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Discontinuity, Nonlinearity, and Complexity 12(4) (2023) 723--735 | DOI:10.5890/DNC.2023.12.002

Thomas Kotoulas, E. Meletlidou

Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

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Abstract

One of the major problems in classical mechanics is to determine the mean field potential in which the motion of a test particle takes place. In the light of the inverse problem of dynamics, we study three-dimensional genuine potentials $V=V(x,y,z)$ producing a set of two-parametric families of regular orbits $f(x,y,z)=c_{1}$, $g(x,y,z)=c_{2}$ ($c_{1}, \; c_{2}=$const.). We focus on homogeneous and axisymmetric potentials which have many physical applications. Then, we establish three differential conditions to be fulfilled by the given two-parametric families of orbits (traced in 3D space by a material point) so that these families can result in the presence of such a potential. All possible cases for the ``\emph{given}'' family of orbits are studied and several compatible pairs of families and potentials are found. Finally, some potentials of physical interest are also presented.

References

1. [1]	Goriely, A. (2001), Integrability and Nonintegrability of Dynamical Systems:19, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co Pte Ltd.

2. [2]	Volchenkov, D. (2021), Nonlinear Dynamics, Chaos, and Complexity: In Memory of Professor Valentin Afraimovich, Higher Education Press, Springer: Singapore.

3. [3]	Meletlidou, E. and Ichtiaroglou, S. (1994), On the number of isolating integrals in perturbed Hamiltonian systems with $n \geq$ 3 degrees of freedom, Journal of Physics A: Mathematical and General, 27, 3919-3926.

4. [4]	Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer: Berlin.

5. [5]	Luo, A.C.J. (2004), Nonlinear dynamics theory of stochastic layers in Hamiltonian systems, Applied Mechanics Reviews, 57(3), 161-172.

6. [6]	Patsis, P.A. and Katsanikas, M. (2014), The phase space of boxy-peanut and X-shaped bulges in galaxies - I. Properties of non-periodic orbits, MNRAS, 445(4), 3525-3545.

7. [7]	Galiulin, A.S. (1984), Inverse Problems in Dynamics, Moscow: Mir.

8. [8]	Szebehely, V. (1974), On the determination of the potential by satellite observations, in Proc. of the Int. Meeting on Earth's Rotation by Satellite Observation, G. Proverbio (ed.), The Univ. of Cagliari Bologna Italy, 31-35.

9. [9]	Bozis, G. (1984), Szebehely's inverse problem for finite symmetrical material concetrations, Astronomy and Astrophysics, 134, 360-364.

10. [10]	Erdi, B. (1982), A generalization of Szebehely's equation for three dimensions, Celestial Mechanics, 28, 209-218.

11. [11]	Bozis, G. (1983), Determination of autonomous three-dimensional force fields from a two-parametric family, Celestial Mechanics, 31, 43-51.

12. [12]	V{a}radi, F. and {E}rdi, B. (1983), Existence of the solution of Szebehely's equation in three dimensions using a two-parametric family of orbits, Celestial Mechanics 30, 395-405.

13. [13]	Puel, F. (1984), Intrinsic formulation of Szebehely's equation, Celestial Mechanics, 32, 209-216.

14. [14]	Bozis, G. and Nakhla, A. (1986), Solution of the three-dimensional inverse problem, Celestial Mechanics, 38, 357-375.

15. [15]	Shorokhov S.G. (1988), Solution of an inverse problem of the dynamics of a particle, Celestial Mechanics, 44, 193-206.

16. [16]	Puel, F. (1992), Explicit solutions of the three-dimensional inverse problem of dynamics using the Frenet reference frame, Celestial Mechanics and Dynamical Astronomy, 53, 207-218.

17. [17]	Bozis, G. (1995), The inverse problem of dynamics: Basic facts, Inverse Problems, 11, 687-705.

18. [18]	Anisiu, M.C. (2004), Two- and three-dimensional inverse problem of dynamics, Mathematica, XLIX(4), 13-26.

19. [19]	Anisiu, M-C. (2005), The energy-free equations of the 3D inverse problem of dynamics, Inverse Problems in Science and Engineering, 13, 545-558.

20. [20]	Bozis, G. and Kotoulas, T. (2005), Homogeneous two-parametric families of orbits in three-dimensional homogeneous potentials, Inverse Problems, 21, 343-356.

21. [21]	Kotoulas, T. and Bozis, G. (2006), Two-parametric families of orbits in axisymmetric potentials, Journal of Physics A: Mathematical and General, 39, 9223-9230.

22. [22]	Anisiu, M.C. and Kotoulas, T. (2006), Construction of 3D potentials from a preassigned two-parametric family of orbits, Inverse Problems, 22, 2255-2269

23. [23]	Sarlet, W., Mestdag, T., and Prince, G. (2017), A generalization of Szebehely's inverse problem of dynamics in dimension three, Reports on Mathematical Physics, 79(3), 367-389.

24. [24]	Boccaletti, D. and Puccaco, G. (1996), Theory of Orbits I: Integrable Systems and Non-perturbative Methods, Springer: Berlin.

25. [25]	Contopoulos, G. (1960), A third integral of motion in a Galaxy, Zeitschrift fur Astrophysik, 49, 273-291.

26. [26]	Caranicolas, N. and Barbanis, B. (1982), Periodic orbits in nearly axisymmetric stellar systems, Astronomy and Astrophysics, 114, 360-366.

27. [27]	Contopoulos, G. and Barbanis, B. (1985), Resonant systems with three degrees of freedom, Astronomy and Astrophysics, 153, 44-54.

28. [28]	Binney, J. and Tremaine, S. (1987), Galactic Dynamics, Princeton University Press, Princeton, New Jersey.

29. [29]	Olaya-Castro, A. and Quiroga, L. (2000), Bose-Einstein condensation in an axially symmetric mesoscopic system, Physica Status Solidi B, 220, 761-764.

30. [30]	Heiss, W. D., Nazmitdinov, R.G., and Radu. S. (1994), Chaos in axially symmetric potentials with Octupole deformation Physical Review Letters, 72(15), 2351-2354.

31. [31]	Bozis, G. and Kotoulas, T. (2004), Three-dimensional potentials producing families of straight lines (FSL), Rendiconti del Seminario della Facolta di Scienze dell' Universita di Cagliari, 74(1-2), 83-99.

32. [32]	Contopoulos, G. and Magnenat, P. (1985), Simple three-dimensional periodic orbits in a galactic-type potential, Celestial Mechanics, 37, 387-414.