 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2011/g/1/solution

Let

P = F_{y^{'}}

and

Q = F_{y^{''}}

. For a smooth variation

y + ε η

δ J = \int_{a}^{b} (F_{y} η + P η^{'} + Q η^{''}) d x = \int_{a}^{b} (F_{y} - P^{'} + Q^{''}) η d x + [(P - Q^{'}) η + Q η^{'}]_{a}^{b} .

Here the last expression follows by two integrations by parts. Taking variations supported in the interior and applying the fundamental lemma of the calculus of variations gives the Euler-Lagrange equation for a second-derivative functional:

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

Q = 0, P - Q^{'} = 0 at a, b .

If an endpoint value or slope is fixed instead, its corresponding variation vanishes and the associated natural condition is not required.

For

H = y^{''} Q - y^{'} (Q^{'} - P) - F

, differentiate using the chain rule:

H^{'} = y^{'} (P^{'} - Q^{''} - F_{y}) - F_{x} .

The Euler-Lagrange equation cancels the first term. Consequently

H^{'} = - F_{x}

, and if there is no explicit

x

dependence,

H = y^{''} F_{y^{''}} + y^{'} (F_{y^{'}} - (F_{y^{''}})^{'}) - F is constant .

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