June  2009, 24(2): 213-271. doi: 10.3934/dcds.2009.24.213

Nonholonomic Lagrangian systems on Lie algebroids


Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093, United States


Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, 28006 Madrid, Spain


Departamento de Matemática Fundamental, Unidad Asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Universidad de la Laguna, Tenerife, Canary Islands, Spain


IUMA and Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received  May 2008 Revised  September 2008 Published  March 2009

This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. The proposed novel formalism provides new insights into the geometry of nonholonomic systems, and allows us to treat in a unified way a variety of situations, including systems with symmetry, morphisms, reduction, and nonlinearly constrained systems. Various examples illustrate the results.
Citation: Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

Hernán Cendra, Viviana A. Díaz. Lagrange-d'alembert-poincaré equations by several stages. Journal of Geometric Mechanics, 2018, 10 (1) : 1-41. doi: 10.3934/jgm.2018001


Luca Biasco, Luigi Chierchia. Exponential stability for the resonant D'Alembert model of celestial mechanics. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 569-594. doi: 10.3934/dcds.2005.12.569


Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421


K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87.


José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213


Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527


Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295


Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453


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


Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105


Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014


Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004


Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81


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


Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399


Dmitry V. Zenkov. Linear conservation laws of nonholonomic systems with symmetry. Conference Publications, 2003, 2003 (Special) : 967-976. doi: 10.3934/proc.2003.2003.967


Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469


Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243


Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017


Waldyr M. Oliva, Gláucio Terra. Improving E. Cartan considerations on the invariance of nonholonomic mechanics. Journal of Geometric Mechanics, 2019, 11 (3) : 439-446. doi: 10.3934/jgm.2019022

2018 Impact Factor: 1.143


  • PDF downloads (17)
  • HTML views (0)
  • Cited by (24)

[Back to Top]