Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y β dx² +\w(x)y=0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0, where X is a Lagrange multiplier. Consider the functional S[y] = ay(1)² + | " dx ẞy², y(0) = 0, with a natural boundary condition at x = 1 and subject to the constraint 1 C[y] = ry(1)² + [[" dx w(x) y² = 1, where a, ẞ and y are nonzero constants.
Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y β dx² +\w(x)y=0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0, where X is a Lagrange multiplier. Consider the functional S[y] = ay(1)² + | " dx ẞy², y(0) = 0, with a natural boundary condition at x = 1 and subject to the constraint 1 C[y] = ry(1)² + [[" dx w(x) y² = 1, where a, ẞ and y are nonzero constants.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Show that the stationary paths of this system satisfy the
Euler-Lagrange equation
d²y
β
dx²
+\w(x)y=0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0,
where X is a Lagrange multiplier.
![Consider the functional
S[y] = ay(1)² + | "
dx ẞy², y(0) = 0,
with a natural boundary condition at x = 1 and subject to the constraint
1
C[y] = ry(1)² + [["
dx w(x) y² = 1,
where a, ẞ and y are nonzero constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2F12316b7e-0455-405b-8a90-e7f39c902e94%2F69y3zz_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the functional
S[y] = ay(1)² + | "
dx ẞy², y(0) = 0,
with a natural boundary condition at x = 1 and subject to the constraint
1
C[y] = ry(1)² + [["
dx w(x) y² = 1,
where a, ẞ and y are nonzero constants.
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