Infinite-dimensional Hamiltonian description of a class of dissipative mechanical systems

TL;DR

This paper introduces an infinite-dimensional Hamiltonian framework for dissipative mechanical systems by representing them as fluid-like systems in phase space, linking energy and phase curves through a conservative 'substitute' system.

Contribution

It develops a novel infinite-dimensional Hamiltonian description of dissipative systems using fluid analogy, avoiding decoupling into multiple ODEs, and relates energy to phase curves via a substitute conservative system.

Findings

Dissipative systems can be represented as infinite-dimensional Hamiltonian systems.

The approach links energy to phase curves through a substitute conservative system.

Canonical momentum and coordinates match physical quantities, simplifying equations.

Abstract

In this paper an approach is proposed to represent a class of dissipative mechanical systems by corresponding infinite-dimensional Hamiltonian systems. This approach is based upon the following structure: for any non-conservative classical mechanical system and arbitrary initial conditions, there exists a conservative system; both systems share one and only one common phase curve; and, the value of the Hamiltonian of the conservative system is, up to an additive constant, equal to the total energy of the non-conservative system on the aforementioned phase curve, the constant depending on the initial conditions. We describe in detail this relationship calling the conservative system ``substitute`` conservative system. By considering the dissipative mechanical system as a special fluid in a domain $D$ of the phase space, viz. a collection of particles in this domain, we are prompted to develop this system as an infinite-dimensional Hamiltonian system of an ideal fluid. By comparing the description of the ideal fluid in Lagrangian coordinates, we can consider the Hamiltonian and the Lagrangian as the respective integrals of the Hamiltonian and the Lagrangian of the substitute conservative system over the initial value space and define a new Poisson bracket to express the equations of motion in Hamiltonian form. The advantage of the approach is that the value of the canonical momentum density $\pi$ is identical with that of the mechanical momentum $m\dot{q}$ and the value of canonical coordinate $q$ is identical with that of the coordinate of the dissipative mechanical system. Therefore we need not to decouple the Newtonian equations of motion into several one-dimensional ordinary differential equations.