November  2016, 36(11): 6023-6064. doi: 10.3934/dcds.2016064

Second-order variational problems on Lie groupoids and optimal control applications

1. 

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109

2. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Calle Nicolás Cabrera 15, Campus UAM, Cantoblanco, Madrid, 28049, Spain

Received  June 2015 Revised  May 2016 Published  August 2016

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in a Lagrangian submanifold. We also study the theory of reduction by symmetries and the corresponding Noether theorem.
Citation: Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
References:
[1]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems, from the point of view of lie algebroids,, Int. J. Geom. Methods Mod. Phys., (2014).  doi: 10.1142/S0219887814500388.  Google Scholar

[2]

R. Benito, de León M and D. Martín de Diego, Higher-order discrete lagrangian mechanics,, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421.  doi: 10.1142/S0219887806001235.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (2003).  doi: 10.1007/b97376.  Google Scholar

[4]

A. M. Bloch, L. Colombo, R. Gupta and D. Martín de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems,, Analysis and geometry in control theory and its applications, (2015).  doi: 10.1007/978-3-319-06917-3_2.  Google Scholar

[5]

A. M. Bloch and P. E. Crouch, On the equivalence of higher order variational problems and optimal control problems,, Proceedings of 35rd IEEE Conference on Decision and Control, (1996), 1648.  doi: 10.1109/CDC.1996.572780.  Google Scholar

[6]

A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric structure-preserving optimal control of the rigid body,, Journal of Dynamical and Control Systems, 15 (2009), 307.  doi: 10.1007/s10883-009-9071-2.  Google Scholar

[7]

AI. Bobenko and YB. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations and semidirect products,, Lett. Math. Phys., 49 (1999).  doi: 10.1023/A:1007654605901.  Google Scholar

[8]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups,, Foundations of Computational Mathematics, 9 (2009), 197.  doi: 10.1007/s10208-008-9030-4.  Google Scholar

[9]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles,, J. Phys. A, 48 (2015).  doi: 10.1088/1751-8113/48/20/205203.  Google Scholar

[10]

A. J. Bruce, K. Grabowska, J. Grabowski and P. Urbanski, New Developments In Geometric Mechanics,, To appear in Banach Center Publications. Preprint available at , (2015).   Google Scholar

[11]

F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005).  doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[12]

C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups,, Proc. R. Soc. A., 469 (2013).  doi: 10.1098/rspa.2013.0249.  Google Scholar

[13]

J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393.  doi: 10.1080/00207177008905922.  Google Scholar

[14]

M. Camarinha, P. Crouch and F. Silva-Leite, Splines of class $C^k$ on non-Euclidean spaces,, IMA J. Math. Control Info., 12 (1995), 399.  doi: 10.1093/imamci/12.4.399.  Google Scholar

[15]

D. Chang and J. Marsden, Asymptotic stabilization of the heavy top using controlled Lagrangians,, Proceedings of the 39th IEEE Conference on Decision and Control, (2000), 269.  doi: 10.1109/CDC.2000.912771.  Google Scholar

[16]

L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems,, Ph.D Thesis, (2014).   Google Scholar

[17]

L. Colombo, S. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control,, Preprint, (2014).  doi: 10.1007/s00332-016-9314-9.  Google Scholar

[18]

L. Colombo and D. Martín de Diego, Second-order constrained variational problems on Lie algebroids,, Preprint, (2016).   Google Scholar

[19]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach,, Journal Mathematical Physics, 51 (2010).  doi: 10.1063/1.3456158.  Google Scholar

[20]

L. Colombo and D. Martín de Diego, Quasivelocities and Optimal Control of Underactuated Mechanical Systems, Geometry and Physics: XVIII Fall Workshop on Geometry and Physics,, AIP Conference Proceedings, 1260 (2010), 133.   Google Scholar

[21]

L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications,, J. Geom. Mech., 6 (2014), 451.  doi: 10.3934/jgm.2014.6.451.  Google Scholar

[22]

L. Colombo, F. Jimenez and D. Martín de Diego, Discrete second-order Euler-Poincaré equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012).  doi: 10.1142/S0219887812500375.  Google Scholar

[23]

