변분법 - 편집 역사

Pythagoras0: /* 개요 */

2021-02-24T03:52:06Z

개요

Pythagoras0: /* 개요 */

2021-02-23T11:32:40Z

개요

Pythagoras0: /* 메타데이터 */

2021-02-23T10:51:22Z

메타데이터

Pythagoras0: /* 메타데이터 */ 새 문단

2021-02-23T10:50:29Z

메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2021-02-23T10:50:25Z

노트: 새 문단

2021년 2월 17일 (수) 12:45에 Pythagoras0님의 편집

2021-02-17T12:45:26Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T14:32:03Z

메타데이터: 새 문단

2020년 11월 13일 (금) 11:04에 Pythagoras0님의 편집

2020-11-13T11:04:55Z

Pythagoras0: /* 메모 */

2015-05-07T07:25:42Z

메모

Pythagoras0: /* 관련도서 */

2015-05-07T07:24:44Z

@@ 3번째 줄: / 3번째 줄: @@
 * 미적분학의 중요한 문제는 함수를 가장 크거나 작게 만드는 점을 찾는 것이다
 * 변분법은 함수의 공간을 정의역으로 갖는 함수에 대한 미적분학이라 할 수 있다
 ==역사==

@@ 1번째 줄: / 1번째 줄: @@
 ==개요==
+* 변분법은 특정한 적분의 값을 가장 크거나 작게 하는 함수를 찾는 문제와 관련된 수학의 분야이다
+* 미적분학의 중요한 문제는 함수를 가장 크거나 작게 만드는 점을 찾는 것이다
+* 변분법은 함수의 공간을 정의역으로 갖는 함수에 대한 미적분학이라 할 수 있다
 ==역사==

@@ 43번째 줄: / 43번째 줄: @@
 * Perfect Form:Variational Principles, Methods, and Applications in Elementary Physics, http://books.google.com/books/about/Perfect_Form.html?id=8uWPG0QK0UIC
 [[분류:수리물리학]]
 == 노트 ==

← 이전 판		2021년 2월 23일 (화) 10:50 판
93번째 줄:		93번째 줄:
	===소스===		===소스===
	<references />		<references />
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q216861 Q216861]
		+	===Spacy 패턴 목록===
		+	* [{'LOWER': 'calculus'}, {'LOWER': 'of'}, {'LEMMA': 'variation'}]
		+	* [{'LOWER': 'variational'}, {'LEMMA': 'calculus'}]

