The Differential Equations of Gravity-free Double Pendulum: Lauricella Hypergeometric Solutions and Their Inversion

TL;DR

This paper derives explicit closed-form solutions for the gravity-free double pendulum's motion using Lauricella hypergeometric functions, providing a novel analytical approach to a highly nonlinear dynamical system.

Contribution

It introduces a new method to solve the nonlinear ODEs of a gravity-free double pendulum using hypergeometric functions, enabling explicit trajectory calculations.

Findings

Closed-form solutions for GFDP motion derived

Trajectories computed explicitly in terms of hypergeometric functions

Method validated with sample problems demonstrating effectiveness

Abstract

This paper solves in closed form the system of ODEs ruling the 2D motion of a gravity free double pendulum (GFDP), not subjected to any force. In such a way its movement is governed by the initial conditions only. The relevant strongly non linear ODEs have been put back to hyperelliptic quadratures which, through the Integral Representation Theorem (IRT), are driven to the Lauricella hypergeometric functions. We compute time laws and trajectories of both point masses forming the GFDP in explicit closed form. Suitable sample problems are carried out in order to prove the method effectiveness.