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The unrestricted action principle fails for the normalized nonholonomic relation $z_{s} = x y_{s}$ : the d'Alembert motion is not vakonomic

Physical scope

A worked check in dimensionless normalized coordinates and time, with physical length and time scales declared, on the nonholonomic relation z_s = x y_s: the d'Alembert motion from the stated normalized initial data has strictly positive y_s and nonzero y_{ss} for s>0, so it cannot satisfy the vakonomic (unrestricted-action) equations of the same constraint, which force y_{ss} to vanish wherever x_{ss} does. Thus these two prescriptions differ for this model and data, so no general equivalence holds. The example reports exact model computations, not measurements.

Example

Adopt the constraint model of The velocity constraint $z^{'} = x y^{'}$ : virtual displacements, the d'Alembert law and the vakonomic law in dimensionless normalized variables: let $s = t / T_{0}$ be dimensionless time and $(x, y, z) = r / L_{0} \in R^{3}$ be dimensionless coordinates, where $T_{0} > 0$ , $L_{0} > 0$ and $m_{0} > 0$ are reference quantities with $[T_{0}] = τ$ , $[L_{0}] = ℓ$ and $[m_{0}] = μ$ . Primes below mean differentiation with respect to $s$ . The normalized unit-mass equations have zero applied force and velocity constraint $z_{s} = x y_{s}$ ; in physical variables this is ${\dot{r}}_{z} = (r_{x} / L_{0}) {\dot{r}}_{y}$ , which is dimensionally consistent. The physical kinetic action is $m_{0} L_{0}^{2} / T_{0}$ times the normalized action; scaling the augmented multiplier by $m_{0} L_{0} / T_{0}$ makes its augmented action the same positive multiple of the normalized one, preserving stationarity. Physical reactions scale as $m_{0} L_{0} / T_{0}^{2}$ times normalized reactions, and physical virtual displacements scale as $L_{0}$ , so the zero-virtual-work condition is preserved. In these normalized variables the d'Alembert prescription is $x_{s s} = 0$ , $y_{s s} = - λ x$ , $z_{s s} = λ$ with $λ = x_{s} y_{s} / (1 + x^{2})$ along motions, and the vakonomic prescription is $x_{s s} = - η y_{s}$ , $y_{s s} = η_{s} x + η x_{s}$ , $z_{s s} = - η_{s}$ for the normalized multiplier $η$ . Consider the normalized initial data $(x, y, z) (0) = (0, 0, 0), (x_{s}, y_{s}, z_{s}) (0) = (1, 1, 0) .$

Then:

the curve $x (s) : = s, y (s) : = \int_{0}^{s} \frac{d u}{\sqrt{1 + u^{2}}}, z (s) : = \int_{0}^{s} u y_{s} (u) d u = \sqrt{1 + s^{2}} - 1$ is a d'Alembert motion with these data on $R$ , and it satisfies $y_{s} (s) = (1 + s^{2})^{- 1 / 2} > 0$ and $y_{s s} (s) = - s (1 + s^{2})^{- 3 / 2}$ for every $s \in R$ , so $y_{s s} (s) \neq 0$ whenever $s \neq 0$ ;
$y_{s s} (1 / 2) = - 4 / (5 \sqrt{5}) \neq 0$ ;
consequently this d'Alembert motion is not a vakonomic motion of the same constraint with the same data: on any nondegenerate interval where a vakonomic pair for this curve is stationary, $x_{s s} = 0$ and $y_{s} > 0$ force $η \equiv 0$ , and then the vakonomic equation forces $y_{s s} \equiv 0$ . Every nondegenerate interval contains some $s \neq 0$ , which contradicts clause 1;
hence, for this normalized model and initial data, the vakonomic stationary-action prescription does not reproduce the d'Alembert motion. This conclusion concerns the augmented-action prescription defined by the vakonomic law above; it makes no claim about a separately specified minimization problem or other variational prescriptions.

