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Convex energy levels of Hamiltonian systems

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Abstract

We give a simple necessary and sufficient condition for a non-regular energy level of a Hamiltonian system to be strictly convex. We suppose that the Hamiltonian function is given by kinetic plus potential energy. We also show that this condition holds for several Hamiltonian functions, including the Hénon-Heiles one.

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References

  1. P. Bernard, C. Grotta-Ragazzo, P. A. S. Salomão,Homoclinic orbits near saddle-center fixed points of Hamiltonian systems with two degrees of freedom, Astérisque.286 (2003), 151–165.

    Google Scholar 

  2. H. Busemann,Convex surfaces, Interscience Tracts in Pure and App. Math., (6) Interscience Publishers, Inc., New York; Interscience Publishers Ltd, London, (1958), 196 pp.

    MATH  Google Scholar 

  3. R. C. Churchill, G. Pecelli, D. L. Rod,Hyperbolic structures in Hamiltonian systems, Rocky Mountain Journal of Mathematics7 (3), (1997) 439–444.

    Article  MathSciNet  Google Scholar 

  4. R. C. Churchill, G. Pecelli, D. L. Rod,A survey of the Hénon-Heiles Hamiltonian with applications to related examples, Stochastic behavior in classical and quantum Hamiltonian systems, Volta Memorial Conf., Como, (1977) 76–136, Lecture Notes in Phys.93, Springer, Berlin-New York (1979).

    Chapter  Google Scholar 

  5. J. Franks,Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math.2 (1996), 1–19.

    MATH  MathSciNet  Google Scholar 

  6. M. Ghomi,The problem of optimal smoothing for convex functions, Proc. Amer. Math. Soc.130 (2002), 2255–2259.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Ghomi,Strictly convex submanifolds and hypersurfaces of positive curvature, Journal of Diff. Geom.57 (2001), 239–271.

    MATH  MathSciNet  Google Scholar 

  8. C. Grotta-Ragazzo,Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers, Comm. Pure App. Math.L (1997), 105–147.

    Article  MathSciNet  Google Scholar 

  9. C. Grotta-Ragazzo,On the stability of double homoclinic loops, Comm. Math. Phys.184 (1997), 251–272.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Hénon, C. Heiles,The applicability of the third integral of motion: Some numerical experiments, Astronom. J.69 (1964), 73–79.

    Article  MathSciNet  Google Scholar 

  11. H. Hofer, K. Wysocki andE. Zehnder,The dynamics on a strictly convex energy surface in ℝ4. Ann. of Math. (2)148 (1) (1998), 197–289.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. M. Lerman,Hamiltonian systems with loops of a separatrix of saddle center, Selecta Math. Sov.10 (1991), 297–306.

    MathSciNet  Google Scholar 

  13. J. H. Maddocks,Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles, SIAM J. Math. Anal.16 (1985), 47–68.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Mielke, P. Holmes, O. O'Reilly,Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle center, J. Dyn. Diff. Eqns.4 (1992), 95–126.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Thorpe,Elementary topics in differential geometry, New York: Springer-Verlag, (1979).

    MATH  Google Scholar 

  16. P. A. S. Salomão,On the existence of global surfaces of sections in energy levels with saddle-center equillibria of Hamiltonian systems with two degrees of freedom. Doctoral Thesis. In Portuguese, IMEUSP, (2002).

  17. F. Weissler, T. Cazenave, A. Haraux,Detailed Asymptotics for a Convex Hamiltonian System with Two Degrees of Freedom, J. Dyn. Diff. Eqns.5 (1) (1993), 155–187.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Pedro A. S. Salomão

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  1. Pedro A. S. Salomão
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Correspondence to Pedro A. S. Salomão.

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To Professor Jorge Sotomayor for his 60th birthday

Partially supported by FAPESP

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Salomão, P.A.S. Convex energy levels of Hamiltonian systems. Qual. Th. Dyn. Syst 4, 439–454 (2004). https://doi.org/10.1007/BF02970869

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  • DOI: https://doi.org/10.1007/BF02970869

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