On the transpositional relation for nonholonomic systems

Federico Talamucci

REVIEW 3 major objections 4 minor 23 references

On the transpositional relation for nonholonomic systems

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Under Četaev's condition, variation and time differentiation commute for all admissible displacements exactly when the constraint's Lagrangian derivatives lie in the span of the constraint gradients.

desk verdict A careful, mostly classical re-derivation of the transpositional condition for nonholonomic constraints, with a few genuinely new small propositions, but the advertised 'clean test' is under-specified for nonlinear constraints because the Lagrangian derivative contains accelerations. read the letter →

arxiv 2506.15729 v1 pith:7KTGERUB submitted 2025-06-11 physics.class-ph math-phmath.MP

Federico Talamucci This is my paper · ORCID

classification physics.class-phmath-phmath.MP MSC 70F2570H03

keywords nonholonomicsystemsČetaevconditiontranspositionalrulecommutationrelationsvirtualdisplacementsLagrangianderivativesvakonomicmechanicsnonlinearkinematicconstraints

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks when the operations of variation and time differentiation can be interchanged in nonholonomic mechanics, and answers with a checkable condition. It works from the two standard assumptions on a velocity-dependent constraint $g_\nu(q,\dot q,t)=0$: the Četaev condition $\delta^{(c)}g_\nu=0$ and the first-variation condition $\delta^{(v)}g_\nu=0$. The author proves that if both hold, then the commutation relations $\delta\dot q_i = \frac{d}{dt}(\delta q_i)$ imply the identity $\sum_i D_i g_\nu\,\delta q_i=0$ for every constraint, where $D_i$ is the Lagrangian derivative; and under the Četaev condition alone this identity holds for all admissible displacements exactly when $D_i g_\nu$ is a linear combination of the constraint gradients $\partial g_\mu/\partial\dot q_i$. Exact constraints and constraints integrable through an integrating factor satisfy this condition, while generic nonlinear constraints such as quadratic homogeneous ones do not. This gives a direct test for whether the kinematic assumptions behind d'Alembert–Lagrange equations and those behind vakonomic-style extended variational principles can be consistently combined.

What carries the argument

The load-bearing object is the transpositional rule, the identity $\delta^{(v)}F - \frac{d}{dt}(\delta^{(c)}F) = \sum_i \frac{\partial F}{\partial\dot q_i}\bigl(\delta\dot q_i - \frac{d}{dt}\delta q_i\bigr) - \sum_i D_i F\,\delta q_i$, together with the Lagrangian derivative $D_iF = \frac{d}{dt}\frac{\partial F}{\partial\dot q_i} - \frac{\partial F}{\partial q_i}$. This identity ties the two competing variation operators — the Četaev-type variation $\delta^{(c)}$, which differentiates only through the velocities, and the first variation $\delta^{(v)}$, which differentiates through both coordinates and velocities — to the commutation defect $\delta\dot q_i - \frac{d}{dt}\delta q_i$ and to the Lagrangian derivatives of the constraint functions. When applied to the constraints $g_\nu$ under assumptions (A) and (B), the rule becomes the pivot from which the necessary conditions, the span characterization of Proposition 3, and the classification of constraint types all follow.

What would settle it

Take the fixed-speed constraint $g = \dot q_1^2 + \dot q_2^2 - 1$ and compute $D_j g = 2\ddot q_j$. Under (A) and (B) the commutation relations force $2(\ddot q_1\,\delta q_1 + \ddot q_2\,\delta q_2)=0$ for every $\delta q$ satisfying $\dot q_1\,\delta q_1+\dot q_2\,\delta q_2=0$, which can hold only when $\ddot q$ is parallel to $\dot q$. A mechanical realisation of such a constraint with any motion whose acceleration has a component transverse to the velocity would therefore violate the paper's necessary condition and refute the claimed compatibility of (A), (B) and (C0).

