  We use three dimensional optical spectroscopy observations of a sample of 22
local Luminous Compact Blue Galaxies (LCBGs) to create kinematic maps. By means
of these, we classify the kinematics of these galaxies into three different
classes: rotating disk (RD), perturbed rotation (PR), and complex kinematics
(CK). We find 48% are RDs, 28% are PRs, and 24% are CKs. RDs show rotational
velocities that range between $\sim50$ and $\sim200 km s^{-1}$, and dynamical
masses that range between $\sim1\times10^{9}$ and $\sim3\times10^{10}
M_{\odot}$. We also address the following two fundamental questions through the
study of the kinematic maps: \emph{(i) What processes are triggering the
current starbust in LCBGs?} We search our maps of the galaxy velocity fields
for signatures of recent interactions and close companions that may be
responsible for the enhanced star formation in our sample. We find 5% of
objects show evidence of a recent major merger, 10% of a minor merger, and 45%
of a companion. This argues in favor of ongoing interactions with close
companions as a mechanism for the enhanced star formation activity in these
galaxies. \emph{(ii) What processes may eventually quench the current starbust
in LCBGs?} Velocity and velocity width maps, together with emission line ratio
maps, can reveal signatures of Active Galactic Nuclei (AGN) activity or
supernova (SN) driven galactic winds that could halt the current burst. We find
only 5% of objects with clear evidence of AGN activity, and 27% with kinematics
consistent with SN-driven galactic winds. Therefore, a different mechanism may
be responsible for quenching the star formation in LCBGs. Finally, from our
analysis, we find that the velocity widths of RDs, rather than accounting
exclusively for the rotational nature of these objects, may account as well for
other kinematic components, and may not be good tracers of their dynamical
masses.
