Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 1001–1011 | Cite as

On Stability of Some Newton Systems

  • Marcelo Farias Caetano
  • Manuel Valentim de Pera GarciaEmail author


The aim of this paper is to study the stability of an equilibrium for the second order ordinary differential equation \(\ddot{q}=F(q), \; q \in \mathbb {R}^{2}\), which are the equations of motion of a point of mass under the action of force F. The smooth force F is not supposed to be gradient. We consider two situations separately, the case of systems which have an indefinite quadratic first integral and the situation where the forces point inwards to circumferences with center at the equilibrium point.


Liapunov stability Non-conservative positional forces Central forces 



  1. 1.
    Albouy, A.: Projective dynamics and first integrals. Regul. Chaotic Dyn. 20(3), 247–276 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Appell, P.E.: De l’homographie en mécanique. Am. J. Math. 12, 103–114 (1890)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barone-Netto, A., de Oliveira Cesar, M.: Non-conservative positional systems-stability. Dyn. Stab. Syst. 2(3–4), 213–221 (1988)zbMATHGoogle Scholar
  4. 4.
    Barone-Netto, A., de Oliveira Cesar, M., Gorni, G.: A computational method for the stability of a class of mechanical systems. J. Differ. Equ. 184(1), 1–19 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bertrand, J.M.: Mémoire sur quleuques-unes des forms les plus simples qui puissent présenter les intégrales des équations différentielles du mouvement d’un point matériel. J. Math. Pure Appl. Sér. II, 2, 113–140 (1857)Google Scholar
  6. 6.
    de Oliveira Cesar, M., Barone-Netto, A.: The existence of Liapunov functions for some non-conservative positional mechanical systems. J. Differ. Equ. 91(2), 235–244 (1991)CrossRefGoogle Scholar
  7. 7.
    de Oliveira Cesar, M., Barone-Netto, A.: A necessary and sufficient condition for the stability of the equilibrium. J. Differ. Equ. 96(1), 142–151 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kasner, E.: The Ithaca colloquium. Bull. Am. Math. Soc. 8(1), 22–25 (1901)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Katzin, G.H., Levine, J.: Quadratic first integrals of the geodesics in spaces of constant curvature. Tensor 16(1), 97–103 (1965)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lagrange, J.-L.: Recherches sur le mouvement d’un corps qui est attiré vers deux centres fixes. Second mémoire, où l’on applique la méthode précédentes à différentes hyphotèses d’attraction. Miscellanea Taurinensia, IV:216–243, 1766–1769 (1773)Google Scholar
  11. 11.
    Lundmark, H.: A new class of integrable Newton systems. J. Nonlinear Math. Phys. 8(sup1), 195–199 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rauch-Wojciechowski, S., Marciniak, K., Lundmark, H.: Quasi-Lagrangian systems of Newton equations. J. Math. Phys. 40(12), 6366–6398 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Taliaferro, S.D.: An inversion of the Lagrange–Dirichlet stability theorem. Arch. Ration. Mech. Anal. 73(2), 183–190 (1980)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zampieri, G.: Liapunov stability for some central forces. J. Differ. Equ. 74(2), 254–265 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zampieri, G., Barone-Netto, A.: Attractive Central Forces May Yield Liapunov Instability. Dynamical Systems and Partial Differential Equations, pp. 105–112. Editorial Equinoccio, Caracas (1986)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUSP Departamento de Matemática AplicadaSão PauloBrazil
  2. 2.Instituto de Matemática e EstatísticaUSP Departamento de Matemática AplicadaSão PauloBrazil

Personalised recommendations