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A gentle (without chopping) approach to the full Kostant-Toda lattice. (English) Zbl 1099.37050

The Toda lattice is fundamental and the basis of all finite-dimensional integrable systems. The Hamiltonian of the Toda lattice is given by \[ H(q_1,\dots,q_N,p_1,\dots,p_N)=\sum^N_{i=1}\frac 12 p^2_i+\sum^{N-1}_{i=1}e^{q_i-q_{i+1}}.\tag{1} \] The authors propose a new algorithm for obtaining the rational integrals of the full Kostant-Toda lattice. To this end, they consider a reduction of a bi-Hamiltonian system on gl\((n,\mathbb{R})\). This system was obtained by reducing the space of maps \(\mathbb{Z}_n\) to GL\((u,\mathbb{R})\) endowed with a structure of a pair of Lie algebras.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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