Application of Explicit Symplectic Algorithms to Integration of Damping Oscillators

摘要

In this paper an approach is outlined. With this approach some explicitalgorithms can be applied to solve the initial value problem of $n-$dimensionaldamped oscillators. This approach is based upon following structure: for anynon-conservative classical mechanical system and arbitrary initial conditions,there exists a conservative system; both systems share one and only one commonphase 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-conservativesystem on the aforementioned phase curve, the constant depending on the initialconditions. A key way applying explicit symplectic algorithms to dampingoscillators is that by the Newton-Laplace principle the nonconservative forcecan be reasonably assumed to be equal to a function of a component ofgeneralized coordinates $q_i$ along a phase curve, such that the damping forcecan be represented as a function analogous to an elastic restoring forcenumerically in advance. Two numerical examples are given to demonstrate thegood characteristics of the algorithms.