Celestial Mechanics and Dynamical Astronomy

, Volume 88, Issue 3, pp 245–257 | Cite as

Special Families of Orbits in the Direct Problem of Dynamics

  • M.-C. Anisiu
  • C. Blaga
  • G. Bozis
Article

Abstract

The direct problem of dynamics in two dimensions is modeled by a nonlinear second-order partial differential equation, which is therefore difficult to be solved. The task may be made easier by adding some constraints on the unknown function γ = fy/fx, where f(x, y) = c is the monoparametric family of orbits traced in the xy Cartesian plane by a material point of unit mass, under the action of a given potential V(x, y). If the function γ is supposed to verify a linear first-order partial differential equation, for potentials V satisfying a differential condition, γ can be found as a common solution of certain polynomial equations.

The various situations which can appear are discussed and are then illustrated by some examples, for which the energy on the members of the family, as well as the region where the motion takes place, are determined. One example is dedicated to a Hénon—Heiles type potential, while another one gives rise to families of isothermal curves (a special case of orthogonal families). The connection between the inverse/direct problem of dynamics and the possibility of detecting integrability of a given potential is briefly discussed.

integrability inverse and direct problem of dynamics special families of orbits 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anisiu, M.-C. and Pal, A.: 1999, 'Special families of orbits for the Hénon-Heiles potential', Rom. Astronom. J. 9(2), 179–185.Google Scholar
  2. Bozis, G.: 1983, 'Inverse problem with two-parametric families of planar orbits', Celest. Mech. 31, 129–143.Google Scholar
  3. Bozis, G.: 1984, 'Szebehely's inverse problem for finite symmetrical material concentrations', Astronom. Astrophys. 134(2), 360–364.Google Scholar
  4. Bozis, G.: 1995, 'The inverse problem of dynamics. Basic facts', Inverse Problems 11, 687–708.Google Scholar
  5. Bozis, G. and Anisiu, M.-C.: 2001, 'Families of straight lines in planar potentials', Rom. Astronom. J. 11(1), 27–43.Google Scholar
  6. Bozis, G. and Ichtiaroglou, S.: 1994, 'Boundary curves for families of planar orbits', Celest. Mech. & Dyn. Astr. 58, 371–385.Google Scholar
  7. Bozis, G. and Meletlidou, E.: 1998, 'Nonintegrability detected from geometrically similar orbits', Celest. Mech. & Dyn. Astr. 68, 335–346.Google Scholar
  8. Bozis, G., Anisiu, M.-C. and Blaga, C.: 1997, 'Inhomogeneous potentials producing homogeneous orbits', Astron. Nachr. 318, 313–318.Google Scholar
  9. Bozis, G., Anisiu, M.-C. and Blaga, C.: 2000, 'A solvable version of the direct problem of dynamics', Rom. Astronom. J. 10(1), 59–70.Google Scholar
  10. Courant, R. and Hilbert, D.: 1962, Methods of Mathematical Physics, Vol. II, Partial Differential Equations, Interscience Publishers, New York.Google Scholar
  11. Ichtiaroglou, S. and Meletlidou, E.: 1990, 'On monoparametric families of orbits sufficient for integrability of planar potentials with linear or quadratic invariants', J. Phys. A: Math. Gen. 23, 3673–3679.Google Scholar
  12. Mishina, A. P. and Proskuryakov, I. V.: 1965, Higher Algebra, Pergamon Press, Oxford.Google Scholar
  13. Morales Ruiz, J. J.: 1999, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser, Basel.Google Scholar
  14. Puel, F.: 1999, 'Potentials having two-orthogonal families of curves as trajectories', Celest. Mech. & Dyn. Astr. 74, 199–210.Google Scholar
  15. Szebehely, V.: 1974, 'On the determination of the potential by satellite observation'. In: E. Proverbio (ed.), Proceedings of the International Meeting on Earth's Rotations by Satellite Observations, Cagliari, Bologna, pp. 31–35.Google Scholar
  16. Yoshida, H.: 1987, 'A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential', Physica D 29, 128–142.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • M.-C. Anisiu
    • 1
  • C. Blaga
    • 2
  • G. Bozis
    • 3
  1. 1.T. Popoviciu Institute of Numerical AnalysisRomanian AcademyCluj-NapocaRomania
  2. 2.Faculty of Mathematics and Computer ScienceBabes-Bolyai UniversityCluj-NapocaRomania
  3. 3.Department of PhysicsAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations