  We present an analysis of the Yangian symmetries of various bosonic sectors
of the dilatation operator of $\cal N$$=4$ SYM. The analysis is presented from
the point of view of Hamiltonian matrix models. In the various SU(n) sectors,
we give a modified presentation of the Yangian generators, which are conserved
on states of any size. A careful analysis of the Yangian invariance of the full
SO(6) sector of the scalars is also presented in this paper. We also study the
Yangian invariance of the dilatation operator beyond first order perturbation
theory in the SU(2) sector. Following this, we derive the continuum limits of
the various matrix models and reproduce the sigma model actions for fast moving
strings reported in the recent literature. We motivate the constructions of
continuum sigma models (corresponding to both the SU(n) and SO(n) sectors) as
variational approximations to the matrix model Hamiltonians. These sigma models
retain the semi-classical counterparts of the original Yangian symmetries of
the dilatation operator. The semi-classical Yangian symmetries of the sigma
models are worked out in detail. The zero curvature representation of the
equations of motion and the construction of the transfer matrix for the SO(n)
sigma model obtained as the continuum limit of the one loop bosonic dilatation
operator is carried out, and the similar constructions for the SU(n) case are
also discussed.
