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.