|
|
| Line 1: |
Line 1: |
| In [[mathematics]], more precisely in [[functional analysis]], an '''energetic space''' is, intuitively, a subspace of a given [[real number|real]] [[Hilbert space]] equipped with a new "energetic" [[Inner product space|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.
| | Greetings. The writer's title is Phebe and she feels comfy when people use the full name. The factor she adores most is body developing and now she is attempting to make cash with it. Hiring is my profession. Years in the past we moved to North Dakota.<br><br>Feel free to surf to my website: [http://www.articlecube.com/profile/Nydia-Lyons/532548 std testing at home] |
| | |
| ==Energetic space==
| |
| Formally, consider a real Hilbert space <math>X</math> with the [[Inner product space|inner product]] <math>(\cdot|\cdot)</math> and the [[norm (mathematics)|norm]] <math>\|\cdot\|</math>. Let <math>Y</math> be a linear subspace of <math>X</math> and <math>B:Y\to X</math> be a [[strongly monotone]] [[symmetric operator|symmetric]] [[linear operator]], that is, a linear operator satisfying
| |
| | |
| * <math>(Bu|v)=(u|Bv)\, </math> for all <math>u, v</math> in <math>Y</math>
| |
| * <math>(Bu|u) \ge c\|u\|^2</math> for some constant <math>c>0</math> and all <math>u</math> in <math>Y.</math>
| |
| | |
| The '''energetic inner product''' is defined as
| |
| :<math>(u|v)_E =(Bu|v)\,</math> for all <math>u,v</math> in <math>Y</math>
| |
| and the '''energetic norm'''{{anchor|energetic norm}} is | |
| :<math>\|u\|_E=(u|u)^\frac{1}{2}_E \, </math> for all <math>u</math> in <math>Y.</math>
| |
| | |
| The set <math>Y</math> together with the energetic inner product is a [[pre-Hilbert space]]. The '''energetic space''' <math>X_E</math> is defined as the [[complete metric space|completion]] of <math>Y</math> in the energetic norm. <math>X_E</math> can be considered a subset of the original Hilbert space <math>X,</math> since any [[Cauchy sequence]] in the energetic norm is also Cauchy in the norm of <math>X</math> (this follows from the strong monotonicity property of <math>B</math>). | |
| | |
| The energetic inner product is extended from <math>Y</math> to <math>X_E</math> by
| |
| : <math> (u|v)_E = \lim_{n\to\infty} (u_n|v_n)_E</math>
| |
| where <math>(u_n)</math> and <math>(v_n)</math> are sequences in ''Y'' that converge to points in <math>X_E</math> in the energetic norm.
| |
| | |
| ==Energetic extension==
| |
| The operator <math>B</math> admits an '''energetic extension''' <math>B_E</math>
| |
| | |
| :<math>B_E:X_E\to X^*_E</math>
| |
| | |
| defined on <math>X_E</math> with values in the [[dual space]] <math>X^*_E</math> that is given by the formula
| |
| | |
| :<math>\langle B_E u | v \rangle_E = (u|v)_E</math> for all <math>u,v</math> in <math>X_E.</math>
| |
| | |
| Here, <math>\langle \cdot |\cdot \rangle_E</math> denotes the duality bracket between <math>X^*_E</math> and <math>X_E,</math> so <math>\langle B_E u | v \rangle_E</math> actually denotes <math>(B_E u)(v).</math>
| |
| | |
| If <math>u</math> and <math>v</math> are elements in the original subspace <math>Y,</math> then
| |
| | |
| :<math>\langle B_E u | v \rangle_E = (u|v)_E = (Bu|v) = \langle u|B|v\rangle</math>
| |
| | |
| by the definition of the energetic inner product. If one views <math>Bu,</math> which is an element in <math>X,</math> as an element in the dual <math>X*</math> via the [[Riesz representation theorem]], then <math>Bu</math> will also be in the dual <math>X_E^*</math> (by the strong monotonicity property of <math>B</math>). Via these identifications, it follows from the above formula that <math>B_E u= Bu.</math> In different words, the original operator <math>B:Y\to X</math> can be viewed as an operator <math>B:Y\to X_E^*,</math> and then <math>B_E:X_E\to X^*_E</math> is simply the function extension of <math>B</math> from <math>Y</math> to <math>X_E.</math> <!---
| |
| | |
| 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, <math>B_E</math> is that [[linear functional]] which acts like ''B'' but has a domain of <math>X_E</math>—that is, its domain includes all limit points, ''u'', of the domain of ''B'' for which ''Bu<sub>n</sub>'' is bounded as <math>u_n\to u</math>.
| |
| | |
| --->
| |
| | |
| ==An example from physics==
| |
| [[File:String illust.svg|right|thumb|A string with fixed endpoints under the influence of a force pointing down.]]
