December  2013, 5(4): 433-444. doi: 10.3934/jgm.2013.5.433

A geometric approach to discrete connections on principal bundles

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, Argentina

Received  May 2013 Revised  October 2013 Published  December 2013

This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and relationships. An existence result for discrete connections on principal bundles equipped with appropriate Riemannian metrics is proved.
Citation: Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433
References:
[1]

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

[2]

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

[3]

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

[4]

_______, Lagrangian reduction of discrete mechanical systems by stages,, in preparation., ().   Google Scholar

[5]

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

[6]

R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle,, Proc. Amer. Math. Soc., 11 (1960), 236.  doi: 10.1090/S0002-9939-1960-0112151-4.  Google Scholar

[7]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Reprint of the 1963 original, (1963).   Google Scholar

[8]

M. Leok, Foundations of Computational Geometric Mechanics,, Ph.D. thesis, (2004).   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

J. Milnor, Morse Theory,, Based on lecture notes by M. Spivak and R. Wells, (1963).   Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

_______, Lagrangian reduction of discrete mechanical systems by stages,, in preparation., ().   Google Scholar

[5]

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

[6]

R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle,, Proc. Amer. Math. Soc., 11 (1960), 236.  doi: 10.1090/S0002-9939-1960-0112151-4.  Google Scholar

[7]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Reprint of the 1963 original, (1963).   Google Scholar

[8]

M. Leok, Foundations of Computational Geometric Mechanics,, Ph.D. thesis, (2004).   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

J. Milnor, Morse Theory,, Based on lecture notes by M. Spivak and R. Wells, (1963).   Google Scholar

[1]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[2]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[3]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[4]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[5]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[6]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[7]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[8]

M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202

[9]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[10]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[11]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[12]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[13]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[14]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[15]

Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341

[16]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

[17]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[18]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[19]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

[20]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]