4 - Toda without Hamiltonian Structure

Summary

There are still algebraic surprises lying concealed within the Toda flow that we have not yet described. A simple argument with matrix factorizations gives rise to new inverse variables, the Z-coordinates, together with new commuting vector fields. In these new variables, the Toda flows become explicit, straight line motions in RN for an appropriate dimension N (Toda flows on full matrices are also considered). Moreover, orbit limits, such as diagonal matrices, lie beyond the scope of standard variables (see, e.g., M below for Jacobi matrices), and the asymptotic analysis must proceed using ad hoc methods. Such orbit limits, however, belong to the domain of the new variables, and asymptotic computations are easily performed through local theory. As described in Remark 4.14, the methods in this section extend the purview of the Toda system substantially.