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The "orbits" in this case have no closed-form in terms of circular or elliptic functions. The locus of the body will tend to fill space in an annular region around the central point. This image of a $F=-1/r$ orbit was produced by "central force workbench" by Juan M. Aguirregabiria. The potential is now of the form $V= \ln|r|$. The only ...

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