134 research outputs found
Dirac method and symplectic submanifolds in the cotangent bundle of a factorizable Lie group
In this work we study some symplectic submanifolds in the cotangent bundle of
a factorizable Lie group defined by second class constraints. By applying the
Dirac method, we study many issues of these spaces as fundamental Dirac
brackets, symmetries, and collective dynamics. This last item allows to study
integrability as inherited from a system on the whole cotangent bundle, leading
in a natural way to the AKS theory for integrable systems
The quantum bialgebra associated with the eight-vertex R-matrix
The quantum bialgebra related to the Baxter's eight-vertex R-matrix is found
as a quantum deformation of the Lie algebra of sl(2)-valued automorphic
functions on a complex torus.Comment: 4 page
Bedrock and Surficial Geologic Map of the Red Rock 7.5’ Quadrangle, Beaverhead County, Southwestern Montana
The Red Rock 7.5 minute quadrangle, located in Beaverhead County, southwestern Montana, spans the Red Rock River Valley, an extensional graben formed between the Tendoy mountain front and the western flank of the Blacktail-Snowcrest uplift (Fig. 1). Notable landmarks within the quadrangle include the Clark Canyon Reservoir (Bureau of Reclamation dam number MT00569) located in the northwest area of the quadrangle and Interstate 15 which runs northwest-southeast through the quadrangle. The highest elevations in the map area are located within the Tendoy Mountains and the Red Rock Hills and are underlain by Paleozoic and Cenozoic bedrock. From these points, broad alluvial fans grade down to the Red Rock River Valley. The quadrangle contains about 3,000 ft of relief.
Mapping of the Red Rock quadrangle was done at a scale of 1:12,000 and was compiled at a scale of 1:24,000. Field work was completed in the summer of 2005 in collaboration with the mapping of the adjacent Briggs Ranch and Kidd quadrangles (Figs. 1 and 2). This strategy allowed for the comparison of structure and stratigraphy across quadrangle boundaries and provided a regional context for the mapping of each quadrangle. This new mapping complements previous mapping of the Monument Hill quadrangle (Newton and others, 2005), Dixon Mountain quadrangle (Harkins and others, 2004b), Caboose Canyon quadrangle (Harkins and others, 2004a), and Dell quadrangle (Aschoff and Schmitt, 2005) and collectively provides new detailed mapping and analysis of a portion of the Red Rock River Valley from Lima to the Clark Canyon Dam (Figs. 1 and 2). This report includes a map and cross section for the Red Rock quadrangle as well as a discussion of the stratigraphy and structure of the map area
Multi-Hamiltonian structures for r-matrix systems
For the rational, elliptic and trigonometric r-matrices, we exhibit the links
between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of
matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral
curves and sheaves supported on them; (c) Symmetric products of a surface. We
have, at each level, a linear space of compatible Poisson structures, and the
maps relating the levels are Poisson. This leads in a natural way to Nijenhuis
coordinates for these spaces. At level (b), there are Hamiltonian systems on
these spaces which are integrable for each Poisson structure in the family, and
which are such that the Lagrangian leaves are the intersections of the
symplective leaves over the Poisson structures in the family. Specific examples
include many of the well-known integrable systems.Comment: 26 pages, Plain Te
Three natural mechanical systems on Stiefel varieties
We consider integrable generalizations of the spherical pendulum system to
the Stiefel variety for a certain metric. For the case
of V(n,2) an alternative integrable model of the pendulum is presented.
We also describe a system on the Stiefel variety with a four-degree
potential. The latter has invariant relations on which provide the
complete integrability of the flow reduced on the oriented Grassmannian variety
.Comment: 14 page
Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
The Hamiltonian structure of the monodromy preserving deformation equations
of Jimbo {\it et al } is explained in terms of parameter dependent pairs of
moment maps from a symplectic vector space to the dual spaces of two different
loop algebras. The nonautonomous Hamiltonian systems generating the
deformations are obtained by pulling back spectral invariants on Poisson
subspaces consisting of elements that are rational in the loop parameter and
identifying the deformation parameters with those determining the moment maps.
This construction is shown to lead to ``dual'' pairs of matrix differential
operators whose monodromy is preserved under the same family of deformations.
As illustrative examples, involving discrete and continuous reductions, a
higher rank generalization of the Hamiltonian equations governing the
correlation functions for an impenetrable Bose gas is obtained, as well as dual
pairs of isomonodromy representations for the equations of the Painleve
transcendents and .Comment: preprint CRM-1844 (1993), 28 pgs. (Corrected date and abstract.
Integrable discretizations of some cases of the rigid body dynamics
A heavy top with a fixed point and a rigid body in an ideal fluid are
important examples of Hamiltonian systems on a dual to the semidirect product
Lie algebra . We give a Lagrangian derivation of
the corresponding equations of motion, and introduce discrete time analogs of
two integrable cases of these systems: the Lagrange top and the Clebsch case,
respectively. The construction of discretizations is based on the discrete time
Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian
reduction. The resulting explicit maps on are Poisson with respect to
the Lie--Poisson bracket, and are also completely integrable. Lax
representations of these maps are also found.Comment: arXiv version is already officia
R-matrix approach to integrable systems on time scales
A general unifying framework for integrable soliton-like systems on time
scales is introduced. The -matrix formalism is applied to the algebra of
-differential operators in terms of which one can construct infinite
hierarchy of commuting vector fields. The theory is illustrated by two
infinite-field integrable hierarchies on time scales which are difference
counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer
soliton systems are constructed as related finite-field restrictions.Comment: 21 page
Generalized Drinfeld-Sokolov Reductions and KdV Type Hierarchies
Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local
reductions of Hamiltonian flows generated by monodromy invariants on the dual
of a loop algebra. Following earlier work of De Groot et al, reductions based
upon graded regular elements of arbitrary Heisenberg subalgebras are
considered. We show that, in the case of the nontwisted loop algebra
, graded regular elements exist only in those Heisenberg
subalgebras which correspond either to the partitions of into the sum of
equal numbers or to equal numbers plus one . We prove that the
reduction belonging to the grade regular elements in the case yields
the matrix version of the Gelfand-Dickey -KdV hierarchy,
generalizing the scalar case considered by DS. The methods of DS are
utilized throughout the analysis, but formulating the reduction entirely within
the Hamiltonian framework provided by the classical r-matrix approach leads to
some simplifications even for .Comment: 43 page
Canonical transformations of the extended phase space, Toda lattices and Stackel family of integrable systems
We consider compositions of the transformations of the time variable and
canonical transformations of the other coordinates, which map completely
integrable system into other completely integrable system. Change of the time
gives rise to transformations of the integrals of motion and the Lax pairs,
transformations of the corresponding spectral curves and R-matrices. As an
example, we consider canonical transformations of the extended phase space for
the Toda lattices and the Stackel systems.Comment: LaTeX2e + Amssymb, 22p
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