Qualitative Theory of Dynamical Systems

, Volume 2, Issue 2, pp 429–453 | Cite as

Global dynamics of mechanical systems with cubic potentials



We study the behavior of solutions of mechanical systems with polynomial potentials of degree 3 by using a blow up of McGehee type. We first state some general properties for positive degree homogeneous potentials. In particular, we prove a very general property of transversality of the invariant manifolds of the flow along the homothetic orbit. The paper focuses in the study of global flow in the case of homogeneous polynomial potentials of degree 3 for negative energy. The flow is fairly simple because of its gradient-like structure, although for some values of the polynomial coefficients we have diverse behaviour of the separatrices on the infinity manifold, which are essential to describe the global flow.

Key Words

Hamiltonian Vector Field Homogeneous Polynomial Potentials Invariant Manifolds Global Flow 


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  1. 1.
    J. Delgado-Fernández and E. Pérez-chavela,The rhomboidal four body problem. Global solution on the total collision Manifold in T. Ratiu, (ed.), The Geometry of Hamiltonian Systems, Springer-Verlag (1991), 97–110.Google Scholar
  2. 2.
    F.N. Diacu,On the validity of Mücket-Treder gravitational law, in E. Lacomba and J. Llibre, (eds.), New trends in Hamiltonian Systems and Celestial Mechanics, World Scientific Publ., Singapore (1995), 127–139.Google Scholar
  3. 3.
    F.N. Diacu,Collision/ejection dynamics for particle systems with quasihomogeneous potentials, (To appear)Google Scholar
  4. 4.
    M. Falconi,Estudio asintótico de escapes en sistemas Hamiltonianos Polinomiales. P.H. Dissertation, UNAM (1996).Google Scholar
  5. 5.
    M. Falconi, andE.A. Lacomba,Asymptotic behavior of escape solutions of mechanical systems with homogeneous potentials, Cont. Math.198, (1996), 181–195.MathSciNetGoogle Scholar
  6. 6.
    B. Grammaticos, andB. Dorizzi,Integrability of Hamiltonian with third and fourth degree polynomial potentials, J. Math. Phys.,24 (9), (1983), 2289–2295.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    E.A. Lacomba,Infinity manifold for positive energy in celestial mechanics, Contemporary Math.58, part III, (1987), 193–201.MathSciNetGoogle Scholar
  8. 8.
    E.A. Lacomba, andL.A. Ibort,Origin and infinity manifolds for mechanical systems with homogeneous potentials, Acta App. Math.11, (1988), 259–284.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    E.A. Lacomba, J. Bryant, andL.A. Ibort,Blow up of mechanical systems with a homogeneous energy, Publ. Math.35, (1991), 333–345.MATHMathSciNetGoogle Scholar
  10. 10.
    R. McGehee,Triple collision in the colinear three body problem, Inventiones Math.,27, (1974), 191–227.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    C. Simo, andE. A. Lacomba,Analysis of some degenerate quadraple collisions, Celestial Mechs.28, (1982), 49–62.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser-Verlag 2001

Authors and Affiliations

  1. 1.Depto. de Matemáticas, Fac. de CienciasUNAM.C. UniversitariaMérico
  2. 2.Mathematics DepartmentUAM-IMérico
  3. 3.Departamento de MatematicasUniversidade Federal de PernambucoRecife-PeBrasil

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