  We take a Hamiltonian-based perspective to generalize Nesterov's accelerated
gradient descent and Polyak's heavy ball method to a broad class of momentum
methods in the setting of (possibly) constrained minimization in Euclidean and
non-Euclidean normed vector spaces. Our perspective leads to a generic and
unifying nonasymptotic analysis of convergence of these methods in both the
function value (in the setting of convex optimization) and in norm of the
gradient (in the setting of unconstrained, possibly nonconvex, optimization).
Our approach relies upon a time-varying Hamiltonian that produces generalized
momentum methods as its equations of motion. The convergence analysis for these
methods is intuitive and is based on the conserved quantities of the
time-dependent Hamiltonian.
