Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction

Adopt the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N\ge 1$ particle labels with fixed masses $m_1,\dots ,m_N>0$, a nondegenerate open time window $I_0\subseteq \mathbb{R}$, an open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (a)), and a force map $F:I_0\times Q\times ({\mathbb{R}}^3{)}^N⟶({\mathbb{R}}^3{)}^N$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate. Fix one inertial frame of the model with adapted coordinates $(t,x)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames), identify $x=(x_1,\dots ,x_N)\in ({\mathbb{R}}^3{)}^N={\mathbb{R}}^{3N}$ with the position tuple of the labels (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), and let $\mathsf{M}$ be the mass operator of clause (d) of that definition, $\mathsf{M}(x_1,\dots ,x_N)=(m_1{x}_1,\dots ,m_N{x}_N)$, a symmetric positive definite linear map with block-diagonal matrix and blocks $m_i{I}_3$. All components below are read in this frame, and dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

(a) Constraint datum. A holonomic-constraint datum on this data is a triple $(O,k,G)$ with:

1. an ambient set $O\subseteq \mathbb{R}\times {\mathbb{R}}^{3N}$, open and nonempty;
2. an integer $k$ with $1\le k\le 3N-1$;
3. a map $G=(G_1,\dots ,G_k):O\to {\mathbb{R}}^k$ of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), each component carrying a declared dimension $[G_{\alpha }]={\gamma }_{\alpha }$ fixed by the model (the model may rescale a component by a nonzero constant, which changes ${\gamma }_{\alpha }$ and the component but not its zero set);

subject to the two hypotheses

4. nonempty admissible locus: the constraint locus $\mathrm{\Sigma }:=\{(t,x)\in O:G(t,x)=0\}$ is nonempty and contained in $I_0\times Q$: every constrained configuration is an admissible configuration of the force-branch model and every constrained time lies in its time window;
5. regularity: for every $(t,x)\in \mathrm{\Sigma }$ the Jacobian $D_{x}G(t,x)$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case) is surjective, equivalently has rank $k$.

For $t\in \mathbb{R}$ write ${\mathrm{\Sigma }}_t:=\{x\in {\mathbb{R}}^{3N}:(t,x)\in \mathrm{\Sigma }\}$ for the instantaneous constraint locus. Here a $C^2$ embedded submanifold of dimension $d$ in ${\mathbb{R}}^n$ means a subset $S$, with its inherited topology, such that each $p\in S$ has an open neighbourhood $U\subseteq {\mathbb{R}}^n$ and a $C^2$ diffeomorphism $h:U\to W$ onto an open $W\subseteq {\mathbb{R}}^n$ satisfying $h(S\cap U)=W\cap ({\mathbb{R}}^d\times \{0\})$; both $h$ and its inverse are $C^2$ in the sense of $C^k$ Euclidean maps and diffeomorphisms. This is the finite-regularity slice convention used for these loci; Embedded submanifolds and slice charts instead requires smooth charts.

Each nonempty ${\mathrm{\Sigma }}_t$ is a $C^2$ embedded submanifold of dimension $3N-k$. Indeed, at any $p\in {\mathrm{\Sigma }}_t$ choose $k$ spatial columns of $D_{x}G(t,p)$ giving an invertible minor, and permute coordinates as $x=(u,v)\in {\mathbb{R}}^{3N-k}\times {\mathbb{R}}^k$. The fixed-time map $H(u,v):=G(t,(u,v))$ is $C^2$ on the open slice of $O$, vanishes at $p$, and has invertible $D_{v}H(p)$. Apply The parametrized implicit function theorem with $C^k$ regularity with regularity order $2$ and no parameter block: on open neighbourhoods $P\times Q$ of $p$, its zero set is exactly $v=\phi (u)$ for a $C^2$ map $\phi :P\to Q$. The map $h(u,v)=(u,v-\phi (u))$ is a $C^2$ diffeomorphism onto the open set $W=\{(u,w):u\in P,w+\phi (u)\in Q\}$, with inverse $(u,w)\mapsto (u,w+\phi (u))$, and sends that graph to $W\cap ({\mathbb{R}}^{3N-k}\times \{0\})$. This proves the claimed local slice property. If $G$ is smooth, the smooth regular-value terminology of Regular and critical points and values and the smooth conclusion of A regular level set is an embedded submanifold also apply. When $k=3N-1$ the instantaneous locus is locally a $C^2$ curve, and the number $\dim {\mathrm{\Sigma }}_t=3N-k$ is the instantaneous degrees of freedom of the constrained branch. The set $V(t,x):=\ker D_{x}G(t,x)\subseteq {\mathbb{R}}^{3N}$ has dimension $3N-k$ by the rank-nullity theorem for matrices (For an $m\times n$ matrix $A$, $\mathrm{rank}(A)+\dim N(A)=n$) and is called the space of virtual displacements of the datum at $(t,x)$; its elements are the first-order differences of constrained configurations at fixed time, in the sense of the mathematical definition A regular holonomic constraint, its virtual displacements and the ideal-reaction law. A vector $v\in {\mathbb{R}}^{3N}$ is a permitted velocity at $(t,x)\in O$ when $G_t(t,x)+D_{x}G(t,x)v=0.$

