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)


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.


Equivalence Class Symmetry Group Invariant Condition Structure Tensor Equivalence Problem 
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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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