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

Improving E. Cartan considerations on the invariance of nonholonomic mechanics

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.

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. BarrettR. BiggsC. 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. BarrettR. BiggsC. 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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