(b) Constrained curve families. Let $J\subseteq I_0$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). A constrained curve family of the datum is a map $x:J\to {\mathbb{R}}^{3N}$ of class $C^2$ with $(t,x(t))\in O$ and $G(t,x(t))=0$ for every $t\in J$; writing $x=(x_1,\dots ,x_N)$ with coordinate curves $x_i:J\to {\mathbb{R}}^3$ and $w_i:=x_i^{\mathrm{\prime }}$, the componentwise derivatives of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors; $w_i$ is the coordinate velocity of label $i$), the chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$) applied to $t\mapsto G(t,x(t))$ gives $G_t(t,x(t))+D_{x}G(t,x(t))w(t)=0(t\in J),$ so the velocity of a $C^1$ constrained curve family is permitted at every time. A tuple $(t_0,x_0,v_0)\in O\times {\mathbb{R}}^{3N}$ is consistent data of the datum when $G(t_0,x_0)=0$ and $G_t(t_0,x_0)+D_{x}G(t_0,x_0)v_0=0$; these are the initial data at which constrained motions are posed below.

(c) Reaction field and the constrained equation. A reaction field along a curve family $x:J\to {\mathbb{R}}^{3N}$ is a continuous map $R:J\to {\mathbb{R}}^{3N}$, of force dimension $\mu \mathrm{\ell }{\tau }^{-2}$ in each component. The equation of the constrained datum along $x$ is $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=F(t,x(t),w(t))+R(t)(t\in J),$ componentwise $m_i{x}_i^{\mathrm{\prime }\mathrm{\prime }}(t)=F_i(t,x(t),w(t))+R_i(t)$. A holonomic-constraint model consists of the force-branch data, a holonomic-constraint datum $(O,k,G)$, and the declaration that the position tuple of the labels satisfies $G(t,x(t))=0$ for all times while a reaction field acts: a constrained motion of the model (with reaction $R$) is a constrained curve family $x:J\to {\mathbb{R}}^{3N}$ together with a reaction field $R$ satisfying the displayed equation. The force map $F$ is the model's declared interaction force of the force branch, and $R$ is the additional force contribution exerted by the mechanism that realizes the constraint; the total force along a constrained motion is the pointwise sum $F+R$ in the sense of clause (v) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, read along the curve. The reaction is not part of the force map: the constrained branch is the branch excluded by clause (i) of that postulate, and the equation above is the adopted equation of the constrained model. For any specified constrained curve family $x$, this equation fixes the reaction uniquely as the residual field $R(t)=\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)-F(t,x(t),x^{\mathrm{\prime }}(t))\mathrm{.}$ Without an additional reaction law, the constraint and this equation do not select which constrained curve families are dynamically admissible: every $C^2$ constrained curve can be paired with its continuous residual field to satisfy the equation. The ideal-reaction law of the postulate of this page is the additional model assumption that selects among these candidates by requiring the reaction to annihilate the virtual displacements.

(d) Scleronomic and rheonomic constraints. The datum is scleronomic when the constraint locus is time-independent, in the precise sense that there is a set $N\subseteq {\mathbb{R}}^{3N}$ with $\mathrm{\Sigma }=\{(t,x)\in O:x\in N\};$ equivalently, membership in the locus is a condition on $x$ alone within $O$: if $(t,x)\in \mathrm{\Sigma }$ and $(t^{\mathrm{\prime }},x)\in O$, then $(t^{\mathrm{\prime }},x)\in \mathrm{\Sigma }$ (the two formulations are exchanged by passing between $N$ and the projection of $\mathrm{\Sigma }$ to ${\mathbb{R}}^{3N}$). For the standard product ambient set $O=I_0\times Q$ the condition reads $\mathrm{\Sigma }=I_0\times (N\cap Q)$. The datum is rheonomic otherwise.

If the datum is scleronomic, then the declared presentation $G$, and every $C^2$ presentation of the same locus with the rank hypothesis, satisfy $G_t(t,x)=0\text{for every }(t,x)\in \mathrm{\Sigma },$ because for $x\in N$ every $(t,x)\in O$ lies in $\mathrm{\Sigma }$, so $t\mapsto G(t,x)$ vanishes identically on the open set of times $\{t:(t,x)\in O\}$; this is the form in which the classification is used by the balance theorems of this page.

