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Linearizations of ordinary differential equations by area preserving maps

Published online by Cambridge University Press:  22 January 2016

Tetsuya Ozawa  and
Hajime Sato
Tetsuya Ozawa
Affiliation:
Department of Math. Meijo Univ., Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan, [email protected]
Hajime Sato
Affiliation:
Graduate School of Math. Nagoya Univ., Chikusa-ku, Nagoya 464-8602, Japan, [email protected]
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Abstract

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We clarify the class of second and third order ordinary differential equations which can be tranformed to the simplest equations Y″ = 0 and Y‴ = 0. The coordinate changes employed to transform the equations are respectively area preserving maps for second order equations and contact form preserving maps for third order equations. A geometric explanation of the results is also given by using connections and associated covariant differentials both on tangent and cotangent spaces.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 156 , 1999 , pp. 109 - 122
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

[1] Tresse, M. A., Détermination des invariants: ponctuels de de l’équation différentielle du second order y″ = w(x, y, y′), Hirzel, Leipzig, 1896.Google Scholar
[2] Cartan, É., Sur les Variétés à Connecxion Projective, Bull. Soc. Math. France, 52 (1924), 205–241.Google Scholar
[3] Cartan, É., Lec¸on sur la théorie des espaces à connexion projective, Gauthier-Villars, Paris, 1937.Google Scholar
[4] Cartan, É., La geometria de las ecuaciones diferencials de tercer orden, Rev. Mat. Hispano-Amer., t. 4 (1941), 1–31 (Œuvres complètes, Partie III, Vol. 2, 1535–1565 Gauthier-Villars, Paris (1955)).Google Scholar
[5] Kamran, N., Lamb, K. G. and Shadwick, W. F., The local equivalence problem for d2y/dx2 = F(x, y, dy/dx) and the Painlevé transcendents, J. Diff. Geom., 22 (1985), 139–150.Google Scholar
[6] Chern, S. S., The geometry of a differential equation Y ‘“= F(X, Y, Y’Y”), Sci. Rep. Tsing Hua Univ., 1940, 97–111.Google Scholar
[7] Sato, H. and Yoshikawa, A. Y., Third order ordinary differential equations and Legendre connections, J. Math. Soc. Japan, 50 (1998), No. 4, 993–1013.Google Scholar
[8] Sato, H. and Yoshikawa, A. Y., Projective contact structures on three-Dimensional manifolds, Preprint.Google Scholar