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 $sim1times10^{9}$ and $sim3times10^{10} M_{odot}$. 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.
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