Lie derivative algorithm for preserving geometry on cylindrical manifolds

Abstract

For geometric nonlinear systems with cylindrical characteristics, they play a crucial role in nonlinear dynamics. Such systems can accurately capture the inherent geometric features, while also possessing the geometric structures of cylindrical manifolds. In the traditional numerical method, the geometric characteristics of the system are rarely considered in the calculation process, so some geometric properties might be lost, leading to incorrect results. Therefore, exploring numerical algorithms that preserve geometric structures is a meaningful topic. In this paper, based on the Lie derivative algorithm, a new geometric numerical integration algorithm is proposed. Meanwhile, the geometric constraint equations are also discretized combined with the Newton-Raphson method. A class of nonlinear dynamic systems exhibiting observable three-dimensional cylindrical geometric manifolds is analyzed and calculated. Compared to the traditional fourth-order Runge-Kutta algorithm, the proposed algorithm with geometric manifold-constrained iterations is found to not only possess high computational efficiency but also effectively preserve the geometric characteristics of the system manifold during the discrete process. Moreover, the Hamiltonian energy is also discretized and compared. It can be observed that the Hamiltonian function is a first order small quantity of step size, which has approximately energy-preserving at a certain step size.