False constraints. A toy model for studying dynamical systems with degenerate Hessian form

This paper studies various aspects of the motion of relativistic rotators, both in the presence and absence of external fields, using a toy model which, in a sense, can be regarded as a non-relativistic limit of the rotators. In a simpler setup, this enables one to gain an insight into the principal difference between mechanical systems with singular and non-singular Hessian, whilst avoiding the complications resulting from the more intricate form of the equations of motion in the fully relativistic regime. In particular, one can comprehend the apparent contradiction between Hessian singularity and simultaneous occurrence of unique solutions for the motion of the fundamental relativistic rotator minimally coupled to the electromagnetic field. With the aid of the toy model the author supports and illustrates his thesis put forward elsewhere that the Hessian singularity is a defect that makes physically unviable some geometric models of spinning particles considered in the literature.Comment: v2: 16 pages (in v2 language corrections + minor changes

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

Fundamental Relativistic Rotator. Hessian singularity and the issue of the minimal interaction with electromagnetic field

There are two relativistic rotators with Casimir invariants of the Poincar\'{e} group being fixed parameters. The particular models of spinning particles were studied in the past both at the classical and quantum level. Recently, a minimal interaction with electromagnetic field has been considered. We show that the dynamical systems can be uniquely singled out from among other relativistic rotators by the unphysical requirement that the Hessian referring to the physical degrees of freedom should be singular. Closely related is the fact that the equations of free motion are not independent, making the evolution indeterminate. We show that the Hessian singularity cannot be removed by the minimal interaction with the electromagnetic field. By making use of a nontrivial Hessian null space, we show that a single constraint appears in the external field for consistency of the equations of motion with the Hessian singularity. The constraint imposes unphysical limitation on the initial conditions and admissible motions. We discuss the mechanism of appearance of unique solutions in external fields on an example of motion in the uniform magnetic field. We give a simple model to illustrate that similarly constrained evolution cannot be determinate in arbitrary fields.Comment: 16 pages, in v2: shortened, improved presentation, proofs moved to Appendices, in v3: further text permutations and a comment added concerning hamiltonization, in v4: language corrections, final for

Spinor particle. An indeterminacy in the motion of relativistic dynamical systems with separately fixed mass and spin

We give an argument that a broad class of geometric models of spinning relativistic particles with Casimir mass and spin being separately fixed parameters, have indeterminate worldline (while other spinning particles have definite worldline). This paradox suggests that for a consistent description of spinning particles something more general than a worldline concept should be used. As a particular case, we study at the Lagrangian level the Cauchy problem for a spinor particle and then, at the constrained Hamiltonian level, we generalize our result to other particles.Comment: 10 pages, 1 figur

Transverse gradients of azimuthal velocity in a global disk model of the Milky Way

In this paper, we aim to estimate the vertical gradients in the rotational velocity of the Galaxy. This is carried out in the framework of a global thin disc model approximation. The predicted gradient values coincide with the observed vertical fall-off in the rotation curve of the Galaxy. The gradient is estimated based on a statistical analysis of trajectories of test bodies in the gravitational field of the disc and in an analytical way using a quasi-circular orbit approximation. The agreement of the results with the gradient measurements is remarkable in view of other more complicated, non-gravitational mechanisms used for explaining the observed gradient values. Finally, we find that models with a significant spheroidal component give worse vertical gradient estimates than the simple disc model. In view of these results, we can surmise that, apart from the central spherical bulge and Galactic halo, the gross mass distribution in the Galaxy forms a flattened rather than spheroidal figure.Comment: 11 pages, 18 figures, in v2 added explicit gradient calculation at z<0.1kpc, reorganized/extended intro and summary, in v3 language correction