Empirical equivalence under uniform boosts (${\mathbf{x}}_i^{\prime }={\mathbf{x}}_i+\mathbf{u}t+\mathbf{a}$)

For Newtonian mechanics with forces depending only on relative positions, a common Galilean transformation leaves relative positions and accelerations unchanged. Internal mechanical observations therefore cannot discriminate rival choices of a wholly uncoupled privileged uniform rest state. The qualification concerns uniform motion; it does not imply that rotation or acceleration is unobservable, or that special relativity makes identical predictions.