Skip to main content
SpringerLink
Log in

Point Classification of Second Order ODEs: Tresse Classification Revisited and Beyond

  • Chapter
  • First Online:
Differential Equations - Geometry, Symmetries and Integrability

Part of the book series: Abel Symposia ((ABEL,volume 5))

Abstract

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the second order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.I. Arnold, “Additional chapters of the theory of ordinary differential equations”, Moscow-Nauka (1978) [Russian]; Engl.: “Geometrical methods in the theory of ordinary differential equations”, Springer (1988).

    Google Scholar 

  2. R. Bryant, M. Dunajski, M. Eastwood, Metrisability of two-dimensional projective structures, arXiv: math/0801.0300 (2008).

    Google Scholar 

  3. R. Bryant, G. Manno, V. S. Matveev, A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields, Math. Ann. 340 (2008), no. 2, 437–463.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  5. C.L. Fefferman, Monge-Amperèe equations, the Bergman kernel, and geometry of pseudo-convex domains, Ann. of Math. (2) 103 (1976), no. 2, 395–416.

    Article  MathSciNet  Google Scholar 

  6. M. Fels, P. Olver, On relative invariants, Math. Ann. 308, no. 4, 701–732 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  7. L. Hsu, N. Kamran, Classification of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries, Proc. London Math. Soc. (3) 58 (1989), no. 2, 387–416.

    Article  MATH  MathSciNet  Google Scholar 

  8. N. Ibragimov, F. Magri, Geometric proof of Lie's linearization theorem, Nonlinear Dynamics 36, 41–46 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  9. N. Kamran, W.F. Shadwick, The solution of the Cartan Equivalence Problem for d2y/dx2 − F(x, y, dy/dx) under the pseudo-group x = (x), = ψ;(x, y), Lecture Notes in Physics 246, Springer, 320–336 (1986).

    Article  MathSciNet  Google Scholar 

  10. I. S. Krasilschik, V. V. Lychagin, A. M. Vinogradov, Geometry of jet spaces and differential equations, Gordon and Breach (1986).

    Google Scholar 

  11. B. Kruglikov, Invariant characterization of Liouville metrics and polynomial integrals, Jour. Geom. and Phys. 58 (2008), 979–995.

    Google Scholar 

  12. B. S. Kruglikov, V. V. Lychagin, Invariants of pseudogroup actions: Homological methods and Finiteness theorem, Int. J. Geomet. Meth. Mod. Phys. 3, no. 5 & 6 (2006), 1131–1165.

    Article  MATH  MathSciNet  Google Scholar 

  13. B. S. Kruglikov, V. V. Lychagin, Geometry of Differential equations, prepr. IHES/M/07/04; in: Handbook of Global Analysis, Ed. D. Krupka, D. Saunders, Elsevier (2008), 725-772.

    Google Scholar 

  14. B. S. Kruglikov, V. V. Lychagin, Compatibility, multi-brackets and integrability of systems of PDEs, prepr. Univ. Tromsø 2006-49; ArXive: math.DG/0610930 (2006); Acta Applicandae Math. (2009).

    Google Scholar 

  15. A. Kumpera, Invariants differentiels d'un pseudogroupe de Lie. I-II. J. Differential Geometry 10 (1975), no. 2, 289–345; 10 (1975), no. 3, 347-416.

    MATH  MathSciNet  Google Scholar 

  16. A. Kumpera, D. Spencer, Lie equations. Volume 1: General theory, Princeton University Press and University Tokyo Press (1972).

    Google Scholar 

  17. S. Lie, Ueber Differentialinvarianten, Math. Ann. 24 (1884), no. 4, 537–578.

    Article  MathSciNet  Google Scholar 

  18. S. Lie, Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten. III, Archiv for Mathematik og Naturvidenskab 8 (Kristiania, 1883), 371-458; Gesam. Abh. Bd. 5 (1924), paper XIV, 362-427.

    Google Scholar 

  19. R. Liouville, Sur les invariants de certaines équations différentielles et sur leurs applications, Journal de l'École Polytechnique 59 (1889), 7–76.

    Google Scholar 

  20. J. Levine, Projective scalar differential invariants, Ann. Math. 37 (1936), no. 3, 618–634.

    Article  Google Scholar 

  21. V. Matveev, A solution of another problem of Sophus Lie: 2-dimensional metrics admitting precisely one projective vector field, ArXiv:0802.2344.

    Google Scholar 

  22. P. Nurowski, G. A. Sparling, Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations, Classical Quantum Gravity 20 (2003), no. 23, 4995–5016.

    Article  MATH  MathSciNet  Google Scholar 

  23. P. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge (1995).

    MATH  Google Scholar 

  24. L. V. Ovsiannikov, Group analysis of differential equations, Russian: Nauka, Moscow (1978); Engl. transl.: Academic, New York (1982).

    Google Scholar 

  25. Yu. Romanovskii, Calculation of local symmetries of second-order ordinary differential equations by Cartan's equivalence method, Mat. Zametki 60, no. 1, 75–91, 159; Engl. transl.: Math. Notes 60 (1996), no. 1-2, 56-67 (1996).

    MathSciNet  Google Scholar 

  26. R. A. Sarkisyan, I. G. Shandra, Regularity and Tresses theorem for geometric structures, Izvestiya RAN: Ser. Mat. 72, no. 2, 151–192 (2008); Engl. transl. Izvestiya: Mathematics 72, no. 2, 345-382 (2008).

    MathSciNet  Google Scholar 

  27. T. V. Thomas, The differential invariants of generalized spaces, Cambridge, The University Press (1934).

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. A. Tresse, Détermination des invariants ponctuels de l'équation différentielle ordinaire du second ordre = ω(x, y, y′), Leipzig (1896).

    Google Scholar 

  30. V. Yumaguzhin, Differential invariants of 2-order ODEs, I, arXiv: math.DG/0804.0674 v1 (2008); revised version to appear in Acta Applicandae Math. (2009).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Tromsø, Tromsø, Norway

    Boris Kruglikov

Authors
  1. Boris Kruglikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Kruglikov .

Editor information

Editors and Affiliations

  1. Dept. Mathematics & Statistics, University of Tromsoe, Tromsoe, 9037, Norway

    Boris Kruglikov

  2. Dept. Mathematics & Statistics, University of Tromsoe, Tromsoe, 9037, Norway

    Valentin Lychagin

  3. of Science and Technology, Norwegian University, Trondheim, 7034, Norway

    Eldar Straume

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kruglikov, B. (2009). Point Classification of Second Order ODEs: Tresse Classification Revisited and Beyond. In: Kruglikov, B., Lychagin, V., Straume, E. (eds) Differential Equations - Geometry, Symmetries and Integrability. Abel Symposia, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00873-3_10

Download citation

Publish with us

Policies and ethics

Search

Navigation