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 $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic map $\cal B$ on the 4-dimensional variety ${\cal V}(2,3)$. The map admits two different reductions, namely, to the Lie group SO(3) and to the coalgebra $so^*(3)$. 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 $\cal B$ to $so^*(3)$ 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 $G$ 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 $G \to Q=G/H$, $H$ being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space $Q$ always possess an invariant measure. We study the case $G=SO(n)$, 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 $V(k,n)$ as the corresponding homogeneous spaces. For $k=1$ 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 $S^{n-1}$. 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

Instanton infra-red stabilization in the nonperturbative QCD vacuum

The influence of nonperturbative fields on instantons in quantum chromodynamics is studied. Nonperturbative vacuum is described in terms of nonlocal gauge invariant vacuum averages of gluon field strength. Effective action for instanton is derived in bilocal approximation and it is demonstrated that stochastic background gluon fields are responsible for infra-red (IR)stabilization of instantons. Comparison of obtained instanton size distribution with lattice data is made.Comment: 3 pages, 2 figures. Talk given at 5th International Conference on Quark Confinement and the Hadron Spectrum, Gargnano, Brescia, Italy, 10-14 Sep 200

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 $X$, such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of $X$. The construction is based on a Lax pair with a spectral curve $S$ of genus four admitting an involution $\sigma$ with two fixed points, and the Jacobian of $X$ arises as the Prym variety Prym $(S,\sigma)$

Instanton IR stabilization in the nonperturbative confining vacuum

The influence of nonperturbative fields on instantons in quantum chromodynamics is studied. Nonperturbative vacuum is described in terms of nonlocal gauge invariant vacuum averages of gluon field strength. Effective action for instanton is derived in bilocal approximation and it is demonstrated that stochastic background gluon fields are responsible for infra-red (IR) stabilization of instantons. Dependence of characteristic instanton size on gluon condensate and correlation length in nonperturbative vacuum is found. It is shown that instanton size in QCD is of order of 0.25 fm. Comparison of obtained instanton size distribution with lattice data is made.Comment: 10 pages, 4 figures; v2: minor corrections, to appear in JHE