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We construct a finite-dimensional Hamiltonian lift of covariance dynamics on Sym$^+_n$ equipped with the Bures--Wasserstein metric. The natural mechanical Lagrangian yields canonical momentum $\\Pi=\\tfrac12 L_\\Sigma[\\dot\\Sigma]$, where $L_\\Sigma$ is the Lyapunov operator, and explicit Hamiltonian $\\mathcal{H}(\\Sigma,\\Pi) = 2{\\rm tr}(\\Pi\\Sigma\\Pi)+V(\\Sigma)$. 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