In these notes, we will discuss from a purely geometric point of view classical mechanics on certain type of Grassmannians and discs. We will briefly discuss a superversion which in some sense combines these two models, and corresponds to the large-$N_c$ limit of $SU(N_c)$ gauge theory with fermionic and bosonic matter fields, both in the fundamental representation, in $1 + 1$ dimensions . This result is a natural extension of ideas in . There it has been shown that the large-$N_c$ phase space of $1+1$ dimensional QCD is given by an infinite dimensional Grassmannian. The complex scalar field version of this theory is worked out in  and it is shown that the phase space is an infinite dimensional disc.