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1.
对Loop代数L(Dr)上具有长程相互作用的Toda力学系统进行推广 ,用一组有序整数对 (X ,Y)来表示Toda链 ,构造出Loop代数L (Dr)的LaxPair,并给出了系统在 (3,2 )Toda链情况下的运动方程和Hamiltonian结构 .在此模型中 ,标准的Toda变量之间和附加的坐标变量之间的泊松括号都非零 ,部分附加的坐标变量之间的泊松结构构成李代数 . 相似文献
2.
We construct a class of integrable generalization of Toda
mechanics with long-range interactions. These systems are
associated with the loop algebras L(Cr) and L(Dr) in the sense that their Lax matrices can be realized in
terms of the c=0 representations of the affine Lie algebras
C(1)r and
D(1)r and the interactions pattern involved
bears the typical characters of the corresponding root systems. We
present the equations of motion and the Hamiltonian structure.
These generalized systems can be identified unambiguously by
specifying the underlying loop algebra together with an ordered
pair of integers (n,m). It turns out that different systems
associated with the same underlying loop algebra but with
different pairs of integers
(n1,m1) and (n2,m2) with
n2<n1 and
m2<m1 can be related by a nested Hamiltonian
reduction procedure. For all nontrivial generalizations, the extra
coordinates besides the standard Toda variables are Poisson
non-commute, and when either $n$ or
m≥3, the Poisson
structure for the extra coordinate variables becomes some Lie
algebra (i.e. the extra variables appear linearly on the
right-hand side of the Poisson brackets). In the quantum case, such
generalizations will become systems with noncommutative variables
without spoiling the integrability. 相似文献
3.
In this paper we consider gradient structures in the dynamics and geometry of the asymmetri nonperiodic tridiagonal and full Toda flow equations. We compare and contrast a number of formulations of the nonperiodic Toda equations. In the case of the full Kostant (asymmetric) Toda flow we explain the role of noncommutative integrability in its qualitative behavior. We describe the relationship between the asymmetric Toda flows and the symmetric and indefinite Toda flows, and prove in particular that one may conjugate from the full Kostant Toda flows to the full symmetric Toda flows via a Poisson map. 相似文献
4.
《Physics letters. A》1999,251(3):177-183
Under three relations connecting the field variables of Toda flows and that of KdV flows, we present three new sequences of combinations of the equations in the Toda hierarchy which have the KdV hierarchy as a continuous limit. The relation between the Poisson structures of the KdV hierarchy and the Toda hierarchy in the continuous limit is also studied. 相似文献
5.
Yu. B. Suris 《Physics letters. A》1993,180(6):419-429
We introduce a general quadratic Poisson bracket on the associative algebra equipped with non-degenerate scalar product. With the help of this bracket we obtain the interpretation of the Toda and relativistic Toda lattices as the restrictions of one and the same bi-Hamiltonian system to two different low-dimensional manifolds, which are Poisson submanifolds with respect to two brackets simultaneously. 相似文献
6.
For any classical Lie algebra $\mathfrak{g}$,
we construct a family of
integrable generalizations of Toda mechanics labeled a pair of ordered
integers $(m,n)$. The universal form of the Lax pair, equations of motion,
Hamiltonian as well as Poisson brackets are provided, and explicit examples
for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all
$m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda
chains are natural reductions of that of the $(m,n)$-chain, and for $m=n$,
there is also a family of symmetrically reduced Toda systems, the
$(m,m)_{\mathrm{Sym}}$-Toda systems, which are also integrable. In the quantum
case, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics of
standard Toda variables coupled to noncommutative variables. Except for the
symmetrically reduced cases, the integrability for all $(m,n)$-Toda systems
survive after quantization. 相似文献
7.
Luen-Chau Li 《Communications in Mathematical Physics》2006,265(2):333-372
We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary dynamical Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars-Schneider equations which correspond to the N-soliton solutions of
affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid. 相似文献
8.
A geometrical and neat framework is established to derive both Toda and periodic Toda systems from the geodesics of symmetric spaces. The counterpart of the Iwasawa decomposition of a semisimple Cie group in the case of a loop group is also derived. By these, we get a Lie su balgebra with Lie bracket [,]R, and the corresponding Poisson bracket {,}R gives the Hamiltonian form of the periodic Toda chains. 相似文献
9.
10.
