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 $B:Y\to X$ be a strongly monotone symmetric linear operator, that is, a linear operator satisfying

• $(Bu|v)=(u|Bv)\,$ for all $u,v$ in $Y$
• $(Bu|u)\geq c\|u\|^{2}$ 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{{safesubst:#invoke:anchor|main}} is

$\|u\|_{E}=(u|u)_{E}^{\frac {1}{2}}\,$ 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

$\langle B_{E}u|v\rangle _{E}=(u|v)_{E}=(Bu|v)=\langle u|B|v\rangle$

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 $X_{E}.$

An example from physics

A string with fixed endpoints under the influence of a force pointing down.

Consider a string whose endpoints are fixed at two points $a<b$ on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point $x$ $(a\leq x\leq b)$ 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 twice continuously differentiable functions $u:[a,b]\to \mathbb {R}$ with the boundary conditionss $u(a)=u(b)=0.$ 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' inequality

$\|u\|^{2}=\int _{a}^{b}u^{2}(x)\,dx\leq C\int _{a}^{b}u'(x)^{2}\,dx=C\,(Bu|u)$

for some $C>0.$

The energetic space in respect to the operator $B$ is then the Sobolev space $H_{0}^{1}(a,b).$ 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 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.

See also

• Inner product space
• Positive definite kernel

References

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