Regular constraints and moving frames

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.

Adopt as a physical assumption of the framework, in the setting of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction:

(i) The ideal-reaction law. Along every constrained motion $x:J\to {\mathbb{R}}^{3N}$ of a holonomic-constraint model, with reaction field $R:J\to {\mathbb{R}}^{3N}$, the reaction annihilates every virtual displacement:

$R(t)\cdot \delta =0\text{for every }t\in J\text{ and every }\delta \in V(t,x(t))=\ker D_{x}G(t,x(t)),$

the pairing being the Euclidean inner product of ${\mathbb{R}}^{3N}$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). This is the d'Alembert condition (the condition of zero virtual work of the reaction) in the sense of A regular holonomic constraint, its virtual displacements and the ideal-reaction law. By the multiplier lemma Reactions annihilating the virtual displacements are transpose multiples, with a positive definite multiplier matrix, applied at each time with the surjective Jacobian $D_{x}G(t,x(t))$, the condition holds at time $t$ if and only if there is a unique $\lambda (t)\in {\mathbb{R}}^k$ with $R(t)=D_{x}G(t,x(t){)}^{\mathsf{T}}\lambda (t);$ this representation is a mathematical consequence of the condition and the rank hypothesis, not a separate assumption, and the multiplier has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}\mathrm{/}{\gamma }_{\alpha }$ in component $\alpha$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The condition is imposed only along the motions of the model, and it does not by itself determine $R$: it constrains $R$ at each time to the $k$-dimensional conormal space, and which element occurs is fixed by the twice-differentiated constraint through the balance theorem of this page.

(ii) What is adopted. Clause (i) is an adopted restriction on the mechanism that realizes the constraint, not a theorem about bodies, tracks or contacts. It is the idealization of a mechanism that exerts only the reaction needed to maintain the constraint, distributed normally to the admissible directions at each instant; the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance excludes constraint reactions from its force maps, and this postulate is the additional law that closes the constrained branch.

(iii) Intended model identifications. The framework intends clause (i) to model, subject to the qualifications of clause (iv): frictionless hinges and joints; smooth bilateral contacts and tracks; a taut light rod, and a taut string while it stays taut; and no-slip rolling in the planar holonomic case, provided the contact can supply the required tangential reaction. Each of these identifications is an application-level statement made and recorded with the apparatus; the framework asserts no measurement, no material parameter and no numerical value here.

(iv) Limits of the idealization. The law is not asserted for:

1. constraints with dry sliding friction along an admissible direction, whose force has a component doing virtual work on admissible displacements — such friction must be declared as a separate force contribution with its own model;
2. unilateral contacts and mechanisms that can release the constraint (a bead on the outside of a surface, a string that can go slack, a block that can separate), which are maintained only while the required reaction satisfies a one-sided capability condition; where the condition fails, the constraint is released and a contact-change model is required, and none is supplied here;
3. points where the Jacobian $D_{x}G$ drops rank, which lie outside the regularity hypothesis of the datum and outside the representation above;
4. direct active forces on the modelled labels with a component along an admissible virtual displacement: these must be declared as additional force contributions. Prescribed motion of a support is instead encoded in the time-dependent constraint $G$ and is compatible with clause (i); its ideal reaction may do actual work as in clause (v). The dynamics and energy budget of equipment driving that support are not modelled here;
5. any claim that the mechanism can mechanically supply the computed reaction: the law computes the reaction the ideal constraint exerts; whether a physical contact, rod or string can supply it (friction cone, sign of a tensile force, strength limits, no-slip conditions) is an additional premise of an application.

(v) The law is not a work statement. Clause (i) concerns virtual displacements at fixed time; it does not assert that the reaction performs no work along the motion. For a scleronomic constraint the reaction power vanishes identically, but for a rheonomic constraint the reaction can do work, with $R\cdot x^{\mathrm{\prime }}=-\lambda \cdot G_t$; the precise statement, its static case and worked models are the content of the rheonomic-power theorem and the companion examples of this page.

(vi) Status and empirical content. The clause is adopted, not deduced; the sources record the ideal-reaction condition as an additional physical hypothesis attached to a constraint, not as a consequence of writing the constraint down. The passage from clause (i) to the constrained equation of motion is a theorem with explicit regularity hypotheses, proved on this page. Whether the idealization is adequate for a given apparatus is decided by comparing the model's conditional predictions with reported observations under the stated regime, with the qualifications of clause (iv) carried along; agreement of such a comparison is evidence about the combined model and apparatus, not a proof of the idealization.

• Adopted, not derived. The Lagrange-multiplier representation of the reaction is mathematical; the identification of the mechanism's reaction with the conormal subspace is the adopted part. Without clause (i), the constraint and balance equation assign the residual reaction $R(t)=\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)-F(t,x(t),x^{\mathrm{\prime }}(t))$ to each chosen constrained curve, but do not select which curves are admissible motions. Clause (i) supplies the additional conormal condition, and the balance theorem derives the resulting ideal dynamics.

