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Part CCSIR NET December 2023

Euler–Lagrange equation and standard functionals

Among the curves connecting the points (1, 2) and (2, 8), let γ\gamma be the curve on which an extremal of J[y]=∫12(1+x3y′)y′J[y] = \int_{1}^{2} (1 + x^{3}y') y' dx can be attained. Which of the following points lie on the curve γ\gamma?

  1. A.(2,3)(\sqrt{2}, 3)
  2. B.(2,6)(\sqrt{2}, 6)✓ Correct
  3. C.(3,22/3)(\sqrt{3}, 22/3)✓ Correct
  4. D.(3,23/3)(\sqrt{3}, 23/3)

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.

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50 are analysed free — try those first.

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

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