A solvable version of the direct problem of Dynamics

Given any two-dimensional potential \(V(x,y)\), we find (if there exists, of course) without integration of the pertinent equation of motion a monoparametric family of orbits of the form \(f(x,y)=y+h(x)=c\) (whose members are shifted paralelly to the \(y\)-axis) traced by a unit mass material point.The main tool to this end is the non-linear second order partial differential equation of the inverse problem of Dynamics (Bozis, 1995). In general, a condition on the given potential is derived: in case this condition is fulfilled, the function gamma can be obtained as the common root of two polynomial equations. In certain particular cases (isotach orbits, one-dimensional potentials) the differential equations become ordinary and the solution is found to completion in a different manner.

George Bozis
University of Thessaloniki, Department of Theoretical Mechanics, Thessaloniki, Greece

Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy, Romania

Cristina Blaga
Astronomical Institute of the Romanian Academy, Astronomical observatory Cluj-Napoca, Romania

monoparametric family of orbits; direct and inverse problem; isotach orbits.

G.Bozis, M-C, Anisiu, C.Blaga, A solvable version of the direct problem of Dynamics, Rom. Astron. J., 10 (1) (2000), 59-70, https://www.astro.ro/~roaj/volume/abs10-1.html

Book of abstracts: https://www.astro.ro/~roaj/volume/abs10-1.html#A%20Solvable%20Version%20Of%20The%20Direct%20Problem%20Of

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