Analog Raychaudhuri equation in mechanics

TL;DR

This paper introduces a novel mechanical analog of the Raychaudhuri equation, analyzing trajectory convergence in systems like pendula, supported by experiments, to elucidate complex physics concepts through elementary models.

Contribution

It develops a mechanics analog of the Raychaudhuri equation and demonstrates its application and experimental validation in simple pendulum systems.

Findings

Trajectories of pendula can be analyzed using the Raychaudhuri equation analogy.

Experimental results agree with the theoretical model.

The approach offers an elementary perspective on advanced physics concepts.

Abstract

Usually, in mechanics, we obtain the trajectory of a particle in a given force field by solving Newton's second law with chosen initial conditions. In contrast, through our work here, we first demonstrate how one may analyse the behaviour of a suitably defined family of trajectories of a given mechanical system. Such an approach leads us to develop a mechanics analog following the well-known Raychaudhuri equation largely studied in Riemannian geometry and general relativity. The idea of geodesic focusing, which is more familiar to a relativist, appears to be analogous to the meeting of trajectories of a mechanical system within a finite time. Applying our general results to the case of simple pendula, we obtain relevant quantitative consequences. Thereafter, we set up and perform a straightforward experiment based on a system with two pendula. The experimental results on this system are found to tally well with our proposed theoretical model. In summary, the simple theory, as well as the related experiment, provides us with a way to understand the essence of a fairly involved concept in advanced physics from an elementary standpoint.