41 research outputs found
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
Modeling vertical structure in circular velocity of spiral galaxy NGC 4244
We study the vertical gradient in azimuthal velocity of spiral galaxy NGC
4244 in a thin disk model. With surface density accounting for the rotation
curve, we model the gradient properties in the approximation of quasi-circular
orbits and find the predictions to be consistent with the gradient properties
inferred from measurements. This consistency may suggest that the mass
distribution in this galaxy is flattened.Comment: 5 pages, 6 figure
Velocity-density twin transforms in thin disk model
Ring mass density and the corresponding circular velocity in thin disk model
are known to be integral transforms of one another. But it may be less familiar
that the transforms can be reduced to one-fold integrals with identical weight
functions. It may be of practical value that the integral for the surface
density does not involve the velocity derivative, unlike the equivalent and
widely known Toomre's formula.Comment: 3 pages; 1 figure (a separate file); v2: added a derivation of w(x)
from an infinitely flattened spheroid; v3 improvement/correction of the
derivation in v2 + minor corr. in the tex