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The modular hierarchy of the Toda lattice. (English) Zbl 1146.53053
The modular vector field of a Poisson manifold is a notion introduced by Koszul and defines a class in the first Poisson cohomology space of the manifold, the so-called modular class. In this work, the authors consider the finite, nonperiodic Toda lattice in the Flaschka variables \((a,b)\) as well as in the natural \((q,p)\) variables. In both cases, it is known that the hierarchy admits multiple Hamiltonian structures. Consequently, there is a family of modular vector fields in each case. The authors show that the modular vector fields are Hamiltonian vector fields with respect to the corresponding structures. Moreover, in the natural \((q,p)\) variables, they show that starting from the modular vector field of the second Poisson structure, the other modular vector fields can be generated by using the recursion operator.

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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