← 이전 판		2021년 2월 23일 (화) 10:50 판
49번째 줄:		49번째 줄:
	===Spacy 패턴 목록===		===Spacy 패턴 목록===
	* [{'LOWER': 'hamilton'}, {'LOWER': "'s"}, {'LEMMA': 'principle'}]		* [{'LOWER': 'hamilton'}, {'LOWER': "'s"}, {'LEMMA': 'principle'}]
		+
		+	== 노트 ==
		+
		+	===말뭉치===
		+	# This post is going to describe a specialized type of calculus called variational calculus.<ref name="ref_b40a388c">[http://bjlkeng.github.io/posts/the-calculus-of-variations/ The Calculus of Variations]</ref>
		+	# I'll try to follow Svetitsky's notes to give some intuition on how we arrive at variational calculus from regular calculus with a bunch of examples along the way.<ref name="ref_b40a388c" />
		+	# Calculus of variations, branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible.<ref name="ref_47c86253">[https://www.britannica.com/science/calculus-of-variations-mathematics Calculus of variations \| mathematics]</ref>
		+	# Modern interest in the calculus of variations began in 1696 when Johann Bernoulli of Switzerland proposed a brachistochrone (“least-time”) problem as a challenge to his peers.<ref name="ref_47c86253" />
		+	# This technique, typical of the calculus of variations, led to a differential equation whose solution is a curve called the cycloid.<ref name="ref_47c86253" />
		+	# Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.<ref name="ref_e51e4c74">[https://en.wikipedia.org/wiki/Calculus_of_variations Calculus of variations]</ref>
		+	# Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum).<ref name="ref_2e7d147b">[https://mathworld.wolfram.com/CalculusofVariations.html Calculus of Variations -- from Wolfram MathWorld]</ref>
		+	# A generalization of calculus of variations known as Morse theory (and sometimes called "calculus of variations in the large") uses nonlinear techniques to address variational problems.<ref name="ref_2e7d147b" />
		+	# The calculus of variations addresses the need to optimize certain quantities over sets of functions.<ref name="ref_4cacc6a8">[https://www2.math.uconn.edu/~gordina/NelsonAaronHonorsThesis2012.pdf The calculus of variations]</ref>
		+	# We then introduce the calculus of variations as it applies to classical mechanics, resulting in the Principle of Stationary Action, from which we develop the foundations of Lagrangian mechanics.<ref name="ref_4cacc6a8" />
		+	# Finally, we examine an extension of the calculus of variations in optimal control.<ref name="ref_4cacc6a8" />
		+	# Such a function is called a functional, the focal point of the calculus of variations.<ref name="ref_4cacc6a8" />
		+	# The best way to appreciate the calculus of variations is by introducing a few concrete examples of both mathematical and practical importance.<ref name="ref_a06122a5">[https://www-users.math.umn.edu/~olver/ln_/cv.pdf The calculus of variations]</ref>
		+	# However, a fully rigorous proof of this fact requires a careful development of the mathematical machinery of the calculus of variations.<ref name="ref_a06122a5" />
		+	# The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations.<ref name="ref_d0833315">[https://www.ucl.ac.uk/~ucahmto/latex_html/pandoc_chapter2.html MATH0043 §2: Calculus of Variations]</ref>
		+	# A typical problem in the calculus of variations involve finding a particular function \(y(x)\) to maximize or minimize the integral \(I(y)\) subject to boundary conditions \(y(a)=A\) and \(y(b)=B\).<ref name="ref_d0833315" />
		+	# While predominantly designed as a textbook for lecture courses on the calculus of variations, this book can also serve as the basis for a reading seminar or as a companion for self-study.<ref name="ref_50d128ca">[https://www.springer.com/gp/book/9783319776361 Calculus of Variations]</ref>
		+	# Calculus of variations is used to nd the gradient of a functional (here E(u)) w.r.t.<ref name="ref_2f934ef9">[https://www.ece.iastate.edu/~namrata/EE520/Calculus_of_Variations.pdf Calculus of variations]</ref>
		+	# We use this same methodology for calculus of variations, but now u is a continuous function of a (cid:82) b a (x)2dx = 1).<ref name="ref_2f934ef9" />
		+	# Here we present three useful examples of variational calculus as applied to problems in mathematics and physics.<ref name="ref_4db316e5">[https://courses.physics.ucsd.edu/2010/Fall/physics200a/LECTURES/CH05.pdf Chapter 5]</ref>
		+	# Lagrangian mechanics is based on the calculus of variations, which is the subject of optimization over a space of paths.<ref name="ref_fa59fee2">[http://lavalle.pl/planning/node698.html 13.4.1.1 Calculus of variations]</ref>
		+	# These lecture notes are intented as a straightforward introduction to the calculus of variations which can serve as a textbook for undergraduate and beginning graduate students.<ref name="ref_99fb5427">[https://www.math.uni-leipzig.de/~miersemann/variabook.pdf Calculus of variations]</ref>
		+	# A huge amount of problems in the calculus of variations have their origin in physics where one has to minimize the energy associated to the problem under consideration.<ref name="ref_99fb5427" />
		+	# real , Integration by parts in the formula for g0(0) and the following basic lemma 2 in the calculus of variations imply Eulers equation.<ref name="ref_99fb5427" />
		+	# Since its beginnings, the calculus of variations has been intimately connected with the theory of dieren- tial equations; in particular, the theory of boundary value problems.<ref name="ref_2e7e869f">[https://pages.pomona.edu/~ajr04747/Fall2017/Math188/Notes/Math188Fall2017Notes.pdf Notes on the calculus of variations and]</ref>
		+	# This interplay between the theory of boundary value problems for dierential equations and the calculus of variations will be one of the major themes in the course.<ref name="ref_2e7e869f" />
		+	# We will focus on Euler's calculus of variations, a method applicable to solving the entire class of extremising problems.<ref name="ref_eb033f03">[https://plus.maths.org/content/frugal-nature-euler-and-calculus-variations Frugal nature: Euler and the calculus of variations]</ref>
		+	# It was in his 1744 book, though, that Euler transformed a set of special cases into a systematic approach to general problems: the calculus of variations was born.<ref name="ref_eb033f03" />
		+	# Euler coined the term calculus of variations, or variational calculus, based on the notation of Joseph-Louis Lagrange whose work formalised some of the underlying concepts.<ref name="ref_eb033f03" />
		+	# In their joint honour, the central equation of the calculus of variations is called the Euler-Lagrange equation.<ref name="ref_eb033f03" />
		+	# The calculus of variations underlies a powerful alternative approach to classical mechanics that is based on identifying the path that minimizes an integral quantity.<ref name="ref_ccd0a3bb">[https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Book%3A_Variational_Principles_in_Classical_Mechanics_(Cline)/05%3A_Calculus_of_Variations 5: Calculus of Variations]</ref>
		+	# In general, the calculus of variations is the branch of mathematics that investigates the stationary values of a generalized function, defined in terms of some generalized variables.<ref name="ref_a35e826b">[https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119580294.ch17 CALCULUS of VARIATIONS]</ref>
		+	# In this regard, calculus of variations has found a wide range of applications in science and engineering.<ref name="ref_a35e826b" />
		+	# Ideas from the calculus of variations are commonly found in papers dealing with the finite element method.<ref name="ref_f92932ad">[https://en.wikiversity.org/wiki/Introduction_to_finite_elements/Calculus_of_variations Introduction to finite elements/Calculus of variations]</ref>
		+	# This handout discusses some of the basic notations and concepts of variational calculus.<ref name="ref_f92932ad" />
		+	# The calculus of variations is a sort of generalization of the calculus that you all know.<ref name="ref_f92932ad" />
		+	# The goal of variational calculus is to find the curve or surface that minimizes a given function.<ref name="ref_f92932ad" />
		+	# In calculus of variations the basic problem is to nd a function y for which the functional I(y) is maximum or minimum.<ref name="ref_111223d8">[https://www.iist.ac.in/sites/default/files/people/COVMain.pdf Calculus of variations]</ref>
		+	===소스===
		+	<references />

