88 research outputs found
Integrable flows and Backlund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)
We show that the -dimensional Euler--Manakov top on can be
represented as a Poisson reduction of an integrable Hamiltonian system on a
symplectic extended Stiefel variety , and present its Lax
representation with a rational parameter.
We also describe an integrable two-valued symplectic map on the
4-dimensional variety . The map admits two different reductions,
namely, to the Lie group SO(3) and to the coalgebra .
The first reduction provides a discretization of the motion of the classical
Euler top in space and has a transparent geometric interpretation, which can be
regarded as a discrete version of the celebrated Poinsot model of motion and
which inherits some properties of another discrete system, the elliptic
billiard.
The reduction of to gives a new explicit discretization of
the Euler top in the angular momentum space, which preserves first integrals of
the continuous system.Comment: 18 pages, 1 Figur
Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
We consider a class of dynamical systems on a Lie group with a
left-invariant metric and right-invariant nonholonomic constraints (so called
LR systems) and show that, under a generic condition on the constraints, such
systems can be regarded as generalized Chaplygin systems on the principle
bundle , being a Lie subgroup. In contrast to generic
Chaplygin systems, the reductions of our LR systems onto the homogeneous space
always possess an invariant measure.
We study the case , when LR systems are multidimensional
generalizations of the Veselova problem of a nonholonomic rigid body motion,
which admit a reduction to systems with an invariant measure on the (co)tangent
bundle of Stiefel varieties as the corresponding homogeneous spaces.
For and a special choice of the left-invariant metric on SO(n), we
prove that under a change of time, the reduced system becomes an integrable
Hamiltonian system describing a geodesic flow on the unit sphere .
This provides a first example of a nonholonomic system with more than two
degrees of freedom for which the celebrated Chaplygin reducibility theorem is
applicable. In this case we also explicitly reconstruct the motion on the group
SO(n).Comment: 39 pages, the proof of Lemma 4.3 and some references are added, to
appear in Journal of Nonlinear Scienc
Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties
We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigma-function of genus two. For each sequence there is an associated genus two curve , such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of . The construction is based on a Lax pair with a spectral curve of genus four admitting an involution with two fixed points, and the Jacobian of arises as the Prym variety Prym
The Hydrodynamic Chaplygin Sleigh
We consider the motion of rigid bodies in a potential fluid subject to
certain nonholonomic constraints and show that it is described by
Euler--Poincar\'e--Suslov equations.
In the 2-dimensional case, when the constraint is realized by a blade
attached to the body, the system provides a hydrodynamic generalization of the
Chaplygin sleigh, whose dynamics are studied in detail. Namely, the equations
of motion are integrated explicitly and the asymptotic behavior of the system
is determined. It is shown how the presence of the fluid brings new features to
such a behavior.Comment: 20 pages, 7 figure
Unimodularity and preservation of volumes in nonholonomic mechanics
The equations of motion of a mechanical system subjected to nonholonomic
linear constraints can be formulated in terms of a linear almost Poisson
structure in a vector bundle. We study the existence of invariant measures for
the system in terms of the unimodularity of this structure. In the presence of
symmetries, our approach allows us to give necessary and sufficient conditions
for the existence of an invariant volume, that unify and improve results
existing in the literature. We present an algorithm to study the existence of a
smooth invariant volume for nonholonomic mechanical systems with symmetry and
we apply it to several concrete mechanical examples.Comment: 37 pages, 3 figures; v3 includes several changes to v2 that were done
in accordance to the referee suggestion
Dynamical systems on infinitely sheeted Riemann surfaces
This paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi-valuedness of the solutions as functions of complex time tau. In this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral. The associated Riemann surface R -> C = {tau} is known to be an infinitely sheeted covering of the complex time plane, ramified at an infinite set of points whose projection in the tau-plane is dense. The main novelty of this paper is that the geometrical structure of these infinitely sheeted Riemann surfaces is described in great detail, which allows us to study global properties of the flow such as asymptotic behaviour of the solutions, periodic orbits and their stability or sensitive dependence on initial conditions. The results are then compared with a numerical integration of the equations of motion. Following the recent approach of Calogero, the real time trajectories of the system are given by paths on R that are projected to a circle on the complex plane tau. Due to the branching of R, the solutions may have different periods or may be aperiodic
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