  By using the \vec{k}\cdot\vec{p} method, we propose a first-principles theory
to study the linear dispersions in phononic and photonic crystals. The theory
reveals that only those linear dispersions created by doubly-degenerate states
can be described by a reduced Hamiltonian that can be mapped into the Dirac
Hamiltonian and possess a Berry phase of -\pi. Triply-degenerate states can
also generate Dirac-like cone dispersions, but the wavefunctions transform like
a spin-1 particle and the Berry phase is zero. Our theory is capable of
predicting accurately the linear slopes of Dirac/Dirac-like cones at various
symmetry points in a Brilliouin zone, independent of frequency and lattice
structure.
