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

Invariant nonholonomic Riemannian structures on three-dimensional Lie groups

Dennis I. Barrett 1, , Rory Biggs 1, , Claudiu C. Remsing 1, and  Olga Rossi 2,

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.
Keywords: invariant nonholonomic Riemannian structure, Schouten curvature tensor., nonholonomic connection, isometric invariant, Nonholonomic mechanical system.
Mathematics Subject Classification: Primary: 70G45, 37J60; Secondary: 53A55, 22E3.
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. doi: 10.1007/s10883-012-9133-8. Google Scholar

[2]

A. Bloch, Nonholonomic Mechanics and Control,, Springer, (2003). doi: 10.1007/b97376. Google Scholar

[3]

E. Cartan, On the geometric representation of nonholonomic mechanical systems (in French),, in Proceedings of the International Congress of Mathematicians, (1928), 253. Google Scholar

[4]

M. Čech and J. Musilová, Symmetries and currents in nonholonomic mechanics,, Commun. Math., 22 (2014), 159. Google Scholar

[5]

S. Chaplygin, On the theory of motion of nonholonomic systems. The theorem of the Reducing Multiplier (in Russian),, Math. Sbornik, XXVIII (1911), 303. Google Scholar

[6]

J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Springer, (2002). doi: 10.1007/b84020. Google Scholar

[7]

R. Cushman, H. Duistermaat and J. Śniatycki, Geometry of Nonholonomically Constrained Systems,, World Scientific, (2010). Google Scholar

[8]

V. Dragović and B. Gajić, The Wagner curvature tensor in nonholonomic mechanics,, Regul. Chaotic Dyn., 8 (2003), 105. doi: 10.1070/RD2003v008n01ABEH000229. Google Scholar

[9]

K. Ehlers, Geometric equivalence on nonholonomic three-manifolds,, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, (2003), 246. Google Scholar

[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, 232 (2005), 75. doi: 10.1007/0-8176-4419-9_4. Google Scholar

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Y. Fedorov and L. García-Naranjo, The hydrodynamic Chaplygin sleigh,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/43/434013. Google Scholar

[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. doi: 10.1007/s00332-004-0603-3. Google Scholar

[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. doi: 10.1088/0951-7715/22/9/009. Google Scholar

[14]

B. Jovanović, Geometry and integrability of Euler-Poincaré-Suslov equations,, Nonlinearity, 14 (2001), 1555. doi: 10.1088/0951-7715/14/6/308. Google Scholar

[15]

J. Koiller, P. Rodrigues and P. Pitanga, Non-holonomic connections following Élie Cartan,, An. Acad. Bras. Cienc., 73 (2001), 165. doi: 10.1590/S0001-37652001000200003. Google Scholar

[16]

V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras (in Russian),, Funkt. Anal. Prilozh., 22 (1988), 69. doi: 10.1007/BF01077727. Google Scholar

[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. doi: 10.1023/A:1022382202778. Google Scholar

[18]

O. Krupková, Geometric mechanics on nonholonomic submanifolds,, Commun. Math., 18 (2010), 51. Google Scholar

[19]

B. Langerock, Nonholonomic mechanics and connections over a bundle map,, J. Phys. A: Math. Gen., 34 (2001). doi: 10.1088/0305-4470/34/44/102. Google Scholar

[20]

A. Lewis, Affine connections and distributions with applications to nonholonomic mechanics,, Rep. Math. Phys., 42 (1998), 135. doi: 10.1016/S0034-4877(98)80008-6. Google Scholar

[21]

A. Lewis, Simple mechanical control systems with constraints,, IEEE Trans. Automat. Control, 45 (2000), 1420. doi: 10.1109/9.871752. Google Scholar

[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), (1999), 299. Google Scholar

[23]

G. Mubarakzyanov, On solvable Lie algebras (in Russian),, Izv. Vysš. Učehn. Zaved. Matematika, 32 (1963), 114. Google Scholar

[24]

