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Grigoriadou, Simela; Bozis, George; Elmabsout, Badaoui

doi:10.1023/a:1008333100119

Cited by 9 publications

(3 citation statements)

References 8 publications

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“…The differential condition (18) is the differential condition for the slope function γ, which, if it is satisfied, ensures the existence of a potential (10). On the other hand, we consider that the condition ( 18) is satisfied by the slope function γ.…”

Section: One Condition On the Slope Function γmentioning

confidence: 99%

“…A review on basic facts of the inverse problem in dynamics was made by [9]. Solvable cases of the planar inverse problem were given in [10]. In a series of papers ( [11][12][13]), the authors presented methodologies; with the aid of these methodologies, one can obtain monoparametric families of planar orbits in the direct problem of Newtonian dynamics.…”

Section: Introductionmentioning

confidence: 99%

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Monoparametric Families of Orbits Produced by Planar Potentials

Kotoulas

2023

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We study the motion of a test particle on the xy−plane. The particle trajectories are given by a one-parameter family of orbits f(x,y) = c, where c = const. By using the tools of the 2D inverse problem of Newtonian dynamics, we find two-dimensional potentials that produce a pre-assigned monoparametric family of regular orbits f(x,y)=c that can be represented by the “slope function” γ=fyfx uniquely. We apply a new methodology in order to find potentials depending on specific arguments, i.e., potentials of the form V(x,y)=P(u) where u=x2+y2,xy,x3−y3,xy (x,y≠ 0). Then, we establish one differential condition for the family of orbits f(x,y) = c. If it is satisfied, it guarantees the existence of such a potential, generating the above family of planar orbits. Then, the potential function V=V(x,y) is found by quadratures. For known families of curves, e.g., ellipse, the logarithmic spiral, the lemniscate of Bernoulli, and circles, we find homogeneous and polynomial potentials that are compatible with this family of orbits. We offer pertinent examples that cover all of the cases, and we examine which of these potentials are integrable. We also study one-dimensional potentials. The families of straight lines in 2D space are also examined.

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Section: One Condition On the Slope Function γmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Monoparametric Families of Orbits Produced by Planar Potentials

Kotoulas

2023

Axioms

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show abstract

“…A review on the basic facts of the inverse problem in dynamics was made by [10]. Other solvable cases of the planar inverse problem were produced by [11][12][13]. Studying the direct problem of Newtonian dynamics [14][15][16], the authors presented methodologies by which one can obtain monoparametric families of planar orbits if the potential V = V(x, y) is given.…”

Section: Introductionmentioning

confidence: 99%

Real and Complex Potentials as Solutions to Planar Inverse Problem of Newtonian Dynamics

Kotoulas

2024

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We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits f(x,y)=c on the xy-plane (where c is the parameter of the family of orbits). This family of orbits can be represented by the “slope function” γ=fyfx uniquely. A new methodology is applied to the basic equation of the planar inverse problem in order to find potentials of a special form, i.e., V=F(x+y)+G(x−y), V=F(x+iy)+G(x−iy) and V=P(x)+Q(y), and polynomial ones. According to this methodology, we impose differential conditions on the family of orbits f(x,y) = c. If they are satisfied, such a potential exists and it is found analytically. For known families of curves, e.g., circles, parabolas, hyperbolas, etc., we find potentials that are compatible with them. We offer pertinent examples that cover all the cases. The case of families of straight lines is referred to.

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Cited by 9 publications

References 8 publications

Monoparametric Families of Orbits Produced by Planar Potentials

Monoparametric Families of Orbits Produced by Planar Potentials

Real and Complex Potentials as Solutions to Planar Inverse Problem of Newtonian Dynamics

Symbolic Computations

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