In this paper, we use Flaschka's change of variables of the open Toda lattice
and its interpretation in term of the group structure of the LU factorisation
as a coadjoint motion on a certain dual of Lie algebra to implement a structure
preserving noise and dissipation. Both preserve the structure of coadjoint
orbit, that is the space of symmetric tri-diagonal matrices and arise as a new
type of multiplicative noise and nonlinear dissipation of the Toda lattice. We
investigate some of the properties of these deformations and in particular the
continuum limit as a stochastic Burger equation with a nonlinear viscosity.
This work is meant to be exploratory, and open more questions that we can
answer with simple mathematical tools and without numerical simulations.