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| − | + | ==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 | ||
| + | * 변분법이 사용된 고전적인 예로 [[최단시간강하곡선 문제(Brachistochrone problem)]]가 있다 | ||
| − | + | ==역사== | |
| + | * [[수학사 연표]] | ||
| − | + | ==메모== | |
| − | + | * 스넬의 법칙 | |
| − | + | * 페르마의 원리 | |
| − | + | * 모페르튀 최소 작용의 원칙 | |
| − | + | * 베르누이 | |
| − | + | * 오일러-라그랑지 변분법 | |
| − | + | * 해밀턴의 원리 http://en.wikipedia.org/wiki/Hamilton%27s_principle | |
| − | + | * 파인만 경로적분 | |
| − | + | * IMA Public Lectures : The Best of All Possible Worlds: The Idea of Optimization; Ivar Ekeland https://www.youtube.com/watch?v=1qlz2M1URno | |
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| + | ==관련된 항목들== | ||
| + | * [[오일러-라그랑지 방정식]] | ||
* [[사이클로이드]] | * [[사이클로이드]] | ||
| + | ** [[최단시간강하곡선 문제(Brachistochrone problem)]] | ||
| + | ** [[등시강하곡선 문제 (Tautochrone problem)]] | ||
* [[측지선]] | * [[측지선]] | ||
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| − | + | ==사전 형태의 자료== | |
| − | + | * http://ko.wikipedia.org/wiki/변분법 | |
| + | * http://en.wikipedia.org/wiki/History_of_variational_principles_in_physics | ||
| + | |||
| − | + | ==관련논문== | |
| − | * [http://www.jstor.org/stable/27646267 Do Dogs Know Calculus of Variations?] | + | * [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 | ** Leonid A. Dickey, The College Mathematics Journal, Vol. 37, No. 1 (Jan., 2006), pp. 20-23 | ||
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| − | * | + | ==관련도서== |
| − | * | + | * Ekeland, Ivar. The Best of All Possible Worlds: Mathematics and Destiny. Reprint edition. Chicago: University Of Chicago Press, 2007. |
| − | * | + | * 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]] | ||
| − | + | == 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 /> | ||
| − | + | == 메타데이터 == | |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q216861 Q216861] |
| − | + | ===Spacy 패턴 목록=== | |
| − | * [ | + | * [{'LOWER': 'calculus'}, {'LOWER': 'of'}, {'LEMMA': 'variation'}] |
| − | * [ | + | * [{'LOWER': 'variational'}, {'LEMMA': 'calculus'}] |
2021년 2월 23일 (화) 19:52 기준 최신판
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
- 변분법이 사용된 고전적인 예로 최단시간강하곡선 문제(Brachistochrone problem)가 있다
역사
메모
- 스넬의 법칙
- 페르마의 원리
- 모페르튀 최소 작용의 원칙
- 베르누이
- 오일러-라그랑지 변분법
- 해밀턴의 원리 http://en.wikipedia.org/wiki/Hamilton%27s_principle
- 파인만 경로적분
- IMA Public Lectures : The Best of All Possible Worlds: The Idea of Optimization; Ivar Ekeland https://www.youtube.com/watch?v=1qlz2M1URno
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/변분법
- http://en.wikipedia.org/wiki/History_of_variational_principles_in_physics
관련논문
- Do Dogs Know Calculus of Variations?
- Leonid A. Dickey, The College Mathematics Journal, Vol. 37, No. 1 (Jan., 2006), pp. 20-23
관련도서
- Ekeland, Ivar. The Best of All Possible Worlds: Mathematics and Destiny. Reprint edition. Chicago: University Of Chicago Press, 2007.
- 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
노트
말뭉치
- This post is going to describe a specialized type of calculus called variational calculus.[1]
- 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.[1]
- 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.[2]
- 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.[2]
- This technique, typical of the calculus of variations, led to a differential equation whose solution is a curve called the cycloid.[2]
- Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.[3]
- 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).[4]
- 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.[4]
- The calculus of variations addresses the need to optimize certain quantities over sets of functions.[5]
- 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.[5]
- Finally, we examine an extension of the calculus of variations in optimal control.[5]
- Such a function is called a functional, the focal point of the calculus of variations.[5]
- The best way to appreciate the calculus of variations is by introducing a few concrete examples of both mathematical and practical importance.[6]
- However, a fully rigorous proof of this fact requires a careful development of the mathematical machinery of the calculus of variations.[6]
- The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations.[7]
- 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 \(y(b)=B\).[7]
- 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.[8]
- Calculus of variations is used to nd the gradient of a functional (here E(u)) w.r.t.[9]
- 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).[9]
- Here we present three useful examples of variational calculus as applied to problems in mathematics and physics.[10]
- Lagrangian mechanics is based on the calculus of variations, which is the subject of optimization over a space of paths.[11]
- 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.[12]
- 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.[12]
- real , Integration by parts in the formula for g0(0) and the following basic lemma 2 in the calculus of variations imply Eulers equation.[12]
- 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.[13]
- 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.[13]
- We will focus on Euler's calculus of variations, a method applicable to solving the entire class of extremising problems.[14]
- 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.[14]
- 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.[14]
- In their joint honour, the central equation of the calculus of variations is called the Euler-Lagrange equation.[14]
- The calculus of variations underlies a powerful alternative approach to classical mechanics that is based on identifying the path that minimizes an integral quantity.[15]
- 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.[16]
- In this regard, calculus of variations has found a wide range of applications in science and engineering.[16]
- Ideas from the calculus of variations are commonly found in papers dealing with the finite element method.[17]
- This handout discusses some of the basic notations and concepts of variational calculus.[17]
- The calculus of variations is a sort of generalization of the calculus that you all know.[17]
- The goal of variational calculus is to find the curve or surface that minimizes a given function.[17]
- In calculus of variations the basic problem is to nd a function y for which the functional I(y) is maximum or minimum.[18]
소스
- ↑ 1.0 1.1 The Calculus of Variations
- ↑ 2.0 2.1 2.2 Calculus of variations | mathematics
- ↑ Calculus of variations
- ↑ 4.0 4.1 Calculus of Variations -- from Wolfram MathWorld
- ↑ 5.0 5.1 5.2 5.3 The calculus of variations
- ↑ 6.0 6.1 The calculus of variations
- ↑ 7.0 7.1 MATH0043 §2: Calculus of Variations
- ↑ Calculus of Variations
- ↑ 9.0 9.1 Calculus of variations
- ↑ Chapter 5
- ↑ 13.4.1.1 Calculus of variations
- ↑ 12.0 12.1 12.2 Calculus of variations
- ↑ 13.0 13.1 Notes on the calculus of variations and
- ↑ 14.0 14.1 14.2 14.3 Frugal nature: Euler and the calculus of variations
- ↑ 5: Calculus of Variations
- ↑ 16.0 16.1 CALCULUS of VARIATIONS
- ↑ 17.0 17.1 17.2 17.3 Introduction to finite elements/Calculus of variations
- ↑ Calculus of variations
메타데이터
위키데이터
- ID : Q216861
Spacy 패턴 목록
- [{'LOWER': 'calculus'}, {'LOWER': 'of'}, {'LEMMA': 'variation'}]
- [{'LOWER': 'variational'}, {'LEMMA': 'calculus'}]