Hamiltonian Lift of Bures--Wasserstein Covariance Dynamics with a Spectral Floor

Covariance dynamics on the positive-definite cone are commonly described by gradient flows, which encode dissipative relaxation but obscure the underlying phase-space structure. 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 $\mathrm{\Pi }=\frac{1}{2}{L}_{\mathrm{\Sigma }}[\dot{\mathrm{\Sigma }}]$, where $L_{\mathrm{\Sigma }}$ is the Lyapunov operator, and explicit Hamiltonian $\mathcal{H}(\mathrm{\Sigma },\mathrm{\Pi })=2\mathrm{t}\mathrm{r}(\mathrm{\Pi }\mathrm{\Sigma }\mathrm{\Pi })+V(\mathrm{\Sigma })$. Adding Rayleigh dissipation recovers the Bures–Wasserstein gradient flow in the overdamped limit. For a spectral-floor and trace-control potential, the quadratic fluctuation Hamiltonian around the isotropic equilibrium separates trace and traceless modes; the baseline stiffness diverges as $(s-\nu {)}^{-2}$ as the equilibrium covariance approaches the floor. The construction identifies a conservative parent system for constrained Bures–Wasserstein covariance relaxation and fixes the local stiffness scale induced by the spectral floor.