Questions: Calculus of Variations and Euler-Lagrange Equations

Question 1

The Euler-Lagrange equation for the functional J[u] = ∫(½|∇u|² - fu)dx is:

• A. -Δu = f (Poisson's equation)
• B. Δu = f
• C. -Δu + u = f
• D. u = f

Answer: (A)

With L = ½|∇u|² - fu, we have ∂L/∂(∇u) = ∇u and ∂L/∂u = -f. The Euler-Lagrange equation is -div(∇u) + (-f) = 0, which gives -Δu = f.

Question 2

The minimal surface equation arises from minimizing the area functional.

• T. True
• F. False

Answer: True

The area of a graph z = u(x,y) is ∫∫√(1+|∇u|²)dxdy. The Euler-Lagrange equation is div(∇u/√(1+|∇u|²)) = 0, the minimal surface equation. Solutions describe soap films and surfaces of zero mean curvature.

Question 3

What is the principle of least action in physics?

Model answer: Physical trajectories are critical points of the action functional S = ∫L dt, where L is the Lagrangian (kinetic minus potential energy)

Hamilton's principle states that the equations of motion (Newton's laws, Maxwell's equations, Einstein's field equations) are all Euler-Lagrange equations for appropriate Lagrangians. This variational formulation is the foundation of modern theoretical physics.

Question 4

A minimizer of a convex functional always satisfies the Euler-Lagrange equation.

• T. True
• F. False

Answer: True

For convex functionals, the Euler-Lagrange equation is both necessary and sufficient for a minimum. For non-convex functionals, the Euler-Lagrange equation characterizes all critical points (minima, maxima, and saddle points), and additional analysis is needed to determine which are minimizers.