132,888 research outputs found
The Euler top and canonical lifts
In this note, we prove a finiteness result for fibers that are canonical
lifts in a given elliptic fibration. The question was motivated by the authors'
construction of an arithmetic Euler top, and it highlights an interesting
discrepancy between the arithmetic and the classical case: in the former, it is
impossible to extend the flows to a compactification of the phase space, viewed
as an elliptic fibration over the space of action variables
Asymptotic stabilization of the heavy top using controlled Lagrangians
In this paper we extend the previous work on the
asymptotic stabilization of pure Euler-PoincarĂŠ mechanical
systems using controlled Lagrangians to the
study of asymptotic stabilization of Euler-PoincarĂŠ mechanical
systems such as the heavy top
General Solution of 7D Octonionic Top Equation
The general solution of a 7D analogue of the 3D Euler top equation is shown
to be given by an integration over a Riemann surface with genus 9. The 7D model
is derived from the 8D invariant self-dual Yang-Mills equation
depending only upon one variable and is regarded as a model describing
self-dual membrane instantons. Several integrable reductions of the 7D top to
lower target space dimensions are discussed and one of them gives 6, 5, 4D
descendants and the 3D Euler top associated with Riemann surfaces with genus 6,
5, 2 and 1, respectively.Comment: 13 pages, Latex, 3 eps.files. Minor changes, eq.(4) adde
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
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
Integrable Top Equations associated with Projective Geometry over Z_2
We give a series of integrable top equations associated with the projective
geometry over Z_2 as a (2^n-1)-dimensional generalisation of the 3D Euler top
equations. The general solution of the (2^n-1)D top is shown to be given by an
integration over a Riemann surface with genus (2^{n-1}-1)^2.Comment: 8 pages, Late
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