3. Let A := {u= (U₁, U2, U3) : [0, 1] → R³ | u € C¹, u(0) = A, u(1) = B) and consider the holonomicproblem:minimize F[u(-)] := √ √ų4 (t)² + (u₂(t), uz(t))Q(us(t), u²(t))ª dtsubject to: u € A, G(u₁ (t), u₂(t), uz(t)) := u₁(t) + u₂(t)² + uz(t)² - 1 = 0.Here Q is a 2x2 diagonal matrix with eigenvalues 0 < A₁ ≤ 2. Assume that u₁ (t)²+(u₂(t), uś (t))Q(u'₂ (t), uz (t))Tis positive for t € [0, 1].(a) Thinking of this problem as a holonomic problem, find the Euler-Lagrange equations (you do notneed to solve them). (b) Formulate the above problem as a problem without constraints and and find the Euler-Lagrangeequation. Hint: note that the holonomic constraint is just saying that u₁(t) = 1 − u₂(t)² — uz(t)².(c) Show that parts (a) and (b) give the same answer.

We need to find such a function u→=(u1,u2,u3)∈A, such that we need to minimize the functional…

Answered: 3. Let A = {u= (u₁, U2, U3): [0, 1] →…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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