  Thermodynamic uncertainty relation (TUR) provides a stricter bound for
entropy production (EP) than that of the thermodynamic second law. This
stricter bound can be utilized to infer the EP and derive other trade-off
relations. Though the validity of the TUR has been verified in various
stochastic systems, its application to general Langevin dynamics has not been
successful in a unified way, especially for underdamped Langevin dynamics,
where odd parity variables in time-reversal operation such as velocity get
involved. Previous TURs for underdamped Langevin dynamics is neither
experimentally accessible nor reduced to the original form of the overdamped
Langevin dynamics in the zero-mass limit. Here, we find an operationally
accessible TUR for underdamped Langevin dynamics with an arbitrary
time-dependent protocol. We show that the original TUR is a consequence of our
underdamped TUR in the zero-mass limit. This indicates that the TUR formulation
presented here can be regarded as the universal form of the TUR for general
Langevin dynamics. The validity of our result is examined and confirmed for
three prototypical underdamped Langevin systems and their zero-mass limits;
free diffusion dynamics, charged Brownian particle in a magnetic field, and
molecular refrigerator.
