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December  2020, 12(4): 607-639. doi: 10.3934/jgm.2020029

Lagrangian reduction of nonholonomic discrete mechanical systems by stages

Javier Fernández 1, , Cora Tori 2, and  Marcela Zuccalli 3,

1. 

Instituto Balseiro, Universidad Nacional de Cuyo – C.N.E.A., Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP, Argentina
2. 

Depto. de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata, Calle 116 entre 47 y 48, 2$ ^{\underline{o}} $ piso, La Plata, Buenos Aires, 1900, Argentina, Centro de Matemática de La Plata (CMaLP)
3. 

Depto. de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calles 50 y 115, La Plata, Buenos Aires, 1900, Argentina, Centro de Matemática de La Plata (CMaLP)

Received  July 2019 Revised  July 2020 Published  November 2020

Fund Project: This research was partially supported by grants from Universidad Nacional de Cuyo (#06/C567 and #06/C574) and Universidad Nacional de La Plata

In this work we introduce a category $ \mathfrak{L D P}_{d} $ of discrete-time dynamical systems, that we call discrete Lagrange–D'Alembert–Poincaré systems, and study some of its elementary properties. Examples of objects of $ \mathfrak{L D P}_{d} $ are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincaré systems. We also introduce a notion of symmetry group for objects of $ \mathfrak{L D P}_{d} $ and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange–Poincaré systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in $ \mathfrak{L D P}_{d} $ to the system obtained by a one-stage reduction by the full symmetry group.

Keywords: Geometric mechanics, discrete nonholonomic mechanical systems, symmetry and reduction.
Mathematics Subject Classification: Primary: 37J06, 70G45; Secondary: 70G75.
Citation: Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co. Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.   Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, vol. 24, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[4]

M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236 (2003), 223-250.  doi: 10.1007/s00220-003-0797-5.  Google Scholar

[5]

H. Cendra and V. A. Díaz, Lagrange-d'Alembert-Poincaré equations by several stages, J. Geom. Mech., 10 (2018), 1-41.  doi: 10.3934/jgm.2018001.  Google Scholar

[6]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction and nonholonomic systems, in Mathematics Unlimited—2001 and Beyond, Springer, Berlin, 2001,221–273.  Google Scholar

[7]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), 108 pp. doi: 10.1090/memo/0722.  Google Scholar

[8]

J. Cortés and S. Martínez, Non-holonomic integrators, Nonlinearity, 14 (2001), 1365-1392.  doi: 10.1088/0951-7715/14/5/322.  Google Scholar

[9]

J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, vol. 1793, Springer-Verlag, Berlin, 2002. doi: 10.1007/b84020.  Google Scholar

[10]

D. M. de Diego and R. S. M. de Almagro, Variational order for forced Lagrangian systems, Nonlinearity, 31 (2018), 3814-3846.  doi: 10.1088/1361-6544/aac5a6.  Google Scholar

[11]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241.  doi: 10.1088/0951-7715/18/5/017.  Google Scholar

[12]

J. Fernández, S. Graiff Zurita and S. Grillo, Error analysis of forced discrete mechanical systems, work-in-progress, 2020. Google Scholar

[13]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, J. Geom. Mech., 2 (2010), 69-111.  doi: 10.3934/jgm.2010.2.69.  Google Scholar

[14]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of discrete mechanical systems by stages, J. Geom. Mech., 8 (2016), 35-70.  doi: 10.3934/jgm.2016.8.35.  Google Scholar

[15]

J. Fernández and M. Zuccalli, A geometric approach to discrete connections on principal bundles, J. Geom. Mech., 5 (2013), 433-444.  doi: 10.3934/jgm.2013.5.433.  Google Scholar

[16]

L. C. García-Naranjo and F. Jiménez, The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators, Discrete Contin. Dyn. Syst., 37 (2017), 4249-4275.  doi: 10.3934/dcds.2017182.  Google Scholar

[17]

H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley Publishing Co., Reading, Mass., 1980  Google Scholar

[18]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, 2nd edition, Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006  Google Scholar

[19]

D. Iglesias, J. C. Marrero, D. M. de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 221-276.  doi: 10.1007/s00332-007-9012-8.  Google Scholar

[20]

D. Iglesias, J. C. Marrero, D. Martín de Diego, E. Martínez and E. Padrón, Reduction of symplectic Lie algebroids by a Lie subalgebroid and a symmetry Lie group, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 049, 28 pp. Google Scholar

[21]

S. M. Jalnapurkar, M. Leok, J. E. Marsden and M. West, Discrete Routh reduction, J. Phys. A, 39 (2006), 5521-5544.  doi: 10.1088/0305-4470/39/19/S12.  Google Scholar

[22]

J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21752-9.  Google Scholar

[23]

