Koopman Generator Decomposition for Port-Hamiltonian System

We study how the vector-field structure of nonlinear port-Hamiltonian systems is reflected in the infinitesimal Koopman generator. The generator admits a natural bracket decomposition into a conservative interconnection-bracket derivation, a dissipative metric-bracket derivation, and an input-port derivation. The conservative component is formally skew-adjoint on a test space whenever the conservative flow preserves the reference measure and the relevant boundary terms vanish. The dissipative component is not claimed to be a positive operator on arbitrary observables; rather, the positive semidefinite object is the metric bracket $[f,f]_R=\nabla f^\intercal\mathbf{R}\nabla f\ge 0$, which yields the exact port-Hamiltonian energy balance for the Hamiltonian observable: $$\mathcal{K}_{\mathbf{u}}\mathcal{H} =-[\mathcal{H},\mathcal{H}]_R +\mathbf{y}^\intercal\mathbf{u} \le \mathbf{y}^\intercal\mathbf{u}.$$ We use these bracket identities to motivate finite-dimensional weak Galerkin and data-driven lifted models: when the Galerkin measure is conservative for the Hamiltonian interconnection flow and boundary terms vanish, the conservative contribution is skew in the Galerkin mass metric, while the dissipative bracket induces a positive semidefinite Dirichlet matrix. These identities motivate structure-preserving lifted port-Hamiltonian surrogates that are passive and support damping injection in the lifted coordinates, while distinguishing exact bracket identities, projection residuals, finite-data estimation error, and the residual and injectivity assumptions needed to transfer lifted conclusions back to the original nonlinear state.