Energetic space

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. An example of this will be given later in the article.

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 $B:Y\; o\; X$ 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 $c>0$ 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 norm is:$|u|\_E=(Bu|u)^frac\{1\}\{2\}\_E\; ,$ 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: $(u|v)\_E\; =\; lim\_\{n\; oinfty\}\; (u\_n|v\_n)\_E$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 $B\_E$

:$B\_E:X\_E\; o\; X^*\_E$

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

:$langle\; B\_E\; u\; |\; v\; angle\_E\; =\; (u|v)\_E$ for all $u,v$ in $X\_E.$

Here, $langle\; cdot\; |cdot\; angle\_E$ denotes the duality bracket between $X^*\_E$ and $X\_E,$ so $langle\; B\_E\; u\; |\; v\; angle\_E$ actually denotes $(B\_E\; u)(v).$

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

:$langle\; B\_E\; u\; |\; v\; angle\_E\; =\; (u|v)\_E\; =\; (Bu|v)$

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 $B:Y\; o\; X$ can be viewed as an operator $B:Y\; o\; X\_E^*,$ and then $B\_E:X\_E\; o\; X^*\_E$ is simply the function extension of $B$ from $Y$ to $X\_E.$

An example from physics

Consider a string whose endpoints are fixed at two points

: $frac\{1\}\{2\}\; int\_a^b!\; u\text{'}(x)^2,\; dx$

and the total potential energy of the string is

: $F(u)\; =\; frac\{1\}\{2\}\; int\_a^b!\; u\text{'}(x)^2,dx\; -\; int\_a^b!\; u(x)f(x),dx.$

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

: $-u"=f,$

with boundary conditionss

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

To study this equation, consider the space $X=L^2(a,\; b),$ that is, the Lp space of all square integrable functions $u:\; [a,\; b]\; o\; mathbb\; R$ 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 twice continuously differentiable functions $u:\; [a,\; b]\; o\; mathbb\; R$ with the boundary conditionss $u(a)=u(b)=0.$ Then $Y$ is a linear subspace of $X.$

Consider the operator $B:Y\; o\; X$ 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

: $(Bu|v)=-int\_a^b!\; u"(x)v(x),\; dx=int\_a^b\; u\text{'}(x)v\text{'}(x)\; =\; (u|Bv)$

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

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

: $|u|^2\; =\; int\_a^b\; u^2(x),\; dx\; le\; C\; int\_a^b\; u\text{'}(x)^2,\; dx\; =\; C,(Bu|u)$

for some $C>0.$

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

: $frac\{1\}\{2\}\; int\_a^b!\; u\text{'}(x)^2,\; dx\; =\; frac\{1\}\{2\}\; (u|u)\_E,$

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 linear system of 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.

ee also

* Inner product space
* Positive definite kernel

References

*cite book
last = Zeidler
first = Eberhard
title = Applied functional analysis: applications to mathematical physics
publisher = New York: Springer-Verlag
date = 1995
pages =
isbn = 0387944427

*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 = 0521345146