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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Planar and Three-Dimensional Potentials Producing Two-Parametric Families of Orbits

Journal of Applied Nonlinear Dynamics 14(3) (2025) 535--550 | DOI:10.5890/JAND.2025.09.004

Thomas Kotoulas

Aristotle University of Thessaloniki, Department of Physics, Thessaloniki, Post code:54124, Greece

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Abstract

We study the motion of a test particle in a conservative force field. We find three-dimensional and planar potentials producing a pre-defined two-parametric family of spatial orbits given in the solved form $f(x,y,z)$=$c_{1}$, $g(x,y,z)$=$c_{2}$ which can be represented by a pair of functions $\alpha(x,y,z)$ and $\beta(x,y,z)$ uniquely. We apply a \textit{new} methodology in order to find potentials depending on the orbital functions $\alpha(x,y,z)$ and $\beta(x,y,z)$ , i.e., potentials of the form $V(x,y,z)=F(w(\alpha, \; \beta))$ where $w$ is an specific combination of the pair ($\alpha, \; \beta$). For an appropriate pair ($\alpha, \; \beta$), we find homogeneous potentials of zero degree, axially symmetric potentials and other results. Moreover, we focus our interest on 2D potentials such as Newtonian, cored, logarithmic and quartic potentials which have many astronomical applications. Families of straight lines are also examined.

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