J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids,, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 213.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[24]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, SIAM J. Control Optim., 41 (2002), 1389.   Google Scholar

[25]

A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques,, Pub. Dep. Math. Lyon, 2/A (1987), 1.   Google Scholar

[26]

P. Crouch and P. Silva Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces,, J. Dynam. Control Systems, 1 (1995), 177.  doi: 10.1007/BF02254638.  Google Scholar

[27]

F. Gay-Balmaz, D. D Holm, D. Meier, T. Ratiu and F. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413.  doi: 10.1007/s00220-011-1313-y.  Google Scholar

[28]

F. Gay-Balmaz, D. D. Holm, D. Meier, T. Ratiu, F. Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553.  doi: 10.1007/s00332-012-9137-2.  Google Scholar

[29]

F. Gay-Balmaz, D. D. Holm and T. Ratiu, Higher-order Lagrange-Poincaré and Hamilton-Poincaré reductions,, Bulletin of the Brazilian Math. Soc., 42 (2011), 579.  doi: 10.1007/s00574-011-0030-7.  Google Scholar

[30]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,, Phys. Lett. A, 133 (1988), 134.  doi: 10.1016/0375-9601(88)90773-6.  Google Scholar

[31]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, 31 (2002).  doi: 10.1007/978-3-662-05018-7.  Google Scholar

[32]

P. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids,, J. Algebra, 129 (1990), 194.  doi: 10.1016/0021-8693(90)90246-K.  Google Scholar

[33]

D. D. Holm, T. Schamah and C. Stoica, Geometry Mechanics and Symmetry,, Oxford Text in Applied Mathematics, (2009).   Google Scholar

[34]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martinez, Discrete nonholonomic Lagrangian systems on Lie groupoids,, Journal of Nonlinear Science, 18 (2008), 221.  doi: 10.1007/s00332-007-9012-8.  Google Scholar

[35]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids,, Dyn. Syst., 23 (2008), 351.  doi: 10.1080/14689360802294220.  Google Scholar

[36]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Padrón, Discrete dynamics in implicit form,, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1117.  doi: 10.3934/dcds.2013.33.1117.  Google Scholar

[37]

A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods,, Acta Numerica, (2005).  doi: 10.1017/S0962492900002154.  Google Scholar

[38]

M. Jóźwikowski and M. Rotkiewicz, Bundle-theoretic methods for higher-order variational calculus,, J. Geom. Mech. 6 (2014), 6 (2014), 99.  doi: 10.3934/jgm.2014.6.99.  Google Scholar

[39]

M. Kobilarov and J. Marsden, Discrete geometric optimal control on Lie groups,, to appear in IEEE Transactions on Robotics, (2010).  doi: 10.1109/TRO.2011.2139130.  Google Scholar

[40]

M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles,, PhD thesis, (2008).   Google Scholar

[41]

T. Lee, M. Leok and N. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465.  doi: 10.1007/s10883-008-9047-7.  Google Scholar

[42]

T. Lee, N. H. McClamroch and M. Leok, Attitude maneuvers of a rigid spacecraft in a circular orbit,, In American Control Conference, (2006), 1742.   Google Scholar

[43]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985).   Google Scholar

[44]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A, (2005).   Google Scholar

[45]

L. Machado, F. Silva Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds,, J. Dyn. Control Syst., 16 (2010), 121.  doi: 10.1007/s10883-010-9080-1.  Google Scholar

[46]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical society Lecture Notes, 213 (2005).  doi: 10.1017/CBO9781107325883.  Google Scholar

[47]

J. C. Marrero, E. Martínez and D. Martín de Diego, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids,, Nonlinearity, 19 (2006), 1313.  doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[48]

J. C. Marrero, E. Martínez and D Martín de Diego, The local description of discrete mechanics,, Geometry, (2015), 285.  doi: 10.1007/978-1-4939-2441-7_13.  Google Scholar

[49]

J. C. Marrero, D. Martín de Diego and A. Stern, Symplectic groupoids and discrete constrained Lagrangian mechanics,, Discrete and Continuous Mechanical Systems, 35 (2015), 367.   Google Scholar

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E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356.  doi: 10.1051/cocv:2007056.  Google Scholar

