2,687 research outputs found
Hamiltonian flows on null curves
The local motion of a null curve in Minkowski 3-space induces an evolution
equation for its Lorentz invariant curvature. Special motions are constructed
whose induced evolution equations are the members of the KdV hierarchy. The
null curves which move under the KdV flow without changing shape are proven to
be the trajectories of a certain particle model on null curves described by a
Lagrangian linear in the curvature. In addition, it is shown that the curvature
of a null curve which evolves by similarities can be computed in terms of the
solutions of the second Painlev\'e equation.Comment: 14 pages, v2: final version; minor changes in the expositio
Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
An infinite family of quasi-maximally superintegrable Hamiltonians with a
common set of (2N-3) integrals of the motion is introduced. The integrability
properties of all these Hamiltonians are shown to be a consequence of a hidden
non-standard quantum sl(2,R) Poisson coalgebra symmetry. As a concrete
application, one of this Hamiltonians is shown to generate the geodesic motion
on certain manifolds with a non-constant curvature that turns out to be a
function of the deformation parameter z. Moreover, another Hamiltonian in this
family is shown to generate geodesic motions on Riemannian and relativistic
spaces all of whose sectional curvatures are constant and equal to the
deformation parameter z. This approach can be generalized to arbitrary
dimension by making use of coalgebra symmetry.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Superintegrable Deformations of the Smorodinsky-Winternitz Hamiltonian
A constructive procedure to obtain superintegrable deformations of the
classical Smorodinsky-Winternitz Hamiltonian by using quantum deformations of
its underlying Poisson sl(2) coalgebra symmetry is introduced. Through this
example, the general connection between coalgebra symmetry and quasi-maximal
superintegrability is analysed. The notion of comodule algebra symmetry is also
shown to be applicable in order to construct new integrable deformations of
certain Smorodinsky-Winternitz systems.Comment: 17 pages. Published in "Superintegrability in Classical and Quantum
Systems", edited by P.Tempesta, P.Winternitz, J.Harnad, W.Miller Jr.,
G.Pogosyan and M.A.Rodriguez, CRM Proceedings & Lecture Notes, vol.37,
American Mathematical Society, 200
On the Reliability of the Langevin Pertubative Solution in Stochastic Inflation
A method to estimate the reliability of a perturbative expansion of the
stochastic inflationary Langevin equation is presented and discussed. The
method is applied to various inflationary scenarios, as large field, small
field and running mass models. It is demonstrated that the perturbative
approach is more reliable than could be naively suspected and, in general, only
breaks down at the very end of inflation.Comment: 7 pages, 3 figure
Binary trees, coproducts, and integrable systems
We provide a unified framework for the treatment of special integrable
systems which we propose to call "generalized mean field systems". Thereby
previous results on integrable classical and quantum systems are generalized.
Following Ballesteros and Ragnisco, the framework consists of a unital algebra
with brackets, a Casimir element, and a coproduct which can be lifted to higher
tensor products. The coupling scheme of the iterated tensor product is encoded
in a binary tree. The theory is exemplified by the case of a spin octahedron.Comment: 15 pages, 6 figures, v2: minor correction in theorem 1, two new
appendices adde
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