Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter August 27, 2008

Equivalence and symmetries of second-order differential equations

  • V. Tryhuk EMAIL logo and O. Dlouhý
From the journal Mathematica Slovaca

Abstract

In this article we investigate the equivalence of underdetermined differential equations and differential equations with deviations of second order with respect to the pseudogroup of transformations $$ \bar x $$ = φ(x), ȳ = ȳ($$ \bar x $$) = L(x) + y(x), $$ \bar z $$ = $$ \bar z $$($$ \bar x $$) = M(x) + z(x). Our main aim is to determine such equations that admit a large pseudogroup of symmetries. Instead the common direct calculations, we use some more advanced tools from differential geometry, however, our exposition is self-contained and only the most fundamental properties of differential forms are employed.

[1] AWANE, A.— GOZE, M.: Pfaffian Systems, k-symplectic Systems, Kluwer Academic Publischers, Dordrecht-Boston-London, 2000. 10.1007/978-94-015-9526-1Search in Google Scholar

[2] BRYANT, R.— CHERN, S. S.— GOLDSCHMIDT, H.— GRIFFITHS, P. A.: Exterior Differential Systems. Math. Sci. Res. Inst. Publ. 18, Cambridge Univ. Press, Cambridge, 1991. 10.1007/978-1-4613-9714-4Search in Google Scholar

[3] CARTAN, E.: Les systémes différentiels extérieurs et leurs applications géometriques. Actualités Sci. Indust. 994, Hermann, Paris, 1945. Search in Google Scholar

[4] CARTAN, E.: Sur la structure des groupes infinis de transformations, Ann. Sci. École Norm. Ser. 3 21, 1904; Oeuvres Complètes, Partie II, Vol 2, Gauthier-Villars, Paris 1953. 10.24033/asens.538Search in Google Scholar

[5] CHRASTINA, J.: Transformations of differential equations. In: Equadiff 9 CD ROM, Papers, Masaryk Univerzity, Brno 1997, pp. 83–92. Search in Google Scholar

[6] CHRASTINA, J.: The Formal Theory of Differential Equations. Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. 6, Masaryk Univ., Brno, 1998. Search in Google Scholar

[7] GARDNER, R. B.: The Method of Equivalence and its Applications. CBMS-NSF Regional Conf. Ser. in Appl. Math. 58, SIAM, Philadelphia, PA, 1989. 10.1137/1.9781611970135Search in Google Scholar

[8] NEUMAN, F.: Global Properties of Linear Ordinary Differential Equations. Math. Appl. (East European Series) 52, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991. Search in Google Scholar

[9] SHARPE, R. V.: Differential Geometry. Grad. Texts in Math. 166, Springer Verlag, New York, 1997. Search in Google Scholar

[10] TRYHUK, V.— DLOUHÝ, O.: The moving frames for differential equations: Part I.: The change of independent variable; Part II.: Underdetermined and functional equations, Arch. Math. (Brno) 39 (2003), 317–333; 40 (2004), 69–88. Search in Google Scholar

Published Online: 2008-8-27
Published in Print: 2008-10-1

© 2008 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 9.12.2023 from https://www.degruyter.com/document/doi/10.2478/s12175-008-0093-0/html
Scroll to top button