In this paper, the authors address the challenge of finding a Hamiltonian formulation for a viscous fluid system, like the Navier-Stokes equations, which are non-Hamiltonian. By considering a higher-order problem and providing additional conditions, they were able to transform a non-Hamiltonian problem into one with Hamiltonian structure. This surprising method, demonstrated through a simple example, shows that every non-Hamiltonian problem has equivalent Hamiltonian variants. The paper also applies this concept to the isotropic Navier-Stokes problem, highlighting the potential for a second-order formulation that is mathematically equivalent to the original problem. The approach offers a new perspective on tackling complex fluid dynamics challenges.
https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/canonical-hamiltonian-formulation-of-the-navierstokes-problem/B3EB9389AE700867A6A3EA63A45E69C6