Breathing Relativistic Rotators and Fundamental Dynamical Systems

Recently, it was shown, that the mechanical model of a massive spinning particle proposed by Kuzenko, Lyakhovich and Segal in 1994, which is also the fundamental relativistic rotator rediscovered independently 15 years later by Staruszkiewicz in quite a different context, is defective as a dynamical system, that is, its Cauchy problem is not well posed. This dynamical system is fundamental, since its mass and spin are parameters, not arbitrary constants of motion, which is a classical counterpart of quantum irreducibility. It is therefore desirable to find other objects which, apart from being fundamental, would also have well posed Cauchy problem. For that purpose, a class of breathing rotators is considered. A breathing rotator consists of a single null vector associated with position and moves in accordance with some relativistic laws of motion. Surprisingly, breathing rotators which are fundamental, are also defective as dynamical systems. More generally, it has been shown, that the necessary condition for a breathing rotator to be similarly defective, is functional dependence of its Casimir invariants of the Poincar{\'e} group.