Regularization of Toda lattices by Hamiltonian reduction

The Toda lattice defined by the Hamiltonian H = 12 ∑n i=1 p 2 i + ∑n−1 i=1 νie qi−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,RI ). 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 2 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 RI . 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.