# American Institute of Mathematical Sciences

June  2016, 8(2): 139-167. doi: 10.3934/jgm.2016001

## Invariant nonholonomic Riemannian structures on three-dimensional Lie groups

 1 Department of Mathematics (Pure and Applied), Rhodes University, 6140 Grahamstown, South Africa, South Africa, South Africa 2 Department of Mathematics, Faculty of Science, University of Ostrava, 701 03 Ostrava, Czech Republic and Department of Mathematics, Ghent University, B-9000 Ghent, Belgium

Received  September 2015 Revised  March 2016 Published  June 2016

We consider Riemannian manifolds endowed with a nonholonomic distribution. These structures model mechanical systems with a (positive definite) quadratic Lagrangian and nonholonomic constraints linear in velocities. We classify the left-invariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups, and describe the equivalence classes in terms of some basic isometric invariants. The classification naturally splits into two cases. In the first case, it reduces to a classification of left-invariant sub-Riemannian structures. In the second case, we find a canonical frame with which to directly compare equivalence classes.
Citation: Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139-167. doi: 10.3934/jgm.2016001
##### References:
 [1] A. Agrachev and D. Barilari, Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst., 18 (2012), 21-44. doi: 10.1007/s10883-012-9133-8. [2] A. Bloch, Nonholonomic Mechanics and Control, Springer, New York, 2003. doi: 10.1007/b97376. [3] E. Cartan, On the geometric representation of nonholonomic mechanical systems (in French), in Proceedings of the International Congress of Mathematicians, vol. 4, Bologna, Italy, 1928, 253-261. [4] M. Čech and J. Musilová, Symmetries and currents in nonholonomic mechanics, Commun. Math., 22 (2014), 159-184. [5] S. Chaplygin, On the theory of motion of nonholonomic systems. The theorem of the Reducing Multiplier (in Russian), Math. Sbornik, XXVIII (1911), 303-314. [6] J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Springer, Berlin, 2002. doi: 10.1007/b84020. [7] R. Cushman, H. Duistermaat and J. Śniatycki, Geometry of Nonholonomically Constrained Systems, World Scientific, Singapore, 2010. [8] V. Dragović and B. Gajić, The Wagner curvature tensor in nonholonomic mechanics, Regul. Chaotic Dyn., 8 (2003), 105-123. doi: 10.1070/RD2003v008n01ABEH000229. [9] K. Ehlers, Geometric equivalence on nonholonomic three-manifolds, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA, 2003, 246-255. [10] K. Ehlers, J. Koiller, R. Montgomery and P. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry, Festschrift in Honor of Alan Weinstein (eds. J. Marsden and T. Ratiu), Birkhäuser, 232 (2005), 75-120. doi: 10.1007/0-8176-4419-9_4. [11] Y. Fedorov and L. García-Naranjo, The hydrodynamic Chaplygin sleigh, J. Phys. A: Math. Theor., 43 (2010), 434013, 18 pp. doi: 10.1088/1751-8113/43/43/434013. [12] Y. Fedorov and B. Jovanović, Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces, J. Nonlinear Sci., 14 (2004), 341-381. doi: 10.1007/s00332-004-0603-3. [13] Y. Fedorov, A. