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U. Helmke
2.3
TODA FLOW
The Toda flow is one of the earliest examples of an isospectral flow which yields a continuous
analogue of a numerical matrix eigenvalue problem. The original motivation for the flow
stems from statistical mechanics; cf. Flaschka (1974, 1975), Moser (1975), Kostant (1979).
Consider the one-dimensional lattice of n idealized mass points Zl < ... < xn, located
on the real axis I~. We assume that the potential energy of the system is described by
Y(xl,..., z~) = ~ e~-~+ ~
k=92
?%
Xo'= -oo, xn+l"=
+oo; while the kinetic energy has the usual form ] ~] x~.
k=l
Thus, ...