  Convergence to equilibrium of underdamped Langevin dynamics is studied under
general assumptions on the potential $U$ allowing for singularities. By
modifying the direct approach to convergence in $L^2$ pioneered by F. H\'erau
and developped by Dolbeault, Mouhot and Schmeiser, we show that the dynamics
converges exponentially fast to equilibrium in the topologies $L^2(d\mu)$ and
$L^2(W^* d\mu)$, where $\mu$ denotes the invariant probability measure and
$W^*$ is a suitable Lyapunov weight. In both norms, we make precise how the
exponential convergence rate depends on the friction parameter $\gamma$ in
Langevin dynamics, by providing a lower bound scaling as $\min(\gamma,
\gamma^{-1})$. The results hold for usual polynomial-type potentials as well as
potentials with singularities such as those arising from pairwise Lennard-Jones
interactions between particles.
