Nonlinear Calculus in Practice 2: Rigidity, Lebesgue, Solid Deformation, and Multibody Dynamics

This sequel to Nonlinear Calculus in Practice develops mathematical connections and worked applications of the Matrix-of-Error framework. Lebesgue integration provides the common foundation linking the earlier probability and quantum-measurement studies to time-dependent budgets, deformation measures, and interacting-body mechanics. A general weighted-budget lemma, abstracted from the earlier enstrophy envelope, supplies equality and quantitative control, with a finite threshold under a persistent decay margin; the periodic flow identifies its rigid configuration. The connection to measured group theory is examined through the work of Margulis, Bader, and Furman, while geometric rigidity supplies a separate, direct connection to solid deformation. An explicit volume-preserving warping map is integrated without prescribing an untested material law. A separate proposed test model couples solid motion, internal strain, and heat through seven stress blocks and companion evolution balances. Worked gravitational and articulated five-body examples retain pair interactions, configuration-dependent inertia, and applied contributions through integration by parts. A controlled comparison finds that replacing resolved bodies by point centers can shift fitted masses by up to 3.205% without improving wider-range held-out gravity predictions. An additional quadrupole correction reduces that mass error to 0.236%; a separate in-range test distinguishes local transfer from extrapolation. Retaining the full volume contribution recovers the known parameters near the observed numerical resolution. A separate leading-quadrupole calculation accounts coherently for radiation from all five bodies. The result is a continuation of the shared integration method through further mathematical systems, with analytic identities, numerical consistency checks, and prospective physical applications distinguished explicitly.