Geometric Constructions of Toda and Periodic Toda Chains

A geometrical and neat framework is established to derive both Toda and periodic Toda systems from the geodesics of symmetric spaces. The counterpart of the Iwasawa decomposition of a semisimple Cie group in the case of a loop group is also derived. By these, we get a Lie su balgebra with Lie bracket ,]R, and the corresponding Poisson bracket {,}R gives the Hamiltonian form of the periodic Toda chains.