March  2016, 8(1): 35-70. doi: 10.3934/jgm.2016.8.35

Lagrangian reduction of discrete mechanical systems by stages

1. 

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

2. 

Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, 50 y 115, La Plata, Buenos Aires, 1900

Received  February 2015 Revised  November 2015 Published  February 2016

In this work we introduce a category of discrete Lagrange--Poincaré systems $\mathfrak{L}\mathfrak{P}_d$ and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete dynamical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in $\mathfrak{L}\mathfrak{P}_d$. We introduce a notion of symmetry group for objects of $\mathfrak{L}\mathfrak{P}_d$ as well as a reduction procedure that is closed in the category $\mathfrak{L}\mathfrak{P}_d$. Furthermore, under some conditions, we show that the reduction in two steps (first by a closed normal subgroup of the symmetry group and then by the residual symmetry group) is isomorphic in $\mathfrak{L}\mathfrak{P}_d$ to the reduction by the full symmetry group.
Citation: Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of discrete mechanical systems by stages. Journal of Geometric Mechanics, 2016, 8 (1) : 35-70. doi: 10.3934/jgm.2016.8.35
References:
[1]

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. doi: 10.5802/aif.233.

[2]

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.

[3]

H. Cendra and V. A. Díaz, Lagrange-d'Alembert-Poincaré equations by several stages, arXiv:1406.7271, 2014.

[4]

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.

[5]

________, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), x+108pp. doi: 10.1090/memo/0722.

[6]

V. A. Díaz, Reducción por etapas de sistemas noholónomos, Ph.D. thesis, Universidad Nacional del Sur, Bahía Blanca, Argentina, 2008.

[7]

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.

[8]

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.

[9]

V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974.

[10]

D. Husemoller, Fibre Bundles, Third ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4757-2261-1.

[11]

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, 28pp. doi: 10.3842/SIGMA.2007.049.

[12]

S. 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.

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1996.

[14]

J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Vol. 218, Springer-Verlag, New York, 2003.

[15]

M. Leok, J. E. Marsden and A. Weinstein, A discrete theory of connections on principal bundles, arXiv:math/0508338, 2005.

[16]

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.

[17]

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

[18]

J. E. 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.

[19]

J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and Tudor S. Ratiu, Hamiltonian Reduction by Stages, Lecture Notes in Mathematics, Vol. 1913, Springer, Berlin, 2007.

[20]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Second ed., Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.

[21]

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

[22]

K. Meyer, Symmetries and integrals in mechanics, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 259-272.

[23]

P. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/093.

[24]

E. Routh, Stability of a Given State of Motion, Macmillan and Co., London, 1877.

[25]

S. Smale, Topology and mechanics. I, Invent. Math., 10 (1970), 305-331.

[26]

________, Topology and mechanics. II. The planar $n$-body problem, Invent. Math., 11 (1970), 45-64.

show all references

References:
[1]

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. doi: 10.5802/aif.233.

[2]

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.

[3]

H. Cendra and V. A. Díaz, Lagrange-d'Alembert-Poincaré equations by several stages, arXiv:1406.7271, 2014.

[4]

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.

[5]

________, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), x+108pp. doi: 10.1090/memo/0722.

[6]

V. A. Díaz, Reducción por etapas de sistemas noholónomos, Ph.D. thesis, Universidad Nacional del Sur, Bahía Blanca, Argentina, 2008.

[7]

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.

[8]

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.

[9]

V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974.

[10]

D. Husemoller, Fibre Bundles, Third ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4757-2261-1.

[11]

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, 28pp. doi: 10.3842/SIGMA.2007.049.

[12]

S. 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.

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1996.

[14]

J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Vol. 218, Springer-Verlag, New York, 2003.

[15]

M. Leok, J. E. Marsden and A. Weinstein, A discrete theory of connections on principal bundles, arXiv:math/0508338, 2005.

[16]

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.

[17]

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

[18]

J. E. 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.

[19]

J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and Tudor S. Ratiu, Hamiltonian Reduction by Stages, Lecture Notes in Mathematics, Vol. 1913, Springer, Berlin, 2007.

[20]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Second ed., Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.

[21]

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

[22]

K. Meyer, Symmetries and integrals in mechanics, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 259-272.

[23]

P. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/093.

[24]

E. Routh, Stability of a Given State of Motion, Macmillan and Co., London, 1877.

[25]

S. Smale, Topology and mechanics. I, Invent. Math., 10 (1970), 305-331.

[26]

________, Topology and mechanics. II. The planar $n$-body problem, Invent. Math., 11 (1970), 45-64.

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