Extension of Hamiltonian Mechanics to Non-Conservative Systems Via Higher-Order Dynamics

Abstract This paper presents a detailed review of the emerging topic of higher-order dynamics and their intrinsic variational structure, which has enabled—for the very first time in history—the general application of Hamiltonian formalism to non-conservative systems. Here the general theory is presented alongside several interesting applications that have been discovered to date. These include the direct modal analysis of non-proportionally damped dynamical systems, a new and more efficient algorithm for computing the resonant frequencies of damped systems with many degrees-of-freedom, and a canonical Hamiltonian formulation of the Navier–Stokes problem. A significant merit of the Hamiltonian formalism is that it leads to the transformation theory of Hamilton and Jacobi, and specifically the Hamilton–Jacobi equation, which reduces even the most complicated of problems to the search for a single scalar function (or functional, for problems in continuum mechanics). With the extension of the Hamiltonian framework to non-conservative systems, now every problem in classical mechanics can be reduced to the search for a single scalar. This discovery provides abundant opportunities for further research, and here we list just a few potential ideas. Indeed, the present authors believe there may be many more applications of higher-order dynamics waiting to be discovered.