Geometric Methods in Variable-Mass Dynamics with Darboux-Frame

This study bridges classical mechanics and differential geometry by providing a comprehensivegeometric analysis of momentum, force, and their higher-order derivatives for particles moving along spacecurves. Focusing on both constant-mass and variable-mass systems, we develop a unified theoretical frame-work using the Frenet and Darboux frames. The core contribution is the resolution of the force derivative, oryank, into its intrinsic geometric components, extending Siacci’s classical theorem and recent advancementsto include this underexplored quantity. We derive formulations for the radial decomposition of yank in theosculating and rectifying planes, offering insights into the transient dynamics of mechanical systems. The the-oretical model is applied to specific trajectories, including slant helices, to demonstrate its utility in predictingthe behavior of variable-mass systems.