January  2019, 39(1): 309-344. doi: 10.3934/dcds.2019013

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

1. 

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

2. 

Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Calle Nicolás Cabrera 13-15, Campus UAM, Madrid, 28049, Spain

3. 

Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kindom

Received  January 2018 Revised  July 2018 Published  October 2018

Fund Project: A. Bloch was supported by NSF grant DMS-1613819 and AFOSR grant FA 9550-18-0028. L. Colombo was supported by MINECO (Spain) grant MTM2016-76072-P. F. Jiménez was supported by the EPSRC project: "Fractional Variational Integration and Optimal Control"; ref: EP/P020402/1. We thank the reviewers for their valuable comments, that have helped to improve this work

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations.

Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.

Citation: Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
References:
[1]

R. Benito and D. Martín de Diego, Discrete Vakonomic Mechanics, Journal of Mathematical Physics, 46 (2005), 083521, 18pp. doi: 10.1063/1.2008214.

[2]

R. BenitoM. de León and D. Martín de Diego, Higher-order discrete lagrangian mechanics, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421-436. doi: 10.1142/S0219887806001235.

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Vol. 24 of Interdisciplinary Appl. Math. Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[4]

A. BlochL. ColomboR. 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, 11 (2015), 35-64. doi: 10.1007/978-3-319-06917-3_2.

[5]

A. M. Bloch and P. E. Crouch, Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems, Proceedings of 33rd IEEE Conference on Decision and Control, (1994), 2584-2590.

[6]

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), 1648-1653.

[7]

A. BlochP. CrouchN. Nordkvist and A. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups, J. Geom. Mech., 3 (2011), 197-223. doi: 10.3934/jgm.2011.3.197.

[8]

A. M. BlochI. I. HusseinM. Leok and A. K. Sanyal, Geometric structure-preserving optimal control of a rigid body, Journal of Dynamical and Control Systems, 15 (2009), 307-330. doi: 10.1007/s10883-009-9071-2.

[9]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42. doi: 10.1007/BF02101622.

[10]

M. BruverisD. EllisF. Gay-Balmaz and D. D. Holm, Un-reduction, Journal of Geometric Mechanics, 3 (2011), 363-387. doi: 10.3934/jgm.2011.3.363.

[11]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.

[12]

C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups, Proc. R. Soc. A., 469 (2013), 20130249.

[13]

J. A. Cadzow, Discrete Calculus of Variations, Int. J. Control, 11 (1970), 393-407.

[14]

M. CamarinhaF. Silva Leite and P. E. Crouch, Splines of class Ck on non-Euclidean spaces, IMA Journal of Mathematical Control & Information, 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.

[15]

M. CamarinhaF. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials, Differential Geometry and its Applications, 15 (2001), 107-135. doi: 10.1016/S0926-2245(01)00054-7.

[16]

C. M. Campos, O. Junge and S. Ober-Blobaum, Higher Order Variational Time Discretization of Optimal Control Problems, 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne, 2012.

[17]

C. M. CamposS. Ober-Blobaum and E. Trelat, High order variational integrators in the optimal control of mechanical systems, Discrete and Continuous Dynamical Systems - Series A, 359 (2015), 4193-4223. doi: 10.3934/dcds.2015.35.4193.

[18]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memories of the American Mathematical Society, 152 (2001), ⅹ+108 pp. doi: 10.1090/memo/0722.

[19]

L. Colombo, Second-order constrained variational problems on Lie algebroids: Applications to optimal control, Journal Geometric Mechanics, 9 (2017), 1-45. doi: 10.3934/jgm.2017001.

[20]

L. ColomboR. Gupta and A. Bloch, Higher-order constrained variational problems on principal bundles with applications to optimal control of underactuated systems, IFAC-PapersOnLine, 48 (2015), 87-92.

[21]

L. ColomboS. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control, Journal of Nonlinear Science, 26 (2016), 1615-1650. doi: 10.1007/s00332-016-9314-9.