• No conservation claimed. The ideality says nothing about energy; energy statements require the potential presentation of the ambient force map and the rheonomic analysis of this page.

• Empirical qualifications are downstream. Any application names the apparatus, the regime, the constraint datum and the evidence used to assess the idealization; the postulate itself carries no observed value.

Adopt a holonomic-constraint model of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction with datum $(O,k,G)$, mass operator $\mathsf{M}$, force map $F$, and the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Suppose, in addition to the hypotheses of those items:

(h1) $G$ is of class $C^3$ on $O$, and $D_{x}G(t,x)$ has rank $k$ for every $(t,x)\in O$ (not only on the constraint locus);

(h2) the force map is of class $C^1$ on $I_0\times Q\times ({\mathbb{R}}^3{)}^N$ ($C^k$ Euclidean maps and diffeomorphisms).

Set $O^{\mathrm{\prime }}:=O\cap (I_0\times Q)$. This is an open nonempty set because $\mathrm{\Sigma }\subseteq O^{\mathrm{\prime }}$ and $\mathrm{\Sigma }$ is nonempty. The matrix-valued map $C:O\to M_k(\mathbb{R})$ below is defined on $O$, while $\mathrm{\Lambda }:O^{\mathrm{\prime }}\times ({\mathbb{R}}^3{)}^N\to {\mathbb{R}}^k$ and $a:O^{\mathrm{\prime }}\times ({\mathbb{R}}^3{)}^N\to {\mathbb{R}}^{3N}$ are defined on the force-map domain restricted to the constraint datum; this is the time-state domain of the first-order system. Every constrained motion in the statement has $(t,x(t))\in \mathrm{\Sigma }\subseteq O^{\mathrm{\prime }}$, so these functions are defined along it.

Let $x:J\to Q$ be a constrained motion of the model with reaction field $R$ satisfying the ideal-reaction law, on an interval $J\subseteq I_0$ with nonempty interior, and write $w=x^{\mathrm{\prime }}$ for its velocity. Then:

1. multiplier form. For every $t\in J$ there is a unique $\lambda (t)\in {\mathbb{R}}^k$ with $R(t)=D_{x}G(t,x(t){)}^{\mathsf{T}}\lambda (t)$; the map $\lambda :J\to {\mathbb{R}}^k$ so defined is continuous, and with

$C(t,x):=D_{x}G(t,x){\mathsf{M}}^{-1}{D}_{x}G(t,x{)}^{\mathsf{T}},\mathrm{\Lambda }(t,x,v):=-C(t,x{)}^{-1}[D_{x}G{\mathsf{M}}^{-1}F(t,x,v)+G_{tt}+2G_{tq}[v]+G_{qq}[v,v]],$

the terms being evaluated at $(t,x)$ or $(t,x,v)$, the $k\times k$ matrix $C(t,x)$ is symmetric positive definite and invertible and $\lambda (t)=\mathrm{\Lambda }(t,x(t),w(t))(t\in J):$ the multiplier, and hence the reaction, is determined by the model data and the motion, not a free or measured quantity.

2. determined dynamics. With $a(t,x,v):={\mathsf{M}}^{-1}(F(t,x,v)+D_{x}G(t,x{)}^{\mathsf{T}}\mathrm{\Lambda }(t,x,v))$, the tuple $(x,w)$ solves the first-order system $x^{\mathrm{\prime }}=v,v^{\mathrm{\prime }}=a(t,x,v)$ on $J$; conversely every solution $(x,v)$ of this system on an interval whose initial datum is consistent, with $R(t):=D_{x}G(t,x(t){)}^{\mathsf{T}}\mathrm{\Lambda }(t,x(t),v(t))$, is a constrained motion of the model satisfying the ideal-reaction law; and through every consistent datum $(t_0,x_0,v_0)$ there is a unique maximal solution of the system. Hence the holonomic-constraint model with ideal reactions has determined dynamics on its constraint locus.

3. preservation. Along every solution of clause 2 with consistent data, $G(t,x(t))=0,G_t(t,x(t))+D_{x}G(t,x(t))v(t)=0$ for every $t$ of its maximal interval.

4. reaction power. Along every ideal constrained motion as in clause 1, $R(t)\cdot w(t)=-\lambda (t)\cdot G_t(t,x(t))(t\in J),$ the pairings being the Euclidean inner products.

The statement is conditional on the adopted force branch and ideal-reaction law and on hypotheses (h1)-(h2); it asserts no empirical value, and the multiplier is a representation tied to the declared components of $G$.

• The multiplier is a representation. The unique $\lambda (t)$ is attached to the declared components $G_{\alpha }$; multiplying $G$ by an invertible matrix-valued function $\rho (t,x)$ leaves the reaction $R$ and the constraint locus unchanged but changes $\lambda$ to ${\rho }^{-\mathsf{T}}\lambda$. Only $R$ enters the equation of motion, so no statement of this page may be read as attributing an independent measured meaning to $\lambda$.

