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. doi: 10.5802/aif.233. Google Scholar

[2]

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

[3]

H. Cendra and V. A. Díaz, Lagrange-d'Alembert-Poincaré equations by several stages,, , (2014). Google Scholar

[4]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems,, in Mathematics Unlimited-2001 and Beyond, (2001), 221. Google Scholar

[5]

________, Lagrangian reduction by stages,, Mem. Amer. Math. Soc., 152 (2001). doi: 10.1090/memo/0722. Google Scholar

[6]

V. A. Díaz, Reducción por etapas de sistemas noholónomos,, Ph.D. thesis, (2008). Google Scholar

[7]

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

[8]

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

[9]

V. Guillemin and A. Pollack, Differential Topology,, Prentice-Hall Inc., (1974). Google Scholar

[10]

D. Husemoller, Fibre Bundles,, Third ed., (1994). doi: 10.1007/978-1-4757-2261-1. Google Scholar

[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). doi: 10.3842/SIGMA.2007.049. Google Scholar

[12]

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

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Wiley Classics Library, (1996). Google Scholar

[14]

J. M. Lee, Introduction to Smooth Manifolds,, Graduate Texts in Mathematics, (2003). Google Scholar

[15]

M. Leok, J. E. Marsden and A. Weinstein, A discrete theory of connections on principal bundles,, , (2005). Google Scholar

[16]

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

[17]

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

[18]

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

[19]

J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and Tudor S. Ratiu, Hamiltonian Reduction by Stages,, Lecture Notes in Mathematics, (1913). Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, Second ed., (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar

[21]

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

[22]

K. Meyer, Symmetries and integrals in mechanics,, Dynamical systems (Proc. Sympos., (1971), 259. Google Scholar

[23]

P. Michor, Topics in Differential Geometry,, Graduate Studies in Mathematics, (2008). doi: 10.1090/gsm/093. Google Scholar

[24]

E. Routh, Stability of a Given State of Motion,, Macmillan and Co., (1877). Google Scholar

[25]

S. Smale, Topology and mechanics. I,, Invent. Math., 10 (1970), 305. Google Scholar

[26]

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

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. doi: 10.5802/aif.233. Google Scholar

[2]

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

[3]

H. Cendra and V. A. Díaz, Lagrange-d'Alembert-Poincaré equations by several stages,, , (2014). Google Scholar

[4]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems,, in Mathematics Unlimited-2001 and Beyond, (2001), 221. Google Scholar

[5]

________, Lagrangian reduction by stages,, Mem. Amer. Math. Soc., 152 (2001). doi: 10.1090/memo/0722. Google Scholar

[6]

V. A. Díaz, Reducción por etapas de sistemas noholónomos,, Ph.D. thesis, (2008). Google Scholar

[7]

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

[8]

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

[9]

V. Guillemin and A. Pollack, Differential Topology,, Prentice-Hall Inc., (1974). Google Scholar

[10]

D. Husemoller, Fibre Bundles,, Third ed., (1994). doi: 10.1007/978-1-4757-2261-1. Google Scholar

[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). doi: 10.3842/SIGMA.2007.049. Google Scholar

[12]

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

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Wiley Classics Library, (1996). Google Scholar

[14]

J. M. Lee, Introduction to Smooth Manifolds,, Graduate Texts in Mathematics, (2003). Google Scholar

[15]

M. Leok, J. E. Marsden and A. Weinstein, A discrete theory of connections on principal bundles,, , (2005). Google Scholar

[16]

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

[17]

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

[18]

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

[19]

J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and Tudor S. Ratiu, Hamiltonian Reduction by Stages,, Lecture Notes in Mathematics, (1913). Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, Second ed., (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar

[21]

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

[22]

K. Meyer, Symmetries and integrals in mechanics,, Dynamical systems (Proc. Sympos., (1971), 259. Google Scholar

[23]

P. Michor, Topics in Differential Geometry,, Graduate Studies in Mathematics, (2008). doi: 10.1090/gsm/093. Google Scholar

[24]

E. Routh, Stability of a Given State of Motion,, Macmillan and Co., (1877). Google Scholar

[25]

S. Smale, Topology and mechanics. I,, Invent. Math., 10 (1970), 305. Google Scholar

[26]

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

[1]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69

[2]

Kathrin Flasskamp, Sebastian Hage-Packhäuser, Sina Ober-Blöbaum. Symmetry exploiting control of hybrid mechanical systems. Journal of Computational Dynamics, 2015, 2 (1) : 25-50. doi: 10.3934/jcd.2015.2.25

[3]

Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595

[4]

Anthony M. Bloch, Melvin Leok, Jerrold E. Marsden, Dmitry V. Zenkov. Controlled Lagrangians and stabilization of discrete mechanical systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 19-36. doi: 10.3934/dcdss.2010.3.19

[5]

Viviana Alejandra Díaz, David Martín de Diego. Generalized variational calculus for continuous and discrete mechanical systems. Journal of Geometric Mechanics, 2018, 10 (4) : 373-410. doi: 10.3934/jgm.2018014

[6]

E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1

[7]

Serena Dipierro. Geometric inequalities and symmetry results for elliptic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3473-3496. doi: 10.3934/dcds.2013.33.3473

[8]

Miguel Rodríguez-Olmos. Book review: Geometric mechanics and symmetry, by Darryl D. Holm, Tanya Schmah and Cristina Stoica. Journal of Geometric Mechanics, 2009, 1 (4) : 483-488. doi: 10.3934/jgm.2009.1.483

[9]

Gianne Derks. Book review: Geometric mechanics. Journal of Geometric Mechanics, 2009, 1 (2) : 267-270. doi: 10.3934/jgm.2009.1.267

[10]

Andrew D. Lewis. The physical foundations of geometric mechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 487-574. doi: 10.3934/jgm.2017019

[11]

François Gay-Balmaz, Darryl D. Holm. Predicting uncertainty in geometric fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-14. doi: 10.3934/dcdss.2020071

[12]

C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7

[13]

Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist. Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems. Journal of Geometric Mechanics, 2013, 5 (1) : 1-38. doi: 10.3934/jgm.2013.5.1

[14]

Firdaus E. Udwadia, Thanapat Wanichanon. On general nonlinear constrained mechanical systems. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 425-443. doi: 10.3934/naco.2013.3.425

[15]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873

[16]

Deepak Kumar, Ahmad Jazlan, Victor Sreeram, Roberto Togneri. Partial fraction expansion based frequency weighted model reduction for discrete-time systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 329-337. doi: 10.3934/naco.2016015

[17]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367

[18]

Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149

[19]

Manuel Falconi, E. A. Lacomba, C. Vidal. The flow of classical mechanical cubic potential systems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 827-842. doi: 10.3934/dcds.2004.11.827

[20]

Franco Cardin, Alberto Lovison. Finite mechanical proxies for a class of reducible continuum systems. Networks & Heterogeneous Media, 2014, 9 (3) : 417-432. doi: 10.3934/nhm.2014.9.417

2018 Impact Factor: 0.525

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]