Abstract
We present a geometric exposition of S. Lie's and E. Cartan's theory of explicit integration of finite-type (in particular, ordinary) differential equations. Numerous examples of how this theory works are given. In one of these, we propose a method of hunting for particular solutions of partial differential equations via symmetry preserving overdetermination.
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References
Bluman G. W. and Kumei S.: Symmetries and Differential Equations. Springer-Verlag, New York, 1989.
Cartan E.: Lecons sur les invariants intégraux, Hermann, Paris, 1922.
Cassidy P. J.: J. Algebra 121 (1989), 169–238.
Ibragimov N. H.: Transformation Groups Applied to Mathematical Physics, D. Reidel, Dordrecht, 1985.
Kamke E.: Differentialgleichungen. Lösungsmethoden und Lösungen: 1. Genwönliche Differentialgleichungen, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1961.
Kobayashi S. and Nomizu K.: Foundations of Differential Geometry, Interscience, New York, 1963.
Krasilshchik I. S., Lychagin V. V. Vinogradov A. M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.
Kolchin E. R.: Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.
Lie S.: Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1891.
Olver P.: Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
Ovsiannikkov L. V.: Group Analysis of Differential Equations, Academic Press, New York, 1982.
Painlevé P.: Acta Math. 25 (1902), 1–85.
Pommaret J. F.: Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, Paris, 1978.
Pommaret J. F.: Differential Galois Theory, Gordon and Breach, New York, 1983.
Polischuk E. M.: Sophus Lie, Nauka, Leningrad, 1983 (in Russian).
Sherring J. and Prince G.: Geometric aspects of reduction of order, Research report, The University of New England, Australia, August 1990.
Stephani, H.: Differential Equations: Their Solution Using Symmetries, Cambridge University Press, 1989.
Sternberg S.: Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs, 1964.
Tsujishita, T.: Homological method of computing invariants of systems of differential equations, to appear in Diff. Geom. Appl. 1 (1991).
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Duzhin, S.V., Lychagin, V.V. Symmetries of distributions and quadrature of ordinary differential equations. Acta Appl Math 24, 29–57 (1991). https://doi.org/10.1007/BF00047361
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DOI: https://doi.org/10.1007/BF00047361