[22]

L. ColomboF. Jiménez and D. Martín de Diego, Variational integrators for mechanical control systems with symmetries, Journal of Computational Dynamics, 2 (2015), 193-225. doi: 10.3934/jcd.2015003.

[23]

L. Colombo, F. Jiménez 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), 1250037, 20 pp. doi: 10.1142/S0219887812500375.

[24]

L. Colombo, D. Martín de Diego and M. Zuccalli, On variational integrators for optimal control of mechanical systems, RACSAM Rev. R. Acad. Cienc. Ser A. Mat, 2011.

[25]

L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications, J. Geom. Mech., 6 (2014), 451-478. doi: 10.3934/jgm.2014.6.451.

[26]

L. Colombo L, D. Martín de Diego and M. Zuccalli, Higher-order variational problems with constraints, Journal of Mathematical Physics., 54 (2013), 093507, 17pp. doi: 10.1063/1.4820817.

[27]

L. Colombo and D. Martín de Diego, Second order variational problems on Lie groupoids and optimal control applications, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 6023-6064. doi: 10.3934/dcds.2016064.

[28]

L. Colombo and P. D. Prieto Martínez, Regularity properties of fiber derivatives associated with higher-order mechanical systems, Journal of Mathematical Physics, 57(2016), 082901, 25pp. doi: 10.1063/1.4960822.

[29]

J CortésM de LeónD. Martín de Diego and S Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM Journal on Control and Optimization, 41 (2002), 1389-1412. doi: 10.1137/S036301290036817X.

[30]

A. FernándezP. García and C. Rodrigo, Variational integrators in discrete vakonomic mechanics, Rev. R. Acad. A, 106 (2012), 137-159. doi: 10.1007/s13398-011-0030-x.

[31]

J. FernándezC. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, Journal of Geometric Mechanics, 2 (2010), 69-111. doi: 10.3934/jgm.2010.2.69.

[32]

F. Gay-BalmazD. Holm and T. Ratiu, Higher order Lagrange-Poincaré, and Hamilton-Poincaré, reductions, Bulletin of the Brazialian Mathematical Society, 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.

[33]

F. Gay-BalmazD. D. HolmD. M. MeierT. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[34]

F. Gay-BalmazD. D. HolmD. M. MeierT. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems Ⅱ, J. Nonlin. Sci., 22 (2012), 553-597. doi: 10.1007/s00332-012-9137-2.

[35]

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

[36]

D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford University Press, 2009.

[37]

D. IglesiasJC MarreroD Mart n de Diego and D Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dynamical Systems, 23 (2008), 351-397. doi: 10.1080/14689360802294220.

[38]

F. Jiménez, M. de León and D. Martín de Diego, Hamiltonian dynamics and constrained variational calculus: Continuous and discrete settings, J. Phys A, 45 (2012), 205204, 29pp. doi: 10.1088/1751-8113/45/20/205204.

[39]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Mathematical Studies 112, North-Holland, Amsterdam, 1985.

[40]

T. LeeM. Leok and H. McClamroch, Optimal attitude control of a rigid body using geometrically exact computations on SO(3), Journal of Dynamical and Control Systems, 14 (2008), 465-487. doi: 10.1007/s10883-008-9047-7.

[41]

T. Lee, M. Leok and H. McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds: A Geometric Approach to Modeling and Analysis, Springer, Interaction of Mechanics and Mathematics, 2018. doi: 10.1007/978-3-319-56953-6.

[42]

M. Leok, J. E. Marsden and A. Weinstein, A Discrete Theory of Connections on Principal Bundles, Preprint, https://arXiv.org/abs/math/0508338, 2005.

[43]

M. Leok and T. Shingel, Prolongation-collocation variational integrators, IMA J. Numer. Anal., 32 (2012), 1194-1216. doi: 10.1093/imanum/drr042.

[44]

A. Lewis, Reduction of Simple Mechanical Systems, Mechanics and Symmetry Seminars, University of Warwick, https://mast.queensu.ca/~andrew/, 1997.

