222 research outputs found
Euler equations on homogeneous spaces and Virasoro orbits
We show that the following three systems related to various hydrodynamical
approximations: the Korteweg--de Vries equation, the Camassa--Holm equation,
and the Hunter--Saxton equation, have the same symmetry group and similar
bihamiltonian structures. It turns out that their configuration space is the
Virasoro group and all three dynamical systems can be regarded as equations of
the geodesic flow associated to different right-invariant metrics on this group
or on appropriate homogeneous spaces. In particular, we describe how Arnold's
approach to the Euler equations as geodesic flows of one-sided invariant
metrics extends from Lie groups to homogeneous spaces. We also show that the
above three cases describe all generic bihamiltonian systems which are related
to the Virasoro group and can be integrated by the translation argument
principle: they correspond precisely to the three different types of generic
Virasoro orbits.Comment: 26 pages, 4 figures, LaTeX. Advances in Mathematics (to appear
Pseudo-Riemannian geodesics and billiards
Many classical facts in Riemannian geometry have their pseudo-Riemannian
analogs. For instance, the spaces of space-like and time-like geodesics on a
pseudo-Riemannian manifold have natural symplectic structures (just like in the
Riemannian case), while the space of light-like geodesics has a natural contact
structure. We discuss the geometry of these structures in detail, as well as
introduce and study pseudo-Euclidean billiards. In particular, we prove
pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the
integrability of the billiard in the ellipsoid and the geodesic flow on the
ellipsoid in a pseudo-Euclidean space.Comment: title abbreviated, text edited; to appear in Advances in Mathematic
Diffeomorphic random sampling using optimal information transport
In this article we explore an algorithm for diffeomorphic random sampling of
nonuniform probability distributions on Riemannian manifolds. The algorithm is
based on optimal information transport (OIT)---an analogue of optimal mass
transport (OMT). Our framework uses the deep geometric connections between the
Fisher-Rao metric on the space of probability densities and the right-invariant
information metric on the group of diffeomorphisms. The resulting sampling
algorithm is a promising alternative to OMT, in particular as our formulation
is semi-explicit, free of the nonlinear Monge--Ampere equation. Compared to
Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when
a large number of samples from a low dimensional nonuniform distribution is
needed.Comment: 8 pages, 3 figure
Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications
We develop the necessary tools, including a notion of logarithmic derivative
for curves in homogeneous spaces, for deriving a general class of equations
including Euler-Poincar\'e equations on Lie groups and homogeneous spaces.
Orbit invariants play an important role in this context and we use these
invariants to prove global existence and uniqueness results for a class of PDE.
This class includes Euler-Poincar\'e equations that have not yet been
considered in the literature as well as integrable equations like Camassa-Holm,
Degasperis-Procesi, CH and DP equations, and the geodesic equations
with respect to right invariant Sobolev metrics on the group of diffeomorphisms
of the circle
Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size
We construct affinization of the algebra of ``complex size''
matrices, that contains the algebras for integral values of the
parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra
results in the quadratic Gelfand--Dickey structure on the
Poisson--Lie group of all pseudodifferential operators of fractional order.
This construction is extended to the simultaneous deformation of orthogonal and
simplectic algebras that produces self-adjoint operators, and it has a
counterpart for the Toda lattices with fractional number of particles.Comment: 29 pages, no figure
Particle dynamics inside shocks in Hamilton-Jacobi equations
Characteristics of a Hamilton-Jacobi equation can be seen as action
minimizing trajectories of fluid particles. For nonsmooth "viscosity"
solutions, which give rise to discontinuous velocity fields, this description
is usually pursued only up to the moment when trajectories hit a shock and
cease to minimize the Lagrangian action. In this paper we show that for any
convex Hamiltonian there exists a uniquely defined canonical global nonsmooth
coalescing flow that extends particle trajectories and determines dynamics
inside the shocks. We also provide a variational description of the
corresponding effective velocity field inside shocks, and discuss relation to
the "dissipative anomaly" in the limit of vanishing viscosity.Comment: 15 pages, no figures; to appear in Philos. Trans. R. Soc. series
Hamiltonian evolutions of twisted gons in \RP^n
In this paper we describe a well-chosen discrete moving frame and their
associated invariants along projective polygons in \RP^n, and we use them to
write explicit general expressions for invariant evolutions of projective
-gons. We then use a reduction process inspired by a discrete
Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the
space of projective invariants, and we establish a close relationship between
the projective -gon evolutions and the Hamiltonian evolutions on the
invariants of the flow. We prove that {any} Hamiltonian evolution is induced on
invariants by an evolution of -gons - what we call a projective realization
- and we give the direct connection. Finally, in the planar case we provide
completely integrable evolutions (the Boussinesq lattice related to the lattice
-algebra), their projective realizations and their Hamiltonian pencil. We
generalize both structures to -dimensions and we prove that they are
Poisson. We define explicitly the -dimensional generalization of the planar
evolution (the discretization of the -algebra) and prove that it is
completely integrable, providing also its projective realization
The Sato Grassmannian and the CH hierarchy
We discuss how the Camassa-Holm hierarchy can be framed within the geometry
of the Sato Grassmannian.Comment: 10 pages, no figure
- …