Velocity-constraint (nonholonomic) models of the force branch: constant-rank constraint laws, the d'Alembert reaction and holonomic equivalence

Adopt the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N\ge 1$ labels with masses $m_1,\dots ,m_N>0$, a nondegenerate open time window $I_0\subseteq \mathbb{R}$, an open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ and a force map $F$ with the standing regularity (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Fix one inertial frame with adapted coordinates $(t,x)$ and write $x=(x_1,\dots ,x_N)\in {\mathbb{R}}^{3N}$ with coordinate velocity $w=x^{\mathrm{\prime }}$ (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

(a) Velocity-constraint datum. A velocity-constraint datum on this data is a quadruple $(O_v,k^{\mathrm{\prime }},A,b)$ with:

1. an ambient set $O_v\subseteq I_0\times Q$, open and nonempty;
2. an integer $k^{\mathrm{\prime }}$ with $1\le k^{\mathrm{\prime }}\le 3N-1$;
3. maps $A:O_v\to M_{k^{\mathrm{\prime }}\times 3N}(\mathbb{R})$ and $b:O_v\to {\mathbb{R}}^{k^{\mathrm{\prime }}}$ of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), of declared dimensions chosen so that every row satisfies $[A_{\alpha j}{w}_j]=[b_{\alpha }]$ — the output of each row of the law below is a single declared quantity;
4. the constant-rank hypothesis: $A(t,x)$ has rank $k^{\mathrm{\prime }}$ for every $(t,x)\in O_v$, equivalently the linear map $A(t,x):{\mathbb{R}}^{3N}\to {\mathbb{R}}^{k^{\mathrm{\prime }}}$ is surjective (Linear map between vector spaces over the same field, For an $m\times n$ matrix $A$, $\mathrm{rank}(A)+\dim N(A)=n$).

The constraint law of the datum is $A(t,x)w+b(t,x)=0.$

(b) Admissible velocities, virtual displacements, admissible curves. For $(t,x)\in O_v$ the admissible velocities are the vectors $v\in {\mathbb{R}}^{3N}$ with $A(t,x)v+b(t,x)=0$. They form the nonempty affine subspace $v_0+\ker A(t,x)$ (nonempty because $A(t,x)$ is surjective), with direction the linear space $V(t,x):=\ker A(t,x)\subseteq {\mathbb{R}}^{3N},\dim V(t,x)=3N-k^{\mathrm{\prime }}$ by rank-nullity (For an $m\times n$ matrix $A$, $\mathrm{rank}(A)+\dim N(A)=n$). The elements of $V(t,x)$ are the virtual displacements of the datum at $(t,x)$: the admissible velocity differences at frozen time and configuration, in the Chetaev prescription of The velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$: virtual displacements, the d'Alembert law and the vakonomic law. A kinematically admissible curve family is a map $x:J\to Q$ of class $C^2$ on an interval $J\subseteq I_0$ with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) such that $(t,x(t))\in O_v$ and $A(t,x(t))x^{\mathrm{\prime }}(t)+b(t,x(t))=0$ for every $t\in J$; its velocity at each time is admissible by definition.

(c) d'Alembert velocity-constraint model. A velocity-constraint model consists of the force-branch data, a velocity-constraint datum $(O_v,k^{\mathrm{\prime }},A,b)$ and the declaration that the labels' motion satisfies the constraint law while a reaction field acts: a motion of the model is a kinematically admissible curve family $x:J\to Q$ together with a continuous reaction field $R:J\to {\mathbb{R}}^{3N}$ of force dimension satisfying $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=F(t,x(t),x^{\mathrm{\prime }}(t))+R(t)(t\in J),$ with $\mathsf{M}$ the mass operator of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (d). The model is a d'Alembert model when, in addition, the reaction is ideal: along every motion, $R(t)\cdot \delta =0\text{for every }\delta \in V(t,x(t))=\ker A(t,x(t)),$ the pairing being the Euclidean inner product (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). This is the adopted idealization of the mechanism that realizes the velocity constraint — the analogue for velocity constraints of the ideal reaction law of the holonomic branch — and it is not a consequence of the constraint being imposed. By the multiplier lemma (A linear functional annihilating the kernel of a surjection is a unique transpose multiple), applied at each time to the surjective $A(t,x(t))$, the ideal condition holds at time $t$ if and only if there is a unique $\mu (t)\in {\mathbb{R}}^{k^{\mathrm{\prime }}}$ with $R(t)=A(t,x(t){)}^{\mathsf{T}}\mu (t);$ the representation is a mathematical consequence of the condition and the rank hypothesis, while the ideal condition itself is the adopted part. The equation of the d'Alembert model in multiplier form is therefore $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}=F+A^{\mathsf{T}}\mu$ along an ideal motion.