[45]

S. LeyendeckerS. Ober-BlobaumJ. Marsden and M. Ortiz, Discrete mechanics and optimal control for constrained systems, Optim. Control, Appl. Methods, 316 (2010), 505-528. doi: 10.1002/oca.912.

[46]

J. C. MarreroD. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348. doi: 10.1088/0951-7715/19/6/006.

[47]

J. C. MarreroD. Martín de Diego and A. Stern, Symplectic groupies and discrete constrained Lagrangian mechanics, Discrete and Continuous Mechanical Systems, Serie A, 35 (2015), 367-397. doi: 10.3934/dcds.2015.35.367.

[48]

J. E. Marsden and J. Scheurle, The Reduced Euler-Lagrange equations, Fields Institute Communications, 1 (1993), 139-164.

[49]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[50]

J. Marsden and J. Wendlandt, Mechanical Integrators Derived from a Discrete Variational Principle, Physica D, 106 (1997), 223-246. doi: 10.1016/S0167-2789(97)00051-1.

[51]

J. E. MarsdenS. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662. doi: 10.1088/0951-7715/12/6/314.

[52]

J. E. MarsdenS. Pekarsky and S. Shkoller, Symmetry reduction of discrete Lagrangian mechanics on Lie groups, Journal of Geometry and Physics, 36 (2000), 140-151. doi: 10.1016/S0393-0440(00)00018-8.

[53]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.

[54]

S. Ober-BlobaumO. Junge and J. E. Marsden, Discrete mechanics and optimal control: An analysis, ESAIM Control Optim. Calc. Var., 17 (2011), 322-352. doi: 10.1051/cocv/2010012.

[55]

S. Ober-Blobaum, Galerkin variational integrators and modified symplectic Runge-Kutta methods, IMA J. Numer. Anal., 37 (2017), 375-406. doi: 10.1093/imanum/drv062.

[56]

T. Ohsawa, Symmetry reduction of optimal control systems and principal connections, SIAM Journal on Control and Optimization, 51 (2013), 96-120. doi: 10.1137/110835219.

[57]

J. P. Ostrowski, Computing Reduced Equations for Robotic Systems with Constraints and Symmetries, IEEE Transactions on robotic and automation, 1999.

[58]

G. W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numer. Math., 113 (2009), 243-264. doi: 10.1007/s00211-009-0245-3.

show all references

References:
[1]

R. Benito and D. Martín de Diego, Discrete Vakonomic Mechanics, Journal of Mathematical Physics, 46 (2005), 083521, 18pp. doi: 10.1063/1.2008214.

[2]

R. BenitoM. de León and D. Martín de Diego, Higher-order discrete lagrangian mechanics, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421-436. doi: 10.1142/S0219887806001235.

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Vol. 24 of Interdisciplinary Appl. Math. Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[4]

A. BlochL. ColomboR. 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, 11 (2015), 35-64. doi: 10.1007/978-3-319-06917-3_2.

[5]

A. M. Bloch and P. E. Crouch, Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems, Proceedings of 33rd IEEE Conference on Decision and Control, (1994), 2584-2590.

[6]

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), 1648-1653.

[7]

A. BlochP. CrouchN. Nordkvist and A. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups, J. Geom. Mech., 3 (2011), 197-223. doi: 10.3934/jgm.2011.3.197.

[8]

A. M. BlochI. I. HusseinM. Leok and A. K. Sanyal, Geometric structure-preserving optimal control of a rigid body, Journal of Dynamical and Control Systems, 15 (2009), 307-330. doi: 10.1007/s10883-009-9071-2.

[9]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42. doi: 10.1007/BF02101622.

[10]

M. BruverisD. EllisF. Gay-Balmaz and D. D. Holm, Un-reduction, Journal of Geometric Mechanics, 3 (2011), 363-387. doi: 10.3934/jgm.2011.3.363.

[11]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.