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259. doi: 10.1088/0951-7715/22/9/009. [14] B. Jovanović, Geometry and integrability of Euler-Poincaré-Suslov equations, Nonlinearity, 14 (2001), 1555-1567. doi: 10.1088/0951-7715/14/6/308. [15] J. Koiller, P. Rodrigues and P. Pitanga, Non-holonomic connections following Élie Cartan, An. Acad. Bras. Cienc., 73 (2001), 165-190. doi: 10.1590/S0001-37652001000200003. [16] V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras (in Russian), Funkt. Anal. Prilozh., 22 (1988), 69-70 (Engl. transl. Funct. Anal. Appl., 22 (1988), 58-59). doi: 10.1007/BF01077727. [17] A. Krasiński, C. Behr, E. Schücking, F. Estabrook, H. Wahlquist, G. Ellis, R. Jantzen and W. Kundt, The Bianchi classification in the Schücking-Behr approach, Gen. Relativ. Gravit., 35 (2003), 475-489. doi: 10.1023/A:1022382202778. [18] O. Krupková, Geometric mechanics on nonholonomic submanifolds, Commun. Math., 18 (2010), 51-77. [19] B. Langerock, Nonholonomic mechanics and connections over a bundle map, J. Phys. A: Math. Gen., 34 (2001), L609-L615. doi: 10.1088/0305-4470/34/44/102. [20] A. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys., 42 (1998), 135-164. doi: 10.1016/S0034-4877(98)80008-6. [21] A. Lewis, Simple mechanical control systems with constraints, IEEE Trans. Automat. Control, 45 (2000), 1420-1436. doi: 10.1109/9.871752. [22] M. MacCallum, On the classification of the real four-dimensional Lie algebras, in On Einstein's Path: Essays in Honour of E. Schücking (ed. A. Harvey), Springer, 1999, 299-317. [23] G. Mubarakzyanov, On solvable Lie algebras (in Russian), Izv. Vysš. Učehn. Zaved. Matematika, 32 (1963), 114-123. [24] J. Neimark and N. Fufaev, Dynamics of Nonholonomic Systems, American Mathematical Society, Providence, 1972. [25] X. Pennec and V. Arsigny, Exponential barycenters of the canonical Cartan connection and invariant means on Lie groups, in Matrix Information Geometry (eds. F. Nielsen and R. Bhatia), Springer, 2013, 123-166. doi: 10.1007/978-3-642-30232-9_7. [26] M. Postnikov, Geometry VI: Riemannian Geometry, Springer, New York, 2001. [27] O. Rossi and J. Musilová，, The relativistic mechanics in a nonholonomic setting: A unified approach to particles with non-zero mass and massless particles, J. Phys. A: Math. Theor., 45 (2012), 255202, 27 pp. doi: 10.1088/1751-8113/45/25/255202. [28] W. Sarlet, A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems, Extracta Math., 11 (1996), 202-212. [29] G. Suslov, Theoretical Mechanics (in Russian), Gostekhizdat, Moscow, 1946. [30] M. Swaczyna, Several examples of nonholonomic mechanical systems, Commun. Math., 19 (2011), 27-56. [31] M. Swaczyna and P. Volný, Uniform projectile motion: Dynamics, symmetries and conservation laws, Rep. Math. Phys., 73 (2014), 177-200. doi: 10.1016/S0034-4877(14)60039-2. [32] J. Tavares, About Cartan geometrization of non-holonomic mechanics, J. Geom. Phys., 45 (2003), 1-23. doi: 10.1016/S0393-0440(02)00118-3. [33] A. Vershik, Classical and non-classical dynamics with constraints, in Global Analysis. Studies and Applications I (eds. Y. Borisovich and Y. Gliklikh), Springer, 1108 (1984), 278-301. doi: 10.1007/BFb0099563. [34] A. Vershik and L. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Sov. Phys. Dokl., 17 (1972), 34-36. [35] A. Vershik and V. Gershkovich, Nonholonomic problems and the theory of distributions, Acta Appl. Math., 12 (1988), 181-209. doi: 10.1007/BF00047498. [36] A. Vershik and V. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, in Dynamical Systems VII (eds. V. Arnol'd and S. Novikov), Springer, 1994, 1-81. [37] A. Veselov and L. Veselova, Integrable nonholonomic systems on Lie groups, Math. Notes, 44 (1988), 810-819. doi: 10.1007/BF01158420.

