  We study the physics of interacting spin-$1$ bosons in an optical lattice
using a variational Gutzwiller technique. We compute the mean-field ground
state wave-function and discuss the evolution of the condensate, spin, nematic,
and singlet order parameters across the superfluid-Mott transition. We then
extend the Gutzwiller method to derive the equations governing the dynamics of
low energy excitations in the lattice. Linearizing these equations, we compute
the excitation spectra in the superfluid and Mott phases for both ferromagnetic
and antiferromagnetic spin-spin interactions. In the superfluid phase, we
recover the known excitation spectrum obtained from Bogoliubov theory. In the
nematic Mott phase, we obtain gapped, quadratically dispersing particle and
hole-like collective modes, whereas in the singlet Mott phase, we obtain a
non-dispersive gapped mode, corresponding to the breaking of a singlet pair.
For the ferromagnetic Mott insulator, the Gutzwiller mean-field theory only
yields particle-hole like modes but no Goldstone mode associated with long
range spin order. To overcome this limitation, we supplement the Gutzwiller
theory with a Schwinger boson mean-field theory which captures super-exchange
driven fluctuations. In addition to the gapped particle-hole-like modes, we
obtain a gapless quadratically dispersing ferromagnetic spin-wave Goldstone
mode. We discuss the evolution of the singlet gap, particle-hole gap, and the
effective mass of the ferromagnetic Goldstone mode as the superfluid-Mott phase
boundary is approached from the insulating side. We discuss the relevance and
validity of Gutzwiller mean-field theories to spinful systems, and potential
extensions of this framework to include more exotic physics which appears in
the presence of spin-orbit coupling or artificial gauge fields.