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 [[분류:수리물리학]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q1145653 Q1145653]

← 이전 판		2020년 12월 28일 (월) 14:32 판
43번째 줄:		43번째 줄:
	* Perfect Form:Variational Principles, Methods, and Applications in Elementary Physics, http://books.google.com/books/about/Perfect_Form.html?id=8uWPG0QK0UIC		* Perfect Form:Variational Principles, Methods, and Applications in Elementary Physics, http://books.google.com/books/about/Perfect_Form.html?id=8uWPG0QK0UIC
	[[분류:수리물리학]]		[[분류:수리물리학]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q1145653 Q1145653]

@@ 34번째 줄: / 34번째 줄: @@
 ==관련논문==
-* [http://www.jstor.org/stable/27646267 Do Dogs Know Calculus of Variations?]<br>
+* [http://www.jstor.org/stable/27646267 Do Dogs Know Calculus of Variations?]
 ** Leonid A. Dickey, The College Mathematics Journal, Vol. 37, No. 1 (Jan., 2006), pp. 20-23

@@ 15번째 줄: / 15번째 줄: @@
 * 해밀턴의 원리 http://en.wikipedia.org/wiki/Hamilton%27s_principle
 * 파인만 경로적분
 ==관련된 항목들==

@@ 38번째 줄: / 38번째 줄: @@
 ==관련도서==
 * Basdevant, Jean-Louis. Variational Principles in Physics. 2007 edition. New York, NY: Springer, 2007.
 * Perfect Form:Variational Principles, Methods, and Applications in Elementary Physics, http://books.google.com/books/about/Perfect_Form.html?id=8uWPG0QK0UIC
 [[분류:수리물리학]]