in the language of Lie, the flow associated to this problem is locally similar to a translation.

Now we can tackle at last the question of giving intrinsic geometrical meaning to . It turns out to correspond to a Lie–Scheffers system on a Lie group. For an arbitrary curve $t\mapsto
g(t)$ on the $r$-dimensional Lie group $G$, we have the vector field along the curve given by $\dot g(\cdot)$ as discussed at the end of Section 1. Then we *define* the left hand side of  as $$\dot g(t)\,g^{-1}(t) := T_{g(t)}R_{g^{-1}(t)}\dot g(t).
\label{eq:madre-del-cordero}$$ By construction, for each value of the parameter $t$, this vector lies in $T_1G\equiv{\mathfrak{g}}$, the tangent Lie algebra of $G$. We obtain in this way a curve on ${\mathfrak{g}}$. Note that, if $g(t)=\exp(t\eta)$, then simply $\dot g(t)\,g^{-1}(t)=\eta$. Now, if ${\gamma}^a$ is a basis for ${\mathfrak{g}}$, then $$\dot g(t)\,g^{-1}(t) = \sum_{a=1}^r Z_a(t){\gamma}^a =: \xi(t) \in {\mathfrak{g}},
\label{eq:eqingr}$$ for some functions $Z_a(t)$. We realize that $g(t)$ is an integral curve of the right invariant vector field along ${\mathbb{R}}_t{\times}G\to G$: $$\xi_G(t,g) := \sum_{a=1}^r Z_a(t){\gamma}^a_G(g) = \sum_{a=1}^r
Z_a(t)X^R_{{\gamma}^a}(g), {\quad\mbox{with}\quad} {\gamma}_G^a(1_G) = {\gamma}^a.$$ Here ${\gamma}^a_G$ is the fundamental vector field or infinitesimal generator of the left group translations generated by ${\gamma}^a$, which is 