Continuous Limit
The Toda chain equations of motion, in the continuous limit in which the distance between neighbors goes to zero, become the KdV equation. Here the index labeling the particle in the chain becomes the new spacial coordinate.
In contrast, the Toda field theory is achieved by introducing a new spacial coordinate which is independent of the chain index label. This is done in a relativistically invariant way, so that time and space are treated on equal grounds. This means that the Toda field theory is not a continuous limit of the Toda chain.
Read more about this topic: Toda Oscillator
Famous quotes containing the words continuous and/or limit:
“There was a continuous movement now, from Zone Five to Zone Four. And from Zone Four to Zone Three, and from us, up the pass. There was a lightness, a freshness, and an enquiry and a remaking and an inspiration where there had been only stagnation. And closed frontiers. For this is how we all see it now.”
—Doris Lessing (b. 1919)
“Moreover, the universe as a whole is infinite, for whatever is limited has an outermost edge to limit it, and such an edge is defined by something beyond. Since the universe has no edge, it has no limit; and since it lacks a limit, it is infinite and unbounded. Moreover, the universe is infinite both in the number of its atoms and in the extent of its void.”
—Epicurus (c. 341–271 B.C.)