The d'Alembert and vakonomic prescriptions are different models: what is proved, what is adopted, and which questions are open

For a velocity-constraint datum $(O_v,k^{\mathrm{\prime }},A,b)$ of Velocity-constraint (nonholonomic) models of the force branch: constant-rank constraint laws, the d'Alembert reaction and holonomic equivalence the framework's model is the d'Alembert model: admissible curves satisfy $A{x}^{\mathrm{\prime }}+b=0$, and the reaction is the ideal (Chetaev) reaction, annihilating the frozen virtual displacements $\ker A(t,x(t))$ at each instant, equivalently $R=A^{\mathsf{T}}\mu$ for a unique multiplier. The vakonomic prescription of The velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$: virtual displacements, the d'Alembert law and the vakonomic law is a different construction: one forms the augmented functional $\mathrm{\Phi }(q,\eta )={\int }_a^b[\frac{1}{2}\mathrm{\mid }{q}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\eta (z^{\mathrm{\prime }}-x{y}^{\mathrm{\prime }})]dt$ for the sample constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ and a multiplier function $\eta$, and asks for pairs $(q,\eta )$ stationary under variations of both the curve and the multiplier. The two prescriptions are definitions of two different laws for the same constraint; neither is a theorem about the other, and the framework adopts the d'Alembert one for its nonholonomic models.

For the constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ 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 the unique maximal d'Alembert motion on $\mathbb{R}$ is not vakonomic. Every other d'Alembert motion with these data is its restriction, and no such restriction is vakonomic either. Thus the set of d'Alembert motions with this initial datum is not a subset of the vakonomic motions with the same datum, so the two solution sets are unequal; the counterexample does not assert that a vakonomic motion with these data exists. Its contradiction follows because the exhibited d'Alembert trajectory has $x^{\mathrm{\prime }\mathrm{\prime }}=0$, which would force the vakonomic multiplier $\eta$ to vanish on an interval and then force $y^{\mathrm{\prime }\mathrm{\prime }}=0$, contrary to its explicit formula. The same constraint is genuinely nonholonomic in the sense of clause (d) of Velocity-constraint (nonholonomic) models of the force branch: constant-rank constraint laws, the d'Alembert reaction and holonomic equivalence: 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 its distribution is not tangent to the level sets of any $C^2$ function with nonvanishing differential. The conclusions are mathematical statements about the two prescriptions; they are read into this page's model through the definitions, with the model's status and exclusions of clause (e) carried along.

The ideal (d'Alembert) condition for the holonomic branch is the adopted postulate Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits, and the corresponding ideal condition for velocity constraints is built into the definition of the d'Alembert model. In particular, no variational principle is claimed for the nonholonomic models of this page: inserting the velocity constraint into a Lagrangian and varying without constraint is the vakonomic construction and generally produces different equations, and substituting the constraint into the kinetic energy of a constrained system is justified only in the holonomic-equivalent case, where the substitution parametrizes the position locus rather than imposing a velocity law. The sources record both warnings (SICM §1.10.3; Stewart, printed page 25; Cline §§5.7 and 6.9; Cortes et al.).

The following are outside this page and remain open here: the behaviour of either prescription at a point where the rank of $A$ drops; unilateral constraints and contact-mode changes (a body that leaves a surface or begins to slip), which require mode-change models with impact or friction laws; the determination of which prescription describes a given physical mechanism; and the quantitative comparison of either model with reported observations. Rolling is a family of models rather than one model: the reduced planar two-coordinate no-slip law used on the companion page is holonomic-equivalent, and the straight-rolling exception is noted in the sources. Tong's cited example is the specific coin of radius $R$ rolling without slipping down a slope, with coordinates $(x,y,\theta ,\varphi )$ and constraints $x^{\mathrm{\prime }}=R{\varphi }^{\mathrm{\prime }}\sin \theta$, $y^{\mathrm{\prime }}=R{\varphi }^{\mathrm{\prime }}\cos \theta$ (Section 2.3.2, Eq. (2.41)); it is reported there as nonholonomic. This remark asserts no classification of all three-dimensional rolling models and supplies no local proof for Tong's example. Friction, rolling resistance, material deformation and the ability of a contact to supply the required reaction are application-level premises; no measured value and no empirical adequacy claim is made by any item of this page.