  In convex optimization, there is an {\em acceleration} phenomenon in which we
can boost the convergence rate of certain gradient-based algorithms. We can
observe this phenomenon in Nesterov's accelerated gradient descent, accelerated
mirror descent, and accelerated cubic-regularized Newton's method, among
others. In this paper, we show that the family of higher-order gradient methods
in discrete time (generalizing gradient descent) corresponds to a family of
first-order rescaled gradient flows in continuous time. On the other hand, the
family of {\em accelerated} higher-order gradient methods (generalizing
accelerated mirror descent) corresponds to a family of second-order
differential equations in continuous time, each of which is the Euler-Lagrange
equation of a family of Lagrangian functionals. We also study the exponential
variant of the Nesterov Lagrangian, which corresponds to a generalization of
Nesterov's restart scheme and achieves a linear rate of convergence in discrete
time. Finally, we show that the family of Lagrangians is closed under time
dilation (an orbit under the action of speeding up time), which demonstrates
the universality of this Lagrangian view of acceleration in optimization.