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##### References:
 [1] A. Agrachev and D. Barilari, Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst., 18 (2012), 21-44. doi: 10.1007/s10883-012-9133-8. [2] A. Bloch, Nonholonomic Mechanics and Control, Springer, New York, 2003. doi: 10.1007/b97376. [3] E. Cartan, On the geometric representation of nonholonomic mechanical systems (in French), in Proceedings of the International Congress of Mathematicians, vol. 4, Bologna, Italy, 1928, 253-261. [4] M. Čech and J. Musilová, Symmetries and currents in nonholonomic mechanics, Commun. Math., 22 (2014), 159-184. [5] S. Chaplygin, On the theory of motion of nonholonomic systems. The theorem of the Reducing Multiplier (in Russian), Math. Sbornik, XXVIII (1911), 303-314. [6] J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Springer, Berlin, 2002. doi: 10.1007/b84020. [7] R. Cushman, H. Duistermaat and J. Śniatycki, Geometry of Nonholonomically Constrained Systems, World Scientific, Singapore, 2010. [8] V. Dragović and B. Gajić, The Wagner curvature tensor in nonholonomic mechanics, Regul. Chaotic Dyn., 8 (2003), 105-123. doi: 10.1070/RD2003v008n01ABEH000229. [9] K. Ehlers, Geometric equivalence on nonholonomic three-manifolds, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA, 2003, 246-255. [10] K. Ehlers, J. Koiller, R. Montgomery and P. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry, Festschrift in Honor of Alan Weinstein (eds. J. Marsden and T. Ratiu), Birkhäuser, 232 (2005), 75-120. doi: 10.1007/0-8176-4419-9_4. [11] Y. Fedorov and L. García-Naranjo, The hydrodynamic Chaplygin sleigh, J. Phys. A: Math. Theor., 43 (2010), 434013, 18 pp. doi: 10.1088/1751-8113/43/43/434013. [12] Y. Fedorov and B. Jovanović, Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces, J. Nonlinear Sci., 14 (2004), 341-381. doi: 10.1007/s00332-004-0603-3. [13] Y. Fedorov, A. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259. doi: 10.1088/0951-7715/22/9/009. [14] B. Jovanović, Geometry and integrability of Euler-Poincaré-Suslov equations, Nonlinearity, 14 (2001), 1555-1567. doi: 10.1088/0951-7715/14/6/308. [15] J. Koiller, P. Rodrigues and P. Pitanga, Non-holonomic connections following Élie Cartan, An. Acad. Bras. Cienc., 73 (2001), 165-190. doi: 10.1590/S0001-37652001000200003. [16] V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras (in Russian), Funkt. Anal. Prilozh., 22 (1988), 69-70 (Engl. transl. Funct. Anal. Appl., 22 (1988), 58-59). doi: 10.1007/BF01077727. [17] A. Krasiński, C. Behr, E. Schücking, F. Estabrook, H. Wahlquist, G. Ellis, R. Jantzen and W. Kundt, The Bianchi classification in the Schücking-Behr approach, Gen. Relativ. Gravit., 35 (2003), 475-489. doi: 10.1023/A:1022382202778. [18] O. Krupková, Geometric mechanics on nonholonomic submanifolds, Commun. Math., 18 (2010), 51-77. [19] B. Langerock, Nonholonomic mechanics and connections over a bundle map, J. Phys. A: Math. Gen., 34 (2001), L609-L615. doi: 10.1088/0305-4470/34/44/102. [20] A. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys., 42 (1998), 135-164. doi: 10.1016/S0034-4877(98)80008-6. [21] A. Lewis, Simple mechanical control systems with constraints, IEEE Trans. Automat. Control, 45 (2000), 1420-1436. doi: 10.1109/9.871752. [22] M. MacCallum, On the classification of the real four-dimensional Lie algebras, in On Einstein's Path: Essays in Honour of E. Schücking (ed. A. Harvey), Springer, 1999, 299-317. [23] G. Mubarakzyanov, On solvable Lie algebras (in Russian), Izv. Vysš. Učehn. Zaved. Matematika, 32 (1963), 114-123. [24] J. Neimark and N. Fufaev, Dynamics of Nonholonomic Systems, American Mathematical Society, Providence, 1972. [25] X. Pennec and V. Arsigny, Exponential barycenters of the canonical Cartan connection and invariant means on Lie groups, in Matrix Information Geometry (eds. F. Nielsen and R. Bhatia), Springer, 2013, 123-166. doi: 10.1007/978-3-642-30232-9_7. [26] M. Postnikov, Geometry VI: Riemannian Geometry, Springer, New York, 2001. [27] O. Rossi and J. Musilová，, The relativistic mechanics in a nonholonomic setting: A unified approach to particles with non-zero mass and massless particles, J. Phys. A: Math. Theor., 45 (2012), 255202, 27 pp. doi: 10.1088/1751-8113/45/25/255202. [28] W. Sarlet, A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems, Extracta Math., 11 (1996), 202-212. [29] G. Suslov, Theoretical Mechanics (in Russian), Gostekhizdat, Moscow, 1946. [30] M. Swaczyna, Several examples of nonholonomic mechanical systems, Commun. Math., 19 (2011), 27-56. [31] M. Swaczyna and P. Volný, Uniform projectile motion: Dynamics, symmetries and conservation laws, Rep. Math. Phys., 73 (2014), 177-200. doi: 10.1016/S0034-4877(14)60039-2. [32] J. Tavares, About Cartan geometrization of non-holonomic mechanics, J. Geom. Phys., 45 (2003), 1-23. doi: 10.1016/S0393-0440(02)00118-3. [33] A. Vershik, Classical and non-classical dynamics with constraints, in Global Analysis. Studies and Applications I (eds. Y. Borisovich and Y. Gliklikh), Springer, 1108 (1984), 278-301. doi: 10.1007/BFb0099563. [34] A. Vershik and L. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Sov. Phys. Dokl., 17 (1972), 34-36. [35] A. Vershik and V. Gershkovich, Nonholonomic problems and the theory of distributions, Acta Appl. Math., 12 (1988), 181-209. doi: 10.1007/BF00047498. [36] A. Vershik and V. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, in Dynamical Systems VII (eds. V. Arnol'd and S. Novikov), Springer, 1994, 1-81. [37] A. Veselov and L. Veselova, Integrable nonholonomic systems on Lie groups, Math. Notes, 44 (1988), 810-819. doi: 10.1007/BF01158420.
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