  Recently, it has been shown that there is a trade-off relation between
thermodynamic cost and current fluctuations, referred to as the thermodynamic
uncertainty relation (TUR). The TUR has been derived for various processes,
such as discrete-time Markov jump processes and overdamped Langevin dynamics.
For underdamped dynamics, it has recently been reported that some modification
is necessary for application of the TUR. In this study, we present a more
generalized TUR, applicable to a system driven by a velocity-dependent force in
the context of underdamped Langevin dynamics, by extending the theory of Vu and
Hasegawa [preprint arXiv:1901.05715]. We show that our TUR accurately describes
the trade-off properties of a molecular refrigerator (cold damping), Brownian
dynamics in a magnetic field, and an active particle system.
