en>Marc André Miron: /* Born approximation to the Lippmann–Schwinger equation */ 0 -> ε

2013-05-15T01:48:11Z

Born approximation to the Lippmann–Schwinger equation: 0 -> ε

← Older revision		Revision as of 03:48, 15 May 2013
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	~~Greetings. The author~~'s ~~title~~ is ~~Phebe and she feels comfortable when individuals use~~ the ~~full~~ title. ~~North Dakota~~ is ~~our birth place~~. ~~What I love doing~~ is ~~doing ceramics but I haven~~'~~t produced~~ a ~~dime~~ with it. ~~Since she was eighteen she~~'~~s been working as~~ a ~~receptionist but her promotion never arrives~~.<br><br>~~Look~~ at ~~my web site~~: [~~http~~://~~www~~.~~eddysadventurestore~~.nl/~~nieuws~~/~~eliminate~~-~~candida~~-~~one~~-~~these~~-~~tips~~ http://~~www~~.~~eddysadventurestore~~.nl/]		In [[mathematics]], specifically in the [[calculus of variations]], the '''fundamental lemma of the calculus of variations''' states that if the [[definite integral]] of the product of a [[continuous function]] f(x) and h(x) is zero, for all continuous functions h(x) that vanish at the endpoints of the range of integration and have their first two [[derivative]]s continuous, then f(x)=0. This [[Lemma (mathematics)\|lemma]] is used in deriving the [[Euler–Lagrange equation]] of the [[calculus of variations]].<ref name='Courant&Hilbert'>{{cite book \| last1 = Courant \| first1 = R. \| authorlink1 = Richard Courant \| last2 = Hilbert \| first2 = D. \| authorlink2 = David Hilbert \| title = Methods of Mathematical Physics \| volume = Vol. 1 \| publisher = Interscience Publishers, Inc \| year = 1953 \| location = New York, New York \| page = 185 \| isbn = 978-0470179529}}</ref> It is a lemma that is typically used to transform a problem from its [[weak formulation]] (variational form) into its strong formulation ([[differential equation]]).

			==Statement==
			A function is said to be of class [[Smooth function\|<math>C^k</math>]] if it is ''k''-times continuously differentiable. For example, class <math>C^0</math> consists of continuous functions, and class <math>C^\infty</math> consists of infinitely [[smooth function]]s.

			Let ''f'' be of class <math>C^k</math> on the interval [''a'',''b'']. Assume furthermore that
			:<math> \int_a^b f(x) \, h(x) \, dx = 0 </math>
			for ''every'' function ''h'' that is of class <math>C^k</math> on [''a'',''b''] with ''h''(''a'') = ''h''(''b'') = 0. Then the '''fundamental lemma of the calculus of variations''' states that <math>f (x)</math> is identically zero on <math>[a, b]</math>.

			In other words, the test functions ''h'' (<math>C^k</math> functions vanishing at the endpoints) ''separate <math>C^k</math> functions'': <math>C^k[a,b]</math> is a [[Hausdorff space]] in the [[weak topology]] of pairing against <math>C^k</math> functions that vanish at the endpoints.

			==Proof==
			Let ''f'' satisfy the hypotheses. Let ''r'' be any smooth function that is 0 at ''a'' and ''b'' and positive on the open interval (''a'', ''b''); for example, ''r'' = - (''x'' - ''a'') (''x'' - ''b''). Let ''h'' = ''r'' ''f''. Then ''h'' is of class ''C''<sup>''k''</sup> on [''a'',''b'']. By the hypotheses,
			:<math>0 = \int_a^b f(x) h(x) \; dx = \int_a^b r(x) f(x)^2 \; dx.</math>
			The integrand now must be 0 except perhaps on a subset of [''a'', ''b''] of measure 0. However, by continuity if there are points where the integrand is non-zero, there is also some interval around that point where the integrand is non-zero, which has non-zero measure. Thus the integrand must be identically 0 over the entire interval. Since ''r'' is positive on (''a'', ''b''), ''f'' is 0 there and hence on all of [''a'', ''b''].

			== The du Bois-Reymond lemma ==
			The ''du Bois-Reymond lemma'' (named after [[Paul du Bois-Reymond]]) is a more general version of the above lemma. It defines a sufficient condition to guarantee that a function vanishes [[almost everywhere]]. Suppose that <math>f</math> is a [[locally integrable function]] defined on an [[open set]] <math>\Omega \subset \mathbb{R}^n</math>. If

			:<math>\int_\Omega f(x) h(x) dx = 0\,</math>

			for all <math>h \in C^\infty_0(\Omega),</math> then ''f''(''x'') = 0 for almost all ''x'' in Ω. Here, <math>C^\infty_0(\Omega)</math> is the space of all infinitely differentiable functions defined on Ω whose [[support (mathematics)\|support]] is a [[compact set]] contained in Ω.

			==Applications==
			This lemma is used to prove that [[maxima and minima\|extrema]] of the [[functional (mathematics)\|functional]]
			:<math> J[y] = \int_{x_0}^{x_1} L(t,y(t),\dot y(t)) \, dt </math>
			are [[weak solution]]s <math>y:[x_0,x_1]\to V</math> (for an appropriate vector space <math>V</math>) of the [[Euler-Lagrange equation]]
			:<math> {\partial L(t,y(t),\dot y(t)) \over \partial y} = {d \over dt} {\partial L(t,y(t),\dot y(t)) \over \partial \dot y} .</math>
			The Euler-Lagrange equation plays a prominent role in [[classical mechanics]] and [[differential geometry]].

			==Footnotes==
			{{reflist}}

			==References==
			* L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer; 2nd edition (September 1990) ISBN 0-387-52343-X.
			*{{cite book\|last=Lang\|first=Serge\|title=Analysis II\|publisher=Addison-Wesley\|year=1969}}
			*{{cite book\|last = Leitmann\|first = George\|title = The Calculus of Variations and Optimal Control: An Introduction\|publisher = Springer\|year = 1981\|isbn = 0-306-40707-8\|url = http://books.google.com/books?id=ActDgnJsvvoC\|accessdate = 2007-04-17}}

			{{PlanetMath attribution\|id=6745\|title=Fundamental lemma of calculus of variations}}

			{{Fundamental theorems}}

			[[Category:Classical mechanics]]
			[[Category:Calculus of variations]]
			[[Category:Smooth functions]]
			[[Category:Lemmas]]
			[[Category:Fundamental theorems\|Calculus of variations]]

en>Nsda: /* References */ rm very special reference (self promotion?)

2011-11-26T20:10:02Z

References: rm very special reference (self promotion?)

New page

Greetings. The author's title is Phebe and she feels comfortable when individuals use the full title. North Dakota is our birth place. What I love doing is doing ceramics but I haven't produced a dime with it. Since she was eighteen she's been working as a receptionist but her promotion never arrives.<br><br>Look at my web site: [http://www.eddysadventurestore.nl/nieuws/eliminate-candida-one-these-tips http://www.eddysadventurestore.nl/]

Born approximation - Revision history

en>Marc André Miron: /* Born approximation to the Lippmann–Schwinger equation */ 0 -> ε

en>Nsda: /* References */ rm very special reference (self promotion?)