associated with the action of a finite-dimensional Lie group on the manifold on which $Y(t;x)$ is defined; and this symmetry of the differential equation can be powerfully exploited through the action of the group of curves on the group manifold on a set of systems of the same type. This variant of Lie’s reduction method is of wide applicability; it is explained in Appendix B.

Let us finally note than in control theory, say on $M\equiv{\mathbb{R}}^n$, business is often with equations of a form not unrelated to : $$\dot x(t) = X_1(x(t)) + \sum_{a=2}^r Z_a(t)\,X_a(x(t));$$ the functions $Z_2,\ldots,Z_r$ being the controls. In the most important cases the $X_1,\ldots,X_r$ vector fields close to a finite-dimensional Lie algebra, or $X_2,\ldots,X_r$ close to a finite-dimensional Lie algebra.

Differential operators on Lie–Rinehart algebras
===============================================

Let $R$ be a commutative, unital ring and ${\mathcal{A}}$ a commutative algebra over $R$ be given. A derivation ${\delta}$ of ${\mathcal{A}}$ is a $R$-linear map from ${\mathcal{A}}$ to itself, such that ${\delta}(ab)={\delta}a\,b+a\,{\delta}b$. Since ${\mathcal{A}}$ is commutative, the linear space $\operatorname{Der}({\mathcal{A}})$ of such maps becomes an ${\mathcal{A}}$-module when we define $(a{\delta})b=a\,{\delta}b$. Moreover, with the usual bracket given by the commutator $\operatorname{Der}({\mathcal{A}})$ is a Lie algebra. In this paper $R={\mathbb{C}}$ nearly always.

