371 research outputs found
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
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