  We develop a theory of Lagrangian reduction on loop groups for completely
integrable systems after having exchanged the role of the space and time
variables in the multi-time interpretation of integrable hierarchies. We then
insert the Sobolev norm $H^1$ in the Lagrangian and derive a deformation of the
corresponding hierarchies. The integrability of the deformed equations is
altered and a notion of weak integrability is introduced. We implement this
scheme in the AKNS and SO(3) hierarchies and obtain known and new equations.
Among them we found two important equations, the Camassa-Holm equation, viewed
as a deformation of the KdV equation, and a deformation of the NLS equation.
