Abstract
The equations for the (finite, nonperiodic) Toda lattice can, as is well known, be written in Lax form,
Here X is a symmetric tridiagonal matrix and Π is the projection onto the skew-symmetric summand in the decomposition of X into skew-symmetric plus upper triangular. The eigenvalues of X are constants of motion of (1), and the Toda lattice turns out to be a completely integrable Hamiltonian system.
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Ercolani, N.M., Flaschka, H., Singer, S. (1993). The Geometry of the Full Kostant-Toda Lattice. In: Babelon, O., Kosmann-Schwarzbach, Y., Cartier, P. (eds) Integrable Systems. Progress in Mathematics, vol 115. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0315-5_9
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