최단시간강하곡선 문제(Brachistochrone problem)

개요

• 중력을 받고 있는 물체가 정지상태에서 출발하여 가장 짧은 시간내에 하강하기 위해서 따라야 하는 곡선
• 1697년에 베르누이에 의하여 답이 출판

Classical Mechanics

곡선의 시작점을 \((x_0,y_0)=(0,0)\), 끝점을 \((x_1,y_1)\)라 두자.

곡선을 따라 내려올때 걸리는 시간은 다음과 같이 구할 수 있다.

\(t=\int \frac{1}{v} \, ds\)(v는 속력, ds 는 길이요소, t는 시간)

에너지 보존 법칙 \(mgy=\frac{1}{2}mv^2\) 에서\(v=\sqrt{2gy}\).

이제 곡선의 x좌표를 y의 함수로 생각하자. 곡선을 따라 내려올 때 걸리는 시간은 \[T=\int \frac{1}{v} \, ds=\frac{1}{\sqrt{2g}}\int_{0}^{y} \frac{\sqrt{1+x'(y)^2}}{\sqrt{y}} \, dy\]

문제의 정의에 따라 이 적분값을 최소가 되게 하는 곡선을 찾아야 한다.

\(F(y,x,x')=\frac{\sqrt{1+(x')^2}}{\sqrt{y}}\) 에 대하여 오일러-라그랑지 방정식 을 적용하면, \[0 =\frac{\partial F}{\partial x} - \frac{d}{dy} \frac{\partial F}{\partial x'}=-\frac{d}{dy}(\frac{x'(y)}{\sqrt{y(1+x'(y)^2)}})\]

적당한 상수 a에 대하여 \(\frac{x'(y)}{\sqrt{y(1+x'(y)^2)}}=\frac{1}{\sqrt{2a}}\)라 두자.

이를 풀면 다음의 미분방정식을 얻는다. \[\frac{dx}{dy}=\sqrt{\frac{y}{2a-y}}\]

(미분방정식의 여러 해에 대한 논의는 http://whistleralley.com/brachistochrone/brachistochrone.htm)

\(x=\int_{0}^{y}\sqrt{\frac{y}{2a-y}}dy\), \(y=2a\sin^2\frac{\theta}{2}=a(1-\cos\theta)\)로 치환하면, \(x=a(\theta-\sin\theta)\)를 얻는다.

여기서 상수 a는 주어진 점 \((x_1,y_1)\)를 지날 수 있는 값으로 결정된다.

따라서 사이클로이드를 얻었다.■

재미있는 사실

• http://en.wikipedia.org/wiki/Half-pipe ?
• Half-Pipe Skateboarding ?

수학용어번역

• Brachistochrone curve
  • brachistos - the shortest, chronos - time
  • 최단시간강하 곡선, 최속강하선, 최단강하선

관련된 항목들

• 오일러-라그랑지 방정식

매스매티카 파일 및 계산 리소스

• http://demonstrations.wolfram.com/TheGreatBrachistochroneRace/

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Brachistochrone_problem
• http://curvebank.calstatela.edu/brach/brach.htm
• http://mathworld.wolfram.com/BrachistochroneProblem.html

관련논문

• The Brachistochrone Problem
  • Nils P. Johnson, The College Mathematics Journal, Vol. 35, No. 3 (May, 2004), pp. 192-197
• Exploring the Brachistochrone Problem
  • LaDawn Haws, Terry Kiser, The American Mathematical Monthly, Vol. 102, No. 4 (Apr., 1995), pp. 328-336

관련도서

• http://books.google.com/books?id=dptKVr-5LJAC&pg=PA223&sig=PVA7Q1U_MyXinobyhOf54BwjShQ&hl=en#v=onepage&q&f=false

링크

• The Brachistochrone

노트

말뭉치

1. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp.
2. According to Newtonian scholar Tom Whiteside, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem.
3. Johann Bernoulli's direct method is historically important as it was the first proof that the brachistochrone is the cycloid.
4. In this blog post, we demonstrate how to use built-in mathematical expressions and the Optimization Module in COMSOL Multiphysics to solve for the brachistochrone curve.
5. The brachistochrone curve is an idealized curve that provides the fastest descent possible.
6. Next, we use an interpolation function to approximate the brachistochrone curve.
7. The brachistochrone curve is a classic physics problem, that derives the fastest path between two points A and B which are at different elevations.
8. There is no better way to learn than through STEM, so follow on to make your very own working brachistochrone model.
9. Before I end I must voice once more the admiration I feel for the unexpected identity of Huygens' tautochrone and my brachistochrone.
10. This article presents the problem of quickest descent, or the Brachistochrone curve, that may be solved by the calculus of variations and the Euler-Lagrange equation.
11. The rst step in the solution of the Euler-Lagrange equation for the brachistochrone problem: is to reduce it to a rst-order equation.
12. the red cycloid beats the other two It can also be asked what the brachistochrone curve among the curves joining two points and having a given shape would be.
13. For example, for two points at the same altitude and V-shaped curves, the brachistochrone curve is the one for which the angle of the V is a right angle, as is shown in the animation opposite.
14. The problem of the brachistochrone with given length is studied on this page.
15. One can also try to find the brachistochrone "with friction".
16. The brachistochrone problem was one of the earliest problems posed in the calculus of variations.
17. Brachistochrone, the planar curve on which a body subjected only to the force of gravity will slide (without friction) between two points in the least possible time.
18. The name brachistochrone comes from two Greek words, brachistos meaning shortest, and chronos meaning time.
19. The brachistochrone curve can be generated by tracking a point on the rim of a wheel as it rolls on the ground.
20. This mathematical challenge is known as the problem of the brachistochrone.
21. Figure 3: Newton’s handwritten solution to the brachistochrone problem (source).
22. The classical problem in calculus of variation is the so called brachistochrone problem 1 posed (and solved) by Bernoulli in 1696.
23. Abstract: This article presents the problem of quickest descent, or the Brachistochrone curve, that may be solved by the calculus of variations and the Euler-Lagrange equation.
24. Historically and pedagogically, the prototype problem introducing the cal- culus of variations is the brachistochrone, from the Greek for shortest time.