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Part CCSIR NET June 2023coupled-euler-lagrange

Coupled euler lagrange

Let y(x) and z(x) be the stationary functions (extremals) of J(y,z)=∫01[(y′)2+(z′)2+y′z′]J(y, z) = \int_{0}^{1} [(y')^{2} + (z')^{2} + y'z'] dx subject to y(0) = 1, y(1) = 0, z(0) = −1, z(1) = 2. Which of the following statements are correct?

  1. A.z(x) + 3y(x) = 2 for x ∈ [0,1].✓ Correct
  2. B.3z(x) + y(x) = 2 for x ∈ [0,1].
  3. C.y(x) + z(x) = 2x for x ∈ [0,1].✓ Correct
  4. D.y(x) + z(x) = x for x ∈ [0,1].

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: A weak minimum of a functional is a strong minimum

More on this topic

The chapter behind this: Euler–Lagrange, first integrals and null Lagrangians — free to read

From Calculus of Variations › Euler–Lagrange equation and standard functionals

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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