1. High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics
- Author
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de Almagro, Rodrigo T. Sato Mart��n
- Subjects
Mathematics - Differential Geometry ,49Mxx, 65L80, 70G45, 70Hxx, 14M17, 22F30 ,Differential Geometry (math.DG) ,FOS: Mathematics ,FOS: Physical sciences ,Numerical Analysis (math.NA) ,Mathematical Physics (math-ph) ,Mathematics - Numerical Analysis ,Mathematical Physics - 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 $M \cong G/H$, where $H$ a closed Lie subgroup, is the isotropy group of each point of $M$. The Lie algebra of $G$ may be decomposed into $\mathfrak{g} = \mathfrak{m} \oplus \mathfrak{h}$, where $\mathfrak{h}$ is the subalgebra that generates $H$ and $\mathfrak{m}$ is a subspace. 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 tend to preserve several properties of their purely variational counterparts., 14 figures. Part of NUMDIFF16
- Published
- 2022