American Institute of Mathematical Sciences

Advanced
September  2019, 11(3): 439-446. doi: 10.3934/jgm.2019022

Improving E. Cartan considerations on the invariance of nonholonomic mechanics

Waldyr M. Oliva 1,2, and  Gláucio Terra 3,

1. 

Universidade de Lisboa, Instituto Superior Técnico, Center for Mathematical Analysis, Geometry and Dynamical Systems, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
2. 

Universidade de São Paulo, Instituto de Matemática e Estatística, Departamento de Matemática Aplicada, Rua do Matão, 1010, 05508-090 São Paulo, Brazil
3. 

Universidade de São Paulo, Instituto de Matemática e Estatística, Departamento de Matemática, Rua do Matão, 1010, 05508-090 São Paulo, Brazil

Received  February 2019 Published  August 2019

This paper concerns an intrinsic formulation of nonholonomic mechanics. Our point of departure is the paper [6], by Koiller et al., revisiting E. Cartan's address at the International Congress of Mathematics held in 1928 at Bologna, Italy ([3]). Two notions of equivalence for nonholonomic mechanical systems $ ( {\mathsf{{M}}}, {{\mathsf{{g}}}}, {\mathscr{D}}) $ are introduced and studied. According to [6], the notions of equivalence considered in this paper coincide. A counterexample is presented here showing that this coincidence is not always true.

Keywords: Nonholonomic systems, Cartan's equivalence method.
Mathematics Subject Classification: Primary: 37J60, 70F25; Secondary: 70G45.
Citation: 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
References:
[1]

A. Bakša, The geometrization of the motion of certain nonholonomic systems, Mat. Vesnik, 12 (1975), 233-240. 

[2]

D. I. Barrett, R. Biggs, C. C. Remsing and O. Rossi, Invariant nonholonomic Riemannian structures on three-dimensional Lie groups, J. Geom. Mech., 8 (2016), 139-167.  doi: 10.3934/jgm.2016001.

[3]

É. Cartan, Sur la represéntation géométrique des systmes matériels non holonomes, in Proc Int Congr Math, 4, Bologna, 1928, 253–261.

[4]

V. Dragović and B. Gajić, The Wagner curvature tensor in nonholonomic mechanics, Regul. Chaotic Dyn., 8 (2003), 105-123.  doi: 10.1070/RD2003v008n01ABEH000229.

[5]

K. Ehlers and J. Koiller, Cartan meets Chaplygin, Theoretical and Applied Mechanics, 46 (2019), 15-46.  doi: 10.2298/TAM190116006E.

[6]

J. Koiller, P. R. Rodrigues and P. Pitanga, Non-holonomic connections following Élie Cartan, Anais da Academia Brasileira de Cincias, 73 (2001), 165–190, http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652001000200003&nrm=iso. doi: 10.1590/S0001-37652001000200003.

[7]

W. M. Oliva, Geometric Mechanics, vol. 1798 of Lecture Notes in Mathematics, Springer-Verlag, 2002. doi: 10.1007/b84214.

[8]

J. N. Tavares, About Cartan geometrization of non-holonomic mechanics, J. Geom. Phys., 45 (2003), 1-23.  doi: 10.1016/S0393-0440(02)00118-3.

[9]

G. Terra, The parallel derivative, Revista Matemática Contemporânea, 29 (2005), 157-170. 

show all references

References:
[1]

A. Bakša, The geometrization of the motion of certain nonholonomic systems, Mat. Vesnik, 12 (1975), 233-240. 

[2]

D. I. Barrett, R. Biggs, C. C. Remsing and O. Rossi, Invariant nonholonomic Riemannian structures on three-dimensional Lie groups, J. Geom. Mech., 8 (2016), 139-167.  doi: 10.3934/jgm.2016001.

[3]

É. Cartan, Sur la represéntation géométrique des systmes matériels non holonomes, in Proc Int Congr Math, 4, Bologna, 1928, 253–261.

[4]

V. Dragović and B. Gajić, The Wagner curvature tensor in nonholonomic mechanics, Regul. Chaotic Dyn., 8 (2003), 105-123.  doi: 10.1070/RD2003v008n01ABEH000229.

[5]

K. Ehlers and J. Koiller, Cartan meets Chaplygin, Theoretical and Applied Mechanics, 46 (2019), 15-46.  doi: 10.2298/TAM190116006E.

[6]

J. Koiller, P. R. Rodrigues and P. Pitanga, Non-holonomic connections following Élie Cartan, Anais da Academia Brasileira de Cincias, 73 (2001), 165–190, http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652001000200003&nrm=iso. doi: 10.1590/S0001-37652001000200003.

[7]

W. M. Oliva, Geometric Mechanics, vol. 1798 of Lecture Notes in Mathematics, Springer-Verlag, 2002. doi: 10.1007/b84214.

[8]

J. N. Tavares, About Cartan geometrization of non-holonomic mechanics, J. Geom. Phys., 45 (2003), 1-23.  doi: 10.1016/S0393-0440(02)00118-3.

[9]

G. Terra, The parallel derivative, Revista Matemática Contemporânea, 29 (2005), 157-170. 

[1]

Kurt Ehlers. Geometric equivalence on nonholonomic three-manifolds. Conference Publications, 2003, 2003 (Special) : 246-255. doi: 10.3934/proc.2003.2003.246
[2]

Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61
[3]

Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137
[4]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213
[5]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204
[6]

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
[7]

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
[8]

Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. On the number of weakly Noetherian constants of motion of nonholonomic systems. Journal of Geometric Mechanics, 2009, 1 (4) : 389-416. doi: 10.3934/jgm.2009.1.389
[9]

María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1
[10]

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
[11]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[12]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001
[13]

Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143
[14]

B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558
[15]

Andrey Tsiganov. Poisson structures for two nonholonomic systems with partially reduced symmetries. Journal of Geometric Mechanics, 2014, 6 (3) : 417-440. doi: 10.3934/jgm.2014.6.417
[16]

Michał Jóźwikowski, Witold Respondek. A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems. Journal of Geometric Mechanics, 2019, 11 (1) : 77-122. doi: 10.3934/jgm.2019005
[17]

Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006
[18]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure and Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703
[19]

J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467
[20]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (333)
  • HTML views (220)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy