{"id":"d2306e10-37f1-4ff2-b488-faf66c8ace12","arxiv_id":"2506.15729","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For nonholonomic systems, commuting the variation and time derivative forces the Lagrangian derivatives of the constraints to lie in the span of the constraint gradients.","lead":"This paper works out the precise conditions under which the time derivative and the variation of coordinates commute for systems with nonholonomic constraints that can be nonlinear in velocities. It connects the Četaev condition and the first variation of constraints to a simple linear-algebra condition on the Lagrangian derivatives of the constraint functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central necessary-condition test is under-specified for nonlinear constraints: D_i g_nu contains q-ddot terms, so the advertised constraint-only check depends on the dynamics unless (42) is read as an off-shell identity.","rationale":"The reader's verdict is CONDITIONAL, centered on the physical status of the Cetaev condition (A). I agree that the unproven Cetaev postulate is a legitimate source of conditionality, and the paper itself concedes the lack of a derivation. However, the most load-bearing technical concern is different: even if one fully accepts Cetaev's condition, the proposed 'clean test' is not well-posed for nonlinear constraints because Dg_nu contains q-ddot. Proposition 3 is correct as a pointwise linear-algebra statement, but its use as a constraint-data test requires deciding whether the membership condition (42) is required for all accelerations (off-shell identity) or only for the accelerations realized by the actual motion. The paper does not make this explicit, and the two readings disagree. The explicit counterexample with g = q-dot_1^2 - q-dot_2 shows that the off-shell reading makes the test overly restrictive, while the on-shell reading makes the test depend on the Lagrangian and initial conditions, so it is not a clean constraint-only criterion. This does not invalidate the conditional theorems, so I would not change the reader's CONDITIONAL verdict; it reinforces the need to state precisely the quantifier over q-ddot and over admissible motions. The paper deserves credit for transparent derivations and for explicitly flagging the unproven status of (A), and the algebra in Propositions 1, 3, and 5 appears internally consistent.","tokens_in":20015,"tokens_out":20348,"duration_ms":239808,"concrete_test":"Evaluate the paper's test on the explicit system g = q-dot_1^2 - q-dot_2 = 0, L = (q-dot_1^2 + q-dot_2^2)/2, using equations (23) of the paper. Solve: mu = 0, q-ddot_1 = q-ddot_2 = 0, so solutions with constant q-dot_1 and q-dot_2 = q-dot_1^2 exist. For variations delta-q_2 = 2 q-dot_1 delta-q_1 with delta-q-dot = d/dt delta-q, verify that (A), (B), and C0 all hold and that sum_i D_i g delta-q_i = 0. If confirmed, this shows a nonlinear constraint can access the commutation relation on-shell, directly contradicting the 'only linear constraints' heuristic of Remark 3 and forcing the paper to specify whether (42) is an off-shell identity or an on-shell dynamical condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The algebra in Propositions 1, 3, and 5 is internally sound, but the paper's central message presents condition (39)/(42) as a clean way to test the compatibility of commutation assumptions with the Cetaev d'Alembert-Lagrange kinematics. For nonlinear constraints this test is not determined by the constraint data alone. In Proposition 3, D_i g_nu is defined as the Lagrangian derivative (14); whenever g_nu is nonlinear in the velocities, D_i g_nu contains second accelerations q-ddot through d/dt(partial g_nu/partial q-dot_i). The proof of Proposition 3 concludes that (39) holds for all admissible delta-q iff Dg_nu lies in the span of the constraint gradients, with coefficients of the form rho_mu(q, q-dot, t). But if Dg_nu contains q-ddot, the coefficients recovered from the orthogonal-complement argument may depend on q-ddot as well; the restriction to coefficients independent of q-ddot is not justified by the linear algebra unless (42) is required to hold as an identity in the second-jet variables. The paper does not state which reading is intended. The distinction matters: a genuinely nonlinear constraint can satisfy (39) on-shell even though (42) fails as an off-shell identity. Example: n=2, g = q-dot_1^2 - q-dot_2 = 0, L = (q-dot_1^2 + q-dot_2^2)/2. The Cetaev equations (23) give mu = 0 and q-ddot_1 = q-ddot_2 = 0, so every motion with constant q-dot_1 and q-dot_2 = q-dot_1^2 is a solution. For any admissible variation delta-q_2 = 2 q-dot_1 delta-q_1 with delta-q-dot = d/dt delta-q, both (A) and (B) hold and sum_i D_i g delta-q_i = 2 q-ddot_1 delta-q_1 = 0. Thus the necessary condition is satisfied by a nonlinear constraint on this dynamics. Section 3.2.1 and Remark 3 assert that quadratic homogeneous constraints cannot satisfy (42) because the relation 'cannot be an identity'; that assertion is correct only in the off-shell reading, and it is not the reading needed for the advertised test on a specific mechanical system.