NETMaths
Log inStart free
Part BCSIR NET June 2023null-lagrangian-term

Null lagrangian term

Consider the variational problem |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.✓
  4. D.(P) has infinitely many stationary functions (extremals).

Solution

The term y|y|y′ is an exact derivative ((|y| up to sign), so it does not affect the Euler–Lagrange equation, which reduces to 2y″ = x. With y(0) = y(1) = 0 this gives the unique extremal , not identically zero.

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

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

ShareWhatsAppTelegram