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a sharp characterization of when the commutation relation $\delta\dot q_i = \frac{d}{dt}(\delta q_i)$ is compatible with the two conventional variational postulates for nonholonomic constraints. Combining the Četaev condition (A), $\delta^{(c)}g_\nu=0$, with the first-variation condition (B), $\delta^{(v)}g_\nu=0$, the transpositional rule reduces to $\sum_i (\partial g_\nu/\partial\dot q_i)(\delta\dot q_i - \frac{d}{dt}\delta q_i) = \sum_i D_i g_\nu\,\delta q_i$, so the commutation relations force $\sum_i D_i g_\nu\,\delta q_i=0$ (Proposition 1). Proposition 3 shows that, under (A) alone, this identity holds for every admissible $\delta q$ if and only if $D_i g_\nu = \sum_\mu \varrho_\mu^{(\nu)}(q,\dot q,t)\,\partial g_\mu/\partial\dot q_i$. Applying this test, the paper verifies the condition for exact constraints, where $D_i g_\nu=0$, and for constraints admitting an integrating factor, where $\varphi_\nu D_j g_\nu = -\dot\varphi_\nu\,\partial g_\nu/\partial\dot q_j$, and demonstrates that for typical nonlinear constraints the condition fails, making the commutation assumptions behind vakonomic mechanics inconsistent with Četaev-based ideal kinematics.

Load-bearing premise

The main propositions all take the Četaev condition $\delta^{(c)}g_\nu=0$ as the definition of ideal virtual displacements for velocity-dependent constraints, including nonlinear ones, and the author explicitly notes that no derivation of this condition from the constraint equation itself is known; if the physically correct ideality condition for nonlinear constraints differs from Četaev's, the necessary conditions proven here need not hold for real systems.

Editorial extensions

If this is right

Under (A) and (B), the commutation relations (C0) are possible only when $\sum_i D_i g_\nu\,\delta q_i=0$; this is a finite check on data already present in the problem.
Whenever $D_i g_\nu$ is not a linear combination of the constraint gradients $\partial g_\mu/\partial\dot q_i$, the d'Alembert–Lagrange kinematics based on the Četaev condition and the vakonomic hypothesis of commuting variations are provably inconsistent.
Exact constraints and constraints integrable through an integrating factor satisfy the condition, so for these classes the two variational routes to the equations of motion can be harmonised.
For generic nonlinear constraints — quadratic homogeneous forms, fixed-speed constraints — the condition fails, so the incompatibility is the rule rather than the exception.
The coefficients $W_{i,j}$ of the general transpositional hypothesis (C) are not determined by the transpositional rule; imposing the stronger termwise relations of type (30) is an additional postulate, not a consequence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step is the paper's own open inverse problem: if $\sum_i D_i g_\nu\,\delta q_i=0$ for all Četaev-admissible displacements, must the constraint admit an integrating factor? A positive answer would make the integrating-factor class exactly the commutation-compatible class.
Proposition 5's affine-space description of $\delta\dot q$ suggests a geometric reading of the compatibility question as the vanishing of a bilinear form on the orthogonal complement of the constraint gradients, which could be evaluated numerically for any model.
For the fixed-speed constraint $g=|\dot q|^2-C(t)$, the necessary condition forces the acceleration to be parallel to the velocity; a realization of that constraint with transverse acceleration would directly falsify the compatibility of (A)+(B)+(C0).
The result can serve as a selection rule for modified vakonomic models: only constraints satisfying the span condition allow the extended Hamilton principle to agree with the d'Alembert–Lagrange equations, and for other constraints the $W_{i,j}$ coefficients must be treated as independent dynamical input.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper is a modest but careful piece of classical analytical mechanics. It does not produce new equations of motion; it produces necessary conditions for the commutation of variation and time-derivative under the Četaev and first-variation assumptions. The main new content I can verify: Proposition 6 (n=2 homogeneous linear constraints), Proposition 7 (integrating-factor sufficient condition), and the explicit characterization of condition (39) via the span condition (42). Those are correct under the stated rank assumption, and the algebra of the transpositional rule (13) checks out. The paper is also unusual in being honest about what it does not know: it says plainly that a derivation of the Četaev condition from the constraint itself is lacking, and that the W_{i,j} hypothesis from [13] is an additional assumption whose meaning is unclear. That transparency is worth crediting.

