Prescribing natural trajectories from scale factors

We pose and solve a classical inverse problem suitable for advanced mechanics courses. Given a two-dimensional orthogonal coordinate system in the plane, we ask how to construct a scalar potential such that, on a chosen energy shell, one set of coordinate curves becomes an exact set of mechanical trajectories. More precisely, if a particle is launched tangent to one of the prescribed curves with the chosen energy, then the resulting natural motion remains on that curve. Working directly with the scale factors of the coordinate system, we obtain a simple closed-form expression for the potential in terms of the scale factor along the prescribed direction of motion. The construction is elementary, constructive, and visual. In parabolic coordinates, imposing central symmetry forces the remaining free function to be constant and recovers an inverse-radius potential of Kepler form. In polar and confocal elliptic–hyperbolic coordinates, the same method produces inverse-square-type and more exotic potentials tied to the geometry of the coordinates. A short geometric coda, based on the Jacobi metric and geodesic curvature, recasts the same condition in coordinate-free form. We conclude with concrete suggestions for classroom use, including analytic derivations, numerical projects, and links to the optics–mechanics analogy.