  A trade-off between the precision of an arbitrary current and the
dissipation, known as the thermodynamic uncertainty relation, has been
investigated for various Markovian systems. Here, we study the thermodynamic
uncertainty relation for underdamped Langevin dynamics. By employing
information inequalities, we prove that, for such systems, the relative
fluctuation of a current at a steady state is constrained by both the entropy
production and the average dynamical activity. We find that, unlike what is the
case for overdamped dynamics, the dynamical activity plays an important role in
the bound. We illustrate our results with two systems, a single-well potential
system and a periodically driven Brownian particle model, and numerically
verify the inequalities.
