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 Π=tfrac12 L_Σ\[dotΣ\], where L_Σ is the Lyapunov operator, and explicit Hamiltonian mathcal{H}(Σ,Π) = 2{rm tr}(ΠΣΠ)+V(Σ). 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-ν)^-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.