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3. Let A = {u= (u₁, U2, U3): [0, 1] → R³ | u € C¹, u(0) = A, u(1) = B} and consider the holonomic problem: minimize F[u()] := √ √u₁(t)² + (u₂(t), u²(t))Q(u₂(t), u²(t))ª dt subject to: u € A, G(u₁(t), u₂(t), uz (t)) := u₁(t) + u₂(t)² + us(t)² - 1 = 0. Here Q is a 2x2 diagonal matrix with eigenvalues 0 < A₁ ≤ 2. Assume that u₁ (t)²+(u₂(t), uz (t))Q(u'₂ (t), uz (t)) T is positive for t = [0, 1]. (a) Thinking of this problem as a holonomic problem, find the Euler-Lagrange equations (you do not need to solve them).

3. Let A = {u= (u₁, U2, U3): [0, 1] → R³ | u € C¹, u(0) = A, u(1) = B} and consider the holonomic problem: minimize F[u()] := √ √u₁(t)² + (u₂(t), u²(t))Q(u₂(t), u²(t))ª dt subject to: u € A, G(u₁(t), u₂(t), uz (t)) := u₁(t) + u₂(t)² + us(t)² - 1 = 0. Here Q is a 2x2 diagonal matrix with eigenvalues 0 < A₁ ≤ 2. Assume that u₁ (t)²+(u₂(t), uz (t))Q(u'₂ (t), uz (t)) T is positive for t = [0, 1]. (a) Thinking of this problem as a holonomic problem, find the Euler-Lagrange equations (you do not need to solve them).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. Let A := {u= (U₁, U2, U3) : [0, 1] → R³ | u € C¹, u(0) = A, u(1) = B) and consider the holonomic problem: minimize F[u(-)] := √ √ų4 (t)² + (u₂(t), uz(t))Q(us(t), u²(t))ª dt subject 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))T is positive for t € [0, 1]. (a) Thinking of this problem as a holonomic problem, find the Euler-Lagrange equations (you do not need to solve them).
Transcribed Image Text:3. Let A := {u= (U₁, U2, U3) : [0, 1] → R³ | u € C¹, u(0) = A, u(1) = B) and consider the holonomic problem: minimize F[u(-)] := √ √ų4 (t)² + (u₂(t), uz(t))Q(us(t), u²(t))ª dt subject 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))T is positive for t € [0, 1]. (a) Thinking of this problem as a holonomic problem, find the Euler-Lagrange equations (you do not need to solve them).
(b) Formulate the above problem as a problem without constraints and and find the Euler-Lagrange equation. 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.
Transcribed Image Text:(b) Formulate the above problem as a problem without constraints and and find the Euler-Lagrange equation. 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.
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