Skip to main content
SpringerLink
Log in

Local relative integral invariants determined by the phase portrait of a vector field on the plane

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We fix some invariant measure for a given vector field on the plane. The pair (the phase portrait of the vector field, the invariant measure of the vector field) determines a Lie subalgebra in the algebra of all smooth vector fields on the plane, namely the stationary subalgebra of the pair. An element of the subalgebra has a relative integral invariant, namely: the integral of the above measure along sets bounded by phase curves of the initial vector field and the element of subalgebra. The main result of the paper is the following:

Theorem. Relative integral invariants in the general situation (more exactly, for the finite-modal case) are expressed in terms of elementary functions of suitable phase coordinates.

Degenerate relative integral invariants, for which the above theorem is not valid, appear in twoparametric families of the above objects in the general situation.

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

Literature Cited

  1. A.A. Andronov, A.A. Vitt, and S.E. Khaikin,Theory of Oscillation, Fizmatgiz, Moscow, 1959. (in Russian)

    Google Scholar 

  2. V.I. Arnold,Ordinary Differential Equations, Nauka, Moscow, 1984. (in Russian)

    Google Scholar 

  3. V.I. Arnold,Complementary Chapters in the Theory of Ordinary Differential Equations, Nauka, Moscow, 1984. (in Russian)

    Google Scholar 

  4. R.I. Bogdanov,Local orbital normal forms of vector fields on the plane. Tr. Sem. im. I.G. Petrovskogo (1979), no. 5, 51–84.

    MATH  MathSciNet  Google Scholar 

  5. R.I. Bogdanov,Singularities of vector Fields on the Plane with Pointed Direction. Invent. math.54 (1979), 247–259.

    Article  MATH  MathSciNet  Google Scholar 

  6. R.I. Bogdanov,Invariants of elementary singularities on the plane, Usp. Mat. Nauk 40 (1985), no. 3, 199–200.

    MATH  MathSciNet  Google Scholar 

  7. R.I. Bogdanov,Symplectic orbital equivalence of vector fields on the plane (elementary singularities), Mathematics i Modeling (compendium of scientific works), Puschino, 1990, pp. 32–45. ([in Russian])

  8. R.I. Bogdanov,Integrals of semiintegrable K-tuples of germs of vector fields on the plane Tr. Sem. im. I.G. Petrovskogo (1992), no. 16, 70–105.

    MATH  Google Scholar 

  9. H. Poincaré,Memoire sur les courbes définie par une équation différentielle, I, II, III, IV. J. Math. Pures Appl. 3 Ser. 7 (1881), 75–422; 3 Ser.8 (1882), 251–296; 4 Ser.1 (1885), 167–244; 4 Ser.2 (1886), 151–217.

    Google Scholar 

  10. S. Sternberg,Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.

    Google Scholar 

Download references

Authors
  1. R. I. Bogdanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 249–278, 1994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bogdanov, R.I. Local relative integral invariants determined by the phase portrait of a vector field on the plane. J Math Sci 75, 1773–1784 (1995). https://doi.org/10.1007/BF02368676

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02368676

Keywords

Search

Navigation