J. Neimark and N. Fufaev, Dynamics of Nonholonomic Systems,, American Mathematical Society, (1972). Google Scholar

[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), (2013), 123. doi: 10.1007/978-3-642-30232-9_7. Google Scholar

[26]

M. Postnikov, Geometry VI: Riemannian Geometry,, Springer, (2001). Google Scholar

[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). doi: 10.1088/1751-8113/45/25/255202. Google Scholar

[28]

W. Sarlet, A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems,, Extracta Math., 11 (1996), 202. Google Scholar

[29]

G. Suslov, Theoretical Mechanics (in Russian),, Gostekhizdat, (1946). Google Scholar

[30]

M. Swaczyna, Several examples of nonholonomic mechanical systems,, Commun. Math., 19 (2011), 27. Google Scholar

[31]

M. Swaczyna and P. Volný, Uniform projectile motion: Dynamics, symmetries and conservation laws,, Rep. Math. Phys., 73 (2014), 177. doi: 10.1016/S0034-4877(14)60039-2. Google Scholar

[32]

J. Tavares, About Cartan geometrization of non-holonomic mechanics,, J. Geom. Phys., 45 (2003), 1. doi: 10.1016/S0393-0440(02)00118-3. Google Scholar

[33]

A. Vershik, Classical and non-classical dynamics with constraints,, in Global Analysis. Studies and Applications I (eds. Y. Borisovich and Y. Gliklikh), 1108 (1984), 278. doi: 10.1007/BFb0099563. Google Scholar

[34]

A. Vershik and L. Faddeev, Differential geometry and Lagrangian mechanics with constraints,, Sov. Phys. Dokl., 17 (1972), 34. Google Scholar

[35]

A. Vershik and V. Gershkovich, Nonholonomic problems and the theory of distributions,, Acta Appl. Math., 12 (1988), 181. doi: 10.1007/BF00047498. Google Scholar

[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), (1994), 1. Google Scholar

[37]

A. Veselov and L. Veselova, Integrable nonholonomic systems on Lie groups,, Math. Notes, 44 (1988), 810. doi: 10.1007/BF01158420. Google Scholar

show all references

References:
[1]

A. Agrachev and D. Barilari, Sub-Riemannian structures on 3D Lie groups,, J. Dyn. Control Syst., 18 (2012), 21. doi: 10.1007/s10883-012-9133-8. Google Scholar

[2]

A. Bloch, Nonholonomic Mechanics and Control,, Springer, (2003). doi: 10.1007/b97376. Google Scholar

[3]

E. Cartan, On the geometric representation of nonholonomic mechanical systems (in French),, in Proceedings of the International Congress of Mathematicians, (1928), 253. Google Scholar

[4]

M. Čech and J. Musilová, Symmetries and currents in nonholonomic mechanics,, Commun. Math., 22 (2014), 159. Google Scholar

[5]

S. Chaplygin, On the theory of motion of nonholonomic systems. The theorem of the Reducing Multiplier (in Russian),, Math. Sbornik, XXVIII (1911), 303. Google Scholar

[6]

J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Springer, (2002). doi: 10.1007/b84020. Google Scholar

[7]

R. Cushman, H. Duistermaat and J. Śniatycki, Geometry of Nonholonomically Constrained Systems,, World Scientific, (2010). Google Scholar

[8]

V. Dragović and B. Gajić, The Wagner curvature tensor in nonholonomic mechanics,, Regul. Chaotic Dyn., 8 (2003), 105. doi: 10.1070/RD2003v008n01ABEH000229. Google Scholar

[9]

K. Ehlers, Geometric equivalence on nonholonomic three-manifolds,, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, (2003), 246. Google Scholar

[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, 232 (2005), 75. doi: 10.1007/0-8176-4419-9_4. Google Scholar

[11]

Y. Fedorov and L. García-Naranjo, The hydrodynamic Chaplygin sleigh,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/43/434013. Google Scholar

