MATH2401: Problem Sheet 3

MATH2401: Problem Sheet 31. ∗ Minimize a functional: Write down and solve Euler’s equation for the extremal f(x)of∫ 1[I[y] = (y ′ ) 2 + y 2 + 2xy ] dx,0if y(0) = 0 and y(1) = −1. By writing y(x) = f(x) + g(x) with g(0) = g(1) = 0, showthatI[f + g] ≥ I[f],and hence that the extremal f(x) minimizes the integral.2. Extremal functions: Find extremal functions of the following functionals:(b)(c ∗ )(a)∫ π/20∫ 21(y ′ ) 2x 3 dx, y(1) = 2, y(2) = 17.[y 2 − (y ′ ) 2 − 2y sin x] dx, y(0) = 1, y(π/2) = 2.∫ π0(y ′ ) 2 + 2y sin x dx, y(0) = y(π) = 0.State (without proof) whether you think you have found a maximum or minimum in eachcase.3. Sailing boat problem: Variations in the wind and water depth in a channel mean thata boat can sail at maximum speed λy, where λ > 0 is a constant. The boat must sailfrom (0, 1) to (2, 1) following a smooth path y(x) in the fastest possible time. You mayuse the fact that the time T for the journey along a path y(x) can be expressed asT [y] = 1 λ∫ 20[1 + (y ′ ) 2 ] 1/2Show first that the extremal function must satisfyy[1 + (y ′ ) 2 ] 1/2 = constant,and use this to show that the path of the boat is an arc of a circle of radius √ 2. Henceshow that the fastest possible time isyT min = 2 λ ln (1 + √ 2).4. ∗ Extremals for functionals with F = F (y, y ′ ): Find extremal functions of the followingfunctionals:(c)(a)∫ 10∫ 2dx.(y ′ ) 2dx, y(0) = 0, y(1) = 2.1 + y2 (y ′ ) 2(b)dx, y(0) = 1, y(2) = 4.0 y 3∫ 2[ ]12 (y′ ) 2 + yy ′ + y + y ′ dx, y(0) = 0, y(2) = 2.0

5. ∗ Minimize an integral: Find the minimum value taken by the integralif y(π) = 1, y(2π) = 3.∫ 2ππ(y ′ ) 2 [1 + (y ′ ) 2 ] 3/2 dx,6. ∗ Find another extremal: The function y = φ(t) is an extremal of the functionalI[y] = 1 2∫ 10[(y ′ ) 2 + y 2 ] dt,and thus satisfies Euler’s equation together with the end conditions y(0) = y(1) = 1.Show thatI(φ) = 1 2 [φ(1)φ′ (1) − φ(0)φ ′ (0).]If f(t) is any twice differentiable function on 0 ≤ t ≤ 1 for which f(0) = f(1) = 0 showthatI[φ + f] ≥ I[φ],and thus deduce that φ(t) minimizes the functional I.Determine φ(t) and deduce that the minimum value that can be taken by I is (e−1)/(e+1).7. ∗ Aircraft problem (Generalisation of Q3 above): An aircraft flight path is givenby a curve y(x) on the (x, y) plane. The aircraft flies with a speed u(y) between two fixedpoints, and flight path is assumed to be smooth. Show that the equation of the aircraft’sflight path resulting in the minimum possible flight time iswhere A and B are constants.∫x = ±u√ dy + B,A2 − u2 Solutions to questions 2 and 3 (unstarred only) to be handed in on Friday 27th October2006.AnswersQuestion 1. f(x) = −x.Question 2c. f(x) = − sin x.Question 4a. f(x) = sinh (x sinh −1 2).Question 4b. f(x) = (1 − x/4) −2 .Question 4c. f(x) = x 2 /2.Question 5. 4(π 2 + 4) 3/2 /π 4 .