1,721,073 research outputs found
The R-matrix theory and the reduction of Poisson manifolds
No full textFrom the MR review by W.Oevel: "Three different constructions of multi-Hamiltonian structures associated with classical R-matrices are reviewed. The equivalence between the tensor product formulation and the algebraic operator approach is discussed. The main point of the work is the discussion of the reduction properties of the general Poisson brackets defined on the underlying algebra. In order to obtain the multi-Hamiltonian structure for a specific integrable system, one has to restrict or reduce the brackets to an orbit of suitable Lax matrices defining a submanifold of the entire algebra. A variety of examples are used to illustrate the possible complications arising in the reduction process. For the Gelfand-Dikii hierarchy and the finite gl(n)-Toda lattice the author reviews how the corresponding R-matrices successfully lead to the multi-Hamiltonian formulations. However, for the restriction of the Toda lattice to Lax matrices in the subalgebra sl(n) no systematic construction of the higher Poisson brackets is available, as this restriction is not compatible with the R-matrix approach to the higher brackets. Similar problems arise for the relativistic Toda hierarchy. Thus, it is explicitly shown that the reduction of the Poisson structures associated with R-matrices may become an obstruction in constructing the bi-Hamiltonian structures of a specific hierarchy of integrable equations.
Soluzioni particolari delle equazioni indefinite della Meccanica per varieta' Riemanniane con gruppi di moto
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Some remarks on the construction of bihamiltonian manifolds
No full textFrom the MR review by M.Modugno: "The paper deals with different constructions of bi-Hamiltonian manifolds, i.e., manifolds M equipped with two skew-symmetric contravariant 2-tensors P and Q , such that [P,P]=[Q,Q]=[P,Q]=0, where [ , ] is the Schouten bracket. The first construction can arise from the infinitesimal deformation of a Poisson manifold (M,P) with respect to a given vector field of the manifold. A second construction is a reduction based on the search for suitable bi-Hamiltonian submanifolds of a given bi-Hamiltonian manifold (M,P,Q). The two approaches are compared and discussed in the cases when M is the dual of a Lie algebra. An application is obtained by considering the above methods for a Lie algebra of formal series, which is related to the so called polynomial spectral problem.
Sulla congruenza e il tensore a divergenza nulla per spazi-tempo del tipo di Schwarzschild, Parte I
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On the reduction theory of the Nijenhuis operator and its applications to Gel'fand-Dikii equations, Proc.Iutam-Isimm Symposium on Modern Developments in Analitical Mechanics, ATTI ACC.SCIENZE DI TORINO, Suppl.Vol.117
No full textThe aim of this paper is twofold: to develop the basic elements of the reduction theory of the Nijenhuis operators, and to explicitly construct the recursion operator of the general Gelfand-Dickii equations. To deal with the first problem, the geometric properties of the Nijenhuis operator are pointed out: they mainly concern a remarkable integrable distribution defined by such operators. The two “reduction theorems” presented in the paper give the conditions allowing to obtain a reduced kernel-free Nijenhuis operator either on any fixed leaf or on the space of the leaves of this distribution, respectively. In the second part of the paper, a remarkable class of matrix Nijenhuis operators is introduced, depending on an arbitrary matrix with constant entries; the recursion operator of the Gelfand-Dickii equations is then obtained as a particular application of the reduction theory to this class of operators
On the continuous limit of integrable lattices III. Kupershmidt systems and sl(n+1) KdV theories
No full textWe discuss the connection between the zero-spacing limit of the N- fields Kupershmidt lattice and the KdV-type theory corresponding to the Lie algebra sl(N+1). The case N = 2 is worked out in detail, recovering from the limit process the Boussinesq theory with its infinitely many commuting vector fields, their Lax pairs and Hamiltonian formulations. Actually, the ‘recombination method’ proposed here to derive the Boussinesq hierarchy from the limit of the N = 2 Kupershmidt system works, in principle, for arbitrary N
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