Regularization of Toda lattices by Hamiltonian reduction

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 $\nu_i=1$ for all $i$, we prove for $n=2,3,4$ that the Toda phase space associated with $T^*G_e$ is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented.