  The recent interest into the Brownian gyrator has been confined chiefly to
the analysis of Brownian dynamics both in theory and experiment despite the
applicability of general cases with definite mass. Considering mass explicitly
in the solution of the Fokker--Planck equation and Langevin dynamics
simulations, we investigate how inertia can change the dynamics and energetics
of the Brownian gyrator. In the Langevin model, the inertia reduces the
nonequilibrium effects by diminishing the declination of the probability
density function and the mean of a specific angular momentum, $j_\theta$, as a
measure of rotation. Another unique feature of the Langevin description is that
rotation is maximized at a particular anisotropy while the stability of the
rotation is minimized at a particular anisotropy or mass. Our results suggest
that the Langevin dynamics description of the Brownian gyrator is intrinsically
different from that with Brownian dynamics. In addition, $j_\theta$ is proven
to be essential and convenient for estimating stochastic energetics such as
heat currents and entropy production even in the underdamped regime.
