Regularization of Toda lattices by Hamiltonian reduction

TL;DR

This paper explores how Hamiltonian reduction techniques regularize Toda lattices, revealing multiple consistent systems and phase space structures, especially for different matrix signs and Lie group generalizations.

Contribution

It demonstrates that Toda lattices can be viewed as reduced systems from a larger regular phase space, uncovering multiple lattice configurations and their geometric properties.

Findings

The full reduced system contains 2^{n-1} Toda lattices.

For odd n, all sign configurations are represented.

For even n, two non-isomorphic reduced systems exist.

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 $\nu_i=1$ 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.