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The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient

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, let $q : [a, b] \to U$ be a $C^{2}$ admissible curve and let $η$ be a variation of $q$ . Then:

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

$δ S (q) [η] = \int_{a}^{b} [L_{q} (t, q (t), q^{'} (t)) \cdot η (t) + L_{v} (t, q (t), q^{'} (t)) \cdot η^{'} (t)] d t;$

the Euler–Lagrange expression $E_{L} (q) : [a, b] \to R^{n}$ of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations is continuous.

Facts & Assumptions

Given: The chart data $(O, L)$ with $O$ open satisfying $[a, b] \times U \times R^{n} \subseteq O$ , a $C^{2}$ curve $q : [a, b] \to U$ , and a $C^{2}$ variation $η$ with $η (a) = η (b) = 0$ . Write $ψ (ε, t) : = (t, q (t) + ε η (t), q^{'} (t) + ε η^{'} (t))$ .

[F1]

The set $[a, b] \subseteq R$ is compact by 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 (it is closed and bounded), the curve $q$ is continuous on $[a, b]$ , and a continuous image of a compact set is compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); hence $q [a, b] \subseteq U$ is a nonempty compact subset. If $U \neq R^{n}$ , then the complement $R^{n} ∖ U$ is a nonempty closed set disjoint from $q [a, b]$ , and A compact set and a disjoint closed set have a positive norm-distance gap supplies $d > 0$ with $∥ q (t) - c ∥_{2} \geq d$ for all $t \in [a, b]$ and all $c \notin U$ .

[F2]

The continuous function $η : [a, b] \to R^{n}$ attains a maximum of $∥ η (t) ∥_{2}$ (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value); write $m_{η} : = \max_{t \in [a, b]} ∥ η (t) ∥_{2} \geq 0$ .

[F3]

The map $ψ$ is continuous on $R \times [a, b]$ and is continuously partially differentiable in $ε$ , with $\frac{\partial ψ}{\partial ε} (ε, t) = (0, η (t), η^{'} (t)),$ because $η$ is $C^{1}$ ; hence $ψ$ is totally differentiable in $(ε, t)$ with the displayed derivatives and the derivative depends continuously on $(ε, t)$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[F4]

The chain rule computes the derivative of a composite through the total derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). In particular, if $g (ε, t) : = L (ψ (ε, t))$ on the domain where $ψ$ takes values in $O$ , then $\frac{\partial g}{\partial ε} (ε, t) = L_{q} (ψ (ε, t)) \cdot η (t) + L_{v} (ψ (ε, t)) \cdot η^{'} (t),$ which is continuous in $(ε, t)$ because $L_{q}, L_{v}, η, η^{'}$ are continuous and $ψ$ is continuous.

[F5]

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 $ε \mapsto \int_{a}^{b} g (ε, t) d t$ has derivative $\int_{a}^{b} h (ε, t) d t$ at every $ε \in [α, β]$ .

[F6]

Integrals used below are Darboux integrals of the continuous integrands over $[a, b]$ (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); the integral is linear and preserves sums (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ); norms and distances are those of $R^{n}$ (Absolute value in an ordered field, Open ball, closed ball and sphere in a metric space, 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, Metric continuity characterisations, with countable choice for the sequential converse, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, 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).

Proof

Proof technique: direct; trap the curve in a positive distance shell of $U$ , then differentiate under the integral on a compact rectangle.

1.1givenF3

Clause 1, the case $U = R^{n}$ : take $ε_{0} : = 1$ ; then for every $ε \in (- 1, 1)$ the map $q + ε η$ is $C^{2}$ by the algebra of $C^{2}$ maps on an open interval and takes values in $R^{n} = U$ .

1.2F1F2algebra

Clause 1, the case $U \neq R^{n}$ : by [F1] choose $d > 0$ with $∥ q (t) - c ∥_{2} \geq d$ for all $t \in [a, b]$ and all $c \notin U$ , and by [F2] put $m_{η} : = \max ∥ η ∥_{2}$ . If $m_{η} = 0$ then $η = 0$ identically and $ε_{0} : = 1$ works. Otherwise take $ε_{0} : = d / (2 m_{η})$ . For $∣ ε ∣ < ε_{0}$ and $t \in [a, b]$ , $∥ q (t) + ε η (t) - q (t) ∥_{2} = ∣ ε ∣ ∥ η (t) ∥_{2} < d / 2 < d,$ so if $q (t) + ε η (t)$ were outside $U$ , the distance from $q (t)$ to the complement would be $< d$ , contradicting the choice of $d$ . Hence $q + ε η$ takes values in $U$ ; it is $C^{2}$ by the algebra of $C^{2}$ maps. This proves clause 1.

2.1F3F4F5F6step 1.2algebra

Clause 2: fix $0 < δ < ε_{0}$ and consider $R : = [- δ, δ] \times [a, b]$ ; by clause 1, $ψ (R) \subseteq {(t, x, w) : (t, x, w) \in O}$ , so $g (ε, t) : = L (ψ (ε, t))$ is defined and continuous on $R$ . By [F3], [F4] the partial derivative $h : = \partial g / \partial ε$ exists and is continuous on $R$ , and for each fixed $t$ it is the derivative of $ε \mapsto g (ε, t)$ . Hence [F5] applies and gives $\frac{d}{d ε} \int_{a}^{b} g (ε, t) d t = \int_{a}^{b} h (ε, t) d t (∣ ε ∣ \leq δ) .$ Evaluating at $ε = 0$ and inserting the description of $h$ from [F4] gives the displayed formula for $δ S (q) [η]$ .

3.1F3F4givenalgebra∎

Clause 3: the map $t \mapsto ψ (0, t) = (t, q (t), q^{'} (t))$ is $C^{1}$ on an open interval around $[a, b]$ and takes values in $O$ ; since $L_{v}$ is $C^{1}$ , the composite $t \mapsto L_{v} (t, q (t), q^{'} (t))$ is $C^{1}$ by the chain rule, with continuous derivative because the derivative is a composite of continuous maps; and $t \mapsto L_{q} (t, q (t), q^{'} (t))$ is continuous. The difference $E_{L} (q)$ of these two continuous maps is continuous.

Remarks

The formula is stated for the chartwise finite-interval setting. The autonomous smooth version on a manifold, where the same computation is performed in every coordinate chart and assembled by a partition argument, is the published theorem on the Hamiltonian-mechanics page of the differential-geometry track; the present statement is the one needed when the Lagrangian depends explicitly on time and $C^{2}$ data on a fixed interval are used.
Only the differentiability of $ε \mapsto S (q + ε η)$ at $ε = 0$ is used later. The proof establishes differentiability on the whole small interval, which is the natural output of the compact-rectangle theorem.

The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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