The Toda lattice, old and new

Carlos Tomei

doi:10.3934/jgm.2013.5.511

Article Contents

2013, Volume 5, Issue 4: 511-530. Doi: 10.3934/jgm.2013.5.511

This issue Previous Article Higher-order mechanics: Variational principles and other topics Next Article

The Toda lattice, old and new

Carlos Tomei^1,

1.
Departamento de Matemática, PUC-Rio, R. Mq. S. Vicente 225, Rio de Janeiro 22451-900

Received: May 31, 2013

Revised: September 30, 2013

Published: November 2013

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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.

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Mathematics Subject Classification: Primary: 65F15, 37S35; Secondary: 53D05.

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