 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2014/5/a/solution

y_{ε} = y + ε η

. Differentiating the functional at zero gives the first variation

δ I = \int_{a}^{b} (F_{y} η + F_{y^{'}} η^{'} + F_{y^{''}} η^{''}) d x .

Integrate the second term once and the third term twice by integration by parts:

δ I = \int_{a}^{b} (F_{y} - \frac{d}{d x} F_{y^{'}} + \frac{d ^{2}}{d x ^{2}} F_{y^{''}}) η d x + [(F_{y^{'}} - \frac{d}{d x} F_{y^{''}}) η + F_{y^{''}} η^{'}]_{a}^{b} .

Choose variations supported away from the endpoints. Stationarity and the fundamental lemma of the calculus of variations force

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

This is the Euler-Lagrange equation for a second-derivative functional. With the interior equation imposed, the remaining boundary expression must vanish for every admissible variation.

At each endpoint, a freely varying value

y

requires

F_{y^{'}} - (F_{y^{''}})^{'} = 0

, and a freely varying slope

y^{'}

requires

F_{y^{''}} = 0

. A fixed value or slope instead makes its corresponding variation zero and supplies no natural condition for that variable. These are the natural boundary conditions for second-derivative functionals, with the value and slope variations treated independently.

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