In this paper, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned Runge-Kutta Munthe-Kaas (RKMK) methods on Lie groups. A homogeneous space $M$ is a manifold where a group $G$ acts transitively. Such a space can be understood as a quotient $M \cong G/H$, where $H$ a closed Lie subgroup, is the isotropy group of each point of $M$. The Lie algebra of $G$ may be decomposed into $\mathfrak{g} = \mathfrak{m} \oplus \mathfrak{h}$, where $\mathfrak{h}$ is the subalgebra that generates $H$ and $\mathfrak{m}$ is a subspace. Thus, variational problems on $M$ can be treated as nonholonomically constrained problems on $G$, by requiring variations to remain on $\mathfrak{m}$. Nonholonomic partitioned RKMK integrators are derived as a modification of those obtained by a discrete variational principle on Lie groups, and can be interpreted as obeying a discrete Chetaev principle. These integrators tend to preserve several properties of their purely variational counterparts.

翻译：在本文中,平质空间的高顺序数字集成器将展示为对 Lie 组采用非hoolomic 分割的龙格- 库塔蒙特- 卡亚斯( RKMK) 方法。 平质空间 $M 是一个元组过渡运行的方块。 这样的空间可以被理解为一个商数 $M\ cong G/H$, 即一个闭合的 Lie 分组, 是每点的异质组 $。 $G 的利位代数可能会被解成 $\ mathfrak{ g} =\ mathfrak{h} 。 一个同质空间 $\ m$\ may 是一个元组, 产生$和 $mathfrak{m} 。 因此, $M 的变异项问题可以被处理为$G $ 上的非血压约束问题, 需要对 $\ machraqraqraq 原则进行变换 。