The soft spots are real but not fatal. The main one is the stress-test concern you sent over: for nonlinear velocity constraints, D_i g_nu contains q-ddot via d/dt of ∂g/∂qdot. Proposition 3 concludes that if (39) holds for all admissible δq then D_g_nu is a linear combination of the constraint gradients with coefficients depending only on (q,qdot,t). But if D_g_nu carries q-ddot, the linear algebra alone does not justify dropping q-ddot from the coefficients. The paper never states whether (42) is meant as an off-shell identity in the second-jet variables or as an on-shell condition. The text's Remark 3 and the homogeneous-section discussion read as if (42) is intended as an identity, but that is not the reading that supports the advertised 'test' on a specific system. I checked the stress-test example: g = qdot_1^2 - qdot_2, L = (qdot_1^2+qdot_2^2)/2, and it does satisfy (39) on-shell for the constant-velocity solutions while (42) fails off-shell. So the paper's claim that quadratic homogeneous constraints cannot satisfy the necessary condition is only true in the off-shell reading, and that reading is not justified by the derivation.

A second soft spot: the paper's practical applicability rests on the Četaev condition for nonlinear constraints, and the author explicitly flags that this is an assumption, not a theorem. That is the right thing to say, but it means the central physical message is conditional. The paper would be stronger if it either restricted the main claims to linear constraints or developed the off-shell reading of (42) systematically.

Who is this for? A reader working in vakonomic versus d'Alembert-Lagrange foundations, especially on the transpositional rule and the Četaev condition, will get value. It deserves a serious referee: the derivations are formal and reproducible, the references to Neimark-Fufaev and the recent vakonomic literature are appropriate, and the small propositions are new enough. I would send it to a knowledgeable referee, but I would flag the q-ddot ambiguity as the thing to force the author to clarify. My own verdict is conditional: the paper is sound within its stated assumptions, but the advertised nonlinear-constraint test needs the on-shell/off-shell distinction made explicit.

Referee Report

3 major / 4 minor

Summary. This paper studies the transpositional identity δ^(v)F - d/dt(δ^(c)F) = Σ (∂F/∂qdot_i)(δqdot_i - d/dt δq_i) - Σ D_i F δq_i (Eq. 13) for nonholonomic constraints gν(q,qdot,t)=0, possibly nonlinear in velocities. Under the Četaev condition (A), the first-variation condition (B), and commutation assumptions (C0)/(C), it derives necessary conditions: in particular, (AB) plus (C0) forces Σ D_i gν δq_i = 0 (Eq. 39). Proposition 3 claims this is equivalent to Dgν being a linear combination of the constraint gradients ∂gμ/∂qdot, and the paper uses this to argue that only linear constraints can be compatible with the commutation relations. It also compares d'Alembert-Lagrange equations with vakonomic and 'modificated vakonomic' equations, and analyzes linear, homogeneous, and velocity-only constraints.

Significance. The paper's algebraic core is transparent and reproducible: the transpositional rule (13) is derived from the definitions, and Propositions 1, 4, 5, and 7 are correct under the stated rank and ideality assumptions. The treatment of exact constraints and constraints with an integrating factor is a useful complement to the literature, and the paper is honest about the fact that the Četaev condition (A) is assumed rather than derived. If the central characterization could be made precise, it would supply a simple necessary condition for commutation and thereby clarify the gap between d'Alembert-Lagrange and vakonomic formulations. At present, however, the main conclusion about nonlinear constraints rests on an unstated quantifier over accelerations, and the advertised 'clean test' is not a constraint-only test in the nonlinear case.

major comments (3)