Facts & Assumptions

Given: The normalized constraint model and initial data of the Example, the scales $L_{0}, T_{0}, m_{0}$ and the displayed curve.

[F1]

The constraint model (The velocity constraint $z^{'} = x y^{'}$ : virtual displacements, the d'Alembert law and the vakonomic law): in normalized variables the allowed velocities and virtual displacements are $D (q) = {(w_{x}, w_{y}, w_{z}) : w_{z} = x w_{y}}$ ; the normalized sample Lagrangian is $\frac{1}{2} ∣ q_{s} ∣^{2}$ . The displayed d'Alembert equations follow from acceleration orthogonal to $D (q)$ and differentiation of the constraint; the displayed vakonomic equations are necessary for augmented-action stationarity, as derived in the Counterexample paragraphs headed "the d'Alembert system" and "the vakonomic equations" of The d'Alembert and vakonomic motions of $z^{'} = x y^{'}$ differ from the same initial data.

[F2]

The comparison result (The d'Alembert and vakonomic motions of $z^{'} = x y^{'}$ differ from the same initial data): every d'Alembert motion with the stated data is claimed to be a vakonomic motion of the same constraint with the same data; the statement is refuted there by exhibiting the difference. In particular the two prescriptions are not equivalent for this constraint and data. The present example re-derives the discrepancy on explicit quantities.

[F3]

Real-power and integral calculus (Continuity and derivatives of positive-base real powers, Real powers for positive bases, with the zero-base positive-exponent convention, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ , Squares of nonzero elements are positive, Absolute value in an ordered field, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): $s \mapsto (1 + s^{2})^{- 1 / 2}$ is continuously differentiable on $R$ with derivative $- s (1 + s^{2})^{- 3 / 2}$ , the expression $1 + s^{2}$ is positive, and $\sqrt{5} > 0$ . Continuous integrands are integrable on every nondegenerate compact interval. Oriented additivity gives $\int_{u}^{v} f + \int_{v}^{w} f = \int_{u}^{w} f$ for arbitrary endpoints in such an interval. The first part of the fundamental theorem differentiates $v \mapsto \int_{A}^{v} f$ on a fixed $[A, B]$ , and its second part evaluates the integral of an integrable derivative.

[F4]

The model setting (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, clause (e), Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): realize the separately adopted d'Alembert prescription of [F1] by one particle of mass $m_{0}$ with $C^{2}$ position curve $r (t) = L_{0} q (t / T_{0})$ , where $q = (x, y, z) : R \to R^{3}$ . The applied force input $F_{a p p} : R \times R^{3} \times R^{3} \to R^{3}$ is identically zero. Along a d'Alembert motion the ideal reaction is $R (t) = (m_{0} L_{0} / T_{0}^{2}) λ (t / T_{0}) (0, - x (t / T_{0}), 1)$ , a continuous vector of force dimension, so $m_{0} \ddot{r} = F_{a p p} + R$ . Its pairing with every physical virtual displacement $L_{0} δ$ , $δ \in D (q)$ , vanishes because $δ_{z} = x δ_{y}$ . Thus zero applied force does not mean zero total force. This constrained prescription is outside Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, clause (i), which explicitly excludes constrained dynamics; that postulate supplies no equation with zero total force for this curve.

Verification

technique · direct; substitute the displayed curve into the d'Alembert equations, evaluate the curvature terms, and derive the contradiction with the vakonomic system

1.1F3givenalgebra

Clause 1, integral derivatives: for any $s_{0} \in R$ , choose $A < \min (0, s_{0})$ and $B > \max (0, s_{0})$ . For either continuous integrand $f (u) = (1 + u^{2})^{- 1 / 2}$ or $f (u) = u (1 + u^{2})^{- 1 / 2}$ , oriented additivity gives $\int_{0}^{s} f = \int_{A}^{s} f - \int_{A}^{0} f$ for $s \in [A, B]$ . The first part of the fundamental theorem differentiates the first term at the interior point $s_{0}$ and the second term is constant, so the integral has derivative $f (s_{0})$ , including when $s_{0} \leq 0$ . Thus $y_{s} = (1 + s^{2})^{- 1 / 2}$ and $z_{s} = s y_{s}$ on $R$ . The derivative of $\sqrt{1 + s^{2}}$ is $s (1 + s^{2})^{- 1 / 2}$ ; the second part of the fundamental theorem on the interval between $0$ and $s$ , with oriented reversal for $s < 0$ and the zero-integral convention for $s = 0$ , gives $z (s) = \sqrt{1 + s^{2}} - 1$ .

