2022/01/28 by de Almagro, Rodrigo T. Sato Martín
#14M17 #22F30 #49Mxx #65L80 #70G45 #70Hxx #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2201.12022
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 ≅ 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 \mathfrakg = \mathfrakm ⊕ \mathfrakh, where \mathfrakh is the subalgebra that generates H and \mathfrakm is a subspace. Thus, variational problems on M can be treated as nonholonomically constrained problems on G, by requiring variations to remain on \mathfrakm. 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.