| |
| Consider a [[rope|string]]<!-- a piece of wire, so the link to [[rope|string]] is not ambiguous--> whose endpoints are fixed at two points <math>a<b</math> on the real line (here viewed as a horizontal line). Let the vertical outer [[force density]] at each point <math>x</math> <math>(a\le x \le b)</math> on the string be <math>f(x)\mathbf{e}</math>, where <math>\mathbf{e}</math> is a [[unit vector]] pointing vertically and <math>f:[a, b]\to \mathbb R.</math> Let <math>u(x)</math> be the [[Deflection (engineering)|deflection]] of the string at the point <math>x</math> under the influence of the force. Assuming that the deflection is small, the [[elastic energy]] of the string is
| |
| | |
| : <math>\frac{1}{2} \int_a^b\! u'(x)^2\, dx</math>
| |
| | |
| and the total [[potential energy]] of the string is
| |
| | |
| : <math>F(u) = \frac{1}{2} \int_a^b\! u'(x)^2\,dx - \int_a^b\! u(x)f(x)\,dx.</math>
| |
| | |
| The deflection <math>u(x)</math> minimizing the potential energy will satisfy the [[differential equation]]
| |
| | |
| : <math>-u''=f\,</math>
| |
| | |
| with [[boundary conditions]]
| |
| | |
| :<math>u(a)=u(b)=0.\,</math>
| |
| | |
| To study this equation, consider the space <math>X=L^2(a, b), </math> that is, the [[Lp space]] of all [[square integrable function]]s <math>u:[a, b]\to \mathbb R</math> in respect to the [[Lebesgue measure]]. This space is Hilbert in respect to the inner product
| |
| | |
| : <math>(u|v)=\int_a^b\! u(x)v(x)\,dx,</math>
| |
| | |
| with the norm being given by
| |
| | |
| : <math>\|u\|=\sqrt{(u|u)}.</math>
| |
| | |
| Let <math>Y</math> be the set of all [[smooth function|twice continuously differentiable functions]] <math>u:[a, b]\to \mathbb R</math> with the [[boundary conditions]]s <math>u(a)=u(b)=0.</math> Then <math>Y</math> is a linear subspace of <math>X.</math>
| |
| | |
| Consider the operator <math>B:Y\to X</math> given by the formula
| |
| | |
| : <math>Bu = -u'',\,</math> | |
| | |
| so the deflection satisfies the equation <math>Bu=f.</math> Using [[integration by parts]] and the boundary conditions, one can see that
| |
| | |
| : <math>(Bu|v)=-\int_a^b\! u''(x)v(x)\, dx=\int_a^b u'(x)v'(x) = (u|Bv) </math> | |
| | |
| for any <math>u</math> and <math>v</math> in <math>Y.</math> Therefore, <math>B</math> is a symmetric linear operator.
| |
| | |
| <math>B</math> is also strongly monotone, since, by the [[Friedrichs' inequality]]
| |
| | |
| : <math>\|u\|^2 = \int_a^b u^2(x)\, dx \le C \int_a^b u'(x)^2\, dx = C\,(Bu|u)</math>
| |
| | |
| for some <math>C>0.</math>
| |
| | |
| The energetic space in respect to the operator <math>B</math> is then the [[Sobolev space]] <math>H^1_0(a, b).</math> We see that the elastic energy of the string which motivated this study is
| |
| | |
| : <math>\frac{1}{2} \int_a^b\! u'(x)^2\, dx = \frac{1}{2} (u|u)_E,</math>
| |
| | |
| so it is half of the energetic inner product of <math>u</math> with itself.
| |
| | |
| To calculate the deflection <math>u</math> minimizing the total potential energy <math>F(u)</math> of the string, one writes this problem in the form
| |
| | |
| :<math>(u|v)_E=(f|v)\,</math> for all <math>v</math> in <math>X_E</math>.
| |
| | |
| Next, one usually approximates <math>u</math> by some <math>u_h</math>, a function in a finite-dimensional subspace of the true solution space. For example, one might let <math>u_h</math> be a continuous [[piecewise-linear function]] in the energetic space, which gives the [[finite element method]]. The approximation <math>u_h</math> 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 <math>u</math> and <math>u_h</math>, see [[Céa's lemma]].
| |
| | |
| ==See 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 = 0-387-94442-7
| |
| }}
| |
| | |
| *{{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
| |
| }}
| |
| | |
| [[Category:Functional analysis]]
| |
| [[Category:Hilbert space]]
| |
Greetings. The writer's title is Phebe and she feels comfy when people use the full name. The factor she adores most is body developing and now she is attempting to make cash with it. Hiring is my profession. Years in the past we moved to North Dakota.
Feel free to surf to my website: std testing at home