The distribution $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{{\mathrm{\partial }}_x,{\mathrm{\partial }}_y+x{\mathrm{\partial }}_z\}$ is not tangent to any regular level sets

Let $O\subseteq {\mathbb{R}}^3$ be open and nonempty, and consider the derivations $A:={\mathrm{\partial }}_x$ and $B:={\mathrm{\partial }}_y+x{\mathrm{\partial }}_z$ acting on $C^2$ functions on $O$. If $G:O\to \mathbb{R}$ is of class $C^2$ with

$AG=0\text{and}BG=0\text{on }O,$

then $DG=0$ on $O$. Consequently, if $G:O\to \mathbb{R}$ is $C^2$ and $DG(q)\mathrm{\ne }0$ for every $q\in O$ (so that every level set of $G$ is a regular level set with tangent space $\ker DG(q)$, Regular and critical points, regular and critical values, and level sets, The tangent space to a regular level set), then the distribution $D(q)=\mathrm{span}\{A(q),B(q)\}$ of The velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$: virtual displacements, the d'Alembert law and the vakonomic law cannot equal $\ker DG(q)$ for every $q\in O$; that is, no such $G$ has level sets whose tangent planes are everywhere the virtual displacements of the constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$.

• The computation shows that the constraint distribution of $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ is not tangent to a two-dimensional regular level foliation: steps 1.1-1.2 identify the bracket $[A,B]$ on $C^2$ functions as the derivation ${\mathrm{\partial }}_z$, and the hypothesis that $A$ and $B$ annihilate a defining function forces ${\mathrm{\partial }}_{z}G=0$ and hence a vanishing gradient. The conclusion needed by the classical-mechanics development is drawn directly, without invoking a converse of the Frobenius theorem.

• The proposition is the precise sense in which $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ is a non-holonomic constraint: it is not equivalent to $G(q)=\text{constant}$ for any $C^2$ function with nonvanishing differential. Nothing is asserted about constraints that merely look similar or about kinematic mechanisms; those belong to the consuming physical models.