# 변분법

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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

## 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'}]