In this talk, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned 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 $G$ decomposes into $\mathfrak{g}=\mathfrak{m}\oplus \mathfrak{h}$, where $\mathfrak{h}$ is the subalgebra associated with $H$. 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 seem to preserve several properties of their purely variational counterparts.