The Brachistochrone Curve

Transcription

1 The Brachistochrone Curve Paige R MacDonald May 16, 2014 Paige R MacDonald The Brachistochrone Curve May 16, / 1

2 The Problem In 1696, Johann Bernoulli posed the following problem to the scientific community: Find the curve of quickest descent between two points in a vertical plane, and not in the same vertical line, for a particle to slide under only the force of gravity, neglecting friction. Paige R MacDonald The Brachistochrone Curve May 16, / 1

3 Solutions were found by Leibniz, L Hopital, Newton, and Jacob and Johann Bernoulli. Paige R MacDonald The Brachistochrone Curve May 16, / 1

4 Solutions were found by Leibniz, L Hopital, Newton, and Jacob and Johann Bernoulli. The curve is known as The Brachistochrone from the Greek brachistos for shortest chronos for time. Paige R MacDonald The Brachistochrone Curve May 16, / 1

5 Johann Bernoulli s Solution Johann Bernoulli s solution begins by considering a problem in optics. A a v 1 α 1 x P c x α 2 v 2 b c B Paige R MacDonald The Brachistochrone Curve May 16, / 1

6 Using the total time from A to B is time = distance velocity, T = a2 + x 2 v 1 + b2 + (c x) 2 v 2. Paige R MacDonald The Brachistochrone Curve May 16, / 1

7 Using the total time from A to B is time = distance velocity, T = a2 + x 2 v 1 + b2 + (c x) 2 v 2. Taking the derivative of each side leads to dt dx = x v 1 a2 + x 2 c x v 2 b2 + (c x) 2 Paige R MacDonald The Brachistochrone Curve May 16, / 1

8 Assuming the ray of light is able to select the path from A to B that minimizes time T, then dt /dx = 0 and x v 1 a2 + x 2 = c x v 2 b2 + (c x) 2 Paige R MacDonald The Brachistochrone Curve May 16, / 1

9 Assuming the ray of light is able to select the path from A to B that minimizes time T, then dt /dx = 0 and and from the figure, sin α 1 = x v 1 a2 + x 2 = c x v 2 b2 + (c x) 2 x a2 + x 2 and sin α 2 = c x b2 + (c x) 2 so. sin α 1 v 1 = sin α 2 v 2. Paige R MacDonald The Brachistochrone Curve May 16, / 1

10 Assuming the ray of light is able to select the path from A to B that minimizes time T, then dt /dx = 0 and and from the figure, sin α 1 = x v 1 a2 + x 2 = c x v 2 b2 + (c x) 2 x a2 + x 2 and sin α 2 = c x b2 + (c x) 2 so sin α 1 v 1 = sin α 2 v 2.. This is Snell s Law of Refraction and demonstrates Fermat s principle of least time. Paige R MacDonald The Brachistochrone Curve May 16, / 1

11 Now we divide the plane into more layers. v 1 α 1 v 2 α 2 α 2 v 3 α 3 α 3 v 4 α 4 Paige R MacDonald The Brachistochrone Curve May 16, / 1

12 Applying Snell s law to the divided layers, sin α 1 v 1 = sin α 2 v 2 = sin α 3 v 3 = sin α 4 v 4. If we continue dividing the layers into smaller and smaller sections, the path approaches a smooth curve and sinα v = constant. Paige R MacDonald The Brachistochrone Curve May 16, / 1

13 Returning to the brachistochrone problem, we set up a coordinate system with an inverted y-axis. A x y y α β B Paige R MacDonald The Brachistochrone Curve May 16, / 1

14 Due to the assumption that there is no friction, energy is conserved and K i + U i = K f + U f 1 2 mv 2 = mgy. Paige R MacDonald The Brachistochrone Curve May 16, / 1

15 Due to the assumption that there is no friction, energy is conserved and K i + U i = K f + U f 1 2 mv 2 = mgy. Solving for velocity, v = 2gy. Paige R MacDonald The Brachistochrone Curve May 16, / 1

16 Due to the assumption that there is no friction, energy is conserved and K i + U i = K f + U f 1 2 mv 2 = mgy. Solving for velocity, v = 2gy. From the figure, sin α = cos β = 1 sec β 1 = 1 + tan 2 β 1 = 1 + (y ) 2 Paige R MacDonald The Brachistochrone Curve May 16, / 1

17 Substituting v = 2gy and sin α = (y ) 2 into sinα/v = constant and simplifying, y[1 + (y ) 2 ] = c. Paige R MacDonald The Brachistochrone Curve May 16, / 1

18 Substituting v = 2gy and sin α = (y ) 2 into sinα/v = constant and simplifying, y[1 + (y ) 2 ] = c. This is the differential equation of the brachistochrone. Paige R MacDonald The Brachistochrone Curve May 16, / 1

19 Substituting v = 2gy and sin α = (y ) 2 into sinα/v = constant and simplifying, y[1 + (y ) 2 ] = c. This is the differential equation of the brachistochrone. Using Leibniz notation [ ( ) ] 2 dy y 1 + = c dx Paige R MacDonald The Brachistochrone Curve May 16, / 1

20 Substituting v = 2gy and sin α = (y ) 2 into sinα/v = constant and simplifying, y[1 + (y ) 2 ] = c. This is the differential equation of the brachistochrone. Using Leibniz notation [ ( ) ] 2 dy y 1 + = c dx Which simplifies to dy dx = ( c y y ) 1/2 Paige R MacDonald The Brachistochrone Curve May 16, / 1

21 Separating variables and solving for dx dx = ( ) 1/2 y dy C y Paige R MacDonald The Brachistochrone Curve May 16, / 1

22 Separating variables and solving for dx dx = ( ) 1/2 y dy C y Let ( ) 1/2 y = tan φ C y and solving for y y = 2c sin φ cos φ Paige R MacDonald The Brachistochrone Curve May 16, / 1

23 Separating variables and solving for dx dx = ( ) 1/2 y dy C y Let ( ) 1/2 y = tan φ C y and solving for y y = 2c sin φ cos φ Substituting back into the differential equation dx = 2c sin 2 φdφ Paige R MacDonald The Brachistochrone Curve May 16, / 1

24 Integrating both sides and simplifying x = c 2 (2φ sin 2φ) + c 1 Paige R MacDonald The Brachistochrone Curve May 16, / 1

25 Using the initial condition, c 1 = 0 and letting a = c/2 and θ = 2φ, we obtain x = a(θ sinθ) and y = a(1 cosθ). Paige R MacDonald The Brachistochrone Curve May 16, / 1

26 Using the initial condition, c 1 = 0 and letting a = c/2 and θ = 2φ, we obtain x = a(θ sinθ) and y = a(1 cosθ). These are the parametric equations of the cycloid. Paige R MacDonald The Brachistochrone Curve May 16, / 1

27 The Cycloid y θ (x, y) a 2πa x Paige R MacDonald The Brachistochrone Curve May 16, / 1

28 The Calculus of Variations Euler s work on the Brachistochrone problem has been crediting with leading to the Calculus of Variations, which is concerned with finding extrema of functionals. Paige R MacDonald The Brachistochrone Curve May 16, / 1

29 The Calculus of Variations Euler s work on the Brachistochrone problem has been crediting with leading to the Calculus of Variations, which is concerned with finding extrema of functionals. This can be related to optimazation problems in elementary calculus, where the goal is to find a minimum of maximum of a function f (x), where a single variable, x is the quantity that varies. f : R R Paige R MacDonald The Brachistochrone Curve May 16, / 1

30 In the calculus of variations, the quantity that varies is itself a function. Paige R MacDonald The Brachistochrone Curve May 16, / 1

31 In the calculus of variations, the quantity that varies is itself a function. A functional, then, is a function that takes a function as its input, and returns a real number. So the integral to minimize is I = I : f R x2 x 1 f (x, y, y )dx Paige R MacDonald The Brachistochrone Curve May 16, / 1

32 Admissable Functions In order to find the function that yields the smallest value for I, we must select from the family of functions only those satisfying certain conditions: Paige R MacDonald The Brachistochrone Curve May 16, / 1

33 Admissable Functions In order to find the function that yields the smallest value for I, we must select from the family of functions only those satisfying certain conditions: Passes through points A and B Of class C 2 Paige R MacDonald The Brachistochrone Curve May 16, / 1

