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An Inverse Problem in the Calculus of Variations and the Characteristic Curves of Connections on SO(3)-Bundles

Published online by Cambridge University Press:  20 November 2018

Richard Atkins  and
Zhong Ge
Richard Atkins
Affiliation:
The Fields Institute for Research in Mathematical Sciences, 185 Columbia St. West, Waterloo, Ontario N2L 5Z5
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Abstract

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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.

Keywords

53C0553C22

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 2 , 01 June 1995 , pp. 129 - 140
Copyright
Copyright © Canadian Mathematical Society 1995

References

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. Bates, L. and Sniatycki, J., Non-holonomic reduction, preprint.Google Scholar
3. Bryant, R., Chern, S. S., Gardner, R., Goldschmidt, H. and Griffiths, P., Exterior Differential Systems, Springer-Verlag, 1991.Google Scholar
4. Bryant, R. and Hsu, L., Rigidity of integral curves of rank two distributions, 1993, preprint.Google Scholar
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. Douglas, J., Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc. 50(1941), 71–128.Google Scholar
7. Ge, Zhong, On a class of constrained variational problems for systems linear in the controls, preprint.Google Scholar
8. Ge, Zhong, Caustics in constrained variational problems and optimal control, preprint.Google Scholar
9. Griffiths, P. A., Exterior Differential Systems and the Calculus of Variations, Progr. Math. 25, Birkhauser, 1983.Google Scholar
10. Jurdjevic, V., The geometry of the plate-ball problem, preprint.Google Scholar
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. 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. Schmidt, B., Conditions on a connection to be a metric connection, Comm. Math. Phys. 29( 1973), 55–59.Google Scholar