  In this note, we provide an overarching analysis of primal-dual dynamics
associated to linear equality-constrained optimization problems using
contraction analysis. For the well-known standard version of the problem: we
establish convergence under convexity and the contracting rate under strong
convexity. Then, for a canonical distributed optimization problem, we use
partial contractivity to establish global exponential convergence of its
primal-dual dynamics. As an application, we propose a new distributed solver
for the least-squares problem with the same convergence guarantees. Finally,
for time-varying versions of both centralized and distributed primal-dual
dynamics, we exploit their contractive nature to establish bounds on their
tracking error. To support our analyses, we introduce novel results on
contraction theory.
