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Part BCSIR NET June 2023null-lagrangian-term

Null lagrangian term

Consider the variational problem (P):J(y)=∫01[(y′)2−y(P): J(y) = \int_{0}^{1} [(y')^{2} - y|y|·y′ + xy] dx, y(0) = 0, y(1) = 0. Which of the following statements is correct?

  1. A.(P) has no stationary function (extremal).
  2. B.y ≡ 0 is the only stationary function (extremal) for (P).
  3. C.(P) has a unique stationary function (extremal) y not identically equal to 0.✓ Correct
  4. D.(P) has infinitely many stationary functions (extremals).

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 hypothesis dropped.

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50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

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

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