2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56607Examples of non-standard construction of Hamiltonian structures for dynamical systems and the respective Hamilton-Jacobi (H-J) equations, without using Lagrangians, are presented. Alternative H-J equations for Euler top are explicitly exhibited and solved. We demonstrate that some stability criterion, relating the slope of a Casimir function parametrized by the Lagrange multiplier to critical point type, depends on the used Hamiltonian structure and it is inadequate for this reason.13 pages, TeXMathematical Physicstext