Regularization of toda lattices by Hamiltonian reduction

László Fehér, Izumi Tsutsui

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

The Toda lattice defined by the Hamiltonian H = 1/2 ∑ni=1 p2i + ∑n-1i=1 νieqi-qi+1 with νi ∈ {±1}, which exhibits singular (blowing up) solutions if some of the νi = -1, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group G = SL(n, ℝ). The subsystem is T*Ge, where Ge = N+AN- consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group N+ × N-. Using the Bruhat decomposition we show that the full reduced system obtained from T*G, which is perfectly regular, contains 2n-1 Toda lattices. More precisely, if n is odd the reduced system contains all the possible Toda lattices having different signs for the νi. If n is even, there exist two non-isomorphic reduced systems with different constituent Toda lattices. The Toda lattices occupy non-intersecting open submanifolds in the reduced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in ℝ2n-1. If νi = 1 for all i, we prove for n = 2, 3, 4 that the Toda phase space associated with T*Ge is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented.

Original languageEnglish
Pages (from-to)97-135
Number of pages39
JournalJournal of Geometry and Physics
Volume21
Issue number2
DOIs
Publication statusPublished - Jan 1997

Keywords

  • Hamiltonian symmetry reduction
  • Toda lattice

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology

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