4,922 research outputs found
Higher Conservation Law for the Multi-Centre Metrics
The multi-centre metrics are a family of euclidean solutions of the empty
space Einstein equations with self-dual curvature. For this full class, we
determine which metrics do exhibit an extra conserved quantity quadraic in the
momenta, induced by a Killing-Stackel tensor. Our results bring to light
several metrics which correspond to classically integrable dynamical systems.
They include, as particular cases, the Eguchi-Hanson and Taub-NUT metrics.Comment: Latex, 16 pages, 0 figur
Heun functions versus elliptic functions
We present some recent progresses on Heun functions, gathering results from
classical analysis up to elliptic functions. We describe Picard's
generalization of Floquet's theory for differential equations with doubly
periodic coefficients and give the detailed forms of the level one Heun
functions in terms of Jacobi theta functions. The finite-gap solutions give an
interesting alternative integral representation which, at level one, is shown
to be equivalent to their elliptic form.Comment: Communication at the International Conference on Difference
Equations, Special Functions and Applications, Munich, 25-30 july 2005, latex
2e, 20 page
Explicit integrable systems on two dimensional manifolds with a cubic first integral
A few years ago Selivanova gave an existence proof for some integrable
models, in fact geodesic flows on two dimensional manifolds, with a cubic first
integral. However the explicit form of these models hinged on the solution of a
nonlinear third order ordinary differential equation which could not be
obtained. We show that an appropriate choice of coordinates allows for
integration and gives the explicit local form for the full family of integrable
systems. The relevant metrics are described by a finite number of parameters
and lead to a large class of models on the manifolds {\mb S}^2, {\mb H}^2 and
P^2({\mb R}) containing as special cases examples due to Goryachev,
Chaplygin, Dullin, Matveev and Tsiganov
Integrability versus separability for the multi-centre metrics
The multi-centre metrics are a family of euclidean solutions of the empty
space Einstein equations with self-dual curvature. For this full class, we
determine which metrics do exhibit an extra conserved quantity quadratic in the
momenta, induced by a Killing-St\" ackel tensor. Our systematic approach brings
to light a subclass of metrics which correspond to new classically integrable
dynamical systems. Within this subclass we analyze on the one hand the
separation of coordinates in the Hamilton-Jacobi equation and on the other hand
the construction of some new Killing-Yano tensors.Comment: 24 pages, latex, no figur
From asymptotics to spectral measures: determinate versus indeterminate moment problems
In the field of orthogonal polynomials theory, the classical Markov theorem
shows that for determinate moment problems the spectral measure is under
control of the polynomials asymptotics. The situation is completely different
for indeterminate moment problems, in which case the interesting spectral
measures are to be constructed using Nevanlinna theory. Nevertheless it is
interesting to observe that some spectral measures can still be obtained from
weaker forms of Markov theorem. The exposition will be illustrated by
orthogonal polynomials related to elliptic functions: in the determinate case
by examples due to Stieltjes and some of their generalizations and in the
indeterminate case by more recent examples.Comment: Lecture given at the International Mediterranean Congress of
Mathematics, Almeria, 6-10 june 2005, latex2e, 16 page
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