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

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

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

An algorithm for solving the pulsar equation

We present an algorithm of finding numerical solutions of pulsar equation. The problem of finding the solutions was reduced to finding expansion coefficients of the source term of the equation in a base of orthogo- nal functions defined on the unit interval by minimizing a multi-variable mismatch function defined on the light cylinder. We applied the algorithm to Scharlemann & Wagoner boundary conditions by which a smooth solu- tion is reconstructed that by construction passes success- fully the Gruzinov's test of the source function exponent.Comment: 4 pages, 4 figures, accepted for publication in ApSS (a shortened version of the previous one

A new method for reconstructing the density distribution of matter in the disks of spiral galaxies from the rotation velocity curve in it

In this paper we propose a new method for reconstructing the surface density of matter in flat disks of spiral galaxies. The surface density is expressed through observational rotation velocity curves of visible matter in the disks of spiral galaxies. The new method is not based on quadrature of special functions. The found solution is used for processing and analysis of observational data from several spiral galaxies. The new method can be used to more accurately estimate the amount of dark matter in spiral galaxies.Comment: 18 pages, 6 figure

Optimisation of Temperature Fields of Microsystems with Self-Organising Neural Nets

Thermal modelling and optimisation of parameter distributed systems is a rather time-consuming process. In this paper the problem of optimisation of temperature fields of VLSI circuits and systems is attacked by a selforganising neural net. The net directly solves the task generated by a heuristic algorithm. No physical model of thermal phenomena is used. The proposed method is simple. Some examples and statistical results are presented. The proposed method is addressed mostly to large, high-speed system designs

Is dark matter present in NGC4736? An iterative spectral method for finding mass distribution in spiral galaxies

An iterative method for reconstructing mass distribution in spiral galaxies using a thin disk approximation is developed. As an example, the method is applied to galaxy NGC 4736; its rotation curve does not allow one to employ a model with a massive spherical halo. We find a global mass distribution in this galaxy (without non-baryonic dark matter) that agrees perfectly with the high resolution rotation curve of the galaxy. This mass distribution is consistent with the $I$-band luminosity profile with the mean mass-to-light ratio $M/L_I=1.2$, and also agrees with the amount of hydrogen observed in the outermost regions of the galaxy. We predict the total mass of the galaxy to be only 3.43\times10^{10}M_{\sun}. It is very close to the value predicted by the modified gravity models and much less than the currently accepted value of 5.0\times10^{10}M_{\sun} (with $\approx70$ of the mass in a dark matter halo).Comment: in v2 version: 1) changed the reference luminosities of the Sun in different bands - this affects mass-to-light ratio, giving more reliable 1.2 in the I-band, 2) found typos corrected, 3) corrected references to literature, figures and equations 4) text permutations + language corrections, accepted for publication in ApJ, May 200