levi, Geometric phases in the motion of rigid bodies,, Arch. Rational Mech. Anal., 122 (1993), 213. doi: 10.1007/BF00380255. Google Scholar [20] K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact,, T\^ohoku Math. J., 30 (1978), 623. doi: 10.2748/tmj/1178229921. Google Scholar [21] J. W. Robbin and D. A. Salamon, "Introduction to Differential Geometry,", Available from: \url{http://www.math.ethz.ch/\, (). Google Scholar [22] R. W. Sharpe, "Differential Geometry,'', GTM 166, (1997). Google Scholar [23] M. Spivak, "A Comprehensive Introduction to Differential Geometry'',, Volume IV, (1999). Google Scholar [24] I. Zelenko, On variational approach to differential invariants of rank two distributions,, Differential Geom. Appl., 24 (2006), 235. doi: 10.1016/j.difgeo.2005.09.004. Google Scholar [25] I. Zelenko, Fundamental form and the Cartan tensor of $(2,5)$-distributions coincide,, J. Dyn. Control Syst., 12 (2006), 247. doi: 10.1007/s10450-006-0383-1. Google Scholar [26] J. A. Zimmerman, Optimal control of the sphere $S^n$ rolling on $E^n$,, Math. Control Signals Systems, 17 (2005), 14. doi: 10.1007/s00498-004-0143-2. Google Scholar
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