Some Novel Aspects of the Plane Pendulum in Classical Mechanics

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.

💡 Research Summary

This paper presents a novel exploration of the classical plane pendulum problem, uncovering deep mathematical connections and new exact solutions. The central theme is demonstrating that the exact solutions of the plane pendulum, hyperbolic pendulum, and inverted pendulum equations, as well as the static solutions of the sine-Gordon (SG) and sine-hyperbolic-Gordon (SHG) equations, can be derived from the solutions of a few common auxiliary nonlinear equations. By introducing specific transformations (e.g., u = sin(θ/2)), the authors reduce these distinct physical problems to solving a single master equation of the form (1-u²)u_yy + u(u_y)² = ±u(1-u²)². This unified approach yields multiple families of exact solutions expressed in terms of Jacobi elliptic functions (sn, cn, dn) and hyperbolic functions (tanh, sech), including complex PT-invariant solutions.

The paper then investigates the plane pendulum equation within the framework of anharmonic approximations. In the first anharmonic approximation, the resulting equation is shown to be analogous to the static field equation of the symmetric φ⁴ model. Leveraging this correspondence, the authors generate a large number of new exact periodic and hyperbolic solutions for the pendulum, also calculating the time period for each periodic solution. For the more constrained second anharmonic approximation, linked to a φ²-φ⁴-φ⁶ model with specific coefficient relations, only two exact solutions are obtained.

A significant innovative contribution is the introduction of a novel “elliptic pendulum” equation defined as θ_tt = -sn(θ,m)/dn(θ,m), where ’m’ is the modulus of the Jacobi elliptic functions. This equation provides a smooth interpolation between the standard plane pendulum (m=0) and the hyperbolic pendulum (m=1). Analysis reveals that in the harmonic approximation, this elliptic pendulum represents a one-parameter family of isochronous systems for all 0≤m≤1, meaning all oscillatory solutions have the same period independent of amplitude. Remarkably, for the special case of m=1/2, the system remains isochronous even within the first anharmonic approximation, presenting a rare example of an isochronous nonlinear oscillator beyond the harmonic limit.

Finally, the paper briefly examines the hyperbolic pendulum equation, deriving several of its exact solutions in both harmonic and first anharmonic approximations. Overall, this work significantly enriches the understanding of the classical pendulum by revealing hidden solution structures, establishing connections to important field-theoretic models, and discovering new isochronous systems within a generalized elliptic pendulum framework.