20,921 research outputs found
Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: pro or contra?
The phenomenon of a topological monodromy in integrable Hamiltonian and
nonholonomic systems is discussed. An efficient method for computing and
visualizing the monodromy is developed. The comparative analysis of the
topological monodromy is given for the rolling ellipsoid of revolution problem
in two cases, namely, on a smooth and on a rough plane. The first of these
systems is Hamiltonian, the second is nonholonomic. We show that, from the
viewpoint of monodromy, there is no difference between the two systems, and
thus disprove the conjecture by Cushman and Duistermaat stating that the
topological monodromy gives a topological obstruction for Hamiltonization of
the rolling ellipsoid of revolution on a rough plane.Comment: 31 pages, 11 figure
On uniform continuous dependence of solution of Cauchy problem on a parameter
Suppose that an -dimensional Cauchy problem \frac{dx}{dt}=f(t,x,\mu) (t
\in I, \mu \in M), x(t_0)=x^0 satisfies the conditions that guarantee
existence, uniqueness and continuous dependence of solution x(t,t_0,\mu) on
parameter \mu in an open set M. We show that if one additionally requires that
family \{f(t,x,\cdot)\}_{(t,x)} is equicontinuous, then the dependence of
solution x(t,t_0,\mu) on parameter \mu \in M is uniformly continuous.
An analogous result for a linear n \times n-dimensional Cauchy problem
\frac{dX}{dt}=A(t,\mu)X+\Phi(t,\mu) (t \in I, \mu \in M), X(t_0,\mu)=X^0(\mu)
is valid under the assumption that the integrals
\int_I\|A(t,\mu_1)-A(t,\mu_2)\|dt and \int_I \|\Phi(t,\mu_1)-\Phi(t,\mu_2)\|dt
can be made smaller than any given constant (uniformly with respect to \mu_1,
\mu_2 \in M) provided that \|\mu_1-\mu_2\| is sufficiently small
The Hess-Appelrot system and its nonholonomic analogs
This paper is concerned with the nonholonomic Suslov problem and its
generalization proposed by Chaplygin. The issue of the existence of an
invariant measure with singular density (having singularities at some points of
phase space) is discussed
New periodic solutions for three or four identical vortices on a plane and a sphere
In this paper we describe new classes of periodic solutions for point
vortices on a plane and a sphere. They correspond to similar solutions
(so-called choreographies) in celestial mechanics.Comment: 15 pages, 6 figure
Disconjugacy of a second order linear differential equation and periodic solutions
The present paper is devoted to a new criterion for disconjugacy of a second
order linear differential equation. Unlike most of the classical sufficient
conditions for disconjugacy, our criterion does not involve assumptions on the
smallness of the coefficients of the equation. We compare our criterion with
several known criteria for disconjugacy, for which we provide detailed proofs,
and discuss the applications of the property of disconjugacy to the problem of
factorization of linear ordinary differential operators, and to the proof of
the generalized Rolle's theorem. The paper is self-contained, and may serve as
a brief introduction to theory of disconjugacy of a second order linear
differential equation
Hamiltonization of Elementary Nonholonomic Systems
In this paper, we develop the Chaplygin reducing multiplier method; using
this method, we obtain a conformally Hamiltonian representation for three
nonholonomic systems, namely, for the nonholonomic oscillator, for the
Heisenberg system, and for the Chaplygin sleigh. Furthermore, in the case of an
oscillator and the nonholonomic Chaplygin sleigh, we show that the problem
reduces to the study of motion of a mass point (in a potential field) on a
plane and, in the case of the Heisenberg system, on the sphere. Moreover, we
consider an example of a nonholonomic system (suggested by Blackall) to which
one cannot apply the reducing multiplier method
Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories
Dynamical equations are formulated and a numerical study is provided for
self-oscillatory model systems based on the triple linkage hinge mechanism of
Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic
mechanical constraint of three rotators as well as systems, where three
rotators interact by potential forces. We present and discuss some quantitative
characteristics of the chaotic regimes (Lyapunov exponents, power spectrum).
Chaotic dynamics of the models we consider are associated with hyperbolic
attractors, at least, at relatively small supercriticality of the
self-oscillating modes; that follows from numerical analysis of the
distribution for angles of intersection of stable and unstable manifolds of
phase trajectories on the attractors. In systems based on rotators with
interacting potential the hyperbolicity is violated starting from a certain
level of excitation.Comment: 30 pages, 18 figure
On the Routh sphere problem
We discuss an embedding of a vector field for the nonholonomic Routh sphere
into a subgroup of commuting Hamiltonian vector fields on six dimensional phase
space. The corresponding Poisson brackets are reduced to the canonical Poisson
brackets on the Lie algebra e(3). It allows us to relate nonholonomic Routh
system with the Hamiltonian system on cotangent bundle to the sphere with
canonical Poisson structure.Comment: LaTeX with AMSFonts, 11 page
On a mechanical lens
In this paper, we consider the dynamics of a heavy homogeneous ball moving
under the influence of dry friction on a fixed horizontal plane. We assume the
ball to slide without rolling. We demonstrate that the plane may be divided
into two regions, each characterized by a distinct coefficient of friction, so
that balls with equal initial linear and angular velocity will converge upon
the same point from different initial locations along a certain segment. We
construct the boundary between the two regions explicitly and discuss possible
applications to real physical systems
Numerical test for hyperbolicity of chaotic dynamics in time-delay systems
We develop a numerical test of hyperbolicity of chaotic dynamics in
time-delay systems. The test is based on the angle criterion and includes
computation of angle distributions between expanding, contracting and neutral
manifolds of trajectories on the attractor. Three examples are tested. For two
of them previously predicted hyperbolicity is confirmed. The third one provides
an example of a time-delay system with nonhyperbolic chaos.Comment: 7 pages, 5 figure
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