Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T14:41:10.561Z Has data issue: false hasContentIssue false
  • English
  • Français

Submersive second order ordinary differential equations

Published online by Cambridge University Press:  24 October 2008

Marek Kossowski  and
Gerard Thompson
Marek Kossowski
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
Gerard Thompson
Affiliation:
Department of Mathematics, University of Toledo, Toledo, OH 43606, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

The objectives of this paper are to define and to characterize submersive second order ordinary differential equations (ODE) and to examine several situations in which such ODE occur. This definition and characterization is in terms of tangent bundle geometry as developed in [4, 6, 7, 10, 11, 14]. From this viewpoint second order ODE are identified with a special vector field on the tangent bundle. The ODE are said to be submersive when this vector field and the canonical vertical endomorphism [14] define a foliation, relative to which the vector field passes to the local quotient.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 207 - 224
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arnold, V. I.. Mathematical Methods of Classical Mechanics. Graduate Texts in Math. no. 69 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[2]Besse, A.. Einstein Manifolds (Springer-Verlag, 1987).CrossRefGoogle Scholar
[3]Bott, R.. Lectures on characteristic classes and foliations. In Lectures on Algebraic and Differential Topology, Lecture Notes in Math. vol. 279 (ed. Gitler, S.) (Springer-Verlag, 1972), pp. 1–94.CrossRefGoogle Scholar
[4]Brickell, F. and Clark, R. S.. Integrable almost tangent structures. J. Differential Geom. 9 (1974), 557–563.CrossRefGoogle Scholar
[5]Crampin, M.. On the differential geometry of the Euler-Lagrange equations and the inverse problem of Lagrangian dynamics. J. Phys. A 14 (1981), 2567–2575.CrossRefGoogle Scholar
[6]Crampin, M.. Tangent bundle geometry for Lagrangian dynamics. J. Phys. A 16 (1983), 3755–3772.CrossRefGoogle Scholar
[7]Crampin, M. and Thompson, G.. Affine bundles and integrable almost tangent structures. Math. Proc. Cambridge Philos. Soc. 48 (1985), 61–71.CrossRefGoogle Scholar
[8]Douglas, J.. Solution to the inverse problem of the calculus of variations. Trans. Amer. Math. Soc. 50 (1941), 71–128.CrossRefGoogle Scholar
[9]Ferrario, C., Vecchio, G. Lo, Marmo, G., Morandi, G. and Rubano, C.. A separability theorem for dynamical systems admitting Lagrangian descriptions. J. Phys. A 20 (1987), 3225–3236.CrossRefGoogle Scholar
[10]Godbillon, C.. Geometrie Differentielle et Mécanique Analytique (Hermann, 1969).Google Scholar
[11]Grifone, J.. Structures presque-tangent et connexions. Ann. Inst. Fourier (Grenoble) 22 (1972), 287–334.CrossRefGoogle Scholar
[12]Guckenheimer, J. and Holmes, P.. Non-Linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, 1983).CrossRefGoogle Scholar
[13]Helgason, S.. Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, 1978).Google Scholar
[14]Klein, J.. Espaces variationnels et mécanique. Ann. Inst. Fourier (Grenoble) 2 (1962), 1–124.CrossRefGoogle Scholar
[15]Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry vol. 1 (Interscience, 1963).Google Scholar
[16]Marsden, J. and Weinstein, A.. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5 (1974), 121–130.CrossRefGoogle Scholar
[17]Mosee, J.. Three integrable Hamiltonian systems connected with isospectral deformations. Adv. in Math. 16 (1975), 197–220.Google Scholar
[18]O'Neill, B.. Semi-Riemannian Geometry (Academic Press, 1981).Google Scholar
[19]Whittaker, E.. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, 1959).Google Scholar