Variable-mass dynamics made explicit: a rotating falling-chain test case

Classical variational formulations are typically presented for systems of constant total mass, which can obscure how to treat problems where a convenient subsystem gains or loses mass. We consider a guided ‘falling chain’ variant in which a uniform chain of length L interacts with a horizontal disc rotating at constant angular speed ω. A single generalized coordinate r(t) describes the chain length on the disc while the remaining length (Formula presented) L−r(t) hangs vertically. We derive the same equation of motion using fixed-mass Lagrangian, D’Alembert, variable-mass and Lagrange-multiplier formulations, thus comparing energy-based, force-based, subsystem-based and constraint-based viewpoints within a single problem. A compact nondimensional form shows that the dynamics depends on a single parameter (Formula presented) (Formula presented), yields a closed-form equilibrium relation for the on-disc fraction, and allows a simple local analysis showing that this equilibrium is linearly unstable in the present idealized model. The example is intended as a short worked comparison problem for advanced analytical mechanics teaching, illustrating how subsystem choice, generalized forces, constraint reactions, and interface momentum flux must be handled in order for fixed-mass and variable-mass descriptions to remain consistent.