GATE MA: Calculus of Variations

Calculus of Variations in GATE MA asks for extremals of integral functionals, boundary conditions, and second-order sufficiency. Typical problems involve geodesics, isoperimetric constraints, and Rayleigh–Ritz approximations.

Core rules

• Functional J[y]=∫_a^b F(x,y,y′)dx. Euler–Lagrange: ∂F/∂y − d/dx(∂F/∂y′)=0.
• If F does not depend on y, then ∂F/∂y′=constant; Beltrami identity applies when F has no explicit x.
• Fixed endpoints give y(a)=A, y(b)=B; free endpoints give natural boundary conditions ∂F/∂y′=0 at that endpoint.
• Second variation Legendre condition: F_{y′y′}≥0 for a minimum.
• Isoperimetric problems use a Lagrange multiplier: extremize J+λK.
• For F=F(x,y′), Euler–Lagrange reduces to F_{y′}=constant.

Worked example

Minimize J[y]=∫_0^1 (y′²+y²)dx with y(0)=y(1)=0. Here F=y′²+y², so EL gives 2y−d/dx(2y′)=0, i.e. y″−y=0. General solution y=Ae^x+Be^{−x}. Boundary conditions give A+B=0 and Ae+Be^{−1}=0, so A=B=0. Thus y≡0, and since F is convex in y′, this is the minimum.

Traps

• The Euler–Lagrange equation is necessary, not sufficient; check the second variation.
• Natural boundary conditions arise only when the endpoint is free; imposing y=0 there is a different problem.
• For functionals with y′^4, the Legendre condition can fail even at a stationary point.
• Isoperimetric constraints need an extra multiplier; do not substitute the constraint before varying.