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

TL;DR

This paper develops high-order numerical integrators for Lagrangian systems on homogeneous spaces using nonholonomic RKMK methods on Lie groups, enabling structure-preserving simulations of constrained mechanical systems.

Contribution

It introduces a novel class of nonholonomic partitioned RKMK integrators tailored for homogeneous spaces, extending variational integrators to constrained settings.

Findings

Integrators preserve key geometric properties.

Method applicable to systems with nonholonomic constraints.

Enhanced accuracy for Lagrangian systems on homogeneous spaces.

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.