420 research outputs found
Adler-Kostant-Symes systems as Lagrangian gauge theories
It is well known that the integrable Hamiltonian systems defined by the
Adler-Kostant-Symes construction correspond via Hamiltonian reduction to
systems on cotangent bundles of Lie groups. Generalizing previous results on
Toda systems, here a Lagrangian version of the reduction procedure is exhibited
for those cases for which the underlying Lie algebra admits an invariant scalar
product. This is achieved by constructing a Lagrangian with gauge symmetry in
such a way that, by means of the Dirac algorithm, this Lagrangian reproduces
the Adler-Kostant-Symes system whose Hamiltonian is the quadratic form
associated with the scalar product on the Lie algebra.Comment: 10 pages, LaTeX2
Construction of KP Hierarchies in Terms of Finite Number of Fields and their Abelianization
The -boson representations of KP hierarchy are constructed in terms of
mutually independent two-boson KP representations for arbitrary number .
Our construction establishes the multi-boson representations of KP hierarchy as
consistent Poisson reductions of standard KP hierarchy within the -matrix
scheme. As a byproduct we obtain a complete description of any
finitely-many-field formulation of KP hierarchy in terms of Darboux coordinates
with respect to the first Hamiltonian structure. This results in a series of
representations of \Win1\, algebra made out of arbitrary even number of boson
fields.Comment: 12 p., LaTeX, minor typos corrected, BGU-93/2/June-P
Regularization of Toda lattices by Hamiltonian reduction
The Toda lattice defined by the Hamiltonian with , which
exhibits singular (blowing up) solutions if some of the , can be
viewed as the reduced system following from a symmetry reduction of a subsystem
of the free particle moving on the group G=SL(n,\Real ). The subsystem is
, where consists of the determinant one matrices with
positive principal minors, and the reduction is based on the maximal nilpotent
group . Using the Bruhat decomposition we show that the full
reduced system obtained from , which is perfectly regular, contains
Toda lattices. More precisely, if is odd the reduced system
contains all the possible Toda lattices having different signs for the .
If is even, there exist two non-isomorphic reduced systems with different
constituent Toda lattices. The Toda lattices occupy non-intersecting open
submanifolds in the reduced phase space, wherein they are regularized by being
glued together. We find a model of the reduced phase space as a hypersurface in
{\Real}^{2n-1}. If for all , we prove for that the
Toda phase space associated with is a connected component of this
hypersurface. The generalization of the construction for the other simple Lie
groups is also presented.Comment: 42 pages, plain TeX, one reference added, to appear in J. Geom. Phy
Two-Matrix String Model as Constrained (2+1)-Dimensional Integrable System
We show that the 2-matrix string model corresponds to a coupled system of
-dimensional KP and modified KP (\KPm) integrable equations subject to a
specific ``symmetry'' constraint. The latter together with the
Miura-Konopelchenko map for \KPm are the continuum incarnation of the matrix
string equation. The \KPm Miura and B\"{a}cklund transformations are natural
consequences of the underlying lattice structure. The constrained \KPm system
is equivalent to a -dimensional generalized KP-KdV hierarchy related to
graded . We provide an explicit representation of this
hierarchy, including the associated -algebra of the second
Hamiltonian structure, in terms of free currents.Comment: 12+1 pgs., LaTeX, preprint: BGU-94 / 15 / June-PH, UICHEP-TH/94-
Integrable Systems and Poisson-Lie T-duality: a finite dimensional example
We study the deep connection between integrable models and Poisson-Lie
T-duality working on a finite dimensional example constructed on SL(2,C) and
its Iwasawa factors SU(2) and B. We shown the way in which Adler-Kostant-Symes
theory and collective dynamics combine to solve the equivalent systems from
solving the factorization problem of an exponential curve in SL(2,C). It is
shown that the Toda system embraces the dynamics of the systems on SU(2) and B.Comment: 34 page
Dirac quantization of free motion on curved surfaces
We give an explicit operator realization of Dirac quantization of free
particle motion on a surface of codimension 1. It is shown that the Dirac
recipe is ambiguous and a natural way of fixing this problem is proposed. We
also introduce a modification of Dirac procedure which yields zero quantum
potential. Some problems of abelian conversion quantization are pointed out.Comment: 16 page
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