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Euler–Lagrange equation
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===Single function of single variable with higher derivatives=== The stationary values of the functional :<math> I[f] = \int_{x_0}^{x_1} \mathcal{L}(x, f, f', f'', \dots, f^{(k)})~\mathrm{d}x ~;~~ f' := \cfrac{\mathrm{d}f}{\mathrm{d}x}, ~f'' := \cfrac{\mathrm{d}^2f}{\mathrm{d}x^2}, ~ f^{(k)} := \cfrac{\mathrm{d}^kf}{\mathrm{d}x^k} </math> can be obtained from the Euler–Lagrange equation<ref name=Courant>{{cite book | last1=Courant | first1=R | author-link1=Richard Courant | last2=Hilbert | first2=D | author-link2=David Hilbert | title = Methods of Mathematical Physics | volume = I | edition = First English | publisher = Interscience Publishers, Inc | year = 1953 | location = New York | isbn = 978-0471504474}}</ref> :<math> \cfrac{\partial \mathcal{L}}{\partial f} - \cfrac{\mathrm{d}}{\mathrm{d} x}\left(\cfrac{\partial \mathcal{L}}{\partial f'}\right) + \cfrac{\mathrm{d}^2}{\mathrm{d} x^2}\left(\cfrac{\partial \mathcal{L}}{\partial f''}\right) - \dots + (-1)^k \cfrac{\mathrm{d}^k}{\mathrm{d} x^k}\left(\cfrac{\partial \mathcal{L}}{\partial f^{(k)}}\right) = 0 </math> under fixed boundary conditions for the function itself as well as for the first <math>k-1</math> derivatives (i.e. for all <math>f^{(i)}, i \in \{0, ..., k-1\}</math>). The endpoint values of the highest derivative <math>f^{(k)}</math> remain flexible.
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