Fig. 7: Force balance on an ensemble of rigid bodies in steady motion.

a Train of domain wall–skyrmions in steady motion. Here, the number of skyrmions N_s = 3 is taken as an example and the transverse forces are ignored. b Uniformly moving train of cuboid–balls in direct contact. c Uniformly moving train of cuboid–balls linked by springs. d Forces on every entity. For the domain wall–skyrmions train, F^H is the field-induced force, which only acts on the domain wall; \(F^{{{{{\mathcal{D}}}}}}\) are the dissipative forces, which act on every magnetic entity in the ensemble; F_i,i+1 and F_i+1,i (i = 0, 1, 2, … N_s) are a pair of action and reaction forces, which arise from dipolar and/or exchange interactions between the magnetizations in neighboring magnetic entities. For the cuboid–balls train, F^H is an applied pushing or pulling force, which directly acts on the cuboid; \(F^{{{{{\mathcal{D}}}}}}\) are the friction forces, which act on every entity in the ensemble; F_i,i+1 and F_i+1,i (_i = 0, 1, 2, … N_s) are the action and reaction forces, which arise from direct contact of neighboring cuboid and balls or from mediation of the spring. Steady motion means that the entire ensemble moves at a uniform velocity v.