Geometric Decomposition of Force and Yank for Variable-Mass Systems in Minkowski 3-Space

We develop a differential-geometric framework for variable-mass particles moving along non-lightlike curves with non-vanishing curvature in Minkowski 3-space E13, employing the Frenet–Serret apparatus adapted to a Lorentzian signature. The force is defined as the time derivative of momentum, F=d(mv)/dt, incorporating mass variation through a Meshchersky-type reactive term; no covariant four-momentum formulation is assumed. Explicit closed-form expressions are derived for the momentum vector P(t), force F(t), and yank Y(t)=dF/dt for three distinct causal types of regular Frenet curves: spacelike curves with a spacelike principal normal, spacelike curves with a timelike principal normal, and timelike curves. The tangential yank component carries the causal sign factor δB, reflecting the type of curve. A theorem on the evolution of kinetic energy separates the inertial contribution mvv˙ from the reactive contribution 12m˙v2 due to mass variation. A radial decomposition of the force in the osculating plane generalizes Siacci’s classical theorem to Lorentzian geometry and variable-mass systems. When the rectifying coordinate b is non-zero, a corresponding decomposition of the yank is also obtained. Three illustrative physical scenarios are discussed: rocket motion with variable mass (with potential future relevance to trajectory prediction, stability analysis, and motion-anomaly assessment in unmanned systems), a geometric analogy for orbital parameter changes, and particle motion in a magnetic monopole field. Two fully worked examples (a Lorentzian helix and a logarithmic spiral) provide explicit closed-form expressions for all geometric and dynamical quantities, accompanied by numerical plots. The results recover the Euclidean case in the appropriate signature limit.