Some Novel Aspects of the Plane Pendulum in Classical Mechanics

TL;DR

The paper investigates nonlinear plane pendulum dynamics and their connections to SG/SHG and inverted pendulum models by deriving reductions that link these systems via exact solutions. A unifying nonlinear framework is established in which the plane pendulum, inverted pendulum, SG, and SHG share common reduced equations, enabling a large set of exact solutions and cross-model insights. It then introduces an elliptic pendulum based on Jacobi elliptic functions that interpolates between the plane and hyperbolic pendulums, revealing a one-parameter family of isochronous behavior in the harmonic limit and preserving isochrony at $m=1/2$ to first order. By developing first- and second-anharmonic approximations, the authors obtain numerous exact periodic and hyperbolic solutions and provide detailed appendices, offering a coherent framework and valuable benchmarks for nonlinear oscillator dynamics.

Abstract

We obtain a novel connection between the exact solutions of the plane pendulum, hyperbolic plane pendulum and inverted plane pendulum equations as well as the static solutions of the sine-Gordon and the sine hyperbolic-Gordon equations and obtain a few exact solutions of the above mentioned equations. Besides, we consider the plane pendulum equation in the first anharmonic approximation and obtain its large number of exact periodic as well as hyperbolic solutions.In addition, we obtain two exact solutions of the plane pendulum equation in the second anharmonic approximation. Further, we introduce an elliptic plane pendulum equation in terms of the Jacobi elliptic functions $-{\rm sn}(θ,m)/{\rm dn}(θ,m)$ which smoothly goes over to the the plane pendulum equation in the $m=0$ limit and the hyperbolic plane pendulum equation in the $m = 1$ limit where $m$ is the modulus of the Jacobi elliptic functions. We show that in the harmonic approximation, the elliptic pendulum problem represents a one-parameter family of isochronous system. Further, for the special case of $m = 1/2$, we show that one has an isochronous system even in the first anharmonic approximation. Finally, we also briefly discuss the hyperbolic plane pendulum and obtain a few of its exact solutions in the harmonic as well as the first anharmonic approximation.