LemmaStatement: 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 second variation of a time-dependent action: twice differentiability of the fixed-endpoint family and the Hessian integral formula

Statement

Let $(O, L)$ be a time-dependent Lagrangian chart over $[a, b] \times U$ in the sense of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations (so $L$ is $C^{2}$ and $[a, b] \times U \times R^{n} \subseteq O$ ), let $q : [a, b] \to U$ be a $C^{2}$ curve and let $η$ be a variation of $q$ (a $C^{2}$ map with $η (a) = η (b) = 0$ ). Write $ψ (ε, t) : = (t, q (t) + ε η (t), q^{'} (t) + ε η^{'} (t))$ and, for $∣ ε ∣ < ε_{0}$ with $ε_{0}$ as in clause 1 below,

$H (ε, t) : = \sum_{j, k = 1}^{n} [\frac{\partial^{2} L}{\partial q_{j} \partial q_{k}} (ψ (ε, t)) η_{j} (t) η_{k} (t) + 2 \frac{\partial^{2} L}{\partial q_{j} \partial v_{k}} (ψ (ε, t)) η_{j} (t) η_{k}^{'} (t) + \frac{\partial^{2} L}{\partial v_{j} \partial v_{k}} (ψ (ε, t)) η_{j}^{'} (t) η_{k}^{'} (t)] .$

Then:

there is $ε_{0} > 0$ such that for every $ε \in (- ε_{0}, ε_{0})$ the curve $q + ε η$ is a $C^{2}$ admissible curve with values in $U$ , and the map $ε \mapsto S (q + ε η)$ is of class $C^{2}$ on $(- ε_{0}, ε_{0})$ ;
for every $ε \in (- ε_{0}, ε_{0})$ ,

$\frac{d^{2}}{d ε^{2}} S (q + ε η) = \int_{a}^{b} H (ε, t) d t;$

in particular the second variation of The fixed-endpoint second variation of a time-dependent action at a curve is

$δ^{2} S (q) [η, η] = \int_{a}^{b} H (0, t) d t,$

the formula of clause 2 with every second partial of $L$ evaluated at $(t, q (t), q^{'} (t))$ .

Facts & Assumptions

Given: The chart data $(O, L)$ , a $C^{2}$ curve $q : [a, b] \to U$ , a variation $η$ , the map $ψ (ε, t) = (t, q (t) + ε η (t), q^{'} (t) + ε η^{'} (t))$ , and the action $S$ and second variation of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations and The fixed-endpoint second variation of a time-dependent action at a curve.

[L1]

Admissibility: there is $ε_{0} > 0$ such that for $∣ ε ∣ < ε_{0}$ the curve $q + ε η$ is a $C^{2}$ admissible curve with values in $U$ , hence $ψ$ maps $(- ε_{0}, ε_{0}) \times [a, b]$ into $O$ because $[a, b] \times U \times R^{n} \subseteq O$ (The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient, clause 1).

[L2]

Differentiation under the integral sign on a compact rectangle (Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral): if $g, h : [α, β] \times [a, b] \to R$ are continuous and for each fixed $t$ the map $ε \mapsto g (ε, t)$ is differentiable on $(α, β)$ with derivative $h (ε, t)$ , then $ε \mapsto \int_{a}^{b} g (ε, t) d t$ is differentiable on $(α, β)$ with derivative $\int_{a}^{b} h (ε, t) d t$ .

