High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics

Abstract

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 a closed Lie subgroup H, is the isotropy group of each point of M. The Lie algebra of G may be decomposed into g=m⊕h, where h is the subalgebra that generates H and m is a subspace spanning the tangent spaces of M. Thus, variational problems on M can be treated as nonholonomically constrained problems on G, by requiring variations to remain on 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.