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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
##### Keywords:
Poisson manifolds; modular vector fields; Toda lattice
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