Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits

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.