Energetic space

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Mathematical concept of energy in physics

title: "Energetic space" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["functional-analysis", "hilbert-spaces"] description: "Mathematical concept of energy in physics" topic_path: "general/functional-analysis" source: "https://en.wikipedia.org/wiki/Energetic_space" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical concept of energy in physics ::

In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energetic inner product.

Energetic space

Formally, consider a real Hilbert space X with the inner product (\cdot|\cdot) and the norm |\cdot|. Let Y be a linear subspace of X and B:Y\to X be a strongly monotone symmetric linear operator, that is, a linear operator satisfying

(Bu|v)=(u|Bv), for all u, v in Y
(Bu|u) \ge c|u|^2 for some constant c0 and all u in Y.

The energetic inner product is defined as :(u|v)_E =(Bu|v), for all u,v in Y and the energetic normenergetic norm is :|u|_E=(u|u)^\frac{1}{2}_E , for all u in Y.

The set Y together with the energetic inner product is a pre-Hilbert space. The energetic space X_E is defined as the completion of Y in the energetic norm. X_E can be considered a subset of the original Hilbert space X, since any Cauchy sequence in the energetic norm is also Cauchy in the norm of X (this follows from the strong monotonicity property of B).

The energetic inner product is extended from Y to X_E by : (u|v)E = \lim{n\to\infty} (u_n|v_n)_E where (u_n) and (v_n) are sequences in Y that converge to points in X_E in the energetic norm.

Energetic extension

The operator B admits an energetic extension B_E

:B_E:X_E\to X^*_E

defined on X_E with values in the dual space X^*_E that is given by the formula

:\langle B_E u | v \rangle_E = (u|v)_E for all u,v in X_E.

Here, \langle \cdot |\cdot \rangle_E denotes the duality bracket between X^*_E and X_E, so \langle B_E u | v \rangle_E actually denotes (B_E u)(v).

If u and v are elements in the original subspace Y, then

by the definition of the energetic inner product. If one views Bu, which is an element in X, as an element in the dual X^* via the Riesz representation theorem, then Bu will also be in the dual X_E^* (by the strong monotonicity property of B). Via these identifications, it follows from the above formula that B_E u= Bu. In different words, the original operator B:Y\to X can be viewed as an operator B:Y\to X_E^, and then B_E:X_E\to X^_E is simply the function extension of B from Y to X_E.

I commented out the below text, since it is not clear what norm one uses to talk about convergence and boundedness. I will think more about it.

That is, B_E is that linear functional which acts like B but has a domain of X_E—that is, its domain includes all limit points, u, of the domain of B for which Bun is bounded as u_n\to u.

An example from physics

::figure[src="https://upload.wikimedia.org/wikipedia/commons/d/da/String_illust.svg" caption="A string with fixed endpoints under the influence of a force pointing down."] ::

Consider a string whose endpoints are fixed at two points a on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point x (a\le x \le b) on the string be f(x)\mathbf{e}, where \mathbf{e} is a unit vector pointing vertically and f:[a, b]\to \mathbb R. Let u(x) be the deflection of the string at the point x under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is

: \frac{1}{2} \int_a^b! u'(x)^2, dx

and the total potential energy of the string is

: F(u) = \frac{1}{2} \int_a^b! u'(x)^2,dx - \int_a^b! u(x)f(x),dx.

The deflection u(x) minimizing the potential energy will satisfy the differential equation

: -u''=f,

with boundary conditions

:u(a)=u(b)=0.,

To study this equation, consider the space X=L^2(a, b), that is, the Lp space of all square-integrable functions u:[a, b]\to \mathbb R in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product

: (u|v)=\int_a^b! u(x)v(x),dx,

with the norm being given by

: |u|=\sqrt{(u|u)}.

Let Y be the set of all Locally integrable function (\text{L}^1_\text{loc}) on [a, b] that are twice continuously differentiable from u:[a, b]\to \mathbb R, with the boundary conditions u(a)=u(b)=0. Then Y is a linear subspace of X.

Consider the operator B:Y\to X given by the formula

: Bu = -u'',,

so the deflection satisfies the equation Bu=f. Using integration by parts and the boundary conditions, one can see that

: (Bu|v)=-\int_a^b! u''(x)v(x), dx=\int_a^b u'(x)v'(x) = (u|Bv)

for any u and v in Y. Therefore, B is a symmetric linear operator.

B is also strongly monotone, since, by the Friedrichs's inequality

: |u|^2 = \int_a^b u^2(x), dx \le C \int_a^b u'(x)^2, dx = C,(Bu|u)

for some C0.

The energetic space in respect to the operator B is then the Sobolev space H^1_0(a, b). We see that the elastic energy of the string which motivated this study is

: \frac{1}{2} \int_a^b! u'(x)^2, dx = \frac{1}{2} (u|u)_E,

so it is half of the energetic inner product of u with itself.

To calculate the deflection u minimizing the total potential energy F(u) of the string, one writes this problem in the form

:(u|v)_E=(f|v), for all v in X_E.

Next, one usually approximates u by some u_h, a function in a finite-dimensional subspace of the true solution space. For example, one might let u_h be a continuous piecewise linear function in the energetic space, which gives the finite element method. The approximation u_h can be computed by solving a system of linear equations.

The energetic norm turns out to be the natural norm in which to measure the error between u and u_h, see Céa's lemma.

References

{{cite book | last = Zeidler | first = Eberhard | title = Applied functional analysis: applications to mathematical physics | publisher = New York: Springer-Verlag | date = 1995 | pages = | isbn = 0-387-94442-7 | url = https://books.google.com/books?id=bFqRaovfY4MC&q=%22energetic+space%22
{{cite book | last = Johnson | first = Claes | title = Numerical solution of partial differential equations by the finite element method | publisher = Cambridge University Press | date = 1987 | pages = | isbn = 0-521-34514-6

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