Application of Explicit Symplectic Algorithms to Integration of Damping Oscillators

TL;DR

This paper introduces a novel approach using explicit symplectic algorithms to efficiently solve damping oscillator problems by transforming non-conservative systems into conservative ones sharing a common phase curve.

Contribution

It presents a new method to apply explicit symplectic algorithms to damped oscillators by linking non-conservative forces to conservative systems through phase curve analysis.

Findings

Algorithms demonstrate good numerical stability.

Effective in handling non-conservative damping forces.

Numerical examples validate the approach.

Abstract

In this paper an approach is outlined. With this approach some explicit algorithms can be applied to solve the initial value problem of $n-$dimensional damped oscillators. This approach is based upon 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. A key way applying explicit symplectic algorithms to damping oscillators is that by the Newton-Laplace principle the nonconservative force can be reasonably assumed to be equal to a function of a component of generalized coordinates $q_i$ along a phase curve, such that the damping force can be represented as a function analogous to an elastic restoring force numerically in advance. Two numerical examples are given to demonstrate the good characteristics of the algorithms.