• Where the hypotheses enter. The rank hypothesis (h1) on all of $O$ is used to make $C$ invertible and $\mathrm{\Lambda }$ everywhere defined and $C^1$; the $C^1$ hypothesis on $F$ is used for the same purpose; the ideal-reaction law supplies the conormal membership of $R$, without which the motion is not determined. At a point of rank loss the multiplier matrix degenerates, the acceleration is not determined by the data, and the statement does not apply; no continuation across such a point is asserted.

• Consistency is an initial-data condition. A solution of the system through nonconsistent data still exists but does not satisfy $G=0$; the constrained motions are exactly the solutions with consistent data, as clause 2 records. Nothing here asserts existence of a constrained motion through data whose position is off the locus or whose velocity is not permitted.

• No energy statement. Clause 4 is the power identity of the reaction; the reading of its consequences for mechanical energy, including the vanishing of the reaction work for scleronomic constraints and the work done by rheonomic constraints, belongs to the rheonomic-power theorem of this page.

Adopt a holonomic-constraint model of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction with datum $(O,k,G)$ and the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Suppose, in addition to the model hypotheses, that $G$ is of class $C^3$ on $O$ and $D_{x}G(t,x)$ has rank $k$ for every $(t,x)\in O$, and that the force map is of class $C^1$ on $I_0\times Q\times ({\mathbb{R}}^3{)}^N$; these are hypotheses (h1)-(h2) of Ideal holonomic constraints determine the reaction: multiplier form, explicit acceleration, unique constrained motions and the reaction-power identity. Let $x:J\to Q$ be a constrained motion with ideal reaction $R$ on an interval $J\subseteq I_0$ with nonempty interior, with velocity $w=x^{\mathrm{\prime }}$ and multiplier $\lambda :J\to {\mathbb{R}}^k$ as in Ideal holonomic constraints determine the reaction: multiplier form, explicit acceleration, unique constrained motions and the reaction-power identity clause 1. Suppose further that the model's force map is presented through a potential with residual forces in the sense of clause (a) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets: a $C^2$ function $U:I_0\times Q\to \mathbb{R}$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and maps $d_i:I_0\times Q\times ({\mathbb{R}}^3{)}^N\to {\mathbb{R}}^3$ with the standing regularity of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, such that $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)+d_i(t,x,v)(i=1,\dots ,N)$ on the state domain (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework for the dimensions). Write $K$ and $E(t):=K(t)+U(t,x(t))$ for the kinetic and mechanical energy of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame clauses (b),(d) and of clause (b) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, and ${\mathfrak{d}}_i(t):=d_i(t,x(t),w(t))$ for the evaluated residual forces. Then:

1. reaction power and reaction work. For every $t\in J$, $R(t)\cdot w(t)=-\lambda (t)\cdot G_t(t,x(t)),$ and for all $t_0<t_1$ in $J$, ${\int }_{t_0}^{t_1}R(t)\cdot w(t)dt=-{\int }_{t_0}^{t_1}\lambda (t)\cdot G_t(t,x(t))dt$. If the datum is scleronomic, then $R\cdot w\equiv 0$ on $J$ and the reaction performs no work over any subinterval.

2. mechanical-energy balance. $E$ is differentiable on $J$ and $E^{\mathrm{\prime }}(t)={\mathrm{\partial }}_{t}U(t,x(t))+{\sum }_i⟨{\mathfrak{d}}_i(t),w_i(t)⟩-\lambda (t)\cdot G_t(t,x(t))(t\in J):$ the mechanical energy changes at the rate of the explicit time dependence of the potential, the power of the residual forces, and the constraint term $-\lambda \cdot G_t$, which vanishes identically for a scleronomic constraint.

3. conservation in the static scleronomic case. If the datum is scleronomic, ${\mathrm{\partial }}_{t}U\equiv 0$ on $I_0\times Q$, and ${\mathfrak{d}}_i\equiv 0$ for every $i$ along the motion, then $E$ is constant on $J$.

4. integrated work form. For all $t_0<t_1$ in $J$, $K(t_1)-K(t_0)=U(t_0,x(t_0))-U(t_1,x(t_1))+{\int }_{t_0}^{t_1}\left[{\mathrm{\partial }}_{t}U\right(t,x(t))+{\sum }_i⟨{\mathfrak{d}}_i(t),w_i(t)⟩-\lambda (t)\cdot G_t(t,x(t)\left)\right]dt\mathrm{.}$

The constraint term of clause 2 is not a force of the model's force map: it is the mechanical work performed by the ideal mechanism that maintains a time-dependent constraint, and clause 1 records that this work is exactly the virtual-work-free reaction's actual power. The statement is conditional on the adopted model, idealization and potential presentation; it asserts no measured value.