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

Lagrangian reduction of nonholonomic discrete mechanical systems by stages

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  December 2020 Early access  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.

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.

[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. 

[3]

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

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[13]

J. FernándezC. 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.

[14]

J. FernándezC. 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.

[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.

[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.

[17]

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

[18]

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

[19]

D. IglesiasJ. C. MarreroD. 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.

[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.

[21]

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

[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.

[23]

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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

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.

[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. 

[3]

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

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[13]

J. FernándezC. 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.

[14]

J. FernándezC. 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.

[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.

[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.

[17]

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

[18]

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

[19]

D. IglesiasJ. C. MarreroD. 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.

[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.

[21]

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

[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.

[23]

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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

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