  Although Nesterov's accelerated gradient (NAG) methods have been studied from
various perspectives, it remains unclear why the most popular forms of NAG must
handle convex and strongly convex objective functions separately. Motivated by
this inconsistency, we propose an NAG method that unifies the existing ones for
the convex and strongly convex cases. We first design a Lagrangian function
that continuously extends the first Bregman Lagrangian to the strongly convex
setting. As a specific case of the Euler--Lagrange equation for this
Lagrangian, we derive an ordinary differential equation (ODE) model, which we
call the unified NAG ODE, that bridges the gap between the ODEs that model NAG
for convex and strongly convex objective functions. We then design the unified
NAG, a novel momentum method whereby the continuous-time limit corresponds to
the unified ODE. The coefficients and the convergence rates of the unified NAG
and unified ODE are continuous in the strong convexity parameter $\mu$ on $[0,
+\infty)$. Unlike the existing popular algorithm and ODE for strongly convex
objective functions, the unified NAG and the unified NAG ODE always have
superior convergence guarantees compared to the known algorithms and ODEs for
non-strongly convex objective functions. This property is beneficial in
practical perspective when considering strongly convex objective functions with
small $\mu$. Furthermore, we extend our unified dynamics and algorithms to the
higher-order setting. Last but not least, we propose the unified NAG-G ODE, a
novel ODE model for minimizing the gradient norm of strongly convex objective
functions. Our unified Lagrangian framework is crucial in the process of
constructing this ODE. Fascinatingly, using our novel tool, called the
differential kernel, we observe that the unified NAG ODE and the unified NAG-G
ODE have an anti-transpose relationship.
