  We derive bounds on the precision of fluctuating currents, which are valid
for the steady state of underdamped Langevin dynamics. These bounds provide a
generalization of the overdamped thermodynamic uncertainty relation to the
finite-mass regime. In the overdamped case, the precision of a current is
bounded by the entropy production. By contrast, the underdamped bound acquires
two additional positive terms, which characterize the local mean acceleration
and the fluctuations of the velocity. We generalize the bound to the cases of a
magnetic field and anisotropic temperature, and derive a joint bound for
several observables. Finally, we apply our results to biased free diffusion and
the Brownian gyrator (with and without magnetic field), as well as to diffusion
in periodic potentials. In the latter case, we show that the underdamped bound
can be tight when taking into account the correlations between the current and
other observables.
