3,450 research outputs found
On a Lagrangian formulation of the incompressible Euler equation
In this paper we show that the incompressible Euler equation on the Sobolev
space , , can be expressed in Lagrangian coordinates as a
geodesic equation on an infinite dimensional manifold. Moreover the Christoffel
map describing the geodesic equation is real analytic. The dynamics in
Lagrangian coordinates is described on the group of volume preserving
diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism
group. Furthermore it is shown that a Sobolev class vector field integrates to
a curve on the diffeomorphism group
Pseudo-distances on symplectomorphism groups and applications to flux theory
Starting from a given norm on the vector space of exact 1-forms of a compact
symplectic manifold, we produce pseudo-distances on its symplectomorphism group
by generalizing an idea due to Banyaga. We prove that in some cases (which
include Banyaga's construction), their restriction to the Hamiltonian
diffeomorphism group is equivalent to the distance induced by the initial norm
on exact 1-forms. We also define genuine "distances to the Hamiltonian
diffeomorphism group" which we use to derive several consequences, mainly in
terms of flux groups.Comment: 21 pages, no figure; v2. various typos corrected, some references
added. Published in Mathematische Zeitschrif
Integrability from an abelian subgroup of the diffeomorphism group
It has been known for some time that for a large class of non-linear field
theories in Minkowski space with two-dimensional target space the complex
eikonal equation defines integrable submodels with infinitely many conservation
laws. These conservation laws are related to the area-preserving
diffeomorphisms on target space. Here we demonstrate that for all these
theories there exists, in fact, a weaker integrability condition which again
defines submodels with infinitely many conservation laws. These conservation
laws will be related to an abelian subgroup of the group of area-preserving
diffeomorphisms. As this weaker integrability condition is much easier to
fulfil, it should be useful in the study of those non-linear field theories.Comment: 13 pages, Latex fil
Geodesic Flow on the Diffeomorphism Group of the circle
We show that certain right-invariant metrics endow the infinite-dimensional
Lie group of all smooth orientation-preserving diffeomorphisms of the circle
with a Riemannian structure. The study of the Riemannian exponential map allows
us to prove infinite-dimensional counterparts of results from classical
Riemannian geometry: the Riemannian exponential map is a smooth local
diffeomorphism and the length-minimizing property of the geodesics holds.Comment: 15 page
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