Inverse-Cube Gravity

The_Brain

What would happen if gravity obeyed an inverse cube law instead of an inverse square one? With regards to the 2-body problem, would angular momentum still be conserved, or does that tangential acceleration term in polar coordinates not drop out? I'm already thinking a 2-body system like this would inherently be unstable, but whether the instability is from angular momentum not being conserved or it is just inherently unstable such as the 3-body problem I am not sure about. Any thoughts?