[L3]

Chain rule and smoothness of the gradients (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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 total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, $C^{k}$ Euclidean maps and diffeomorphisms, $C^{k}$ maps and multi-index derivative notation in Euclidean space, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , 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): because $L$ is $C^{2}$ on the open set $O$ , its gradient maps $L_{q} : O \to R^{n}$ and $L_{v} : O \to R^{n}$ are $C^{1}$ . For fixed $t$ the curve $ε \mapsto ψ (ε, t)$ is affine in $ε$ , with $ε$ -derivative the constant vector $(0, η (t), η^{'} (t))$ ; hence for $∣ ε ∣ < ε_{0}$ the chain rule gives the $ε$ -derivatives

$\frac{\partial}{\partial ε} [L (ψ)] = L_{q} (ψ) \cdot η (t) + L_{v} (ψ) \cdot η^{'} (t),$

$\frac{\partial}{\partial ε} [L_{q} (ψ)]_{j} = \sum_{k} \frac{\partial^{2} L}{\partial q_{j} \partial q_{k}} (ψ) η_{k} (t) + \sum_{k} \frac{\partial^{2} L}{\partial q_{j} \partial v_{k}} (ψ) η_{k}^{'} (t),$

$\frac{\partial}{\partial ε} [L_{v} (ψ)]_{j} = \sum_{k} \frac{\partial^{2} L}{\partial v_{j} \partial q_{k}} (ψ) η_{k} (t) + \sum_{k} \frac{\partial^{2} L}{\partial v_{j} \partial v_{k}} (ψ) η_{k}^{'} (t),$

and each displayed expression is continuous on every rectangle $[- δ, δ] \times [a, b]$ with $0 < δ < ε_{0}$ , the compositions of the continuous second partials with the continuous map $ψ$ and the continuous functions $η, η^{'}$ being continuous.

[L4]

Clairaut-Schwarz (Clairaut--Schwarz theorem for continuous second partial derivatives): the second partials of the $C^{2}$ function $L$ commute on $O$ , so $\partial^{2} L / \partial v_{j} \partial q_{k} = \partial^{2} L / \partial q_{k} \partial v_{j}$ there.

[L5]

Integrals (The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, Absolute value in an ordered field): a continuous real function on the compact interval $[a, b]$ is Darboux integrable, and the integral of a finite sum of integrable functions with real coefficients is the corresponding sum of integrals.

[L6]

Uniform continuity and uniform integral errors (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , If $f, g$ are integrable on $[a, b]$ then so are $∣ f ∣$ , $f^{2}$ , $f g$ , $\max (f, g)$ and $\min (f, g)$ , and $∣ \int_{a}^{b} f ∣ \leq \int_{a}^{b} ∣ f ∣$ ): the closed bounded rectangle $R = [- δ, δ] \times [a, b]$ is compact, a continuous real function on a compact metric space is uniformly continuous; and if $u, v$ are integrable on $[a, b]$ with $∣ u (t) - v (t) ∣ \leq η$ for every $t \in [a, b]$ and some $η \geq 0$ , then $u - v$ and $∣ u - v ∣$ are integrable and $∣ \int_{a}^{b} u - \int_{a}^{b} v ∣ = ∣ \int_{a}^{b} (u - v) ∣ \leq \int_{a}^{b} ∣ u - v ∣ \leq η (b - a),$ by linearity of the integral, the triangle inequality for the integral and the two-sided bound of monotonicity.

Proof

Proof technique: direct; apply differentiation under the integral sign twice on a compact rectangle, the first derivative being the first-variation integrand and the second being the Hessian quadratic form of $L$ along the family.

1.1givenL1L3

Fix $0 < δ < ε_{0}$ with $ε_{0}$ from [L1], put $R : = [- δ, δ] \times [a, b]$ , and define $g (ε, t) : = L (ψ (ε, t))$ on $R$ . By [L1] the values of $ψ$ on $R$ lie in $O$ , so $g$ is defined and continuous on $R$ ; by [L3] the $ε$ -partial $g_{1} (ε, t) = L_{q} (ψ) \cdot η (t) + L_{v} (ψ) \cdot η^{'} (t)$ exists on $R$ and is continuous there.

2.1L2L5step 1.1

By [L2] applied to the continuous pair $(g, g_{1})$ on $R$ , the map $F (ε) : = S (q + ε η) = \int_{a}^{b} g (ε, t) d t$ is differentiable on $(- δ, δ)$ with derivative $F^{'} (ε) = \int_{a}^{b} g_{1} (ε, t) d t$ .

