Poisson brackets in Hydrodynamics

This paper investigates different Poisson structures that have been proposed to give a Hamiltonian formulation to evolution equations issued from fluid mechanics. Our aim is to explore the main brackets which have been proposed and to discuss the difficulties which arise when one tries to give a rigorous meaning to these brackets. Our main interest is in the definition of a valid and usable bracket to study rotational fluid flows with a free boundary. We discuss some results which have emerged in the literature to solve some of the difficulties that arise. It appears to the author that the main problems are still open

Integrability of invariant metrics on the diffeomorphism group of the circle

Each H^k Sobolev inner product defines a Hamiltonian vector field X_k on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X_0 and X_1 are bi-Hamiltonian relatively to a modified Lie-Poisson structure

Least action principle for an integrable shallow water equation

For an integrable shallow water equation we describe a geometrical approach showing that any two nearby fluid configurations are successive states of a unique flow minimizing the kinetic energy.Comment: arXiv version is already officia

Recurrent Surface Homeomorphisms

An orientation-preserving recurrent homeomorphism of the two-sphere which is not the identity is shown to admit exactly two fixed points. A recurrent homeomorphism of a compact surface with negative Euler characteristic is periodic.Comment: 10 pages LaTeX; fixed some reference

Point fixe lié à une orbite périodique d'un difféomorphisme de R2

in French, 4 pages.International audienceGiven a diffeomorphism of the plane, which has a periodic orbit, we show how Nielsen fixed point theory can be used to establish the existence of a fixed point which is linked with this periodic orbit

Invariant-based approach to symmetry class detection

In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is based on the invariants of the irreducible tensors appearing in the harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that aim we first introduce a geometrical description of the space of elasticity tensors. This framework is used to derive invariant-based conditions that characterize symmetry classes. For low order symmetry classes, such conditions are given on a triplet of quadratic forms extracted from the harmonic decomposition of the elasticity tensor $C$, meanwhile for higher-order classes conditions are provided in terms of elements of $H^{4}$, the higher irreducible space in the decomposition of $C$. Proceeding in such a way some well known conditions appearing in the Mehrabadi-Cowin theorem for the existence of a symmetry plane are retrieved, and a set of algebraic relations on polynomial invariants characterizing the orthotropic, trigonal, tetragonal, transverse isotropic and cubic symmetry classes are provided. Using a genericity assumption on the elasticity tensor under study, an algorithm to identify the symmetry class of a large set of tensors is finally provided.Comment: 32 page