RELATIVISTIC DYNAMICS OF SIMPLE AND COUPLED DOUBLE PENDULUMS

Abstract

The double pendulum is one of the most widely studied systems in classical mechanics because of its rich nonlinear behaviour and its importance in understanding coupled oscillations and deterministic chaos. While its classical dynamics are well established, relativistic extensions have received comparatively little attention. In this work, we develop a relativistic formulation of both the simple and coupled double pendulum within the framework of special relativity using the Lagrangian formalism. Starting from the physical model, the equations of motion are derived by incorporating the relativistic kinetic energy while retaining the classical gravitational potential. The resulting nonlinear equations are transformed into an equivalent first-order state-space system and solved numerically using the classical fourth-order Runge--Kutta method. Numerical simulations are carried out in dimensionless form for small-angle in-phase motion, small-angle out-of-phase motion, and large-angle nonlinear motion. Time evolution, phase-space trajectories, and Lorentz factors are examined to investigate the influence of relativistic effects on the system dynamics. The results show that the relativistic formulation naturally reproduces the classical behaviour in the low-velocity limit while introducing velocity-dependent corrections that become increasingly significant as the system approaches relativistic speeds. Besides extending a familiar nonlinear system into the relativistic regime, the present formulation provides a self-contained example that brings together Lagrangian mechanics, special relativity, and numerical methods within a single framework, making it a useful reference for advanced studies in classical and computational mechanics.