Non-standard Construction of Hamiltonian Structures and of the   Hamilton-Jacobi equation
  Examples 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. 
M. Herrera, S. A. Hojman