Finite-Time Observable and Efficiency-Aware Transfer in Hamiltonian Systems: Operator Duality, Extremal Geometry, and Ratio Selection

We develop a finite-dimensional theory of finite-time observable transfer in Hamiltonian systems on compact energy surfaces, with the transfer of energy itself as a special case. For a given Hamiltonian flow and a chosen receiving observable, the terminal transfer is the change in that observable between the start and end of a trajectory over a fixed time horizon, and the instantaneous rate of transfer is the Poisson bracket of the observable with the Hamiltonian. An observable–measure duality — the pairing between the Koopman (composition) operator acting on observables and the transfer (Perron–Frobenius) operator acting on measures — expresses this transfer equivalently from either side, and it separates two regimes. In the typical regime the transfer is probabilistic: against an invariant ensemble the signed transfer cancels in the mean, while the cumulative positive and negative influx grow with the horizon even as the net endpoint change stays bounded. In the optimal regime the transfer is geometric: the finite-time maximizers satisfy a stationarity condition expressed through a pulled-back covector (an adjoint gradient), and the near-optimal set is controlled by a constrained Hessian on the energy surface. We extend the theory to a multi-observable accounting map — distinguishing useful output, source depletion, auxiliary storage, exposure, and cost — and to efficiency ratios, which require a positive-denominator admissibility condition and admit a Dinkelbach characterization and Pareto scalarizations. In the positive-definite quadratic case every object becomes a symmetric matrix and the optima reduce to Rayleigh–Ritz and symmetric generalized eigenvalue problems after whitening by the energy metric. A verified three-channel example shows that the maximum-output state and the maximum-efficiency state are dynamically distinct even in a conservative linear system, and a Hénon–Heiles study computes the finite-time transfer field on a Poincaré section, exhibiting a sharply localized near-optimal set together with cumulative influx that grows linearly against a bounded net endpoint transfer. The contribution is a finite-time theory of observable transfer and efficiency on energy surfaces, assembled from classical ergodic and spectral ingredients.