Families of Planar Orbits in Polar Coordinates Compatible with Potentials
Year: 2024
Type: article
Abstract: In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits f(r,θ)=c in polar coordinates (where c is the parameter varying along the family of orbits), which are generated by planar potentials V=V(r,θ). The corresponding family of orbits can be uniquely represented by the “slope function” γ=fθfr. By using the basic partial differential equation of the planar inverse problem, which combines families of orbits and potentials, we apply a new methodology in order to find specific potentials, e.g., V=A(r)+B(θ) or V=H(γ) and one-dimensional potentials, e.g., V=A(r) or V=G(θ). In order to determine such potentials, differential conditions on the family of orbits f(r,θ) = c are imposed. If these conditions are fulfilled, then we can find a potential of the above form analytically. For the given families of curves, such as ellipses, parabolas, Bernoulli’s lemniscates, etc., we find potentials that produce them. We present suitable examples for all cases and refer to the case of straight lines.
Source: Mathematics
DOI: 10.3390/math12213435
Author: Thomas Kotoulas
Institution: Aristotle University of Thessaloniki
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,955
OpenAlex ID: W4404044061