The Journal of Geometric Mechanics (JGM)

The Toda lattice, old and new

Pages: 511 - 530, Volume 5, Issue 4, December 2013      doi:10.3934/jgm.2013.5.511

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Carlos Tomei - Departamento de Matemática, PUC-Rio, R. Mq. S. Vicente 225, Rio de Janeiro 22451-900, Brazil (email)

Abstract: Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also non-Jacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra.
    The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.

Keywords:  Toda lattice, completely integrable systems, QR algorithm, isospectral manifolds, inverse scattering.
Mathematics Subject Classification:  Primary: 65F15, 37S35; Secondary: 53D05.

Received: June 2013;      Revised: October 2013;      Available Online: December 2013.