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Free-endpoint normal mode of a coupled variational functional

For

I [y, z] = \int_{0}^{x_{0}} (y^{'2} + z^{'2} + 2 yz) d x, y (0) = z (0) = 0,

with free terminal values, the Euler-Lagrange equations and natural conditions give a nonzero stationary family exactly when

x_{0} = (k + \frac{1}{2}) π

. It is

y = C sin x, z = - C sin x .

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