Extension of Hamiltonian Mechanics to Non-Conservative Systems

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.