[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. doi: 10.1007/s00332-004-0603-3. Google Scholar

[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. doi: 10.1088/0951-7715/22/9/009. Google Scholar

[14]

B. Jovanović, Geometry and integrability of Euler-Poincaré-Suslov equations,, Nonlinearity, 14 (2001), 1555. doi: 10.1088/0951-7715/14/6/308. Google Scholar

[15]

J. Koiller, P. Rodrigues and P. Pitanga, Non-holonomic connections following Élie Cartan,, An. Acad. Bras. Cienc., 73 (2001), 165. doi: 10.1590/S0001-37652001000200003. Google Scholar

[16]

V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras (in Russian),, Funkt. Anal. Prilozh., 22 (1988), 69. doi: 10.1007/BF01077727. Google Scholar

[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. doi: 10.1023/A:1022382202778. Google Scholar

[18]

O. Krupková, Geometric mechanics on nonholonomic submanifolds,, Commun. Math., 18 (2010), 51. Google Scholar

[19]

B. Langerock, Nonholonomic mechanics and connections over a bundle map,, J. Phys. A: Math. Gen., 34 (2001). doi: 10.1088/0305-4470/34/44/102. Google Scholar

[20]

A. Lewis, Affine connections and distributions with applications to nonholonomic mechanics,, Rep. Math. Phys., 42 (1998), 135. doi: 10.1016/S0034-4877(98)80008-6. Google Scholar

[21]

A. Lewis, Simple mechanical control systems with constraints,, IEEE Trans. Automat. Control, 45 (2000), 1420. doi: 10.1109/9.871752. Google Scholar

[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), (1999), 299. Google Scholar

[23]

G. Mubarakzyanov, On solvable Lie algebras (in Russian),, Izv. Vysš. Učehn. Zaved. Matematika, 32 (1963), 114. Google Scholar

[24]

J. Neimark and N. Fufaev, Dynamics of Nonholonomic Systems,, American Mathematical Society, (1972). Google Scholar

[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), (2013), 123. doi: 10.1007/978-3-642-30232-9_7. Google Scholar

[26]

M. Postnikov, Geometry VI: Riemannian Geometry,, Springer, (2001). Google Scholar

[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). doi: 10.1088/1751-8113/45/25/255202. Google Scholar

[28]

W. Sarlet, A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems,, Extracta Math., 11 (1996), 202. Google Scholar

[29]

G. Suslov, Theoretical Mechanics (in Russian),, Gostekhizdat, (1946). Google Scholar

[30]

M. Swaczyna, Several examples of nonholonomic mechanical systems,, Commun. Math., 19 (2011), 27. Google Scholar

[31]

M. Swaczyna and P. Volný, Uniform projectile motion: Dynamics, symmetries and conservation laws,, Rep. Math. Phys., 73 (2014), 177. doi: 10.1016/S0034-4877(14)60039-2. Google Scholar

[32]

J. Tavares, About Cartan geometrization of non-holonomic mechanics,, J. Geom. Phys., 45 (2003), 1. doi: 10.1016/S0393-0440(02)00118-3. Google Scholar

[33]

A. Vershik, Classical and non-classical dynamics with constraints,, in Global Analysis. Studies and Applications I (eds. Y. Borisovich and Y. Gliklikh), 1108 (1984), 278. doi: 10.1007/BFb0099563. Google Scholar

[34]

A. Vershik and L. Faddeev, Differential geometry and Lagrangian mechanics with constraints,, Sov. Phys. Dokl., 17 (1972), 34. Google Scholar

[35]

A. Vershik and V. Gershkovich, Nonholonomic problems and the theory of distributions,, Acta Appl. Math., 12 (1988), 181. doi: 10.1007/BF00047498. Google Scholar

[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), (1994), 1. Google Scholar

[37]

A. Veselov and L. Veselova, Integrable nonholonomic systems on Lie groups,, Math. Notes, 44 (1988), 810. doi: 10.1007/BF01158420. Google Scholar

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