Modified relativistic rotator. Toward classical Fundamental Dynamical Systems consisting of a worldline and a single spinor

The Author shows how to construct a class of Lagrangians for relativistic dynamical systems described by position and a single spinor. One arrives to it by imposing three requirements: 1) Hamilton action should be reparametrization invariant, 2) the number of dimensional parameters should be minimal, 3) the spinor phase should be a cyclic variable. In more detail are discussed those of the the Lagrangians which depend on the spinor's null vector and its worldline. An interesting relation between a Hessian determinant and Casimir invariants for such objects leads to the conclusion that no fundamental objects of this kind exist with worldlines uniquely determinable from the Hamilton action and the initial conditions. This unexpected result poses the general question about existence of classical fundamental dynamical systems with well posed Cauchy problem. [A relativistic dynamical system is said to be fundamental if its Casimir invariants are parameters, not constants of motion]