One fruitful motivating principle of much research on the family of
integrable systems known as ``Toda lattices'' has been the heuristic assumption
that the periodic Toda lattice in an affine Lie algebra is directly analogous
to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper
shows that the analogy is not perfect. A discrepancy arises because the natural
generalization of the structure theory of finite-dimensional simple Lie
algebras is not the structure theory of loop algebras but the structure theory
of affine Kac-Moody algebras. In this paper we use this natural generalization
to construct the natural analog of the nonperiodic Toda lattice. Surprisingly,
the result is not the periodic Toda lattice but a new completely integrable
system on the periodic Toda lattice phase space. This integrable system is
prescribed purely in terms of Lie-theoretic data. The commuting functions are
precisely the gauge-invariant functions one obtains by viewing elements of the
loop algebra as connections on a bundle over S1
