PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The ideal holonomic system in a possibly time-dependent chart: explicit acceleration, constraint preservation, local existence and reaction power

Statement

Let $O \subseteq R \times R^{n}$ be open and nonempty, let $G : O \to R^{k}$ be of class $C^{3}$ with $G_{q} (t, q)$ of rank $k$ for every $(t, q) \in O$ (a regular holonomic constraint, A regular holonomic constraint, its virtual displacements and the ideal-reaction law), let $M \in M_{n} (R)$ be symmetric positive definite, and let $f : O \times R^{n} \to R^{n}$ be of class $C^{1}$ . Define, for $(t, q, v) \in O \times R^{n}$ ,

$C (t, q) : = G_{q} (t, q) M^{- 1} G_{q} (t, q)^{T}, Λ (t, q, v) : = - C (t, q)^{- 1} [G_{q} M^{- 1} f + G_{t t} + 2 G_{t q} [v] + G_{q q} [v, v]], a (t, q, v) : = M^{- 1} (f + G_{q}^{T} Λ),$

where all terms in the bracket are evaluated at $(t, q, v)$ , and $2 G_{t q} [v] : = \sum_{i} 2 \partial_{t} \partial_{q_{i}} G v_{i}$ and $G_{q q} [v, v] : = \sum_{i, j} \partial_{i} \partial_{j} G v_{i} v_{j}$ . Then:

$C$ is symmetric positive definite and $C^{- 1}$ , $Λ$ and $a$ are of class $C^{1}$ on $O \times R^{n}$ ; in particular $a$ is locally Lipschitz in $(q, v)$ ;
through every $(t_{0}, q_{0}, v_{0}) \in O \times R^{n}$ there is a unique maximal solution $(q, v) : I_{t_{0}} \to R^{2 n}$ , with $(t, q (t), v (t)) \in O \times R^{n}$ , of the first-order system

$q^{'} = v, v^{'} = a (t, q, v);$

if the data satisfy the consistency conditions

$G (t_{0}, q_{0}) = 0, G_{t} (t_{0}, q_{0}) + G_{q} (t_{0}, q_{0}) v_{0} = 0,$

then the maximal solution satisfies $G (t, q (t)) = 0$ and $G_{t} (t, q (t)) + G_{q} (t, q (t)) q^{'} (t) = 0$ for all $t \in I_{t_{0}}$ ; consequently $q$ is an ideal-constraint motion of $(G, M, f)$ with multiplier $λ (t) = Λ (t, q (t), v (t))$ , and its reaction $R = G_{q}^{T} Λ$ satisfies the power identity

$R (t) \cdot q^{'} (t) = - Λ (t, q (t), v (t)) \cdot G_{t} (t, q (t));$

conversely, every $C^{2}$ ideal-constraint motion $q : I \to R^{n}$ of $(G, M, f)$ whose velocity is permitted solves this first-order system with $v = q^{'}$ ; hence the ideal-constraint motions with permitted velocity are exactly the solutions of the system with consistent data.

Facts & Assumptions

Given: The $C^{3}$ regular constraint $G$ with rank $k$ everywhere, the symmetric positive definite $M$ , the $C^{1}$ force field $f$ , the functions $C, Λ, a$ , and data $(t_{0}, q_{0}, v_{0}) \in O \times R^{n}$ .

[F1]

The multiplier lemma (Reactions annihilating the virtual displacements are transpose multiples, with a positive definite multiplier matrix): $C$ is symmetric positive definite and invertible, $R ⊥ \ker G_{q}$ iff $R = G_{q}^{T} λ$ for a unique $λ$ , and $λ = C^{- 1} G_{q} M^{- 1} R$ for such $R$ .

[F2]

Regularity: $G$ is $C^{3}$ , so $G_{q}$ is $C^{2}$ , $C$ is $C^{2}$ , and $C^{- 1}$ is $C^{2}$ (Matrix inversion preserves $C^{k}$ regularity where the determinant is nonzero); the bracket in $Λ$ is $C^{1}$ in $(t, q, v)$ (its terms are sums and products of $C^{1}$ and $C^{2}$ maps in $(t, q)$ with polynomials in $v$ ), and $f$ is $C^{1}$ ; hence $Λ$ and $a$ are $C^{1}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[F3]

Chain rule, twice: for a $C^{2}$ curve $q$ with $(t, q (t)) \in O$ , the function $h (t) : = G (t, q (t))$ is $C^{2}$ and $h^{''} = G_{t t} + 2 G_{t q} [q^{'}] + G_{q q} [q^{'}, q^{'}] + G_{q} q^{''},$ the terms being evaluated at $(t, q (t))$ ; this follows by differentiating $h^{'} = G_{t} + G_{q} \cdot q^{'}$ and commuting the mixed partials of $G$ , which are continuous because $G$ is $C^{3}$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Clairaut--Schwarz theorem for continuous second partial derivatives, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[F4]

Picard–Lindelöf and maximal solutions (Picard-Lindelöf local existence and uniqueness for first-order systems, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, First-order systems, initial value problems, and solutions on intervals, Local Lipschitz continuity in the state variable, locally uniform in time and parameters): a $C^{1}$ field on an open set is locally Lipschitz in the state variable, so every initial value problem for the system in clause 2 has a unique maximal solution.

