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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.   Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

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