In 1998 Hans Munthe-Kaas wrote a series of papers developing Runge-Kutta methods on Lie groups (and manifolds) where he uses the exponential map to reformulate ordinary differential equations on the Lie group as ordinary differential equations on the Lie algebra. For general Lie groups where an explicit expression of the differential of the exponential map and its inverse are not readily available, Munthe-Kaas uses a series expansion in terms of iterated Lie brackets to derive Runge-Kutta type integrators that incorporate correction terms expressed in terms of the commutators. In this short note, I will not explore this simplification by assuming that we can evaluate a (generalization of the) exponential map and its differential explicitly. Such formulas are easy to come by when working with low-dimensional matrix Lie groups like \(\newcommand{\SE}{\mathrm{SE}}\SE(3)\). \(\newcommand{\Msf}{\mathsf{M}}\newcommand{\Vsf}{\mathsf{V}}\newcommand{\Esf}{\mathsf{E}}\newcommand{\Fsf}{\mathsf{F}}\newcommand{\Circles}{\mathscr{C}}\newcommand{\Spheres}{\mathscr{S}}\newcommand{\Willmore}{\mathcal{W}}\newcommand{\RP}{\mathbb{R}\mathrm{P}}\newcommand{\RR}{\mathbb{R}}\newcommand{\CC}{\mathbb{C}}\newcommand{\HH}{\mathbb{H}}\newcommand{\CP}{\CC\mathrm{P}}\newcommand{\g}{\mathfrak{g}}\newcommand{\Ad}{\operatorname{Ad}}\newcommand{\SE}{\mathsf{SE}}\newcommand{\se}{\mathsf{se}}\newcommand{\ad}{\operatorname{ad}}\newcommand{\SO}{\mathsf{SO}}\newcommand{\cay}{\operatorname{cay}}\)
To fix some notation let \(G\) be a Lie group with Lie algebra \(\newcommand{\g}{\mathfrak{g}}\g\). We will be interested in reformulating an ordinary differential equation of the form
\[
\begin{cases}
\frac{d}{dt}g(t) = f(t,g(t)),\\
g(t_0) = g_0,
\end{cases}
\]
on \(G\) into an ordinary differential equation on \(\g\); here, \(t\in[t_0,t_1],\) \(g(t)\in G,\) and \(f(t,g(t))\in T_{g(t)}G.\)
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