107 research outputs found
On a family of integrable systems on with a cubic integral of motion
We discuss a family of integrable systems on the sphere with an
additional integral of third order in momenta. This family contains the
Coryachev-Chaplygin top, the Goryachev system, the system recently discovered
by Dullin and Matveev and two new integrable systems. On the non-physical
sphere with zero radius all these systems are isomorphic to each other.Comment: LaTeX, 8 page
Chaplygin gas with non-adiabatic pressure perturbations
Perturbations in a Chaplygin gas, characterized by an equation of state , may acquire non-adiabatic contributions if spatial variations of the
parameter are admitted. This feature is shown to be related to a specific
internal structure of the Chaplygin gas. We investigate how perturbations of
this type modify the adiabatic sound speed and influence the time dependence of
the gravitational potential which gives rise to the Integrated Sachs-Wolfe
effect in the anisotropy spectrum of the cosmic microwave background.Comment: 16 pages, comments and references added, accepted for publication in
Class.Quantum Gra
LR and L+R Systems
We consider coupled nonholonomic LR systems on the product of Lie groups. As
examples, we study -dimensional variants of the spherical support system and
the rubber Chaplygin sphere. For a special choice of the inertia operator, it
is proved that the rubber Chaplygin sphere, after reduction and a time
reparametrization becomes an integrable Hamiltonian system on the
--dimensional sphere. Also, we showed that an arbitrary L+R system
introduced by Fedorov can be seen as a reduced system of an appropriate coupled
LR system.Comment: 18 pages, 1 figur
On integrable system on with the second integral quartic in the momenta
We consider integrable system on the sphere with an additional integral
of fourth order in the momenta. At the special values of parameters this system
coincides with the Kowalevski-Goryachev-Chaplygin system.Comment: LaTeX, 6 page
A fully quantum method of determination of penetrability and reflection coefficients in quantum FRW model with radiation
In the paper the closed Friedmann--Robertson--Walker model with quantization
in the presence of the positive cosmological constant and radiation is studied.
For analysis of tunneling probability for birth of an asymptotically deSitter,
inflationary Universe as a function of the radiation energy a new definition of
a "free" wave propagating inside strong fields is proposed. On such a basis,
tunneling boundary condition is corrected, penetrability and reflection
concerning to the barrier are calculated in fully quantum stationary approach.
For the first time non-zero interference between the incident and reflected
waves has been taken into account which turns out to play important role inside
cosmological potentials and could be explained by non-locality of barriers in
quantum mechanics. Inside whole region of energy of radiation the tunneling
probability for the birth of the inflationary Universe is found to be close to
its value obtained in semiclassical approach. The reflection from the barrier
is determined for the first time (which is essentially differs on 1 at the
energy of radiation close to the barrier height). The proposed method could be
easily generalized on the cosmological models with the barriers of arbitrary
shape, that has been demonstrated for the FRW-model with included Chaplygin
gas. Result is stable for variations of the studied barriers, accuracy are
found to be 11--18 digits for all coefficients and energies below the barrier
height.Comment: 28 pages, 7 figures (20 files in eps format), 4 table
Integrable Euler top and nonholonomic Chaplygin ball
We discuss the Poisson structures, Lax matrices, -matrices, bi-hamiltonian
structures, the variables of separation and other attributes of the modern
theory of dynamical systems in application to the integrable Euler top and to
the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio
The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation
We consider the motion of a planar rigid body in a potential flow with
circulation and subject to a certain nonholonomic constraint. This model is
related to the design of underwater vehicles.
The equations of motion admit a reduction to a 2-dimensional nonlinear
system, which is integrated explicitly. We show that the reduced system
comprises both asymptotic and periodic dynamics separated by a critical value
of the energy, and give a complete classification of types of the motion. Then
we describe the whole variety of the trajectories of the body on the plane.Comment: 25 pages, 7 figures. This article uses some introductory material
from arXiv:1109.321
Quaternion Solution for the Rock'n'roller: Box Orbits, Loop Orbits and Recession
We consider two types of trajectories found in a wide range of mechanical
systems, viz. box orbits and loop orbits. We elucidate the dynamics of these
orbits in the simple context of a perturbed harmonic oscillator in two
dimensions. We then examine the small-amplitude motion of a rigid body, the
rock'n'roller, a sphere with eccentric distribution of mass. The equations of
motion are expressed in quaternionic form and a complete analytical solution is
obtained. Both types of orbit, boxes and loops, are found, the particular form
depending on the initial conditions. We interpret the motion in terms of
epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is
associated with box orbits. The small-amplitude solutions for the symmetric
case, or Routh sphere, are expressed explicitly in terms of epicycles; there is
no recession in this case
FRW Quantum Cosmology with a Generalized Chaplygin Gas
Cosmologies with a Chaplygin gas have recently been explored with the
objective of explaining the transition from a dust dominated epoch towards an
accelerating expansion stage. We consider the hypothesis that the transition to
the accelerated period involves a quantum mechanical process. Three physically
admissible cases are possible. In particular, we identify a minisuperspace
configuration with two Lorentzian sectors, separated by a classically forbidden
region. The Hartle-Hawking and Vilenkin wave functions are computed, together
with the transition amplitudes towards the accelerating epoch. Furthermore, it
is found that for specific initial conditions, the parameters characterizing
the generalized Chaplygin gas become related through an expression involving an
integer . We also introduce a phenomenological association between some
brane-world scenarios and a FRW minisuperspace cosmology with a generalized
Chaplygin gas. The aim is to promote a discussion and subsequent research on
the quantum creation of brane cosmologies from such a perspective. Results
suggest that the brane tension would become related with generalized Chaplygin
gas parameters through another expression involving an integer.Comment: 13 pages, 3 figures, RevTeX
Discrete Nonholonomic LL Systems on Lie Groups
This paper applies the recently developed theory of discrete nonholonomic
mechanics to the study of discrete nonholonomic left-invariant dynamics on Lie
groups. The theory is illustrated with the discrete versions of two classical
nonholonomic systems, the Suslov top and the Chaplygin sleigh. The preservation
of the reduced energy by the discrete flow is observed and the discrete
momentum conservation is discussed.Comment: 32 pages, 13 figure
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