REGULARIZATION OF TODA-LATTICES BY HAMILTONIAN REDUCTION

The Toda lattice defined by the Hamiltonian H = 1/2 Sigma(i = 1)(n) p( i)(2) + Sigma(i = 1)(n-1) v(i)3(qi-qi+1) with v(i) epsilon {+/-1}, whi ch exhibits singular (blowing up) solutions if some of the v(i) = -1, can be viewed as the reduced system following from a symmetry reductio n of a subsystem of the free particle moving on the group G = SL(n, R) . The subsystem is TG(e), where Ge = N(+)AN_ consists of the determin ant one matrices with positive principal miners, and the reduction is based on the maximal nilpotent group N-+ x N_. Using the Bruhat decomp osition we show that the full reduced system obtained from TG, which is perfectly regular, contains 2(n-1) Toda lattices. More precisely, i f n is odd the reduced system contains all the possible Toda lattices having different signs for the vi. If it is even, there exist two non- isomorphic reduced systems with different constituent Toda lattices. T he Toda lattices occupy non-intersecting open submanifolds in the redu ced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in R(2n- 1) If v(i) = 1 for all i, we prove for n = 2, 3, 4 that the Toda phase space associated with TG(e) is a connected component of this hypersu rface. The generalization of the construction for the other simple Lie groups is also presented.