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Suppose we want to make an integral $S$ of the form $$S = \int_{x_1}^{x_2} f\left[y_1(x), \dots, y_n(x), y'_1(x), \dots, y'_n(x), x\right]dx$$ stationary with the constraint $y_1\left(x_1\right) = \tilde{y}_1, \dots, y_n(x_1) = \tilde{y}_n, y_1\left(x_2\right) = \bar{y}_1, \dots, y_n(x_2) = \bar{y}_n$. Is it necessary, sufficient, or necessary and sufficient for the $y_1(x), \dots, y_n(x)$ to satisfy the $n$ Euler-Lagrange equations, $$\frac{\partial f}{\partial y_1} = \frac{d}{dx}\frac{\partial f}{\partial y'_1}, \dots,\frac{\partial f}{\partial y_n} = \frac{d}{dx}\frac{\partial f}{\partial y'_n},$$ in order to make $S$ stationary?

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Necessary and sufficient, because stationary paths are by definition paths where the functional/variational derivative vanishes, i.e. Euler-Lagrange (EL) equations are satisfied.

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