J. C. Marrero, D. 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.  Google Scholar

[24]

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

[25]

J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[26]

J. E. Marsden, G. Misiołek, J.-P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages, Lecture Notes in Mathematics, vol. 1913, Springer, Berlin, 2007.  Google Scholar

[27]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.  doi: 10.1007/s00332-005-0698-1.  Google Scholar

[28]

K. R. Meyer, Symmetries and integrals in mechanics, in Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973,259–272.  Google Scholar

[29]

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.  Google Scholar

[30]

S. Smale, Topology and mechanics. Ⅰ, Invent. Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar

[31]

S. Smale, Topology and mechanics. Ⅱ. The planar $n$-body problem, Invent. Math., 11 (1970), 45-64.  doi: 10.1007/BF01389805.  Google Scholar

[32]

G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moscow, 1946. Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co. Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.   Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, vol. 24, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[4]

M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236 (2003), 223-250.  doi: 10.1007/s00220-003-0797-5.  Google Scholar

[5]

H. Cendra and V. A. Díaz, Lagrange-d'Alembert-Poincaré equations by several stages, J. Geom. Mech., 10 (2018), 1-41.  doi: 10.3934/jgm.2018001.  Google Scholar

[6]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction and nonholonomic systems, in Mathematics Unlimited—2001 and Beyond, Springer, Berlin, 2001,221–273.  Google Scholar

[7]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), 108 pp. doi: 10.1090/memo/0722.  Google Scholar

[8]

J. Cortés and S. Martínez, Non-holonomic integrators, Nonlinearity, 14 (2001), 1365-1392.  doi: 10.1088/0951-7715/14/5/322.  Google Scholar

[9]

J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, vol. 1793, Springer-Verlag, Berlin, 2002. doi: 10.1007/b84020.  Google Scholar

[10]

D. M. de Diego and R. S. M. de Almagro, Variational order for forced Lagrangian systems, Nonlinearity, 31 (2018), 3814-3846.  doi: 10.1088/1361-6544/aac5a6.  Google Scholar

[11]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241.  doi: 10.1088/0951-7715/18/5/017.  Google Scholar

[12]

J. Fernández, S. Graiff Zurita and S. Grillo, Error analysis of forced discrete mechanical systems, work-in-progress, 2020. Google Scholar

[13]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, J. Geom. Mech., 2 (2010), 69-111.  doi: 10.3934/jgm.2010.2.69.  Google Scholar

[14]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of discrete mechanical systems by stages, J. Geom. Mech., 8 (2016), 35-70.  doi: 10.3934/jgm.2016.8.35.  Google Scholar

[15]

J. Fernández and M. Zuccalli, A geometric approach to discrete connections on principal bundles, J. Geom. Mech., 5 (2013), 433-444.  doi: 10.3934/jgm.2013.5.433.  Google Scholar

[16]

L. C. García-Naranjo and F. Jiménez, The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators, Discrete Contin. Dyn. Syst., 37 (2017), 4249-4275.  doi: 10.3934/dcds.2017182.  Google Scholar

[17]

H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley Publishing Co., Reading, Mass., 1980  Google Scholar

[18]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, 2nd edition, Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006  Google Scholar

[19]

D. Iglesias, J. C. Marrero, D. M. de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 221-276.  doi: 10.1007/s00332-007-9012-8.  Google Scholar

[20]

D. Iglesias, J. C. Marrero, D. Martín de Diego, E. Martínez and E. Padrón, Reduction of symplectic Lie algebroids by a Lie subalgebroid and a symmetry Lie group, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 049, 28 pp. Google Scholar

[21]

S. M. Jalnapurkar, M. Leok, J. E. Marsden and M. West, Discrete Routh reduction, J. Phys. A, 39 (2006), 5521-5544.  doi: 10.1088/0305-4470/39/19/S12.  Google Scholar

[22]

J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21752-9.  Google Scholar

[23]

J. C. Marrero, D. 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.  Google Scholar

[24]

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

[25]

J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[26]

J. E. Marsden, G. Misiołek, J.-P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages, Lecture Notes in Mathematics, vol. 1913, Springer, Berlin, 2007.  Google Scholar

[27]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.  doi: 10.1007/s00332-005-0698-1.  Google Scholar

[28]

K. R. Meyer, Symmetries and integrals in mechanics, in Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973,259–272.  Google Scholar

[29]

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.  Google Scholar

[30]

S. Smale, Topology and mechanics. Ⅰ, Invent. Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar

[31]

S. Smale, Topology and mechanics. Ⅱ. The planar $n$-body problem, Invent. Math., 11 (1970), 45-64.  doi: 10.1007/BF01389805.  Google Scholar

[32]

G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moscow, 1946. Google Scholar

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