High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics
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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 𝔤 = 𝔪⊕𝔥, where 𝔥 is the subalgebra that generates H and 𝔪 is a subspace. 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.
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