[12]

C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups, Proc. R. Soc. A., 469 (2013), 20130249.

[13]

J. A. Cadzow, Discrete Calculus of Variations, Int. J. Control, 11 (1970), 393-407.

[14]

M. CamarinhaF. Silva Leite and P. E. Crouch, Splines of class Ck on non-Euclidean spaces, IMA Journal of Mathematical Control & Information, 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.

[15]

M. CamarinhaF. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials, Differential Geometry and its Applications, 15 (2001), 107-135. doi: 10.1016/S0926-2245(01)00054-7.

[16]

C. M. Campos, O. Junge and S. Ober-Blobaum, Higher Order Variational Time Discretization of Optimal Control Problems, 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne, 2012.

[17]

C. M. CamposS. Ober-Blobaum and E. Trelat, High order variational integrators in the optimal control of mechanical systems, Discrete and Continuous Dynamical Systems - Series A, 359 (2015), 4193-4223. doi: 10.3934/dcds.2015.35.4193.

[18]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memories of the American Mathematical Society, 152 (2001), ⅹ+108 pp. doi: 10.1090/memo/0722.

[19]

L. Colombo, Second-order constrained variational problems on Lie algebroids: Applications to optimal control, Journal Geometric Mechanics, 9 (2017), 1-45. doi: 10.3934/jgm.2017001.

[20]

L. ColomboR. Gupta and A. Bloch, Higher-order constrained variational problems on principal bundles with applications to optimal control of underactuated systems, IFAC-PapersOnLine, 48 (2015), 87-92.

[21]

L. ColomboS. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control, Journal of Nonlinear Science, 26 (2016), 1615-1650. doi: 10.1007/s00332-016-9314-9.

[22]

L. ColomboF. Jiménez and D. Martín de Diego, Variational integrators for mechanical control systems with symmetries, Journal of Computational Dynamics, 2 (2015), 193-225. doi: 10.3934/jcd.2015003.

[23]

L. Colombo, F. Jiménez 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), 1250037, 20 pp. doi: 10.1142/S0219887812500375.

[24]

L. Colombo, D. Martín de Diego and M. Zuccalli, On variational integrators for optimal control of mechanical systems, RACSAM Rev. R. Acad. Cienc. Ser A. Mat, 2011.

[25]

L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications, J. Geom. Mech., 6 (2014), 451-478. doi: 10.3934/jgm.2014.6.451.

[26]

L. Colombo L, D. Martín de Diego and M. Zuccalli, Higher-order variational problems with constraints, Journal of Mathematical Physics., 54 (2013), 093507, 17pp. doi: 10.1063/1.4820817.

[27]

L. Colombo and D. Martín de Diego, Second order variational problems on Lie groupoids and optimal control applications, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 6023-6064. doi: 10.3934/dcds.2016064.

[28]

L. Colombo and P. D. Prieto Martínez, Regularity properties of fiber derivatives associated with higher-order mechanical systems, Journal of Mathematical Physics, 57(2016), 082901, 25pp. doi: 10.1063/1.4960822.

[29]

J CortésM de LeónD. Martín de Diego and S Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM Journal on Control and Optimization, 41 (2002), 1389-1412. doi: 10.1137/S036301290036817X.

[30]

A. FernándezP. García and C. Rodrigo, Variational integrators in discrete vakonomic mechanics, Rev. R. Acad. A, 106 (2012), 137-159. doi: 10.1007/s13398-011-0030-x.

[31]

J. FernándezC. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, Journal of Geometric Mechanics, 2 (2010), 69-111. doi: 10.3934/jgm.2010.2.69.

[32]

F. Gay-BalmazD. Holm and T. Ratiu, Higher order Lagrange-Poincaré, and Hamilton-Poincaré, reductions, Bulletin of the Brazialian Mathematical Society, 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.

[33]

F. Gay-BalmazD. D. HolmD. M. MeierT. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[34]

F. Gay-BalmazD. D. HolmD. M. MeierT. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems Ⅱ, J. Nonlin. Sci., 22 (2012), 553-597. doi: 10.1007/s00332-012-9137-2.

