1,143 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
Interaction between Kirchhoff vortices and point vortices in an ideal fluid
We consider the interaction of two vortex patches (elliptic Kirchhoff
vortices) which move in an unbounded volume of an ideal incompressible fluid. A
moment second-order model is used to describe the interaction. The case of
integrability of a Kirchhoff vortex and a point vortex by the variable
separation method is qualitatively analyzed. A new case of integrability of two
Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff
vortices is proposed and used to analyze their regular and chaotic behavior.Comment: 27 pages, 7 figure
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
Motion of vortex sources on a plane and a sphere
The Equations of motion of vortex sources (examined earlier by Fridman and
Polubarinova) are studied, and the problems of their being Hamiltonian and
integrable are discussed. A system of two vortex sources and three
sources-sinks was examined. Their behavior was found to be regular. Qualitative
analysis of this system was made, and the class of Liouville integrable systems
is considered. Particular solutions analogous to the homothetic configurations
in celestial mechanics are given.Comment: 19 pages, 4 figure
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
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
The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration
This paper is concerned with the Chaplygin sleigh with timevarying mass
distribution (parametric excitation). The focus is on the case where excitation
is induced by a material point that executes periodic oscillations in a
direction transverse to the plane of the knife edge of the sleigh. In this
case, the problem reduces to investigating a reduced system of two first-order
equations with periodic coefficients, which is similar to various nonlinear
parametric oscillators. Depending on the parameters in the reduced system, one
can observe different types of motion, including those accompanied by strange
attractors leading to a chaotic (diffusion) trajectory of the sleigh on the
plane. The problem of unbounded acceleration (an analog of Fermi acceleration)
of the sleigh is examined in detail. It is shown that such an acceleration
arises due to the position of the moving point relative to the line of action
of the nonholonomic constraint and the center of mass of the platform. Various
special cases of existence of tensor invariants are found
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
Oriented Area as a Morse Function on Polygon Spaces
We study polygon spaces arising from planar configurations of necklaces with
some of the beads fixed and some of the beads sliding freely. These spaces
include configuration spaces of flexible polygons and some other natural
polygon spaces. We characterise critical points of the oriented area function
in geometric terms and give a formula for the Morse indices. Thus we obtain a
generalisation of isoperimetric theorems for polygons in the plane.Comment: 14 pages, 2 figures The difference from the previous version: minor
changes in the structure of the document; the proof of Lemma 11 (which is now
Lemma 4.7) is corrected; two pictures adde
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