13D (Variational Principles)

y + ε η

with

η = η^{'} = 0

at both endpoints, integrating the

η^{'}

term once and the

η^{''}

term twice gives

δ I = \int_{a}^{b} (f_{y} - \frac{d}{d x} f_{y^{'}} + \frac{d ^{2}}{d x ^{2}} f_{y^{''}}) η, d x .

The fundamental lemma yields the higher-order Euler-Lagrange equation

f_{y} - \frac{d}{d x} f_{y^{'}} + \frac{d ^{2}}{d x ^{2}} f_{y^{''}} = 0 .

b

Differentiate

Q = f - (f_{y^{'}} - \frac{d}{d x} f_{y^{''}}) y^{'} - f_{y^{''}} y^{''} .

The product rule cancels all terms except

Q^{'} = y^{'} (f_{y} - \frac{d}{d x} f_{y^{'}} + \frac{d ^{2}}{d x ^{2}} f_{y^{''}}) + f_{x} .

On an extremal with no explicit

x

dependence both terms vanish, proving the stated conserved quantity.

y^{(4)} - 2 y^{''} - y = 0.

Put

A = 1 + 2

B = 2 - 1

, and

L = π /2

. The conditions at zero reduce the general solution to

y = C (cosh A x - cos B x) + D (sinh A x - \frac{A}{B} sin B x) .

U = cosh (A L) - cos (B L), V = sinh (A L) - \frac{A}{B} sin (B L),

U^{'} = A sinh (A L) + B sin (B L), V^{'} = A (cosh (A L) - cos (B L)) .

C = \frac{L V ^{'} - V}{U V ^{'} - U ^{'} V}, D = \frac{U - L U ^{'}}{U V ^{'} - U ^{'} V},

which determines the unique extremal explicitly.

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