Variational Bures--Wasserstein Dynamics under Capacity Constraint

The Bures--Wasserstein (BW) covariance programme developed in a series of recent preprints presents its dynamics as capacity-constrained gradient flows on the cone of positive-definite matrices. We show that the same dynamics admit a variational formulation with explicit symplectic structure. We construct a conservative action on the BW manifold, extended by capacity parameters $(\numin, \Qmax, \Dfixed)$ entering as couplings, and show that its Euler--Lagrange equations reproduce the BW dynamics after a standard Legendre transform. The symplectic form on the associated phase space restricts cleanly to the double-quantum (DQ) and zero-quantum (ZQ) subspaces previously identified in the algebraic sector classification of the programme. We prove that the quadratic part of the Hamiltonian contains $a^{\dagger}a^{\dagger} + aa$ pairing terms in the DQ sector and bilinear mixing terms in the ZQ sector, and that these algebraic structures force SU(1,1) and SU(2) respectively as the kinematic symmetry groups. The sector classification is therefore not ad hoc but a consequence of the variational structure. The resulting action remains symplectically well posed across admissible capacity couplings and provides the conservative foundation for later dissipative and boundary-theoretic extensions.