Imagine that Flatland habitats go into space. All stars and planets are the circles instead of spheres and gravity law is $F=Gm_1m_2/R$ instead of $F=Gm_1m_2/R^2$.
How the other formulas will be changed in this case?
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1 Answer
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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 formula that is unchanged is Kepler's second law: Equal areas are swept out in equal time, as this is true for all central forces.