(d) Holonomic equivalence and nonholonomic models. The datum is holonomic-equivalent on $O_v$ when there exist a $C^2$ map $\mathrm{\Phi }:O_v\to {\mathbb{R}}^{k^{\mathrm{\prime }}}$ with ${\mathrm{\Phi }}_x$ of rank $k^{\mathrm{\prime }}$ on $O_v$ and a $C^2$ map $\varrho :O_v\to GL(k^{\mathrm{\prime }},\mathbb{R})$ (Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case) with $A(t,x)=\varrho (t,x){\mathrm{\Phi }}_x(t,x),b(t,x)=\varrho (t,x){\mathrm{\Phi }}_t(t,x)\text{on }{O}_v;$ for a physical datum, each component ${\mathrm{\Phi }}_{\beta }$ has a declared dimension ${\varphi }_{\beta }$. With $[x_j]=\mathrm{\ell }$ and $[t]=\tau$, assign $[{\varrho }_{\alpha \beta }]=[b_{\alpha }]\tau \mathrm{/}{\varphi }_{\beta }$; then every term in the two matrix products has the declared dimension of the corresponding entry of $A$ or $b$. These component assignments are part of the model's declaration. The datum is locally holonomic-equivalent at a point when such a pair exists on some open neighbourhood of that point, and locally nonholonomic there when no such neighbourhood and pair exists, as in Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (e). A nonholonomic model in this local sense specifies a point (or region of points) with this failure. Absence of one pair on the whole domain $O_v$ alone is insufficient, since a global obstruction need not be local nonintegrability. In the holonomic-equivalent case the law is the differentiated family of position constraints $\mathrm{\Phi }(t,x)=\text{constant}$: for a $C^1$ curve with $(t,x(t))\in O_v$, the chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$) gives $\frac{d}{dt}\mathrm{\Phi }(t,x(t))={\mathrm{\Phi }}_t+{\mathrm{\Phi }}_x{x}^{\mathrm{\prime }}={\varrho }^{-1}(b+A{x}^{\mathrm{\prime }}),$ so the constraint law holds exactly when $\mathrm{\Phi }(t,x(t))$ is constant; a holonomic-equivalent model is thus one whose constraint is equivalent to a family of $k^{\mathrm{\prime }}$ independent position constraints with regular differentials, presented through the nowhere-vanishing factor $\varrho$. For $k^{\mathrm{\prime }}=1$ the factorization $A=\varrho {\mathrm{\Phi }}_x$, $b=\varrho {\mathrm{\Phi }}_t$ says that the one-form $bdt+A\cdot dx$ is $\varrho d\mathrm{\Phi }$: in this single-row case, local nonholonomicity means that no such integrating factor and regular potential exist on any neighbourhood of the specified point. For $k^{\mathrm{\prime }}>1$, local nonholonomicity is defined by failure of the matrix factorization above, not by a single one-form criterion. In a separate reduced two-coordinate model $q=(s,\theta )\in {\mathbb{R}}^2$ of a disk rolling along a straight line (not the $3N$-coordinate datum of clauses (a)-(c)), let $[s]=[a]=\mathrm{\ell }$, $[\theta ]=1$ and $a>0$. Its no-slip law $s^{\mathrm{\prime }}-a{\theta }^{\mathrm{\prime }}=0$ is holonomic-equivalent with $A=(1,-a)$, $b=0$, $\mathrm{\Phi }(s,\theta )=s-a\theta$ of dimension $\mathrm{\ell }$, and dimensionless $\varrho =1$; the velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ of The velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$: virtual displacements, the d'Alembert law and the vakonomic law is not: an integrating factor would make the distribution $\ker A$ tangent to the level sets of $\mathrm{\Phi }$, and 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 proves that this distribution is not tangent to the level sets of any $C^2$ function with nonvanishing differential. Tong's cited example is a specific coin of radius $R>0$ rolling without slipping down a slope, with generalized coordinates $(x,y,\theta ,\varphi )$ and velocity constraints $x^{\mathrm{\prime }}=R{\varphi }^{\mathrm{\prime }}\sin \theta ,y^{\mathrm{\prime }}=R{\varphi }^{\mathrm{\prime }}\cos \theta ,$ where $x,y,R$ have length dimension and $\theta ,\varphi$ are angles in radians, which he identifies as non-holonomic (Section 2.3.2, Eq. (2.41)). This is a source-reported classification of that rolling-coin model; this item does not prove it or classify all three-dimensional rolling models.

