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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 |
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. |
show all references
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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