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Geometrically similar orbits in homogeneous potentials. (English) Zbl 0774.34009

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.

MSC:

34A55 Inverse problems involving ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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