2.2L2L3L5step 1.1

By [L3] the $ε$ -partial of $g_{1}$ exists on $R$ and equals the four-term sum $g_{2} (ε, t) = [L_{q q} (ψ) η \cdot η + L_{q v} (ψ) η^{'} \cdot η] + [L_{v q} (ψ) η \cdot η^{'} + L_{v v} (ψ) η^{'} \cdot η^{'}],$ written with the block matrices $L_{q q}, L_{q v}, L_{v q}, L_{v v}$ of second partials evaluated at $ψ = ψ (ε, t)$ ; it is continuous on $R$ . Applying [L2] to the pair $(g_{1}, g_{2})$ shows that $ε \mapsto \int_{a}^{b} g_{1} (ε, t) d t$ is differentiable on $(- δ, δ)$ with derivative $\int_{a}^{b} g_{2} (ε, t) d t$ . Hence $F$ is twice differentiable on $(- δ, δ)$ with $F^{''} (ε) = \int_{a}^{b} g_{2} (ε, t) d t$ , and since every $0 < δ < ε_{0}$ is allowed, clause 2 holds for every $ε \in (- ε_{0}, ε_{0})$ once $g_{2}$ is rewritten in the symmetric form $H$ .

3.1L6L5step 2.2

Continuity of the second derivative: the function $g_{2}$ is continuous on the compact rectangle $R$ , so [L6] makes it uniformly continuous. Given $\tilde{ε} > 0$ choose $γ > 0$ such that $∣ g_{2} (ε, t) - g_{2} (ε^{'}, t) ∣ < \tilde{ε} / (b - a)$ for every $t \in [a, b]$ whenever $ε, ε^{'} \in (- δ, δ)$ satisfy $∣ ε - ε^{'} ∣ < γ$ . Then for such $ε, ε^{'}$ , linearity of the integral and [L6] give $∣ F^{''} (ε) - F^{''} (ε^{'}) ∣ = ∣ \int_{a}^{b} (g_{2} (ε, t) - g_{2} (ε^{'}, t)) d t ∣ \leq \frac{\tilde{ε}}{b - a} (b - a) = \tilde{ε},$ so $F^{''}$ is continuous on $(- δ, δ)$ ; since every $0 < δ < ε_{0}$ is allowed, $ε \mapsto S (q + ε η)$ is of class $C^{2}$ on $(- ε_{0}, ε_{0})$ , which is clause 1.

4.1L4step 2.2given∎

Symmetrisation: by [L4], for all $j, k$ one has $(L_{v q} (ψ))_{j k} = \partial^{2} L / \partial v_{j} \partial q_{k} (ψ) = \partial^{2} L / \partial q_{k} \partial v_{j} (ψ) = (L_{q v} (ψ))_{k j}$ , so $η^{'} \cdot L_{v q} (ψ) η = η \cdot L_{q v} (ψ) η^{'}$ after relabelling the summation indices; therefore $g_{2} = H$ and clauses 2 and 3 follow, the latter by the definition of $δ^{2} S (q) [η, η]$ and evaluation at $ε = 0$ .

Remarks

The statement is deliberately integral: the second derivative of the family is identified with the integral of the Hessian form, and no pointwise second-order Taylor hypothesis on $L$ beyond $C^{2}$ is used.
Both differentiations are instances of the same compact-rectangle theorem, the first with the first-variation integrand of The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient and the second with its $ε$ -derivative; no second-order chain rule is needed, because the family is affine in $ε$ and the second derivative of $L \circ ψ$ is obtained by applying the ordinary chain rule to the $C^{1}$ maps $L_{q}$ and $L_{v}$ .

The second variation of a time-dependent action: twice differentiability of the fixed-endpoint family and the Hessian integral formula

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources