Geodesic
An Inverse Problem in the Calculus of Variations and the Characteristic Curves of Connections on SO(3)-Bundles
Canadian mathematical bulletin, Tome 38 (1995) no. 2, pp. 129-140

Voir la notice de l'article provenant de la source Cambridge University Press

This paper concerns an inverse problem in the calculus of variations, namely, when a two-dimensional symmetric connection is globally a Riemannian or pseudo-Riemannian connection. Two new local characterizations of such connections in terms of the Ricci tensor and the Riemann curvature tensor respectively are given, together with a solution to the global problem. As an application, the problem of whether the characteristic curves of a connection on an SO(3)-bundle on a surface are the geodesies of a Riemannian metric on the surface is studied. Some applications to non-holonomic dynamics are discussed.
DOI : 10.4153/CMB-1995-019-x
Mots-clés : 53C05, 53C22 
Atkins, Richard. An Inverse Problem in the Calculus of Variations and the Characteristic Curves of Connections on SO(3)-Bundles. Canadian mathematical bulletin, Tome 38 (1995) no. 2, pp. 129-140. doi: 10.4153/CMB-1995-019-x
@article{10_4153_CMB_1995_019_x,
     author = {Atkins, Richard},
     title = {An {Inverse} {Problem} in the {Calculus} of {Variations} and the {Characteristic} {Curves} of {Connections} on {SO(3)-Bundles}},
     journal = {Canadian mathematical bulletin},
     pages = {129--140},
     year = {1995},
     volume = {38},
     number = {2},
     doi = {10.4153/CMB-1995-019-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1995-019-x/}
}
TY  - JOUR
AU  - Atkins, Richard
TI  - An Inverse Problem in the Calculus of Variations and the Characteristic Curves of Connections on SO(3)-Bundles
JO  - Canadian mathematical bulletin
PY  - 1995
SP  - 129
EP  - 140
VL  - 38
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1995-019-x/
DO  - 10.4153/CMB-1995-019-x
ID  - 10_4153_CMB_1995_019_x
ER  - 
%0 Journal Article
%A Atkins, Richard
%T An Inverse Problem in the Calculus of Variations and the Characteristic Curves of Connections on SO(3)-Bundles
%J Canadian mathematical bulletin
%D 1995
%P 129-140
%V 38
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1995-019-x/
%R 10.4153/CMB-1995-019-x
%F 10_4153_CMB_1995_019_x

[1] 1. Anderson, I. and Thompson, G., The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc. 473(1992). Google Scholar

[2] 2. Bates, L. and Sniatycki, J., Non-holonomic reduction, preprint. Google Scholar

[3] 3. Bryant, R., Chern, S. S., Gardner, R., Goldschmidt, H. and Griffiths, P., Exterior Differential Systems, Springer-Verlag, 1991. Google Scholar

[4] 4. Bryant, R. and Hsu, L., Rigidity of integral curves of rank two distributions, 1993, preprint. Google Scholar

[5] 5. Cartan, E., Les systèmes de Pfaff a cinq variables et les equations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. 27(1910), 109–192. Google Scholar

[6] 6. Douglas, J., Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc. 50(1941), 71–128. Google Scholar

[7] 7. Ge, Zhong, On a class of constrained variational problems for systems linear in the controls, preprint. Google Scholar

[8] 8. Ge, Zhong, Caustics in constrained variational problems and optimal control, preprint. Google Scholar

[9] 9. Griffiths, P. A., Exterior Differential Systems and the Calculus of Variations, Progr. Math. 25, Birkhauser, 1983. Google Scholar

[10] 10. Jurdjevic, V., The geometry of the plate-ball problem, preprint. Google Scholar

[11] 11. Kamran, N., Lamb, K. G. and Shadwick, W., The local equivalence problem for d2y/dx2 - F(x,y, dy/dx) and the Painlevé transcendents, J. Differential Geom. 22(1985), 117—123. Google Scholar

[12] 12. Morandi, G., Ferrario, C. and Vecchio, G., The inverse problem in the calculus of variations and the geometry of the tangent bundle, Phys. Rep. 188(1990), 147–284. Google Scholar

[13] 13. Schmidt, B., Conditions on a connection to be a metric connection, Comm. Math. Phys. 29( 1973), 55–59. Google Scholar

Cité par Sources :