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UBC Mathematics 402(201)—Assignment 10 Due at 09:00 on Thursday 26 March 2015 1. Consider this isoperimetric problem on a fixed interval, assuming γ > 0: Z 2 Z 2 p 2 min x(t) dt = γ . 1 + ẋ(t) dt : x(0) = 0 = x(2), x 0 0 (a) Prove: If a minimizing arc exists, its graph must be an arc of a circle in the (t, x)-plane. π (b) Suppose γ = − 1. Show that a minimizing arc does exist, and describe it in detail. 2 2. Consider the following problem: Z 1 Z 2 2 min ẋ(t) − x(t) dt : x(0) = 1, x(1) = e, 0 1 2 2 ẋ(t) + x(t) 0 dt = e − 1 . 2 (a) Find an abnormal extremal (λ0 = 0). (b) Prove that this extremal provides the problem’s unique global minimum. (c) Show that there are no admissible extremals for which λ0 = 1. 3. Imagine taking a uniform chain of given length s and fixing its ends at the points (0, 0) and (1, 0). Gravity will pull the chain into a certain curve: find this curve. In detail, (a) Present an equation describing the shape of the curve and a clear description of how any constants in that equation are to be determined; and (b) Using computer assistance, find and sketch on the same axes the hanging-chain shapes for the cases s = 1.25, s = 1.50, and s = 2.00. Suggestions: Assume that s ≥ 1, i.e., there is enough chain to link the given points. Using t as the label for a horizontal axis in space, and x as the label for a vertical axis, the problem turns into a search for a particular arc x ∈ P WS[0, 1] satisfying Z 1p 1 + ẋ(t)2 dt. (1) x(0) = 0, x(1) = 0, s = 0 p Here we recognize ds = 1 + ẋ2 dt as the length of chain lying above the horizontal segment dt. This short arc of chain will have mass dm = ρ ds (where ρ > 0 is the linear density of the chain, with units of kg/m), and thus its gravitational potential energy will be p dU = (dm) gx = ρgx(t) 1 + ẋ(t)2 dt. The chain will arrange itself to minimize the total potential, Z 1 p def ρgx(t) 1 + ẋ(t)2 dt. U [x] = 0 def Since both ρ and g are positive constants, it suffices to minimize Λ[x] = U [x]/(ρg) subject to the constraints in line (1). File “hw10”, version of 19 Mar 2015, page 1. Typeset at 11:07 March 19, 2015. 4. Use Lagrange Multipliers to find all possible local minimizers in the problem min Z 1 2 x(t) dt : x(0) = 0, x(1) = 0, 0 Z 1 ẋ(t) dt = 1 . 2 0 Can convexity methods be used to conclude that a minimum is achieved? Explain. [Troutman problem 6.23.] File “hw10”, version of 19 Mar 2015, page 2. Typeset at 11:07 March 19, 2015.