[51]

E. Martínez, Geometric formulation of mechanics on Lie algebroids,, In Proceedings of the VIII Fall Workshop on Geometry and Physics, 2 (1999), 209.   Google Scholar

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show all references

References:
[1]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems, from the point of view of lie algebroids,, Int. J. Geom. Methods Mod. Phys., (2014).  doi: 10.1142/S0219887814500388.  Google Scholar

[2]

R. Benito, de León M and D. Martín de Diego, Higher-order discrete lagrangian mechanics,, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421.  doi: 10.1142/S0219887806001235.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (2003).  doi: 10.1007/b97376.  Google Scholar

[4]

A. M. Bloch, L. Colombo, R. Gupta and D. Martín de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems,, Analysis and geometry in control theory and its applications, (2015).  doi: 10.1007/978-3-319-06917-3_2.  Google Scholar

[5]

A. M. Bloch and P. E. Crouch, On the equivalence of higher order variational problems and optimal control problems,, Proceedings of 35rd IEEE Conference on Decision and Control, (1996), 1648.  doi: 10.1109/CDC.1996.572780.  Google Scholar

[6]

A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric structure-preserving optimal control of the rigid body,, Journal of Dynamical and Control Systems, 15 (2009), 307.  doi: 10.1007/s10883-009-9071-2.  Google Scholar

[7]

AI. Bobenko and YB. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations and semidirect products,, Lett. Math. Phys., 49 (1999).  doi: 10.1023/A:1007654605901.  Google Scholar

[8]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups,, Foundations of Computational Mathematics, 9 (2009), 197.  doi: 10.1007/s10208-008-9030-4.  Google Scholar

[9]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles,, J. Phys. A, 48 (2015).  doi: 10.1088/1751-8113/48/20/205203.  Google Scholar

[10]

A. J. Bruce, K. Grabowska, J. Grabowski and P. Urbanski, New Developments In Geometric Mechanics,, To appear in Banach Center Publications. Preprint available at , (2015).   Google Scholar

[11]

F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005).  doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[12]

C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups,, Proc. R. Soc. A., 469 (2013).  doi: 10.1098/rspa.2013.0249.  Google Scholar

[13]

J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393.  doi: 10.1080/00207177008905922.  Google Scholar

[14]

M. Camarinha, P. Crouch and F. Silva-Leite, Splines of class $C^k$ on non-Euclidean spaces,, IMA J. Math. Control Info., 12 (1995), 399.  doi: 10.1093/imamci/12.4.399.  Google Scholar

[15]

D. Chang and J. Marsden, Asymptotic stabilization of the heavy top using controlled Lagrangians,, Proceedings of the 39th IEEE Conference on Decision and Control, (2000), 269.  doi: 10.1109/CDC.2000.912771.  Google Scholar

[16]

L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems,, Ph.D Thesis, (2014).   Google Scholar

[17]

L. Colombo, S. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control,, Preprint, (2014).  doi: 10.1007/s00332-016-9314-9.  Google Scholar

[18]

L. Colombo and D. Martín de Diego, Second-order constrained variational problems on Lie algebroids,, Preprint, (2016).   Google Scholar

[19]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach,, Journal Mathematical Physics, 51 (2010).  doi: 10.1063/1.3456158.  Google Scholar

[20]

L. Colombo and D. Martín de Diego, Quasivelocities and Optimal Control of Underactuated Mechanical Systems, Geometry and Physics: XVIII Fall Workshop on Geometry and Physics,, AIP Conference Proceedings, 1260 (2010), 133.   Google Scholar

[21]

L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications,, J. Geom. Mech., 6 (2014), 451.  doi: 10.3934/jgm.2014.6.451.  Google Scholar

[22]

L. Colombo, F. Jimenez and D. Martín de Diego, Discrete second-order Euler-Poincaré equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012).  doi: 10.1142/S0219887812500375.  Google Scholar

[23]

J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids,, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 213.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[24]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, SIAM J. Control Optim., 41 (2002), 1389.   Google Scholar

[25]

A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques,, Pub. Dep. Math. Lyon, 2/A (1987), 1.   Google Scholar

[26]

