Gradient Flows on Kač-Moody Algebras and the Periodic Toda Lattice

In this paper, we provide a new formulation for the generalized periodic Toda lattice. Since the work of Kostant, Adler and Symes, it has been known that the Toda lattice is related to the structure of simple Lie algebras. Indeed, the non-periodic and the periodic Toda lattices can be expressed as Hamiltonian systems on coadjoint orbits: the former of a simple Lie group and the latter of the associated loop group. Alternatively, the non-periodic Toda lattice was expressed as a gradient flow on an adjoint orbit of a simple Lie group by Bloch, Brockett and Ratiu. Based on the description of certain gradient flows on adjoint orbits in affine Lie algebras as double bracket equations, we show that the periodic Toda lattice also admits a canonical gradient formulation and relate it to the structure of affine Kač-Moody algebras.