  The high-resolution differential equation framework has been proven to be
tailor-made for Nesterov's accelerated gradient descent method~(\texttt{NAG})
and its proximal correspondence -- the class of faster iterative shrinkage
thresholding algorithms (FISTA). However, the systems of theories is not still
complete, since the underdamped case ($r < 2$) has not been included. In this
paper, based on the high-resolution differential equation framework, we
construct the new Lyapunov functions for the underdamped case, which is
motivated by the power of the time $t^{\gamma}$ or the iteration $k^{\gamma}$
in the mixed term. When the momentum parameter $r$ is $2$, the new Lyapunov
functions are identical to the previous ones. These new proofs do not only
include the convergence rate of the objective value previously obtained
according to the low-resolution differential equation framework but also
characterize the convergence rate of the minimal gradient norm square. All the
convergence rates obtained for the underdamped case are continuously dependent
on the parameter $r$. In addition, it is observed that the high-resolution
differential equation approximately simulates the convergence behavior
of~\texttt{NAG} for the critical case $r=-1$, while the low-resolution
differential equation degenerates to the conservative Newton's equation. The
high-resolution differential equation framework also theoretically
characterizes the convergence rates, which are consistent with that obtained
for the underdamped case with $r=-1$.
