The velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$: virtual displacements, the d'Alembert law and the vakonomic law

Work in ${\mathbb{R}}^3$ with coordinates $(x,y,z)$ on configuration space and velocities $(x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})$; inner products and norms are those of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. All intervals used as motion domains below are open intervals of $\mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and a map on such an interval is said to be $C^r$ when all its components are ($C^k$ Euclidean maps and diffeomorphisms).

The constraint and its virtual displacements. The constraint law is

$z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}\mathrm{.}$

A configuration $q=(x,y,z)$ is written without a subscript, and its velocity is $v=(x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})$. The allowed velocities at $q$ and the virtual displacements at $q$ are both the same linear subspace

$D(q):=\{w=(w_x,w_y,w_z)\in {\mathbb{R}}^3:w_z=x{w}_y\}=\mathrm{span}\{(1,0,0),(0,1,x)\}\subseteq {\mathbb{R}}^3\mathrm{.}$

Thus a virtual displacement $\delta =(\delta x,\delta y,\delta z)$ obeys $\delta z=x\delta y$; this is the Chetaev prescription for the linear velocity constraint $z^{\mathrm{\prime }}-x{y}^{\mathrm{\prime }}=0$, in which the virtual displacements are computed from the constraint function frozen at the current configuration.

Sample Lagrangian and zero applied force. The adopted Lagrangian is

$L_0(q,v):=\frac{1}{2}\mathrm{\mid }v{\mathrm{\mid }}^2=\frac{1}{2}(x^{\mathrm{\prime }2}+y^{\mathrm{\prime }2}+z^{\mathrm{\prime }2}),$

of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times {\mathbb{R}}^3$; no applied force is introduced. Since $L_0$ is a special case of a Lagrangian chart as in A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations (with $n=3$, $U={\mathbb{R}}^3$ and $O=\mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3$), the words stationary, action and Euler--Lagrange expression below have the meanings fixed there whenever they are used.

d'Alembert motions. A $C^2$ curve $q:I\to {\mathbb{R}}^3$ is a d'Alembert motion of the constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ when

1. $z^{\mathrm{\prime }}(t)=x(t)y^{\mathrm{\prime }}(t)$ for every $t\in I$, and
2. $q^{\mathrm{\prime }\mathrm{\prime }}(t)\cdot \delta =0$ for every $\delta \in D(q(t))$ and every $t\in I$.

Clause 2 is the ideal-reaction (d'Alembert) condition for unit mass and zero applied force: the acceleration has no component along any admissible virtual displacement, so the constraint force performs no virtual work.

The augmented functional and the vakonomic law. Let $a<b$ be reals with $[a,b]\subseteq I_0$ for some open interval $I_0$. For a $C^2$ curve $q:I_0\to {\mathbb{R}}^3$ and a $C^2$ function $\eta :I_0\to \mathbb{R}$ put

$\mathrm{\Phi }(q,\eta ):={\int }_a^b[\frac{1}{2}\mathrm{\mid }{q}^{\mathrm{\prime }}(t){\mathrm{\mid }}^2+\eta (t)(z^{\mathrm{\prime }}(t)-x(t)y^{\mathrm{\prime }}(t)\left)\right]dt\mathrm{.}$

The integrand is continuous because $q,q^{\mathrm{\prime }}$ and $\eta$ are continuous, so $\mathrm{\Phi }(q,\eta )$ is a well-defined real number (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$, A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion). An admissible pair variation of a pair $(q,\eta )$ is a pair of $C^2$ maps $(\xi ,\zeta )$ on $I_0$ with $\xi (a)=\xi (b)=0$, and the pair $(q,\eta )$ is a vakonomic stationary pair when the ordinary two-sided directional derivative

$\frac{d}{ds}{\mid }_{s=0}\mathrm{\Phi }(q+s\xi ,\eta +s\zeta )$

exists and equals $0$ for every admissible pair variation $(\xi ,\zeta )$ with $\xi (a)=\xi (b)=0$. A $C^2$ curve $q:I_0\to {\mathbb{R}}^3$ is a vakonomic motion of the constraint when there is a $C^2$ function $\eta$ on $I_0$ such that $(q,\eta )$ is a vakonomic stationary pair on every $[a,b]\subseteq I_0$ with $a<b$, using the same multiplier $\eta$ throughout $I_0$. The functional and its endpoint conditions are taken on each such subinterval.

Scope. Both laws are prescriptions for the same constraint, and the definition deliberately does not identify them: the d'Alembert law is a differential condition on the acceleration along the constraint distribution, while the vakonomic law arises from stationarity of the augmented functional over curves and multipliers. For this constraint the two prescriptions give different motions from the same initial data; that is the content of The d'Alembert and vakonomic motions of $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ differ from the same initial data. The constraint law $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ is not equivalent to any condition $G(q)=\text{const}$ for a $C^2$ function $G$ with nonvanishing differential, as proved in 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; this is the sense in which the constraint is non-holonomic.

• The two spanning fields of $D$ are $A:={\mathrm{\partial }}_x$ and $B:={\mathrm{\partial }}_y+x{\mathrm{\partial }}_z$, in the sense that at each $q$ the vectors $A$ and $B$ evaluated at $q$ span $D(q)$. The non-integrability computation of 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 uses exactly these two fields.

• The stationary-pair definition quantifies over all $C^2$ multipliers $\eta$ and all $C^2$ variations, with no condition on $\eta$ at the endpoints; this is the standard augmented-Lagrangian formulation of the vakonomic equations, recorded here as a definition and not as a statement about which of the two laws describes any physical system.