변분법

둘러보기로 가기 검색하러 가기

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

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.

노트

말뭉치

1. This post is going to describe a specialized type of calculus called variational calculus.
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.
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.
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.
5. This technique, typical of the calculus of variations, led to a differential equation whose solution is a curve called the cycloid.
6. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
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).
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.
9. The calculus of variations addresses the need to optimize certain quantities over sets of functions.
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.
11. Finally, we examine an extension of the calculus of variations in optimal control.
12. Such a function is called a functional, the focal point of the calculus of variations.
13. The best way to appreciate the calculus of variations is by introducing a few concrete examples of both mathematical and practical importance.
14. However, a fully rigorous proof of this fact requires a careful development of the mathematical machinery of the calculus of variations.
15. The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations.
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 \(y(b)=B\).
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.
18. Calculus of variations is used to nd the gradient of a functional (here E(u)) w.r.t.
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).
20. Here we present three useful examples of variational calculus as applied to problems in mathematics and physics.
21. Lagrangian mechanics is based on the calculus of variations, which is the subject of optimization over a space of paths.
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.
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.
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.
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.
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.
27. We will focus on Euler's calculus of variations, a method applicable to solving the entire class of extremising problems.
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.
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.
30. In their joint honour, the central equation of the calculus of variations is called the Euler-Lagrange equation.
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.
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.
33. In this regard, calculus of variations has found a wide range of applications in science and engineering.
34. Ideas from the calculus of variations are commonly found in papers dealing with the finite element method.
35. This handout discusses some of the basic notations and concepts of variational calculus.
36. The calculus of variations is a sort of generalization of the calculus that you all know.
37. The goal of variational calculus is to find the curve or surface that minimizes a given function.
38. In calculus of variations the basic problem is to nd a function y for which the functional I(y) is maximum or minimum.

메타데이터

Spacy 패턴 목록

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