Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-07-20T05:21:20Z https://physics.stackexchange.com/feeds/question/468928 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://physics.stackexchange.com/q/468928 0 Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary PiKindOfGuy https://physics.stackexchange.com/users/113773 2019-03-27T09:35:49Z 2019-03-27T09:56:03Z <p>Suppose we want to make an integral <span class="math-container">$S$</span> of the form <span class="math-container">$$S = \int_{x_1}^{x_2} f\left[y_1(x), \dots, y_n(x), y'_1(x), \dots, y'_n(x), x\right]dx$$</span> stationary with the constraint <span class="math-container">$y_1\left(x_1\right) = \tilde{y}_1, \dots, y_n(x_1) = \tilde{y}_n, y_1\left(x_2\right) = \bar{y}_1, \dots, y_n(x_2) = \bar{y}_n$</span>. Is it <em>necessary</em>, <em>sufficient</em>, or <em>necessary and sufficient</em> for the <span class="math-container">$y_1(x), \dots, y_n(x)$</span> to satisfy the <span class="math-container">$n$</span> Euler-Lagrange equations, <span class="math-container">$$\frac{\partial f}{\partial y_1} = \frac{d}{dx}\frac{\partial f}{\partial y'_1}, \dots,\frac{\partial f}{\partial y_n} = \frac{d}{dx}\frac{\partial f}{\partial y'_n},$$</span> in order to make <span class="math-container">$S$</span> stationary?</p> https://physics.stackexchange.com/questions/468928/-/468933#468933 1 Answer by Qmechanic for Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary Qmechanic https://physics.stackexchange.com/users/2451 2019-03-27T09:56:03Z 2019-03-27T09:56:03Z <p>Necessary and sufficient, because <a href="https://en.wikipedia.org/wiki/Stationary_point" rel="nofollow noreferrer">stationary paths</a> are <em>by definition</em> paths where the <a href="https://en.wikipedia.org/wiki/Functional_derivative" rel="nofollow noreferrer">functional/variational derivative</a> vanishes, i.e. <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation" rel="nofollow noreferrer">Euler-Lagrange (EL) equations</a> are satisfied.</p>