An Examination of the Time-Centered Difference Scheme for Dissipative Mechanical Systems from a Hamiltonian Perspective

TL;DR

This paper explores a Hamiltonian perspective on the time-centered difference scheme for dissipative mechanical systems, revealing its symplectic properties and energy preservation through a novel conservative system substitution approach.

Contribution

It introduces a method to analyze dissipative systems using conservative system analogs and demonstrates the symplectic nature of the time-centered scheme in this context.

Findings

The scheme preserves total energy of dissipative systems.

The method reveals symplectic structure in dissipative system integration.

Numerical examples confirm energy preservation and symplectic properties.

Abstract

On this paper, we have proposed an approach to observe the time-centered difference scheme for dissipative mechanical systems from a Hamiltonian perspective and to introduce the idea of symplectic algorithm to dissipative systems. The dissipative mechanical systems discussed in this paper are finite dimensional.This approach is based upon a proposition: for any nonconservative classical mechanical system and any initial condition, there exists a conservative one; the two systems share one and only one common phase curve; the Hamiltonian of the conservative system is the sum of the total energy of the nonconservative system on the aforementioned phase curve and a constant depending on the initial condition. Hence, this approach entails substituting an infinite number of conservative systems for a dissipative mechanical system corresponding to varied initial conditions. Therefore, first we utilize the time-centered difference scheme directly to solve the original system, after which we substitute the numerical solution for the analytical solution to construct a conservative force equal to the dissipative force on the phase curve, such that we would obtain a substituting conservative system numerically. Finally, we use the time-centered scheme to integrate the substituting system numerically. We will find an interesting fact that the latter solution resulting from the substituting system is equivalent to that of the former. Indeed, there are two transition matrices within time grid points: the first one is unsymplectic and the second symplectic. In fact, the time-centered scheme for dissipative systems can be thought of as an algorithm that preserves the symplectic structure of the substituting conservative systems. In addition, via numerical examples we find that the time-centered scheme preserves the total energy of dissipative systems.