Skip to main content
SpringerLink
Log in

Differential Invariants of Second Order ODEs, I

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

This paper is devoted to differential invariants of equations

$$y''=a^{3}(x,y)y^{\prime3}+a^{2}(x,y)y^{\prime2}+a^{1}(x,y)y'+a^{0}(x,y).$$

w.r.t. point transformations. The natural bundle of these equations and its bundles of k-jets of sections, k=0,1,2,…, are considered. The action of the pseudogroup of all point transformations on these bundles is investigated. Tensor differential invariants distinguishing orbits of this action on jet bundles of second and third orders are constructed. A complete collection of generators and their differential syzygies is obtained for the algebra of all scalar differential invariants of a wide class of considered equations containing generic equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alekseevskiy, D.V., Lychagin, V.V., Vinogradov, A.M.: Fundamental ideas and conceptions of differential geometry. In: Sovremennye Problemy Matematiki. Fundamental’nye Napravleniy. Itogi Nauki i Techniki, vol. 28, pp. 298. VINITI, AN SSSR, Moscow (1988) (in Russian) [English transl.: Encyclopedia of Math. Sciences, vol. 28, pp. 298, Springer, Berlin (1991)]

    Google Scholar 

  2. Arnold, V.I.: Advanced Chapters of the Theory of Ordinary Differential Equations. Nauka, Moscow (1978), p. 400 (in Russian)

    Google Scholar 

  3. Bernshteyn, I.N., Rozenfel’d, B.I.: Homogeneous spaces of infinite dimensional Lie algebras and characteristic classes of foliations. Usp. Mat. Nauk 28(4), 103–138 (1973) (in Russian)

    Google Scholar 

  4. Cartan, E.: Sur les varietes a connexion projective. Bull. Soc. Math. France 52, 205–241 (1924)

    MATH  MathSciNet  Google Scholar 

  5. Chern, S.S.: Projective geometry, contact transformations, and CR-structures. Arch. Math. 38, 1–5 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gardner, R.B.: Differential geometric methods interacting control theory. In: Brockett, R.W. . (eds.) Differential Geometry Control Theory, pp. 117–180. Birkhäuser, Boston (1983)

    Google Scholar 

  7. Grissom, C., Thompson, G., Wilkens, G.: Linearization of second order ordinary differential equations via Cartan’s equivalence method. J. Differ. Equ. 77, 1–15 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Guillemin, V., Sternberg, S.: An algebraic model of transitive differential geometry. Bull. Am. Math. Soc. 70, 16–47 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gusyatnikova, V.N., Yumaguzhin, V.A.: Point transformations and linearisability of 2-order ordinary differential equations. Mat. Zametki 49(1), 146–148 (1991) (in Russian)

    MATH  MathSciNet  Google Scholar 

  10. Krasil’shchik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, p. 336. Gordon & Breach, New York (1986)

    MATH  Google Scholar 

  11. Krasil’shchik, I.S., Vinogradov, A.M. (eds.): Symmetries and conservation laws for differential equations of mathematical physics. In: Translations of Mathematical Monographs, vol. 82, p. 461. Am. Math. Soc., Providence (1999)

    Google Scholar 

  12. Kruglikov, B., Lychagin, V.: On equivalence of differential equations. Acta Comment. Univ. Tartu Math. 3, 7–29 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Lie, S.: Vorlesungen über Continuierliche Gruppen. Teubner, Leipzig (1893)

    Google Scholar 

  14. Lie, S.: Theorie der Transformationsgruppen, vol. III. Teubner, Leipzig (1930)

    Google Scholar 

  15. Liouville, R.: Sur les invariantes de certaines equationes differentielles. J. Ecole Politechnique 59, 7–88 (1889)

    Google Scholar 

  16. Lychagin, V.: Homogeneous geometric structures and homogeneous differential equations. In: The Interplay between Differential Geometry and Differential Equations. AMS Translations, Advances in Math. Sciences, vol. 2, 167, pp. 143–164. AMS, Providence (1995)

    Google Scholar 

  17. Sharipov, R.A.: On the point transformations for the equation y″=P+3Qy′+3Ry′2+Sy′3. Preprint, Electronic Archive at LANL, solv-int/9706003 (1997)

  18. Sharipov, R.A.: Effective procedure of point-classification for the equations y″=P+3Qy′+3Ry′2+Sy′3. Preprint, Electronic Archive at LANL, math/9802027 (1998)

  19. Singer, I.M., Sternberg, S.: On the infinite groups of Lie and Cartan, I. J. Anal. Math. 15, 1–114 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sternberg, S.: Lectures on Differential Geometry, p. 412. Prentice-Hall, New Jersey (1964)

    MATH  Google Scholar 

  21. Thomsen, G.: Uber die topologischen Invarianten der Differentialgleichung y″=f(x,y)y′ 3+g(x,y)y′ 2+h(x,y)y′+k(x,y). Abh. Math. Semin. Hamb. Univ. 7, 301–328 (1930)

    Article  Google Scholar 

  22. Tresse, A.: Sur les invariants differentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)

    Article  MathSciNet  Google Scholar 

  23. Tresse, A.: Détermination des Invariants Ponctuels de l’Equation Differentielle Ordinaire du Second Ordre: y″=ω(x,y,y′), p. 88. Hirzel, Leipzig (1896)

    MATH  Google Scholar 

  24. Vinogradov, A.M.: Scalar differential invariants, diffieties and characteristic classes. In: Francaviglia, M. (ed.) Mechanics, Analysis and Geometry: 200 Years after Lagrange, pp. 379–414. North-Holland, Amsterdam (1991)

    Google Scholar 

  25. Yumaguzhin, V.A.: On the obstruction to linearizability of 2-order ordinary differential equations. Acta Appl. Math. 83(12), 133–148 (2004). arXiv:0804.0306

    Article  MATH  MathSciNet  Google Scholar 

  26. Yumaguzhin, V.A.: Differential invariants of 2-order ODEs, I. arXiv:0804.0674 (2008)

  27. Yumaguzhin, V.A.: On the obstruction to integrability of almost-complex structures. arXiv:0804.0690v1 (2008)

Download references

Author information

Authors and Affiliations

  1. Program Systems Institute of RAS, Pereslavl’-Zalesskiy, 152020, Russia

    Valeriy A. Yumaguzhin

Authors
  1. Valeriy A. Yumaguzhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valeriy A. Yumaguzhin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yumaguzhin, V.A. Differential Invariants of Second Order ODEs, I. Acta Appl Math 109, 283–313 (2010). https://doi.org/10.1007/s10440-009-9454-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-009-9454-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation