  The continuous-time model of Nesterov's momentum provides a thought-provoking
perspective for understanding the nature of the acceleration phenomenon in
convex optimization. One of the main ideas in this line of research comes from
the field of classical mechanics and proposes to link Nesterov's trajectory to
the solution of a set of Euler-Lagrange equations relative to the so-called
Bregman Lagrangian. In the last years, this approach led to the discovery of
many new (stochastic) accelerated algorithms and provided a solid theoretical
foundation for the design of structure-preserving accelerated methods. In this
work, we revisit this idea and provide an in-depth analysis of the action
relative to the Bregman Lagrangian from the point of view of calculus of
variations. Our main finding is that, while Nesterov's method is a stationary
point for the action, it is often not a minimizer but instead a saddle point
for this functional in the space of differentiable curves. This finding
challenges the main intuition behind the variational interpretation of
Nesterov's method and provides additional insights into the intriguing geometry
of accelerated paths.
