Bozis, George; Stefiades, Apostolos Geometrically similar orbits in homogeneous potentials. (English) Zbl 0774.34009 Inverse Probl. 9, No. 2, 233-240 (1993). Summary: If \(V(r,\theta)=r^ m G(\theta)\) is, in polar coordinates, a homogeneous potential, of degree \(m\), which can give rise to a given family \(f(r,\theta)=rg(\theta)=\) constant of geometrically similar planar orbits, a second-order ordinary linear, in \(G(\theta)\), homogeneous differential equation is found for any function \(g(\theta)\) specifying the family. In contrast with the partial differential equation relating potentials with families, this ordinary equation is much easier to handle. For certain choices of \(g(\theta)\) it can be solved to completion, for any \(m\). Examples are offered. Two special cases referring to central potentials and isoenergetic families are studied. Cited in 1 Document MSC: 34A55 Inverse problems involving ordinary differential equations 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations Keywords:homogeneous potential; geometrically similar planar orbits; second-order ordinary, linear, homogeneous differential equation; central potentials; isoenergetic families PDFBibTeX XMLCite \textit{G. Bozis} and \textit{A. Stefiades}, Inverse Probl. 9, No. 2, 233--240 (1993; Zbl 0774.34009) Full Text: DOI