5,258 research outputs found
Generalization of the Goryachev-Chaplygin Case
In this paper we present a generalization of the Goryachev-Chaplygin
integrable case on a bundle of Poisson brackets, and on Sokolov terms in his
new integrable case of Kirchhoff equations. We also present a new analogous
integrable case for the quaternion form of rigid body dynamics' equations. This
form of equations is recently developed and we can use it for the description
of rigid body motions in specific force fields, and for the study of different
problems of quantum mechanics. In addition we present new invariant relations
in the considered problems.Comment: 10 page
Euler-Poisson Equations and Integrable Cases
In this paper we propose a new approach to the study of integrable cases
based on intensive computer methods' application. We make a new investigation
of Kovalevskaya and Goryachev-Chaplygin cases of Euler-Poisson equations and
obtain many new results in rigid body dynamics in absolute space. Also we
present the visualization of some special particular solutions.Comment: 24 pages, 27 figure
Superintegrable systems on sphere
We consider various generalizations of the Kepler problem to
three-dimensional sphere , a compact space of constant curvature. These
generalizations include, among other things, addition of a spherical analog of
the magnetic monopole (the Poincar\'e--Appell system) and addition of a more
complicated field, which itself is a generalization of the MICZ-system. The
mentioned systems are integrable -- in fact, superintegrable. The latter is due
to the vector integral, which is analogous to the Laplace--Runge--Lenz vector.
We offer a classification of the motions and consider a trajectory isomorphism
between planar and spatial motions. The presented results can be easily
extended to Lobachevsky space .Comment: 14 pages, 2 figure
On the spectral stability of kinks in 2D Klein-Gordon model with parity-time-symmetric perturbation
In a series of recent works by Demirkaya et al. stability analysis for the
static kink solutions to the 1D continuous and discrete Klein-Gordon equations
with a -symmetric perturbation has been analysed. We consider the
linear stability problem for the static kink in 2D Klein-Gordon field taking
into account spatially localized -symmetric perturbation. The
perturbation is in the form of viscous friction, which does not affect the
static solutions to the unperturbed Klein-Gordon equation. Small dynamic
perturbation around the static kink solution is considered to formulate the
linear stability problem. The effect of the small perturbation on the solutions
to the corresponding eigenvalue problem is analysed. The main result is
presented in the form of a theorem describing the behavior of the eigenvalues
corresponding to the extended and localised eigenmodes as the functions of the
perturbation parameter
Generalization of the Goraychev--Chaplygin Case
In this paper we present a generalization of the Goraychev--Chaplygin
integrable case on a bundle of Poisson brackets, and on Sokolov terms in his
new integrable case of Kirchhoff equations. We also present a new analogous
integrable case for the quaternion form of rigid body dynamics' equations. This
form of equations is recently developed and we can use it for the description
of rigid body motions in specific force fields, and for the study of different
problems of quantum mechanics. In addition we present new invariant relations
in the considered problems.Comment: 15 page
On the History of the Development of the Nonholonomic Dynamics
The main directions in the development of the nonholonomic dynamics are
briefly considered in this paper. The first direction is connected with the
general formalizm of the equations of dynamics that differs from the Lagrangian
and Hamiltonian methods of the equations of motion's construction. The second
direction, substantially more important for dynamics, includes investigations
concerning the analysis of the specific nonholonomic problems. We also point
out rather promising direction in development of nonholonomic systems that is
connected with intensive use of the modern computer-aided methods.Comment: 5 page
The Rolling Body Motion Of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics
In this paper we consider cases of existence of invariant measure, additional
first integrals, and Poisson structure in a problem of rigid body's rolling
without sliding on plane and sphere. The problem of rigid body's motion on
plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that
the equations of motion are reduced to a second-order linear differential
equation in the case when the surface of dynamically symmetric body is a
surface of revolution. These results were partially generalized by P. Woronetz,
who studied the motion of body of revolution and the motion of round disk with
sharp edge on the surface of sphere. In both cases the systems are Euler-Jacobi
integrable and have additional integrals and invariant measure. It turns out
that after some change of time defined by reducing multiplier, the reduced
system is a Hamiltonian system. Here we consider different cases when the
integrals and invariant measure can be presented as finite algebraic
expressions.
We also consider the generalized problem of rolling of dynamically
nonsymmetric Chaplygin ball. The results of studies are presented as tables
that describe the hierarchy of existence of various tensor invariants:
invariant measure, integrals, and Poisson structure in the considered problems.Comment: 24 pages, 11 figures, 2 table
Absolute and relative choreographies in the problem of point vortices moving on a plane
We obtained new periodic solutions in the problems of three and four point
vortices moving on a plane. In the case of three vortices, the system is
reduced to a Hamiltonian system with one degree of freedom, and it is
integrable. In the case of four vortices, the order is reduced to two degrees
of freedom, and the system is not integrable. We present relative and absolute
choreographies of three and four vortices of the same intensity which are
periodic motions of vortices in some rotating and fixed frame of reference,
where all the vortices move along the same closed curve. Similar choreographies
have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the
n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the
choreographies that appear in vortex dynamics have a number of distinct
features.Comment: 11 pages, 6 figure
Two-body problem on a sphere. Reduction, stochasticity, periodic orbits
We consider the problem of two interacting particles on a sphere. The
potential of the interaction depends on the distance between the particles. The
case of Newtonian-type potentials is studied in most detail. We reduce this
system to a system with two degrees of freedom and give a number of remarkable
periodic orbits. We also discuss integrability and stochastization of the
motion.Comment: 15 pages, 6 figure
Reduction and chaotic behavior of point vortices on a plane and a sphere
We offer a new method of reduction for a system of point vortices on a plane
and a sphere. This method is similar to the classical node elimination
procedure. However, as applied to the vortex dynamics, it requires substantial
modification. Reduction of four vortices on a sphere is given in more detail.
We also use the Poincare surface-of-section technique to perform the reduction
a four-vortex system on a sphere.Comment: 10 pages, 3 figure
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