From Free Bodies to Static Structure

This preprint examines a question that precedes conventional static analysis: how do initially separate, freely movable bodies become an assembly that may legitimately be treated as a static structure? The paper describes formation as a sequence of action-driven motions, encounters, contact changes, dissipation, and the generation of reaction paths. It proposes an action-relative working criterion for the transition to static treatment: transient motion and dynamically relevant inertia must have decayed, the generated contacts and boundaries must permit reactions that balance the prescribed action, and no action-effective admissible first-order motion may remain. This condition does not require complete kinematic fixation. An assembly may retain admissible neutral or action-opposing motions while remaining arrested under the action being considered. A sphere on a horizontal plane, a sphere in a horizontal V-groove, and a tetrahedral four-sphere arrangement serve as deliberately simple reference systems. The four-sphere example demonstrates that the same local contact geometry may be arrested when the lower spheres are restrained, yet become mobile through radial spreading on a smooth, unconfined base. Static treatability therefore depends not only on local contacts, but also on the prescribed action and the reaction capability of the complete body–boundary system. The mathematical ingredients draw on established unilateral-contact mechanics, virtual work, convex duality, rigidity, and feasible-motion analysis. The contribution of this paper is their organisation around the physical transition through which the object of static analysis first comes into existence. This is a preprint and has not yet undergone formal peer review.