Sample Final Exam Problem 5
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Taking the first variation of the functional , we have
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Integrating the first terms of the above expression by parts, we have,
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Integrating by parts again,
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Expanding out,
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Rearranging,
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Using the principle of minimum potential energy, for the functional
to have a minimum, we must have . Therefore, we have
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Since and are arbitrary, the Euler equation
for this problem is
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and the associated boundary conditions are
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and
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