Straight line orbits in Hamiltonian flows

  • J. E. HowardEmail author
  • J. D. Meiss
Original Article


We investigate straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem is proven for natural Hamiltonians quadratic in the momenta for arbitrary dimensions and is considered in more detail for two and three dimensions. Next we specialize to homogeneous potentials and their superpositions, including the familiar Hénon–Heiles problem. It is shown that SLO’s can exist for arbitrary finite superpositions of N-forms. The results are applied to a family of potentials having discrete rotational symmetry as well as superpositions of these potentials.


Natural flows Hamiltonian flows Hénon–Heiles Homogeneous potentials Direct and indirect methods 


  1. Antonov V.A., Timoshkova E.I.: Simple trajectories in a rotationally symmetric gravitational field. Astron. Rep. 37(2), 138–144 (1993)MathSciNetADSGoogle Scholar
  2. Arribas M., Elipe A., Kalvouridis T., Palacios M.: Homographic solutions in the planar n + 1 body problem with quasi-homogeneous potentials. Celest. Mech. Dyn. Astron. 99, 1–12 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. Bozis G., Kotoulas T.A.: Three-dimensional potentials producing families of straight lines (FSL). Rendiconti Seminario Facout Scienze Universat Cagliari 74(1–2), 83–98 (2004)Google Scholar
  4. Bozis G., Kotoulas A.T.: Homogeneous two-parametric families of orbits in three-dimensional homogeneous potentials. Inverse Probl. 21, 343–356 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. Bozis G.: Generalization of Szebehely’s equation. Celest. Mech. 29, 329–334 (1983)zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. Bozis G.: The inverse problem of dynamics: basic facts. Inverse Probl. 11(4), 687–708 (1995)zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. Chenciner A., Gerver J., Montgomery R., Simó C.: Simple choreographic motions of N bodies: a preliminary study. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics, pp. 287–308. Springer, New York (2002)CrossRefGoogle Scholar
  8. Hall L.S.: A theory of exact and approximate configurational invariants. Physica D 8, 90–116 (1983)zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. Hénon M., Heiles C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)CrossRefADSGoogle Scholar
  10. Moeckel R.: On central configurations. Math. Zeit. 205, 499–517 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  11. Maciejewski A.J., Przybylska M.: Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential. J. Math. Phys. 46, 062901 (2005)CrossRefMathSciNetADSGoogle Scholar
  12. Puel F.: Intrinsic formulation of the equation of Szebehely. Celest. Mech. Dyn. Astron. 32, 209 (1984)zbMATHMathSciNetGoogle Scholar
  13. Puel F.: Explicit solutions of the three-dimensional inverse problem of dynamics, using the Frenet reference frame. Celest. Mech. Dyn. Astron. 53(3), 207–218 (1988)CrossRefMathSciNetADSGoogle Scholar
  14. Szebehely V.: Theory of orbits: the restricted problem of three bodies. Academic Press, New York (1967)Google Scholar
  15. Szebehely, V.: On the determination of the potential. In Zagar, F., Proverbio, E. (eds.) Il Problema Della Rotazione Terrestre. Bologna, Universit Di Cagliari, 1974. Rendiconti Del Seminario Della Facoltà Di Scienze dell’Università Di CagliariGoogle Scholar
  16. Van Der Merwe P.D.T.: Solvable forms of a generalized H énon–Heiles system. Phys. Lett. A 156(5), 216–220 (1991)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Laboratory for Atmospheric and Space Physics and Center for Integrated Plasma StudiesUniversity of ColoradoBoulderUSA
  2. 2.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

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