Regularization of Toda lattices by Hamiltonian reduction

Feher, Laszlo; Tsutsui, Izumi

doi:10.1016/S0393-0440(96)00010-1

Regularization of Toda lattices by Hamiltonian reduction

Laszlo Feher
(
- INS, Tokyo
)
,
Izumi Tsutsui
(
- INS, Tokyo
)

Nov, 1995

43 pages

Published in:

J.Geom.Phys. 21 (1997) 97-135

e-Print:

hep-th/9511118 [hep-th]

DOI:

10.1016/S0393-0440(96)00010-1

Report number:

INS-1123

View in:

reference search12 citations

Citations per year

Abstract:

The Toda lattice defined by the Hamiltonian

H={1\over 2} \sum_{i=1}~n p_i~2 + \sum_{i=1}~{n-1} \nu_i e~{q_i-q_{i+1}}

with

\nu_i\in \{ \pm 1\}

, which exhibits singular (blowing up) solutions if some of the

\nu_i=-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,\Real ). The subsystem is

T~*G_e

, where

G_e=N_+ A N_-

consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group

N_+ \times N_-

. Using the Bruhat decomposition we show that the full reduced system obtained from

T~*G

, which is perfectly regular, contains

2~{n-1}

Toda lattices. More precisely, if

n

is odd the reduced system contains all the possible Toda lattices having different signs for the

\nu_i

. 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 {\Real}~{2n-1}. If