[Section 2.1, Proposition 3 and Remark 3] Proposition 3 is under-specified and, on the natural reading, false. The vector D_i g_nu defined in (14) contains qddot_i whenever g_nu is nonlinear in qdot_i, so at a fixed state (q,qdot,t) the value of Dg_nu is not determined until an acceleration qddot is chosen. The proof's linear-algebra step therefore yields coefficients in (42) that may depend on qddot, not only on (q,qdot,t). Concretely, take n=2, g = qdot_1^2 - qdot_2, L = (qdot_1^2 + qdot_2^2)/2. The Četaev equations (23) give mu=0 and qddot_1 = qddot_2 = 0, so every solution has qdot_1 constant and qdot_2 = qdot_1^2. For every admissible variation, δq_2 = 2 qdot_1 δq_1, the left side of (39) is 2 qddot_1 δq_1, which vanishes on-shell. Thus (39) holds for all admissible δq along all solutions, but (42) cannot hold as an identity in qddot because the left side contains 2 qddot_1 and the right side does not. The 'if and only if' in Proposition 3 is therefore valid only if (39) is required to hold identically in the second-jet variables; that reading is not stated and is not the one supplied by the on-shell necessary-condition argument in Property 1.
[Section 3.2.1, after Eq. (79)] Section 3.2.1 uses the identity reading of (39) to conclude, after Eq. (79), that the required relation 'cannot be an identity' and that commutation (C0) cannot be satisfied for degree-2 homogeneous constraints. This is exactly the off-shell reading that Proposition 3 does not justify. Property 1 derives (39) from (AB)+(C0) along actual motions, so as a necessary condition it is on-shell; at a given state the accelerations qddot are fixed by the equations of motion. The paper should either prove that (39) must hold as an identity in qddot, which would require an additional assumption on the class of virtual displacements or variations, or restrict the incompatibility claim to the off-shell/constraint-data reading. As written, the abstract's claim of a clean way to test compatibility is overdrawn: for nonlinear constraints, condition (39)/(42) depends on the dynamics through qddot and is not determined by the constraint functions alone.
[Section 1.2.1 and Section 4] Section 1.2.1 explicitly states that a way to derive the Četaev condition (10) from the constraint (1) is lacking. Since assumption (A) is the basis of Propositions 1, 3, and 5 and of the nonlinear examples in Section 3.2, this caveat should be carried into the conclusions and the abstract. The current wording in Sections 3.2 and 4 often presents (A) as the operative definition of ideality for nonlinear constraints; the paper should state clearly that all results are conditional on accepting (A) for such constraints, and that this acceptance is a physical assumption rather than a consequence of the constraint equations.

minor comments (4)

[Section 2.1, proof of Proposition 3] In the proof of Proposition 3, the expression '∂qν/∂qdot_i' should read '∂gν/∂qdot_i'.
[Section 2.2.1, proof of Proposition 5] In the proof of Proposition 5, 'Dq = 0' should read 'Dδq = 0'.
[Section 3.1.1, equation after (55)] The display after Eq. (55) for condition (B) appears garbled: the coefficient of δq_i should be Σ_r (∂ξ_{ν,r}/∂q_i) qdot_r + ∂η_ν/∂q_i, and the last sum should be Σ_r ξ_{ν,r} δqdot_r.
[General] Equation (39) is referenced in the paragraph immediately before Section 2.1 ('If it is known that (39) holds') before it is defined; reorder or renumber so the reference follows the definition. There are also numerous typos, including 'indipendent', 'traslation', 'Obviuosly', 'fufilled', 'satisifed', 'Exact constaints', 'Papastravidis' in [17], and inconsistent φν/Φν in the proof of Property 4; a careful copyedit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's propositions are explicit deductions from stated assumptions and the transpositional identity; imported hypotheses are flagged as such.

full rationale

The derivation chain is self-contained in the mathematical sense: the transpositional rule (13) is proved in-line by adding and subtracting a term, and Propositions 1, 3, 5, etc. are deductive consequences of the stated assumptions (A), (B), (C0) and (C). The paper does not fit parameters to data, does not rename an empirical regularity as a prediction, and does not invoke any load-bearing self-citation; in fact the only imported objects are the Četaev condition (A), the first variation condition (B), and the W_{i,j} ansatz (27)/(30), and the paper explicitly labels the latter as 'additional hypotheses, whose meaning is unclear to us' rather than deriving anything from it. The admitted statement that 'a way to derive (10) directly from (1) is lacking' is an honest acknowledgement that (A) is an assumption, not evidence of circularity: a conditional theorem does not become circular because its hypothesis is unproven. The possible dependence of the coefficients in (42) on the accelerations for nonlinear constraints is a mathematical under-specification or correctness concern, not a circular-reasoning defect, since the equivalence in Proposition 3 is a direct linear-algebra argument from the stated condition (39) and assumption (A). Accordingly, the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted and no new physical entities are introduced. The paper's results are conditional on standard domain assumptions, chiefly the Četaev ideality condition and the choice of first-variation condition, plus a regularity assumption on the constraint Jacobian. The generalized vakonomic discussion imports the W_i,j ansatz from the literature and explicitly labels the extra conditions as unclear.

