The one-dimensional potential energy function that is analogous to a two-dimensional track

When learning about potential energy functions, students are sometimes told that the potential energy function is analogous to that of a particle sliding along a frictionless roller coaster track or wire confined to a vertical plane with peaks and valleys of the track corresponding to unstable and stable equilibrium points. However, motion along a track with height z(x) is a constrained two-dimensional motion, not a one-dimensional motion, so the exact nature of this analogy may be unclear. We show that the horizontal motion of a point mass sliding along a frictionless track z(x) and subject to a uniform gravitational field is equivalent to the motion of a particle in one dimension characterized by an “analogous potential energy” function UE(x), which generally depends on the total energy of the system (and thus on the initial conditions). We derive a general expression for UE(x) in terms of z(x) and the total energy and show that the equilibrium points of the actual potential energy U(x)=mgz(x) are also static equilibrium points for UE(x) with the same stability. However, UE(x) may have additional dynamic equilibrium points that are not present for U(x). As an example, we derive UE(x) for a double well track and determine the period of oscillations on that track. The results show that in general a single track corresponds to many different analogous potential energy functions, each with a different value for the total energy.