(e) Status and scope. The d'Alembert law of clause (c) is a prescription for the reaction at frozen time and configuration, adopted as the physical idealization of the mechanism; it is not identified with the vakonomic (stationary augmented action) prescription of The velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$: virtual displacements, the d'Alembert law and the vakonomic law. For the sample model $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ with unit mass, zero applied force and initial data $(x,y,z)(0)=(0,0,0)$, $(x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})(0)=(1,1,0)$, The d'Alembert and vakonomic motions of $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ differ from the same initial data proves that there is a unique maximal d'Alembert motion on $\mathbb{R}$ with those data. Every other d'Alembert motion through the datum is its restriction; neither the maximal motion nor any such restriction is vakonomic. Thus the d'Alembert solution set is not a subset of the vakonomic solution set for that datum, and the two sets are unequal; this does not assert that a vakonomic motion with those data exists. The closing remark of this page records the distinction and its scope. The framework excludes from this branch: points where the rank of $A$ drops (the multiplier system degenerates there, and no continuation is asserted); unilateral constraints and contact changes (a skate leaving the ground, a wheel beginning to slip), which require their own mode-change models; impacts; dry friction along admissible directions, which must be declared as a separate force contribution; and active drives, whose driving is an additional force of the model. Whether a concrete mechanism is described by this class, and whether its reaction can supply the computed multiplier, are application-level statements recorded with the apparatus; the definition asserts no measured value.

• Position versus velocity constraints. A holonomic model constrains configurations and its constraint locus is a submanifold of position space; a velocity-constraint model constrains velocities, its configuration remains in the open set $Q$, and its virtual displacements are the frozen kernel of $A$. A holonomic model can be written as a velocity-constraint model by differentiating its constraint, but not conversely; the classification of clause (d) is the precise sense of the converse obstruction.

• The multiplier is not measured. As in the holonomic branch, only the reaction $R=A^{\mathsf{T}}\mu$ enters the equation, and $\mu$ is tied to the declared rows of the law: replacing $(A,b)$ by $(\varrho A,\varrho b)$ with an invertible $\varrho$ leaves the admissible velocities and the reaction unchanged and changes $\mu$ to ${\varrho }^{-\mathsf{T}}\mu$.

• Rolling is several models. Planar straight rolling, no-slip rolling whose contact can supply the required friction, slip with kinetic friction, and prescribed motion of the contact are different models with different predictions; the planar ideal no-slip case is holonomic-equivalent. Tong's rolling-coin-on-a-slope example is a specific source-reported nonholonomic model; no classification of all three-dimensional rolling models is asserted here, and rank or contact changes are outside this definition. The companion page compares the planar ideal no-slip and frictionless sliding models explicitly.