[35]

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

[36]

D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford University Press, 2009.

[37]

D. IglesiasJC MarreroD Mart n de Diego and D Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dynamical Systems, 23 (2008), 351-397. doi: 10.1080/14689360802294220.

[38]

F. Jiménez, M. de León and D. Martín de Diego, Hamiltonian dynamics and constrained variational calculus: Continuous and discrete settings, J. Phys A, 45 (2012), 205204, 29pp. doi: 10.1088/1751-8113/45/20/205204.

[39]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Mathematical Studies 112, North-Holland, Amsterdam, 1985.

[40]

T. LeeM. Leok and H. McClamroch, Optimal attitude control of a rigid body using geometrically exact computations on SO(3), Journal of Dynamical and Control Systems, 14 (2008), 465-487. doi: 10.1007/s10883-008-9047-7.

[41]

T. Lee, M. Leok and H. McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds: A Geometric Approach to Modeling and Analysis, Springer, Interaction of Mechanics and Mathematics, 2018. doi: 10.1007/978-3-319-56953-6.

[42]

M. Leok, J. E. Marsden and A. Weinstein, A Discrete Theory of Connections on Principal Bundles, Preprint, https://arXiv.org/abs/math/0508338, 2005.

[43]

M. Leok and T. Shingel, Prolongation-collocation variational integrators, IMA J. Numer. Anal., 32 (2012), 1194-1216. doi: 10.1093/imanum/drr042.

[44]

A. Lewis, Reduction of Simple Mechanical Systems, Mechanics and Symmetry Seminars, University of Warwick, https://mast.queensu.ca/~andrew/, 1997.

[45]

S. LeyendeckerS. Ober-BlobaumJ. Marsden and M. Ortiz, Discrete mechanics and optimal control for constrained systems, Optim. Control, Appl. Methods, 316 (2010), 505-528. doi: 10.1002/oca.912.

[46]

J. C. MarreroD. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348. doi: 10.1088/0951-7715/19/6/006.

[47]

J. C. MarreroD. Martín de Diego and A. Stern, Symplectic groupies and discrete constrained Lagrangian mechanics, Discrete and Continuous Mechanical Systems, Serie A, 35 (2015), 367-397. doi: 10.3934/dcds.2015.35.367.

[48]

J. E. Marsden and J. Scheurle, The Reduced Euler-Lagrange equations, Fields Institute Communications, 1 (1993), 139-164.

[49]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[50]

J. Marsden and J. Wendlandt, Mechanical Integrators Derived from a Discrete Variational Principle, Physica D, 106 (1997), 223-246. doi: 10.1016/S0167-2789(97)00051-1.

[51]

J. E. MarsdenS. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662. doi: 10.1088/0951-7715/12/6/314.

[52]

J. E. MarsdenS. Pekarsky and S. Shkoller, Symmetry reduction of discrete Lagrangian mechanics on Lie groups, Journal of Geometry and Physics, 36 (2000), 140-151. doi: 10.1016/S0393-0440(00)00018-8.

[53]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.

[54]

S. Ober-BlobaumO. Junge and J. E. Marsden, Discrete mechanics and optimal control: An analysis, ESAIM Control Optim. Calc. Var., 17 (2011), 322-352. doi: 10.1051/cocv/2010012.

[55]

S. Ober-Blobaum, Galerkin variational integrators and modified symplectic Runge-Kutta methods, IMA J. Numer. Anal., 37 (2017), 375-406. doi: 10.1093/imanum/drv062.

[56]

T. Ohsawa, Symmetry reduction of optimal control systems and principal connections, SIAM Journal on Control and Optimization, 51 (2013), 96-120. doi: 10.1137/110835219.

[57]

J. P. Ostrowski, Computing Reduced Equations for Robotic Systems with Constraints and Symmetries, IEEE Transactions on robotic and automation, 1999.

[58]

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