  We study the dynamical state of cores by using a simple analytical model, a
sample of observational massive cores, and numerical simulations of collapsing
massive cores. From the model, we find that, if cores are formed from turbulent
compressions, they evolve from small to large column densities, increasing
their velocity dispersion as they collapse, while they tend to equipartition
between gravity and kinetic energy.
  From the observational sample, we find that: (a) cores with substantially
different column densities in the sample do not follow a Larson-like
linewidth-size relation. Instead, cores with higher column densities tend to be
located in the upper-left corner of the Larson velocity dispersion-size
diagram, a result predicted previously (Ballesteros-Paredes et al. 2011a). (b)
The data exhibit cores with overvirial values.
  Finally, in the simulations we reproduce the behavior depicted by the model
and observational sample: cores evolve towards larger velocity dispersions and
smaller sizes as they collapse and increase their column density. However,
collapsing cores appear to approach overvirial states within a free-fall time.
The cause of this apparent excess of kinetic energy is an underestimation of
the gravitational energy, due to the assumption that the gravitational energy
is given by the energy of an isolated sphere of constant column density. This
excess disappears when the gravitational energy is correctly calculated from
the actual spatial mass distribution, where inhomogeneities, as well as the
potential due to the mass outside of the core, also contribute to the
gravitational energy. We conclude that the observed energy budget of cores in
surveys is consistent with their non-thermal motions being driven by their
self-gravity and in a hierarchical and chaotic collapse.
