Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction.

The back-reaction of a radiated wave on the emitting source is a general problem. In the most general case, back-reaction on moving wave sources depends on their whole history. Here we study a model system in which a pointlike source is piloted by its own memory-endowed wave field. Such a situation is implemented experimentally using a self-propelled droplet bouncing on a vertically vibrated liquid bath and driven by the waves it generates along its trajectory. The droplet and its associated wave field form an entity having an intrinsic dual particle-wave character. The wave field encodes in its interference structure the past trajectory of the droplet. In the present article we show that this object can self-organize into a spinning state in which the droplet possesses an orbiting motion without any external interaction. The rotation is driven by the wave-mediated attractive interaction of the droplet with its own past. The resulting "memory force" is investigated and characterized experimentally, numerically, and theoretically. Orbiting with a radius of curvature close to half a wavelength is shown to be a memory-induced dynamical attractor for the droplet's motion.