The integrability of the geodesic flow for the multi-centre metrics began with the discovery of the generalized Runge–Lenz vector for the Taub–NUT metric [1] and the derivation of its Killing–Stäckel tensor in [2]. It was generalized to the Eguchi–Hanson metric in [3] where the Hamilton–Jacobi equation was separated. A further progress led to the integrability proof of the full 2-centre metric ...

Optimal sample path properties of stochastic processes often involve generalized Hölderor variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of ψ (x) ≡ x/ log log (1/x) near 0+. Such ψ-variation results extend to classes of processes with values in abstract metric spaces. (No Gaussian or Markovian properties are assumed.) To esta...

We study the effects of integrability-breaking perturbations on the nonequilibrium evolution of many-particle quantum systems. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalizations of quantum Boltzmann equations. We benchmark our method against time-dependent density matrix renormalization group compu...

The classical Liouvile integrability means that there exist n independent first integrals in involution for 2n-dimensional phase space. However, in the infinite-dimensional case, an infinite number of independent first integrals in involution don’t indicate that the system is solvable. How many first integrals do we need in order to make the system solvable? To answer the question, we obtain an...

In this work, we are investigating the problem of integrability of Bianchi class A cosmological models. This class of systems is reduced to the form of Hamiltonian systems with exponential potential forms. The dynamics of Bianchi class A models is investigated through the EulerLagrange equations and geodesic equations in the Jacobi metric. On this basis, we have come to some general conclusions...

Contents 1 Integrability in classical mechanics 4 1.

Recently a covariant approach to cold matter universes in the zero-shear hypersurfaces (or longitudinal) gauge has been developed. This approach reveals the existence of an integrability condition, which does not appear in standard non-covariant treatments. A simple derivation and generalization of the integrability condition is given, based on showing that the quasi-Newtonian models are a sub-...

In this paper we present an approach towards the comprehensive analysis of the non-integrability of differential equations in the form ẍ = f(x, t) which is analogous to Hamiltonian systems with 1 + 1/2 degree of freedom. In particular, we analyze the non-integrability of some important families of differential equations such as Painlevé II, Sitnikov and HillSchrödinger equation. We emphasize in...

We describe a recent progress in finding solutions to three-particle evolution equations at leading order in the QCD coupling constant for multiparton correlation functions based on the integrability of corresponding interaction Hamiltonians. Talk given at the 4th Workshop on Continuous Advances in QCD Minneapolis, May 12-14, 2000 INTEGRABILITY OF TWIST-THREE EVOLUTION EQUATIONS IN QCD A. V. BE...

Using a recently developed method, based on a generalization of the zero curvature representation of Zakharov and Shabat, we study the integrability structure in the Abelian Higgs model. It is shown that the model contains integrable sectors, where integrability is understood as the existence of infinitely many conserved currents. In particular, a gauge invariant description of the weak and str...