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==개요==
 
==개요==
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* 변분법은 특정한 적분의 값을 가장 크거나 작게 하는 함수를 찾는 문제와 관련된 수학의 분야이다
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* 미적분학의 중요한 문제는 함수를 가장 크거나 작게 만드는 점을 찾는 것이다
−
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* 변분법은 함수의 공간을 정의역으로 갖는 함수에 대한 미적분학이라 할 수 있다
 +
* 변분법이 사용된 고전적인 예로 [[최단시간강하곡선 문제(Brachistochrone problem)]]가 있다
  
 
==역사==
 
==역사==
44번째 줄: 45번째 줄:
 
[[분류:수리물리학]]
 
[[분류:수리물리학]]
  
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==메타데이터==
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== 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===
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<references />
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== 메타데이터 ==
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===위키데이터===
 
===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1145653 Q1145653]
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* ID :  [https://www.wikidata.org/wiki/Q216861 Q216861]
 
===Spacy 패턴 목록===
 
===Spacy 패턴 목록===
−
* [{'LOWER': 'hamilton'}, {'LOWER': "'s"}, {'LEMMA': 'principle'}]
+
* [{'LOWER': 'calculus'}, {'LOWER': 'of'}, {'LEMMA': 'variation'}]
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* [{'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

관련된 항목들


사전 형태의 자료


관련논문


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

  1. This post is going to describe a specialized type of calculus called variational calculus.[1]
  2. 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]
  3. 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]
  4. 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]
  5. This technique, typical of the calculus of variations, led to a differential equation whose solution is a curve called the cycloid.[2]
  6. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.[3]
  7. 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]
  8. 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]
  9. The calculus of variations addresses the need to optimize certain quantities over sets of functions.[5]
  10. 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]
  11. Finally, we examine an extension of the calculus of variations in optimal control.[5]
  12. Such a function is called a functional, the focal point of the calculus of variations.[5]
  13. The best way to appreciate the calculus of variations is by introducing a few concrete examples of both mathematical and practical importance.[6]
  14. However, a fully rigorous proof of this fact requires a careful development of the mathematical machinery of the calculus of variations.[6]
  15. The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations.[7]
  16. 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]
  17. 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]
  18. Calculus of variations is used to nd the gradient of a functional (here E(u)) w.r.t.[9]
  19. 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]
  20. Here we present three useful examples of variational calculus as applied to problems in mathematics and physics.[10]
  21. Lagrangian mechanics is based on the calculus of variations, which is the subject of optimization over a space of paths.[11]
  22. 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]
  23. 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]
  24. 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]
  25. 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]
  26. 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]
  27. We will focus on Euler's calculus of variations, a method applicable to solving the entire class of extremising problems.[14]
  28. 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]
  29. 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]
  30. In their joint honour, the central equation of the calculus of variations is called the Euler-Lagrange equation.[14]
  31. 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]
  32. 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]
  33. In this regard, calculus of variations has found a wide range of applications in science and engineering.[16]
  34. Ideas from the calculus of variations are commonly found in papers dealing with the finite element method.[17]
  35. This handout discusses some of the basic notations and concepts of variational calculus.[17]
  36. The calculus of variations is a sort of generalization of the calculus that you all know.[17]
  37. The goal of variational calculus is to find the curve or surface that minimizes a given function.[17]
  38. 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. ↑ 1.0 1.1 The Calculus of Variations
  2. ↑ 2.0 2.1 2.2 Calculus of variations | mathematics
  3. ↑ Calculus of variations
  4. ↑ 4.0 4.1 Calculus of Variations -- from Wolfram MathWorld
  5. ↑ 5.0 5.1 5.2 5.3 The calculus of variations
  6. ↑ 6.0 6.1 The calculus of variations
  7. ↑ 7.0 7.1 MATH0043 §2: Calculus of Variations
  8. ↑ Calculus of Variations
  9. ↑ 9.0 9.1 Calculus of variations
  10. ↑ Chapter 5
  11. ↑ 13.4.1.1 Calculus of variations
  12. ↑ 12.0 12.1 12.2 Calculus of variations
  13. ↑ 13.0 13.1 Notes on the calculus of variations and
  14. ↑ 14.0 14.1 14.2 14.3 Frugal nature: Euler and the calculus of variations
  15. ↑ 5: Calculus of Variations
  16. ↑ 16.0 16.1 CALCULUS of VARIATIONS
  17. ↑ 17.0 17.1 17.2 17.3 Introduction to finite elements/Calculus of variations
  18. ↑ Calculus of variations

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'calculus'}, {'LOWER': 'of'}, {'LEMMA': 'variation'}]
  • [{'LOWER': 'variational'}, {'LEMMA': 'calculus'}]
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