  We have extended the semi-classical theory to include a general account of
matrix valued Hamiltonians, i.e. those that describe quantum systems with
internal degrees of freedoms, based on a generalization of the Gutzwiller trace
formula for a $n\times n$ dimensional Hamiltonian $H(\hat{\boldsymbol
p},\hat{\boldsymbol q})$. The classical dynamics is governed by $n$
Hamilton-Jacobi (HJ) equations, that act in a phase space endowed with a
classical Berry curvature encoding anholonomy in the parallel transport of the
eigenvectors of $H(\hat{\boldsymbol p}\to\boldsymbol p,\hat{\boldsymbol
q}\to\boldsymbol q)$, which describe the internal structure of the
semi-classical particles. This Berry curvature is a fully classical object and
is, in that sense, as fundamental to the semi-classical theory of matrix
Hamiltonians as the Hamilton-Jacobi equations. At the $\mathcal{O}(\hbar^1)$
level, it results in an additional semi-classical phase composed of (i) a Berry
phase and (ii) a dynamical phase resulting from the classical particles "moving
through the Berry curvature". We show that the dynamical part of this
semi-classical phase will, generally, only be zero only for the case in which
the Berry phase is topological (i.e. depends only on the winding number). We
illustrate the method by calculating the Landau spectrum for monolayer
graphene, the four-band model of AB bilayer graphene, and for a more
complicated matrix Hamiltonian describing the silicene band structure. Finally
we apply our method to an inhomogeneous system consisting of a strain
engineered one dimensional moir\'e in bilayer graphene, finding localized
states near the Dirac point that arise from electron trapping in a
semi-classical moir\'e potential. The semi-classical density of states of these
localized states we show to be in perfect agreement with an exact quantum
mechanical calculation of the density of states.
