276 research outputs found
The Toda lattice is super-integrable
We prove that the classical, non-periodic Toda lattice is super-integrable.
In other words, we show that it possesses 2N-1 independent constants of motion,
where N is the number of degrees of freedom. The main ingredient of the proof
is the use of some special action--angle coordinates introduced by Moser to
solve the equations of motion.Comment: 8 page
Poisson brackets with prescribed Casimirs
We consider the problem of constructing Poisson brackets on smooth manifolds
with prescribed Casimir functions. If is of even dimension, we achieve
our construction by considering a suitable almost symplectic structure on ,
while, in the case where is of odd dimension, our objective is achieved by
using a convenient almost cosymplectic structure. Several examples and
applications are presented.Comment: 24 page
From the Toda Lattice to the Volterra lattice and back
We discuss the relationship between the multiple Hamiltonian structures of
the generalized Toda lattices and that of the generalized Volterra lattices. We
use a symmtery approach for Poisson structures that generalizes the Poisson
involution theorem.Comment: 15 pages; Final version to appear in Reports on Math. Phy
A Gentle (without Chopping) Approach to the Full Kostant-Toda Lattice
In this paper we propose a new algorithm for obtaining the rational integrals
of the full Kostant-Toda lattice. This new approach is based on a reduction of
a bi-Hamiltonian system on gl(n,R). This system was obtained by reducing the
space of maps from Z_n to GL(n,R) endowed with a structure of a pair of
Lie-algebroids.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
The modular hierarchy of the Toda lattice
The modular vector field plays an important role in the theory of Poisson
manifolds and is intimately connected with the Poisson cohomology of the space.
In this paper we investigate its significance in the theory of integrable
systems. We illustrate in detail the case of the Toda lattice both in Flaschka
and natural coordinates.Comment: 16 pages, 29 references, to appear in Differential Geometry and its
application
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