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20395,"tokens_out":14540,"duration_ms":154163,"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":[{"comment":"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":"Section 2.1, Proposition 3 and Remark 3"},{"comment":"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":"Section 3.2.1, after Eq. (79)"},{"comment":"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.","section":"Section 1.2.1 and Section 4"}],"minor_comments":[{"comment":"In the proof of Proposition 3, the expression '∂qν/∂qdot_i' should read '∂gν/∂qdot_i'.","section":"Section 2.1, proof of Proposition 3"},{"comment":"In the proof of Proposition 5, 'Dq = 0' should read 'Dδq = 0'.","section":"Section 2.2.1, proof of Proposition 5"},{"comment":"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.","section":"Section 3.1.1, equation after (55)"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of physics.class-ph. The main technical risk is the unstated quantifier over accelerations in Proposition 3 and Remark 3; I believe this is fixable in revision by stating the intended reading and adjusting the nonlinear-constraint conclusions accordingly. If the authors cannot resolve the issue, the paper should be narrowed to linear constraints, where the main claims are sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nA 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.\n\nWho 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.","headline":"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.","tokens_in":21041,"tokens_out":809,"would_cite":true,"duration_ms":11100,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["70F25","70H03"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["nonholonomic systems","Četaev condition","transpositional rule","commutation relations","virtual displacements","Lagrangian derivatives","vakonomic mechanics","nonlinear kinematic constraints"],"falsifier":"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).","tokens_in":19801,"feed_emoji":"⚙️","tokens_out":13699,"duration_ms":124984,"temperature":0.7,"pith_summary":"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.","feed_headline":"Commuting variations demand a Lagrangian-derivative condition","feed_subtitle":"Četaev-based and vakonomic kinematics agree only when the constraint's Lagrangian derivatives fall in the right span.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the standard definition of virtual displacements for linear kinematic constraints and the Hamilton–Suslov principle that assumptions (A) and (B) extend.","marker":"[16]"},{"why":"Provides the vakonomic equations of motion (29) that the paper contrasts with the d'Alembert–Lagrange equations (23).","marker":"[1]"},{"why":"Introduces the modified vakonomic equations with coefficients $W_{i,j}$ and the extra postulate (30) whose status the paper analyses.","marker":"[13]"},{"why":"Discusses how to determine the coefficients $W_{i,j}$ and reconciles vakonomic with the traditional nonholonomic method; the paper positions its necessary conditions against that programme.","marker":"[20]"},{"why":"Supports applying the Četaev condition to homogeneous constraints and supplies the nonlinear example the paper uses to test its characterization.","marker":"[5]"},{"why":"Provides the nonholonomic double pendulum, a concrete quadratic homogeneous constraint whose Lagrangian derivatives are used to test the commutation-compatibility claims.","marker":"[2]"},{"why":"Establishes the Hölder-principle reading of combining the Četaev condition with the commutation relations (C0).","marker":"[23]"}],"fun_headline_variants":["Nonholonomic commutation fails for generic nonlinear constraints","When do variations commute? Only if a span condition holds","Vakonomic vs Četaev: a sharp compatibility test","Transpositional rule holds only for constrained Lagrangian derivatives","Nonlinear constraints break variation commutation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Nonholonomic commutation fails for generic nonlinear constraints","When do variations commute? Only if a span condition holds","Vakonomic vs Četaev: a sharp compatibility test","Transpositional rule holds only for constrained Lagrangian derivatives","Nonlinear constraints break variation commutation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000164,"raw_usage":{"total_tokens":1276,"prompt_tokens":1001,"completion_tokens":275,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":199}},"tokens_in":617,"tokens_out":275,"duration_ms":3635,"temperature":1.0,"reasoning_tokens":199,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:42:24.003369+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard definition of virtual displacements for linear kinematic constraints and the Hamilton–Suslov principle that assumptions (A) and (B) extend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the vakonomic equations of motion (29) that the paper contrasts with the d'Alembert–Lagrange equations (23)."},{"cited_title":", Ramírez, R","cited_arxiv_id":null,"evidence_quote":"Introduces the modified vakonomic equations with coefficients $W_{i,j}$ and the extra postulate (30) whose status the paper analyses."},{"cited_title":", Giammarini, A","cited_arxiv_id":null,"evidence_quote":"Discusses how to determine the coefficients $W_{i,j}$ and reconciles vakonomic with the traditional nonholonomic method; the paper positions its necessary conditions against that programme."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supports applying the Četaev condition to homogeneous constraints and supplies the nonlinear example the paper uses to test its characterization."},{"cited_title":"(2011) The non–holonomic double pendulum, an example of non-linear non-holonomic system,Regular and Chaotic Dynamics,1 n","cited_arxiv_id":null,"evidence_quote":"Provides the nonholonomic double pendulum, a concrete quadratic homogeneous constraint whose Lagrangian derivatives are used to test the commutation-compatibility claims."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Hölder-principle reading of combining the Četaev condition with the commutation relations (C0)."}],"review_version":1}