176 research outputs found
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
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
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
A spinor-like representation of the contact superconformal algebra K'(4)
In this work we construct an embedding of a nontrivial central extension of
the contact superconformal algebra K'(4) into the Lie superalgebra of
pseudodifferential symbols on the supercircle S^{1|2}. Associated with this
embedding is a one-parameter family of spinor-like tiny irreducible
representations of K'(4) realized just on 4 fields instead of the usual 16.Comment: 19 pages, TeX. Corrections to the references in the paper to be
published in J. Math. Phys. v 42, no 1, 200
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
Contact complete integrability
Complete integrability in a symplectic setting means the existence of a
Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we
describe complete integrability in a contact set-up as a more subtle structure:
a flag of two foliations, Legendrian and co-Legendrian, and a
holonomy-invariant transverse measure of the former in the latter. This turns
out to be equivalent to the existence of a canonical
structure on the leaves of the co-Legendrian foliation. Further, the above
structure implies the existence of contact fields preserving a special
contact 1-form, thus providing the geometric framework and establishing
equivalence with previously known definitions of contact integrability. We also
show that contact completely integrable systems are solvable in quadratures. We
present an example of contact complete integrability: the billiard system
inside an ellipsoid in pseudo-Euclidean space, restricted to the space of
oriented null geodesics. We describe a surprising acceleration mechanism for
closed light-like billiard trajectories
The Biot-Savart operator and electrodynamics on subdomains of the three-sphere
We study steady-state magnetic fields in the geometric setting of positive
curvature on subdomains of the three-dimensional sphere. By generalizing the
Biot-Savart law to an integral operator BS acting on all vector fields, we show
that electrodynamics in such a setting behaves rather similarly to Euclidean
electrodynamics. For instance, for current J and magnetic field BS(J), we show
that Maxwell's equations naturally hold. In all instances, the formulas we give
are geometrically meaningful: they are preserved by orientation-preserving
isometries of the three-sphere.
This article describes several properties of BS: we show it is self-adjoint,
bounded, and extends to a compact operator on a Hilbert space. For vector
fields that act like currents, we prove the curl operator is a left inverse to
BS; thus the Biot-Savart operator is important in the study of curl
eigenvalues, with applications to energy-minimization problems in geometry and
physics. We conclude with two examples, which indicate our bounds are typically
within an order of magnitude of being sharp.Comment: 24 pages (was 28 pages) Revised to include a new introduction, a
detailed example, and results about helicity; other changes for readabilit
-analogue of modified KP hierarchy and its quasi-classical limit
A -analogue of the tau function of the modified KP hierarchy is defined by
a change of independent variables. This tau function satisfies a system of
bilinear -difference equations. These bilinear equations are translated to
the language of wave functions, which turn out to satisfy a system of linear
-difference equations. These linear -difference equations are used to
formulate the Lax formalism and the description of quasi-classical limit. These
results can be generalized to a -analogue of the Toda hierarchy. The results
on the -analogue of the Toda hierarchy might have an application to the
random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are
adde
Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions
We introduce a notion of a weak Poisson structure on a manifold modeled
on a locally convex space. This is done by specifying a Poisson bracket on a
subalgebra \cA \subeq C^\infty(M) which has to satisfy a non-degeneracy
condition (the differentials of elements of \cA separate tangent vectors) and
we postulate the existence of smooth Hamiltonian vector fields. Motivated by
applications to Hamiltonian actions, we focus on affine Poisson spaces which
include in particular the linear and affine Poisson structures on duals of
locally convex Lie algebras. As an interesting byproduct of our approach, we
can associate to an invariant symmetric bilinear form on a Lie algebra
\g and a -skew-symmetric derivation a weak affine Poisson
structure on \g itself. This leads naturally to a concept of a Hamiltonian
-action on a weak Poisson manifold with a \g-valued momentum map and hence
to a generalization of quasi-hamiltonian group actions
- …