A. V. Tsiganov 《Physics of Atomic Nuclei》2002,65(6):1128-1134
For an integrable system on Poisson manifolds, a construction of separated variables is discussed. We suppose that, for a given integrable system, we know a realization of the corresponding Lagrangian submanifold as the product of plane curves. In this case, we can use properties of the foliation of the initial Poisson manifold on symplectic leaves and values of the Casimir functions in order to construct separated variables. 相似文献
11.
The general conformal affine Toda (CAT) fields are derived as the result of imposing the constraint explicitly on the gauged WZNW action. The reduction procedure naturally indicates the integrability of the resulting system. The action, equations of motion, canonical Poisson brackets for this system are derived from WZNW model. The energy momentum tensor is derived also and is shown to be traceless after being improved. 相似文献
12.
V.S. Gerdjikov A. Kyuldjiev G. Marmo G. Vilasi 《The European Physical Journal B - Condensed Matter and Complex Systems》2002,29(2):177-181
We consider generalizations of the standard Hamiltonian dynamics to complex dynamical variables and introduce the notions
of real Hamiltonian form in analogy with the notion of real forms for a simple Lie algebra. Thus to each real Hamiltonian
system we are able to relate several nonequivalent ones. On the example of the complex Toda chain we demonstrate how starting
from a known integrable Hamiltonian system (e.g. the Toda chain) one can complexify it and then project onto different real forms.
Received 18 October 2001 / Received in final form 24 May 2002 Published online 2 October 2002
RID="a"
ID="a"e-mail: gerjikov@inrne.bas.bg 相似文献
13.
We show that the analog of the Miura maps and Bäcklund-Darboux transformations for a general class of equations of Toda type and for a generalized class of periodic Toda flows are isomorphisms of Poisson Lie groups. 相似文献
14.
Luen-Chau Li 《Communications in Mathematical Physics》1999,203(3):573-592
Given a classical r-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian
formulation of Lax equations. Examples for which our formalism applies include the Benny hierarchy, the dispersionless Toda
lattice hierarchy, the dispersionless KP and modified KP hierarchies, the dispersionless Dym hierarchy, etc.
Received: 10 February 1998 / Accepted: 9 December 1998 相似文献
15.
16.
Yu. B. Chernyakov G. I. Sharygin A. S. Sorin 《Communications in Mathematical Physics》2014,330(1):367-399
In this paper we discuss some geometrical and topological properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagram for the full symmetric Toda system in dimensions n = 3, 4 coincides with the Hasse diagram of the Bruhat order of symmetric groups S 3 and S 4. The method we use is based on the existence of a vast collection of invariant subvarieties of the Toda flow in orthogonal groups. We show how one can extend it to the case of general n. The resulting theorem identifies the set of singular points of dim = n Toda flow with the elements of the permutation group S n , so that points will be connected by a trajectory, if and only if the corresponding elements are Bruhat comparable. We also show that the dimension of the submanifolds, spanned by the trajectories connecting two singular points, is equal to the length of the corresponding segment in the Hasse diagram. This is equivalent to the fact that the full symmetric Toda system is in fact a Morse–Smale system. 相似文献
17.
Martin C Gutzwiller 《Annals of Physics》1981,133(2):304-331
The periodic Toda lattice consists of N particles which move along a closed line and are coupled with an exponential spring to their immediate neighbors. This system, in contrast to the open Toda lattice, has only bound states. In the method of Kac and Van Moerbeke, the classical periodic Toda chain is transformed to a new of set of canonically conjugate variables, μ and ν, which are closely related to the natural coordinates of an open Toda chain with one particle less. The quantum mechanical eigenfunctions for this reduced system are constructed explicitly, and this allows the quantum mechanical analogs of μ and ν to be defined. The bounds states for the periodic Toda chain are then written as linear combinations of functions resembling the wave functions of the reduced open chain. These functions satisfy a set of remarkably simple recursion formulas, and the coefficients in the expansion can be written essentially as a product of as many factors as pairs of conjugate variables μ and ν. Each factor is given as a solution of a second order difference equation which can be recognized as a quantum analog for the equations of motion of one pair μ and ν. The quantization conditions result from cancelling out the exponential growth in the overall wave function, and are phrased in terms of certain phase angles being submultiples of π according as the representation of the group of cyclic permutations. The calculations are simple for N = 3, and moderately tricky for N = 4 although the results are always fairly obvious. 相似文献
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20.
Anindya Ghose Choudhury 《Central European Journal of Physics》2013,11(1):49-58
In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed. 相似文献