This work deals with geometric numerical integration on a Lie group using the Cayley transformation. We investigate a coupled system of differential equations in a Lie group setting that occurs in Lattice Quantum Chromodynamics. To simulate elementary particles, expectation values of some operators are computed using the Hybrid Monte Carlo method. In this context, Hamiltonian equations of motion in a non-Abelian setting are solved with a time-reversible and volume-preserving integration method. Usually, the exponential function is used in the integration method to map the Lie algebra to the Lie group. In this paper, the focus is on geometric numerical integration using the Cayley transformation instead of the exponential function. The geometric properties of the method are shown for the example of the Störmer–Verlet method. Moreover, the advantages and disadvantages of both mappings are discussed.

这项工作使用Cayley变换处理Lie群上的几何数值积分。我们调查在格群量子色动力学中发生的李群设置中的微分方程的耦合系统。为了模拟基本粒子，使用混合蒙特卡洛方法计算了一些算子的期望值。在这种情况下，使用时间可逆且体积保持积分的方法求解非阿贝尔环境中的哈密顿运动方程。通常，在积分方法中使用指数函数将李代数映射到李群。在本文中，重点是使用Cayley变换而不是指数函数的几何数值积分。以Störmer-Verlet方法为例显示了该方法的几何特性。此外，