[F5]

If a $C^{1}$ function on an interval has derivative identically zero, it is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant); and the integral of a continuous derivative recovers the increment (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

technique · direct; verify smoothness, take the twice-differentiated constraint, and use Picard–Lindelöf for the flow statement

1.1F1F2

Clause 1: by [F1], $C$ is symmetric positive definite and invertible; by [F2], $C^{- 1}$ , $Λ$ and $a$ are $C^{1}$ . A $C^{1}$ right-hand side $F (t, q, v) : = (v, a (t, q, v))$ is locally Lipschitz in the state variable: around every $(t_{0}, q_{0}, v_{0})$ choose $h, r > 0$ with $[t_{0} - h, t_{0} + h] \times \overline{B} ((q_{0}, v_{0}), r) \subseteq O \times R^{n}$ . Its continuous state derivative has a finite bound $L$ on this compact product. At fixed time, integration along the straight segment between two states in the ball gives $∥ F (t, z) - F (t, \tilde{z}) ∥_{2} \leq L ∥ z - \tilde{z} ∥_{2}$ . The same argument works on any such compact product (A $C^{1}$ map is locally Lipschitz on compact coordinate subsets, Local Lipschitz continuity in the state variable, locally uniform in time and parameters, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative). This proves clause 1.

2.1F2F4step 1.1

Clause 2: the system $q^{'} = v$ , $v^{'} = a (t, q, v)$ has a $C^{1}$ right-hand side on the open set $O \times R^{n}$ , so by [F4] every initial value problem has a unique maximal solution; this is clause 2.

3.1F1F3F5step 2.1algebra

Clause 3, preservation: let $(q, v)$ be the maximal solution through consistent data and put $h (t) : = G (t, q (t))$ on the maximal interval. By [F3], $h^{''} = G_{t t} + 2 G_{t q} [q^{'}] + G_{q q} [q^{'}, q^{'}] + G_{q} a = G_{t t} + 2 G_{t q} [q^{'}] + G_{q q} [q^{'}, q^{'}] + G_{q} M^{- 1} f + G_{q} M^{- 1} G_{q}^{T} Λ = G_{t t} + 2 G_{t q} [q^{'}] + G_{q q} [q^{'}, q^{'}] + G_{q} M^{- 1} f + C Λ = 0,$ because $G_{q} M^{- 1} G_{q}^{T} = C$ and $Λ = - C^{- 1} [G_{q} M^{- 1} f + G_{t t} + 2 G_{t q} [q^{'}] + G_{q q} [q^{'}, q^{'}]]$ . Thus $h^{''} = 0$ on the interval, so $h^{'}$ is constant and $h$ is affine by [F5]: $h (t) = h (t_{0}) + h^{'} (t_{0}) (t - t_{0})$ . The consistency conditions say $h (t_{0}) = 0$ and $h^{'} (t_{0}) = G_{t} (t_{0}, q_{0}) + G_{q} (t_{0}, q_{0}) v_{0} = 0$ , so $h = 0$ and $h^{'} = G_{t} (t, q (t)) + G_{q} (t, q (t)) q^{'} (t) = 0$ on the interval. In particular $q$ is a constrained curve with permitted velocity, and by construction $M q^{''} = f + G_{q}^{T} Λ$ , so $q$ is an ideal-constraint motion with multiplier $Λ$ along it.

4.1step 3.1algebra

Clause 3, power: since $G_{t} + G_{q} q^{'} = 0$ on the interval, $R \cdot q^{'} = (G_{q}^{T} Λ) \cdot q^{'} = Λ \cdot G_{q} q^{'} = - Λ \cdot G_{t},$ where $R = M q^{''} - f = G_{q}^{T} Λ$ .

5.1F1F3step 3.1algebra∎

Clause 4: let $q : I \to R^{n}$ be an ideal-constraint motion with permitted velocity. Then $G (t, q (t)) = 0$ and $M q^{''} = f + G_{q}^{T} λ$ for some continuous $λ$ ; differentiating the constraint twice along the curve and using [F3] gives $0 = G_{t t} + 2 G_{t q} [q^{'}] + G_{q q} [q^{'}, q^{'}] + G_{q} q^{''}$ , and substituting $q^{''} = M^{- 1} f + M^{- 1} G_{q}^{T} λ$ gives $C λ = - [G_{q} M^{- 1} f + G_{t t} + 2 G_{t q} [q^{'}] + G_{q q} [q^{'}, q^{'}]] = C Λ (t, q, q^{'}),$ so $λ = Λ (t, q, q^{'})$ by invertibility of $C$ from [F1]. Hence $q^{''} = a (t, q, q^{'})$ and the system holds; conversely every solution with consistent data is an ideal-constraint motion by step 3.1. This proves clause 4.

Remarks

The construction is rheonomic: $G$ may depend on $t$ and is never frozen. The time dependence enters the acceleration through $G_{t}$ , $G_{t t}$ and $G_{t q}$ , and the reaction power $R \cdot q^{'} = - Λ \cdot G_{t}$ need not vanish, so an ideal moving constraint can do work on the system. This is the mechanism behind the rotating-hoop example of the companion page.
The hypothesis that $G_{q}$ has rank $k$ on all of $O$ is used twice: to invert $C$ and to make the vector field $a$ everywhere defined and $C^{1}$ . At a point where the rank drops, $C$ becomes singular and the construction stops; the proposition asserts nothing about continuation beyond such a point, and no continuation theorem is implied.
The consistency conditions are exactly the requirements that the initial position lies on the constraint and the initial velocity is permitted. Arbitrary data $(t_{0}, q_{0}, v_{0})$ still determine a unique solution of the system, but not an ideal-constraint motion: the constraint is preserved only for consistent data.

The ideal holonomic system in a possibly time-dependent chart: explicit acceleration, constraint preservation, local existence and reaction power

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources