Skip to main content

Special Families of Orbits in the Direct Problem of Dynamics

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 = f y /f x , 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.

This is a preview of subscription content, access via your institution.

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.

  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 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Anisiu, MC., Blaga, C. & Bozis, G. Special Families of Orbits in the Direct Problem of Dynamics. Celestial Mechanics and Dynamical Astronomy 88, 245–257 (2004). https://doi.org/10.1023/B:CELE.0000017170.88493.e6

Download citation

  • integrability
  • inverse and direct problem of dynamics
  • special families of orbits