2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/50366Starting with Lagrangians, which turn out to be degenerate, the Hamiltonian operators for integrable systems can be constructed using Dirac's theory of constraints. We illustrate this by giving a systematic discussion of the first Hamiltonian structure of KdV. The first symplectic 2-form obtained from Dirac's theory is the time component of the covariant Witten-Zuckerman symplectic 2-form current. We derive the flux of the first symplectic 2-form. Then we turn to a new Lagrangian for KdV recently obtained by Pavlov and present the corresponding new covariant symplectic structure for KdV. This shows that the inverse of Magri's second Hamiltonian operator is also a Hamiltonian operator for KdV.To be published in "Integrable Hierarchies and Modern Physics Theories, NATO ARW-UIC 2000" Editors H. Aratyn and A. SorinHigh Energy Physics - Theorytext