assumptions (5)

domain assumption Constraint regularity: rank(∂g/∂˙q) = κ (condition (2)).
Used throughout to guarantee that the constraint gradients form a κ-dimensional space and that the orthogonal complement has dimension n−κ; invoked in Section 1.1 and Proposition 3.
domain assumption Četaev condition (A) defines ideal virtual displacements for nonlinear constraints.
The paper adopts δ^(c)gν=0 as the definition of virtual displacements even for nonlinear gν, noting in Section 1.2.1 that no derivation from (1) is known. All propositions about commutation are conditional on this.
domain assumption First-variation condition (B) is the appropriate variational constraint in extended Hamilton principles.
The paper takes δ^(v)gν=0 as the stationarity-consistent constraint variation in vakonomic-type principles, following the cited literature; this is an assumption about which variational principle applies, not a theorem.
domain assumption The transpositional relation (27) with coefficients W_i,j is a valid representation of the commutation defect.
In the generalized vakonomic discussion, the paper imports the ansatz δ˙q_i − d/dt(δq_i) = Σ_j W_i,j δq_j from [13] and notes that the added conditions (30) are not derivable from the transpositional rule; see Sections 1.3.2 and 3.2.1.
standard math Euler's theorem on homogeneous functions and the vanishing of v^T A v for skew-symmetric A.
Used in Section 3.2.1 (Theorem 1) and Proposition 6 to connect homogeneity of constraints with the Četaev condition and to prove the n=2 case.

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Cite this review

Pith. "Pith review of On the transpositional relation for nonholonomic systems." pith.science (2026). https://pith.science/paper/7KTGERUB

@misc{pith2026250615729,
  author       = {Pith},
  title        = {Pith review of: On the transpositional relation for nonholonomic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KTGERUB}},
  note         = {Machine review of arXiv:2506.15729}
}

abstract

This paper investigates the dynamics of nonholonomic mechanical systems, with a particular focus on the fundamental variational assumptions and the role of the transpositional rule. We analyze how the $\check Cetaev condition and the first variation of constraints define compatible virtual displacements for systems subject to kinematic constraints, which can be both linear and nonlinear in generalized velocities. The study meticulously explores the necessary conditions for the commutation relations to hold, clarifying their impact on the consistency of the derived equations of motion. By detailing the interplay between these variational identities and the Lagrangian derivatives of the constraint functions, we shed light on the differences between equations of motion formulated via d'Alembert--Lagrange principle and those obtained from extended time-integral variational principles. This work aims to provide a clearer theoretical framework for understanding and applying these core principles in the complex domain of nonholonomic dynamics.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [13]

, Ramírez, R

Llibre, J. , Ramírez, R. , Sadovaskaia, N. (2014) A new approach to vakonomic mechanics,Nonlinear Dynamics 78, 2219–2247

work page 2014

[2] [1]

Arnol’d, V. I. , Kozlov, V. V. , Neishtadt, A. I.(2006)Mathematical Aspects of Classical and Celestial Mechanics, Third Editiom, E. M. S. ,Dynamical Systems III, Springer

work page 2006

[3] [2]

(2011) The non–holonomic double pendulum, an example of non-linear non-holonomic system,Regular and Chaotic Dynamics,1 n

Benenti, S. (2011) The non–holonomic double pendulum, an example of non-linear non-holonomic system,Regular and Chaotic Dynamics,1 n. 5, 417–442

work page 2011

[4] [3]

(2010) On the Compatibility of Nonholonomic Systems and Related Variational Systems,Acta Physica Universitatis comenianae Vol

Cronström, C. (2010) On the Compatibility of Nonholonomic Systems and Related Variational Systems,Acta Physica Universitatis comenianae Vol. L–LI, 1 and 2, 25–36

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ˇCetaev, N. G. (1962) On the Gauss Principles,Papers on Analytical Mechanics323, Science Academy

work page 1962

[6] [5]

Flannery M. R. (2011) d’Alembert–Lagrange analytical dynamics for nonholonomic systems,Journal of Mathe- matical Physics52, 032705 1–29

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[7] [6]

Flannery M. R. (2005) The enigma of nonholonomic constraints,American Journal of Physics73, 265–272

work page 2005

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Golub, G. H. , Van Loan, C. F. (2013)Matrix Computations 4th ed. Johns Hopkins University Press

work page 2013