1. The problem statement, all variables and given/known data Given potential V(r)=−ke−ar/r; k,a>0. Using the method of the equivalent one-dimensional potential discuss the nature of the motion, stating the ranges of l and E appropriate to each type of motion. When are circular orbits possible? Find the period of small radial oscillations about the circular motion. 2. Relevant equations E=K+V L=K-V pq=dL/dq' Taylor series expansion 3. The attempt at a solution After the usual Lagrangian/momentum stuff I get E=0.5mr'²+(l²/2mr²)-(ke-ar/r), giving the effective potential of Veff=(l²/2mr²)-(ke-ar/r) I only have a rough idea of what the graph of Veff looks like, infinity at zero, zero at infinity, and I assume it has only one extrema (a minimum). The only thing that seems obvious is that for E>=0, the motion is unbounded, and for a circular orbit, r'=0, so E=Veff. The problem is the exponential function makes it difficult to solve for zeroes of dV/dr to locate and determine this minimum value. Also, I'm unsure of what the limits of l will be. All the examples I've seen of this process focused on the energy only. Also, if you expand the potential around the minimum at point r0, since V'=0, you get V(r0)+0.5V''(r0)(r-r0)² Other than that, I'm pretty much stuck at where to go next with this.