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 analyze the behaviour of a suitably defined family of trajectories of a given mechanical system. Such an approach leads us to develop a mechanics analogue 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 of this system are found to tally well with our proposed theoretical model. In summary, the simple theory and the related experiment provide us with a way to understand the essence of a fairly involved concept in advanced physics from an elementary standpoint.
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Acknowledgements
We would like to thank Anang Kumar Singh and Kushal Lodha for their help in the experimental part. Rajendra Prasad Bhatt thanks Department of Physics, IIT Kharagpur where he was a student in the Master of Science programme, when this work was carried out.
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Rajendra Prasad Bhatt is currently a research scholar at IUCAA, Pune. The work described here was carried out during his MSc(2yr) tenure at IIT Kharagpur.
Anushree Roy is a faculty member at the Department of Physics, IIT Kharagpur.
Sayan Kar is a faculty member at the Department of Physics, IIT Kharagpur.
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Bhatt, R.P., Roy, A. & Kar, S. Analog Raychaudhuri Equation in Mechanics. Reson 28, 389–410 (2023). https://doi.org/10.1007/s12045-023-1562-z
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DOI: https://doi.org/10.1007/s12045-023-1562-z