  Within mean field Gross-Pitaevskii framework, ultra cold atomic condensates
with long range interaction is predicted to have a supersolid like ground state
beyond a critical interaction strength. Such mean field supersolid like ground
state has periodically modulated superfluid density which implies the
coexistence of superfluid and crystalline order. Ultra cold atomic system in
such mean field ground state can be subjected to artificial gauge field created
either through rotation or by introducing space dependent coupling among
hyperfine states of the atoms using Raman lasers. Starting from this
Gross-Pitaevskii energy functional that describes such systems at zero
temperature, we construct hydrodynamic theory to describe the low energy long
wavelength excitations of such rotating supersolid of weakly interacting ultra
cold atoms in two spatial dimensions for generic type of long range
interaction. We treat the supersolidity in such system within the framework of
well known two fluid approximation. Considering such system in the fast
rotation limit where a vortex lattice in superfluid coexists with the
supersolid lattice, we analytically obtain the dispersion relations of
collective excitations around this equilibrium state. The dispersion relation
gives the modes of the rotating supersolid which can be experimentally measured
within the current technology. We point out that this can clearly identify such
a ultra cold atomic supersolid phase in an unambiguous way.
