An analog of the pendulum/rotor system with elementary solutions

We explore a novel one-dimensional model potential, V (x), which mimics many aspects of the classic pendulum/rotor system, including low- and high-energy limits which are exactly soluble, as well as a divergent period separating the two. Using energy methods, the classical period, T(E), is evaluated in closed form for all values of the energy (0 < E < ∞) in terms of simple logarithmic functions, in contrast to the pendulum/rotor system where the results require elliptic integrals. We find an unexpected duality between the exact expressions for the period on either side of the divergence. Using both energy methods, and a Newton’s law differential equation approach, we present closed form expressions for trajectory solutions, x(t) and p(t) = mv(t) (or the linear momentum), in terms of hyperbolic functions, which explicitly give the same values for T(E).