Skip to main content
SpringerLink
Log in

A Partial Reciprocal of Dirichlet Lagrange Theorem Detected by Jets

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

We study the stability of an equilibrium point in a conservative Hamiltonian system in the case that equilibrium is not a minimum of the potential energy and this fact is shown by a jet of this function. Thanks to a modification of a result of Krasovskii, we prove that for a large class of systems under these conditions equilibrium is unstable and there is an asymptotic trajectory to that point.

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. Painlevé, P.: Sur 1a stabi1ite de l’equi1ibre. C. R. Acad. Sci. Paris, sér. A-B 138, 1555–1557 (1904)

  2. Garcia, M.V.P., Tal, F.A.: Stability of equilibrium of conservative systems with two degrees of freedom. J. Differ. Equ. 194, 364–381 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Moauro, V., Negrini, P.: On the inversion of Lagrange–Dirichlet theorem. Differ. Integral Equ. 2(4), 471–478 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Lyapunov, A.M.: Sur l’instabilité de l’equilibre dans certains cas où la fonction de forces n’est pas un maximun. J. Math. Pures Appl. Ser. V 3, 81–94 (1897)

    MATH  Google Scholar 

  5. Barone Netto: Jet-detectable Extrema. Proceeding AMS (1984)

  6. Maffei, C., Moauro, V., Negrini, P.: On the inversion of the Lagrange–Dirichlet theorem in a case of nonhomogeneous potential. Differ. Integral Equ. 4(4), 767–782 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Freire Jr., R.S., Garcia, M.V.P., Tal, F.A.: Instability of equilibrium points of some Lagrangian systems. J. Differ. Equ. 245, 490–504 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rouche, N., Habets, P., Laloy, M.: Stability Theory by Lyapunov’s Direct Method. Springer, New York (1977)

    Book  MATH  Google Scholar 

  9. Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, Princeton University Press, Princeton

  10. Krasovskii, N.N.: Stability of the Motion. Stanford University Press, Stanford (1963)

    MATH  Google Scholar 

  11. Zampiere, G.: Completely integrable Hamiltonian systems with weak: Lyapunov instability or isochrony. Commun. Math. Phys. 303(1), 73–87 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Barone Netto, A., Gorni, G., Zampieri, G.: Local extrema of analytic functions. Nonlinear Differ. Equ. Appl. 3(3), 287–303 (1996)

  13. Palamodov, V.: Stability of motion and algebraic geometry. Transl. Am. Math. Soc. Ser. 2 168(25), 5–20 (1995)

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, USP, Rua do Matão 1010 Cidade Universitária, São Paulo, SP, CEP 05508-090, Brazil

    Manuel Valentim de Pera Garcia & Gerard John Alva Morales

Authors
  1. Manuel Valentim de Pera Garcia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerard John Alva Morales
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel Valentim de Pera Garcia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Pera Garcia, M.V., Morales, G.J.A. A Partial Reciprocal of Dirichlet Lagrange Theorem Detected by Jets. Qual. Theory Dyn. Syst. 16, 371–389 (2017). https://doi.org/10.1007/s12346-016-0196-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-016-0196-x

Keywords

Search

Navigation