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Energy identity and displacement for Euclidean gradient trajectories

Statement

Let $m \geq 1$ be an integer, let $a, b \in R$ with $a < b$ , let $U \subseteq R^{m}$ be open and $f \in C^{1} (U)$ , and let $x : [a, b] \to U$ be a $C^{1}$ solution of $x^{'} = - \nabla f (x)$ on $(a, b)$ . Here $C^{1}$ on $[a, b]$ means that $x$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and its derivative has a continuous extension to $[a, b]$ , also denoted $x^{'}$ . The gradient is the Euclidean vector of partial derivatives. Then $f (x (b)) - f (x (a)) = - \int_{a}^{b} ∥ \nabla f (x (t)) ∥^{2} d t,$ and $∥ x (b) - x (a) ∥ \leq \int_{a}^{b} ∥ x^{'} (t) ∥ d t$ . We call this last integral the length on $[a, b]$ .

Statement references

Continuously differentiable maps, local inverses, and local diffeomorphisms

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral

Facts & Assumptions

[F1]

The $C^{1}$ hypothesis on the open-domain map $f$ uses Continuously differentiable maps, local inverses, and local diffeomorphisms. The vector derivative and the componentwise vector integral use The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral; the closed-interval $C^{1}$ convention is specified in the Statement.

[F2]

The chain rule The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a) and gradient representation For a differentiable scalar field, Dᵥf(a)=⟨∇ f(a),v⟩ and the unit direction of steepest ascent is the normalized gradient apply at each interior time.

[F3]

A continuous real function on a real interval with $a < b$ is Riemann integrable by A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion. Newton–Leibniz Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative requires continuity on $[a, b]$ , differentiability on $(a, b)$ , and an integrable extension of the interior derivative; it requires no endpoint derivative.

[F4]

For an integrable vector function on $[a, b]$ with $a < b$ , For a ≤ b and f : [a,b] → ℝᵐ integrable when a<b, ‖∫ₐᵇ f‖₂ ≤ ∫ₐᵇ ‖ f‖₂; for a<b, ‖ f‖₂ is integrable gives integrability of its Euclidean norm and bounds the norm of its integral by the integral of its norm.

[F5]

Total differentiability at a point implies continuity there by Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity.

[F6]

Clause 1 of Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous makes composites of continuous maps continuous, including maps between Euclidean subspaces with their inherited topologies.

[F7]

For maps from a metric subspace to $R^{m}$ with $m \geq 1$ , clause 1 of A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions identifies continuity with continuity of every component; clause 3 preserves scalar multiples and inner products.

Proof

Given: The data and hypotheses in the Statement.

Goal: Prove the energy identity and the displacement bound.

1.1F1F2given

At every $t \in (a, b)$ the chain rule and gradient representation give $(f \circ x)^{'} (t) = ⟨ \nabla f (x (t)), x^{'} (t) ⟩ = - ∥ \nabla f (x (t)) ∥^{2}$ .

1.2F1F5F6F7given

The $C^{1}$ hypothesis makes $f$ totally differentiable at every point of $U$ , hence continuous there. It also makes the entries of its derivative matrix continuous; these are the components of $\nabla f$ , so the gradient is continuous by componentwise continuity. Composing with the continuous curve $x$ shows that $f \circ x$ and $\nabla f \circ x$ are continuous on $[a, b]$ . Scalar multiplication also makes $- f \circ x$ continuous. Both $x$ and the stipulated continuous extension $x^{'}$ have continuous components on $[a, b]$ .

2.1F3F7step 1.1step 1.2algebra

The real function $t \mapsto ∥ \nabla f (x (t)) ∥^{2} = ⟨ \nabla f (x (t)), \nabla f (x (t)) ⟩$ is continuous by the inner-product clause of [F7], hence integrable. Newton–Leibniz applied to $- f \circ x$ and this integrable extension of its interior derivative gives $f (x (a)) - f (x (b)) = \int_{a}^{b} ∥ \nabla f (x (t)) ∥^{2} d t$ , which is the asserted energy identity. Applying the same argument on any nondegenerate closed subinterval gives a nonpositive energy increment, since the integrand is nonnegative; thus energy is nonincreasing.

3.1F1F3F4step 1.2∎

Each component of the continuous extension $x^{'}$ is integrable on $[a, b]$ . By the vector-derivative definition, its interior values are the derivatives of the corresponding components of $x$ . Each component of $x$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , so Newton–Leibniz with that component of $x^{'}$ as the integrable extension applies. Reassembling the resulting scalar identities by the componentwise vector-integral definition gives $x (b) - x (a) = \int_{a}^{b} x^{'} (t) d t$ . The vector-integral norm inequality now yields $∥ x (b) - x (a) ∥ \leq \int_{a}^{b} ∥ x^{'} (t) ∥ d t$ , including integrability of the speed.

Depends on

Used by · 3

Dependency tree · 2 levels

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