  Mid-infrared molecular hydrogen (H$_2$) emission is a powerful cooling agent
in galaxy mergers and in radio galaxies; it is a potential key tracer of gas
evolution and energy dissipation associated with mergers, star formation, and
accretion onto supermassive black holes. We detect mid-IR H$_2$ line emission
in at least one rotational transition in 91\% of the 214 Luminous Infrared
Galaxies (LIRGs) observed with Spitzer as part of the Great Observatories
All-sky LIRG Survey (GOALS). We use H$_2$ excitation diagrams to estimate the
range of masses and temperatures of warm molecular gas in these galaxies. We
find that LIRGs in which the IR emission originates mostly from the Active
Galactic Nuclei (AGN) have about 100K higher H$_2$ mass-averaged excitation
temperatures than LIRGs in which the IR emission originates mostly from star
formation. Between 10 and 15\% of LIRGs have H$_2$ emission lines that are
sufficiently broad to be resolved or partially resolved by the high resolution
modules of Spitzer's Infrared Spectrograph (IRS). Those sources tend to be
mergers and contain AGN. This suggests that a significant fraction of the H$_2$
line emission is powered by AGN activity through X-rays, cosmic rays, and
turbulence. We find a statistically significant correlation between the kinetic
energy in the H$_2$ gas and the H$_2$ to IR luminosity ratio. The sources with
the largest warm gas kinetic energies are mergers. We speculate that mergers
increase the production of bulk in-flows leading to observable broad H$_2$
profiles and possibly denser environments.