2.1F1F3F4step 1.1givenalgebra

Clause 1, d'Alembert equations: with $x (s) = s$ one has $x_{s} \equiv 1$ and $x_{s s} \equiv 0$ ; differentiating step 1.1 by [F3] gives $y_{s s} = - s (1 + s^{2})^{- 3 / 2}$ and $z_{s s} = y_{s} + s y_{s s} = (1 + s^{2})^{- 3 / 2}$ , both continuous, so the curve is $C^{2}$ . The d'Alembert multiplier is $λ = x_{s} y_{s} / (1 + x^{2}) = (1 + s^{2})^{- 3 / 2}$ , so $y_{s s} = - λ x$ and $z_{s s} = λ$ ; the constraint holds by step 1.1, and the initial data match. Thus the curve is a d'Alembert motion on $R$ with the stated data, $y_{s} > 0$ everywhere, and $y_{s s}$ vanishes only at $s = 0$ .

3.1F3step 2.1algebra

Clause 2: at $s = 1 / 2$ , $1 + s^{2} = 5 / 4$ , so by [F3] $y_{s s} (1 / 2) = - (1 / 2) (5 / 4)^{- 3 / 2} = - (1 / 2) \cdot 8 / (5 \sqrt{5}) = - 4 / (5 \sqrt{5}) \neq 0$ , because $4 \neq 0$ and $5 \sqrt{5} > 0$ .

3.2F1F2step 2.1algebra

Clause 3, contradiction: suppose the same curve were also vakonomic, with a $C^{2}$ multiplier $η$ making the augmented pair stationary on some nondegenerate interval $[a, b]$ . Its first vakonomic equation reads $x_{s s} = - η y_{s}$ ; since $x_{s s} = 0$ and $y_{s} > 0$ everywhere by step 2.1, this forces $η = 0$ on $[a, b]$ . The second equation then gives $y_{s s} = 0$ on $(a, b)$ . Choose $s_{0} \in (a, b)$ with $s_{0} \neq 0$ , which exists because $(a, b)$ is a nonempty open interval; step 2.1 gives $y_{s s} (s_{0}) \neq 0$ , a contradiction. Hence no interval in the vakonomic definition makes this curve stationary, so it is not a vakonomic motion with these normalized data.

4.1F1F2step 3.2∎

Clause 4: since the d'Alembert motion with these normalized data is not vakonomic by step 3.2, the vakonomic stationary-action prescription does not reproduce that motion for this model and data. The proof asserts no conclusion about minimization or other variational prescriptions, and no universal claim about all velocity constraints follows; the holonomic reduction of this page concerns the separate holonomic branch.

Remarks

The verification uses only the two displayed prescriptions and exact calculus; the underlying separation of the two dynamics is recorded in the cited counterexample, and the model-theoretic warning belongs to the scope remark of this pair's main page.

The unrestricted action principle fails for the normalized nonholonomic relation $z_{s} = x y_{s}$ : the d'Alembert motion is not vakonomic

Example

Facts & Assumptions

Verification

Remarks

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The unrestricted action principle fails for the normalized nonholonomic relation zs=xyszs​=xys​: the d'Alembert motion is not vakonomic

Example

Facts & Assumptions

Verification

Remarks

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Dependency tree · 2 levels

Sources

The unrestricted action principle fails for the normalized nonholonomic relation $z_{s} = x y_{s}$ : the d'Alembert motion is not vakonomic