Some Novel Aspects of the Plane Pendulum in Classical Mechanics

TL;DR

This paper explores novel exact solutions and connections between various forms of the plane pendulum, including elliptic, hyperbolic, and anharmonic approximations, revealing new integrable cases and isochronous properties.

Contribution

It introduces new exact solutions, establishes connections with sine-Gordon equations, and analyzes isochronous behavior in elliptic and anharmonic pendulum models.

Findings

Derived novel exact solutions for plane, hyperbolic, and elliptic pendulums.

Established a connection between pendulum equations and sine-Gordon models.

Identified conditions for isochronous behavior in elliptic pendulum systems.

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 $-{\rm sn}(\theta,m)/{\rm dn}(\theta,m)$ 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.