34 Admissable Functions In order to find the function that yields the smallest value for I, we must select from the family of functions only those satisfying certain conditions: Passes through points A and B Of class C 2 Assuming a function y(x) exists that minimizes the integral, we consider a function η(x) that disturbs y(x) slightly. Paige R MacDonald The Brachistochrone Curve May 16, / 1

35 y (x 1, y 1 ) ȳ(x) = y(x) + αη(x) αη(x) y(x) (x 2, y 2 ) η(x) η(x) x 1 x x 2 x η(x 1 ) = η(x 2 ) = 0, α a small parameter Paige R MacDonald The Brachistochrone Curve May 16, / 1

36 For any value of the function η(x), ȳ(x) represents a one-parameter family of admissable functions whose vertical deviation from the minimizing curve y(x) is αη(x). Paige R MacDonald The Brachistochrone Curve May 16, / 1

37 For any value of the function η(x), ȳ(x) represents a one-parameter family of admissable functions whose vertical deviation from the minimizing curve y(x) is αη(x). For any choice η(x), y(x) belongs to the family and corresponds to α = 0. Substituting ȳ(x) into the integral I (α) = I (α) = x2 x 1 x2 x 1 f (x, ȳ, ȳ )dx f [x, y(x) + αη(x), y (x) + αη (x)]dx. Paige R MacDonald The Brachistochrone Curve May 16, / 1

38 When α = 0, ȳ(x) = y(x) and since y(x) minimizes the the integral, I (α) must have a minimum when α = 0, or when I (α) = 0. Paige R MacDonald The Brachistochrone Curve May 16, / 1

39 When α = 0, ȳ(x) = y(x) and since y(x) minimizes the the integral, I (α) must have a minimum when α = 0, or when I (α) = 0. I (α) = x2 x 1 α f (x, ȳ, ȳ )dx Paige R MacDonald The Brachistochrone Curve May 16, / 1

40 When α = 0, ȳ(x) = y(x) and since y(x) minimizes the the integral, I (α) must have a minimum when α = 0, or when I (α) = 0. 0 = x2 x 1 I (α) = x2 x 1 α f (x, ȳ, ȳ )dx ( f x x α + f ȳ ȳ α + f ȳ ȳ α which integrates to give Euler s equation ( ) d f f dx y y = 0 ) dx Paige R MacDonald The Brachistochrone Curve May 16, / 1

41 Special Case In the case where x is missing from the function f, Euler s equation can be integrated to a less general form y f y f = c which is known as Beltrami s identity. Paige R MacDonald The Brachistochrone Curve May 16, / 1

42 Euler s Solution Returning to the brachistochrone problem So the integral to be minimized is velocity = ds dt dt = ds v x2 x (y ) 2 dx 2gy Since x does not appear explicitly in the function to be minimized, we can use Beltrami s identity. Paige R MacDonald The Brachistochrone Curve May 16, / 1

43 ( y y f y f = c ) 1 + (y ) 2 y 2gy [ ] 1 1 2gy 2 [1 + (y ) 2 ] 1/2 2y y 1 + (y ) 2 2gy = c 1 + (y ) 2 2gy = c Paige R MacDonald The Brachistochrone Curve May 16, / 1

44 Which simplifies to Paige R MacDonald The Brachistochrone Curve May 16, / 1

45 Which simplifies to y[1 + (y ) 2 ] = C Paige R MacDonald The Brachistochrone Curve May 16, / 1

46 The end. Paige R MacDonald The Brachistochrone Curve May 16, / 1

47 Bibliography Till Tantau. Tikz and PGF Packages. David Arnold. Writing Scientific Papers in Latex. George Simmons Differential Equations with Applications and Historical Notes. Douglas S. Shafer The Brachistochrone: Historical Gateway to the Calculus of Variations. Nils P. Johnson The Brachistochrone Problem J J O Connor and E F Robertson 2002 History Topic: The Brachistochrone Problem ac.uk/printht/brachistochrone.html Unknown author. The Brachistochrone Problem Paige R MacDonald The Brachistochrone Curve May 16, / 1