Regular and Chaotic Dynamics

, Volume 20, Issue 3, pp 277–292 | Cite as

Intermediate Toda systems

  • Pantelis A. DamianouEmail author
  • Hervé Sabourin
  • Pol Vanhaecke


We construct a large family of Hamiltonian systems which interpolate between the classical Kostant-Toda lattice and the full Kostant-Toda lattice and we discuss their integrability. There is one such system for every nilpotent ideal I in a Borel subalgebra b+ of an arbitrary simple Lie algebra g. The classical Kostant-Toda lattice corresponds to the case of I = [n+, n+], where n+ is the unipotent ideal of b+, while the full Kostant-Toda lattice corresponds to I = {0}. We mainly focus on the case I = [[n+, n+], n+]. In this case, using the theory of root systems of simple Lie algebras, we compute the rank of the underlying Poisson manifolds and construct a set of (rational) functions in involution, large enough to ensure Liouville integrability. These functions are restrictions of well-chosen integrals of the full Kostant-Toda lattice, except for the case of the Lie algebras of type C and D where a different function (Casimir) is needed. The latter fact, and other ones listed in the paper, point at the Liouville integrability of all the systems we construct, but also at the nontriviality of obtaining the result in full generality.


Toda lattices integrable systems 

MSC2010 numbers

37J35 17B20 17B22 70H06 


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Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Pantelis A. Damianou
    • 1
    Email author
  • Hervé Sabourin
    • 2
  • Pol Vanhaecke
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus
  2. 2.Laboratoire de Mathématiques, UMR 7348 du CNRSUniversité de PoitiersFuturoscope Chasseneuil CedexFrance

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