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The solution of the cartan equivalence problem for \(\frac{{d^2 y}}{{dx^2 }} = F(x,y,\frac{{dy}}{{dx}})\)under the pseudo-group \(\bar x = \varphi (x),\bar y = \psi (x,y)\)

  • N. Kamran
  • W. F. Shadwick
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 246)

Abstract

We give a complete solution to the local equivalence problem for \(\frac{{d^2 y}}{{dx^2 }} = F(x,y,\frac{{dy}}{{dx}})\)under the pseudo-group of coordinate transformations \(\bar x = \varphi (x),\bar y = \psi (x,y)\). Applying Cartan's equivalence method, we obtain an e-structure on J1 (ℝ, ℝ) x G, where G is a certain three-dimensional real Lie group. Vie show that except for the equivalence class of \(\frac{{d^2 y}}{{dx^2 }} = 0\), the G-action can be used to reduce this {e}-structure on J1 (ℝ, ℝ) x G to an e-structure on a lower-dimensional space J1(ℝ, ℝ) x G(1), where the Lie group G(1) is at most one-dimensional. We then show how the invariants obtained by this procedure can be used to obtain necessary and sufficient conditions for equivalence.

Keywords

Equivalence Class Symmetry Group Invariant Condition Structure Tensor Equivalence Problem 
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References

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    E. Cartan, jtAnn. Ecole Normale 25, 1908, p. 57 (collected works part II,p. 719).Google Scholar
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    E. Cartan, Séminaire de Math., expose D, 11 janvier 1937; Selecta, p. 113 (collected works part II, p. 1311).Google Scholar
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    R.B. Gardner, Differential Geometric Control Theory, R. Brockett, R. Millman and H. Sussman eds. Progress in Mathematics, Vol. 27, Birkhaüser, Boston 1983.Google Scholar
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    E.L. Ince, Ordinary Differential Equations, Green and Co. London 1927.Google Scholar
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    N. Kamran, K. Lamb and W. Shadwick, J. Diff. Geom., 1985 (in press).Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • N. Kamran
    • 1
  • W. F. Shadwick
    • 2
  1. 1.Centre de Recherches MathématiquesUniversity de MontréalMontrealCanada
  2. 2.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

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