A group theoretical approach to hydrodynamics considers hydrodynamics to be the
differential geometry of diffeomorphism groups. The principle of least action implies that the …

Preface “... ad alcuno, dico, di quelli, che troppo laconicamente vorrebbero vedere, nei piu
angusti spazii che possibil fusse, ristretti i filosofici insegnamenti, sı che sempre si usasse …

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold.
In this paper we present a collection of open problems along with several new constructions …

In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid.
Under the assumption that the vorticity is concentrated along a smooth curve in R^ 3 R 3, we …

We introduce a geometric framework to study Newton's equations on infinite-dimensional
configuration spaces of diffeomorphisms and smooth probability densities. It turns out that …

Fluctuation theorems specify the non-zero probability to observe negative entropy
production, contrary to a naive expectation from the second law of thermodynamics. For …

We present a Hamiltonian framework for higher-dimensional vortex filaments (or
membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1 …

The skew mean curvature flow is an evolution equation for d dimensional manifolds
embedded in R d+ 2 (or more generally, in a Riemannian manifold). It can be viewed as a …

The Madelung transform is known to relate Schrödinger-type equations in quantum
mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally …

In this paper, we present a novel Lagrangian formulation of the equations of motion for point
vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists …