Exam 10 January 2013, questions PDF

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MATH 265001 MATH 265001 This question paper consists of 4 printed pages, each of which is identified the reference MATH 265001 All calculators must carry an approval sticker issued the School of Mathematics. c UNIVERSITY OF LEEDS Examination for the Module MATH 2650 (January 2013) Calculus of Variat...

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MATH 265001

MATH 265001 All calculators must carry an approval sticker issued by the School of Mathematics.

This question paper consists of 4 printed pages, each of which is identified by the reference MATH 265001

c UNIVERSITY OF LEEDS  Examination for the Module MATH 2650 (January 2013) Calculus of Variations Time allowed: 2 hours Answer no more than four questions. All questions carry equal marks.

1. (a) State and prove the Fundamental Lemma of the Calculus of Variations. (b) The Euler-Lagrange equation for the integral Z x2 I= F (x, y(x), y ′ (x)) dx x1

takes the form

∂F d − ∂y dx



∂F ∂y ′



= 0.

Write down a first integral of the Euler-Lagrange equation when F = F (x, y ′ ). Derive the first integral (the Beltrami identity) y′

∂F −F = C ∂y ′

(where C is an arbitrary constant) of the Euler-Lagrange equation in the case when F = F (y, y ′ ). (c) Find the functions y(x) that are extrema of the following integrals, subject to the given end conditions: Z 2  2 (i) I = x2 y ′ + 2y 2 dx, y(1) = 2, y(2) = 5; Z1 3   y′ ′2 (ii) I = y + 2 dx, y(1) = 1, y (3) = 4; x 1 Z 2  2 (iii) I = y ′ + y 2 dx, y(0) = 0, y (1) = sinh(1). 0

MATH 265001

2. (a) Consider the integral Z x2 I= F (x, y1 (x), y2 (x), . . . , yn (x), y1′ (x), y2′ (x), . . . , yn′ (x))dx, x1

with yi (x1 ) = Ai , yi (x2 ) = Bi (i = 1, . . . n). Derive the necessary conditions (the EulerLagrange equations) for I to be minimised. [You may assume without proof the Fundamental Lemma of the Calculus of Variations.] (b) Find the functions y(x) and z(x) that minimise the integral I=

Z

π/2 0



 y ′2 + z ′2 + y 2 + 8yz + 7z 2 dx,

with y(0) = z(0) = 0, y(π/2) = sinh(3π/2) + 2, z(π/2) = 2 sinh(3π/2) − 1. 3. (a) Consider the integral I=

Z

x2

F (x, y(x), y ′ (x) dx,

x1

subject to the constraint J=

Z

x2

G(x, y(x), y ′ (x) dx = L,

x1

where L is a constant, with boundary conditions y(x1 ) = y1 , y(x2 ) = y2 . Derive the necessary condition (an extended version of the Euler-Lagrange equation) for I to be minimised. [You may assume without proof the Fundamental Lemma of the Calculus of Variations.] (b) The area under the curve described by the function y(x) is given by Z x2 A= ydx.

(∗)

x1

Show that the curve of length L that maximises the area under the curve joining two points P = (x1 , y1 ) and Q = (x2 , y2 ) is the arc of a circle. [You may use, without proof, the Beltrami identity stated in Question 1.] (c) If P has coordinates (−1, 0) and Q has coordinates (1, 0), sketch the curve when L = 3 and L = 2. Explain briefly why the representation (∗) is not appropriate for the solution to this variational problem when L > π .

MATH 265001

4. The problem of finding the geodesic (the shortest arc between two points) on an arbitrary surface, expressed as g(x, y, z) = 0, can be considered using a parametric form of the surface, taking x = x(u, v), y = y(u, v), z = z(u, v). The square of the differential of arc length ds can be written ds2 = dx2 + dy 2 + dz 2 = P (u, v)du2 + 2Q(u, v)du dv + R(u, v)dv 2 . (a) Derive the expressions for P , Q and R. (b) If Q = 0, what does this say about the (u, v) coordinate system? (c) In the special case when Q = 0 and when P and R are functions only of u, show that the geodesic is given by ! Z √ P √ du, v=C R2 − C 2 R

where C is a constant of integration.

(d) The surface of a sphere of radius a can be parametrised by u and v, where x = a sin u cos v and y = a sin u sin v, z = a cos u. First calculate ds and then, using the result in part (c), write down an integral expression for v. Use the substitution cotu =

 2 a C

1/2 −1 sin α

to evaluate the integral, and hence show that the equation for the geodesic on the surface of the sphere takes the form A1 x + A2 y + A3 z = 0, where A1 , A2 and A3 are constants. Give a geometrical interpretation of this result.

MATH 265001

5. Consider the Sturm-Liouville system composed of the ordinary differential equation   d dy p(x) + q(x)y + λr(x)y = 0, dx dx

(∗)

with boundary conditions y(x1 ) = y1 , y(x2 ) = y2 . Define Λ by Λ=

R x2 x1

(†)

(p(x)y ′2 − q(x)y 2 ) dx R x2 . r(x)y 2 dx x1

(a) Show that the problem of determining functions satisfying (†) that make Λ stationary is equivalent to the problem of determining the eigenfunctions of (∗), subject to (†). (b) Now consider the equation   d dy + λy = 0, (1 + x) dx dx with boundary conditions y(0) = y(1) = 0. By using the result in part (a) and adopting the trial function y = cx(1 − x), show that the eigenvalue λ must be less than 15. (c) Explain briefly how you might obtain a better estimate to the eigenvalue.

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