The identity is a consequence of time-independence and not a criterion for it. Write $s(x):=(x_1{)}_1\in \mathbb{R}$ for the first Cartesian component of particle $1$, of length dimension $\mathrm{\ell }$, and fix a nonzero speed $u$ of dimension $\mathrm{\ell }{\tau }^{-1}$. The scalar presentation $G(t,x):=(s(x)-ut{)}^2$, of dimension ${\mathrm{\ell }}^2$, on $O=\mathbb{R}\times {\mathbb{R}}^{3N}$ has the moving zero set $\{s(x)=ut\}$ while $G_t(t,x)=-2u(s(x)-ut)$ vanishes at every point of that set; this presentation violates the rank hypothesis, but the identity can also hold for a rank-regular presentation: on $O=((0,T_{\ast })\cup (2T_{\ast },3T_{\ast }))\times {\mathbb{R}}^{3N}$, with $T_{\ast }>0$ of time dimension, the scalar $C^2$ map equal to $s(x)$ on $(0,T_{\ast })\times {\mathbb{R}}^{3N}$ and to a fixed $L_{\ast }>0$ of length dimension on $(2T_{\ast },3T_{\ast })\times {\mathbb{R}}^{3N}$ has dimension $\mathrm{\ell }$ and the rank-$1$ zero set $\mathrm{\Sigma }=(0,T_{\ast })\times \{x\in {\mathbb{R}}^{3N}:s(x)=0\}$ with $G_t=0$ on $\mathrm{\Sigma }$, yet $\mathrm{\Sigma }$ is not of the form $\{(t,x)\in O:x\in N\}$ for any $N$: the point $(T_{\ast }\mathrm{/}2,0)\in \mathrm{\Sigma }$ would force $0\in N$, and then $(5T_{\ast }\mathrm{/}2,0)\in O$ would lie in $\mathrm{\Sigma }$. Hence the classification is a property of the locus, and the identity is used only in the direction proved above.

Under a nowhere-vanishing $C^2$ rescaling $G^{\mathrm{\prime }}=\rho G$ one has $D_x{G}^{\mathrm{\prime }}=\rho D_{x}G$ and $G_t^{\mathrm{\prime }}={\rho }_{t}G+\rho G_t=\rho G_t$ at the points of $\mathrm{\Sigma }$, so the virtual displacements and permitted velocities are unchanged, the multiplier of a motion transforms as ${\lambda }^{\mathrm{\prime }}={\rho }^{-1}\lambda$, and the reaction $D_x{G}^{\mathrm{\prime }\mathsf{T}}{\lambda }^{\mathrm{\prime }}$ and the product $-{\lambda }^{\mathrm{\prime }}\cdot G_t^{\mathrm{\prime }}=-\lambda \cdot G_t$ of the rheonomic power theorem are unchanged at the locus.

(e) Status. This item defines a model class: it fixes the constraint datum, the constrained curves, the virtual displacements, the reaction field and the equation of the constrained branch. It adopts no ideal-reaction law (that is the content of the ideal-reaction postulate of this page), no particular force law, and no measured value. Whether a concrete apparatus — a frictionless joint, a smooth track, a taut rod or string, a no-slip contact — is described by a holonomic-constraint model is an application-level judgement recorded with the apparatus, together with the regime in which the rigidity, frictionlessness or contact assumptions are adequate; the framework itself attaches no empirical claim to the class.

• The constrained branch is a separate branch. The equation of the force branch, $m_i{x}_i^{\mathrm{\prime }\mathrm{\prime }}=F_i$, is not the equation of a constrained motion unless $R=0$; conversely, a solution of the force-branch equation is a constrained motion with $R=0$ only if it also satisfies $G(t,x(t))=0$ and the permitted-velocity condition. Nothing in this definition asserts that $R$ is small, ideal or absent.

• Presentation of the constraint. The components $G_{\alpha }$ are redundant descriptions of the locus and are fixed only up to multiplication by an invertible matrix-valued function; the dimension count $3N-k$ and the virtual displacements at the locus are unchanged under such a re-presentation, whereas the multiplier of a later motion is tied to the declared components. This is why the multiplier is never a measured quantity in this framework; only the reaction enters the equation.

• Time-dependent loci need not be graphs. Nothing here asserts that the instantaneous loci ${\mathrm{\Sigma }}_t$ are nonempty for every $t$ of $I_0$, nor that they vary continuously in a parameterization; the model is stated on the locus it declares, and questions of continuation in time belong to the solutions of the constrained equation, not to the definition.