P. Crouch and P. Silva Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces,, J. Dynam. Control Systems, 1 (1995), 177.  doi: 10.1007/BF02254638.  Google Scholar

[27]

F. Gay-Balmaz, D. D Holm, D. Meier, T. Ratiu and F. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413.  doi: 10.1007/s00220-011-1313-y.  Google Scholar

[28]

F. Gay-Balmaz, D. D. Holm, D. Meier, T. Ratiu, F. Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553.  doi: 10.1007/s00332-012-9137-2.  Google Scholar

[29]

F. Gay-Balmaz, D. D. Holm and T. Ratiu, Higher-order Lagrange-Poincaré and Hamilton-Poincaré reductions,, Bulletin of the Brazilian Math. Soc., 42 (2011), 579.  doi: 10.1007/s00574-011-0030-7.  Google Scholar

[30]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,, Phys. Lett. A, 133 (1988), 134.  doi: 10.1016/0375-9601(88)90773-6.  Google Scholar

[31]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, 31 (2002).  doi: 10.1007/978-3-662-05018-7.  Google Scholar

[32]

P. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids,, J. Algebra, 129 (1990), 194.  doi: 10.1016/0021-8693(90)90246-K.  Google Scholar

[33]

D. D. Holm, T. Schamah and C. Stoica, Geometry Mechanics and Symmetry,, Oxford Text in Applied Mathematics, (2009).   Google Scholar

[34]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martinez, Discrete nonholonomic Lagrangian systems on Lie groupoids,, Journal of Nonlinear Science, 18 (2008), 221.  doi: 10.1007/s00332-007-9012-8.  Google Scholar

[35]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids,, Dyn. Syst., 23 (2008), 351.  doi: 10.1080/14689360802294220.  Google Scholar

[36]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Padrón, Discrete dynamics in implicit form,, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1117.  doi: 10.3934/dcds.2013.33.1117.  Google Scholar

[37]

A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods,, Acta Numerica, (2005).  doi: 10.1017/S0962492900002154.  Google Scholar

[38]

M. Jóźwikowski and M. Rotkiewicz, Bundle-theoretic methods for higher-order variational calculus,, J. Geom. Mech. 6 (2014), 6 (2014), 99.  doi: 10.3934/jgm.2014.6.99.  Google Scholar

[39]

M. Kobilarov and J. Marsden, Discrete geometric optimal control on Lie groups,, to appear in IEEE Transactions on Robotics, (2010).  doi: 10.1109/TRO.2011.2139130.  Google Scholar

[40]

M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles,, PhD thesis, (2008).   Google Scholar

[41]

T. Lee, M. Leok and N. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465.  doi: 10.1007/s10883-008-9047-7.  Google Scholar

[42]

T. Lee, N. H. McClamroch and M. Leok, Attitude maneuvers of a rigid spacecraft in a circular orbit,, In American Control Conference, (2006), 1742.   Google Scholar

[43]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985).   Google Scholar

[44]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A, (2005).   Google Scholar

[45]

L. Machado, F. Silva Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds,, J. Dyn. Control Syst., 16 (2010), 121.  doi: 10.1007/s10883-010-9080-1.  Google Scholar

[46]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical society Lecture Notes, 213 (2005).  doi: 10.1017/CBO9781107325883.  Google Scholar

[47]

J. C. Marrero, E. Martínez and D. Martín de Diego, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids,, Nonlinearity, 19 (2006), 1313.  doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[48]

J. C. Marrero, E. Martínez and D Martín de Diego, The local description of discrete mechanics,, Geometry, (2015), 285.  doi: 10.1007/978-1-4939-2441-7_13.  Google Scholar

[49]

J. C. Marrero, D. Martín de Diego and A. Stern, Symplectic groupoids and discrete constrained Lagrangian mechanics,, Discrete and Continuous Mechanical Systems, 35 (2015), 367.   Google Scholar

[50]

E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356.  doi: 10.1051/cocv:2007056.  Google Scholar

[51]

E. Martínez, Geometric formulation of mechanics on Lie algebroids,, In Proceedings of the VIII Fall Workshop on Geometry and Physics, 2 (1999), 209.   Google Scholar

[52]

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