Editing Euler–Lagrange equation (section)

==Generalizations ==

===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.

===Several functions of single variable with single derivative===
If the problem involves finding several functions (<math>f_1, f_2, \dots, f_m</math>) of a single independent variable (<math>x</math>) that define an extremum of the functional
:<math>
    I[f_1,f_2, \dots, f_m] = \int_{x_0}^{x_1} \mathcal{L}(x, f_1, f_2, \dots, f_m, f_1', f_2', \dots, f_m')~\mathrm{d}x
    ~;~~ f_i' := \cfrac{\mathrm{d}f_i}{\mathrm{d}x}
 </math>
then the corresponding Euler–Lagrange equations are<ref name=Weinstock>{{cite book |last=Weinstock |first=R. |year=1952 |title=Calculus of Variations with Applications to Physics and Engineering |url=https://archive.org/details/calculusofvariat00wein |url-access=registration |publisher=McGraw-Hill |location=New York }}</ref>
:<math>
   \begin{align}
     \frac{\partial \mathcal{L}}{\partial f_i} - \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\partial \mathcal{L}}{\partial f_i'}\right) = 0 ; \quad i = 1, 2, ..., m
   \end{align}
</math>

===Single function of several variables with single derivative===
A multi-dimensional generalization comes from considering a function on n variables. If <math>\Omega</math> is some surface, then

: <math> 
   I[f] = \int_{\Omega} \mathcal{L}(x_1, \dots , x_n, f, f_{1}, \dots , f_{n})\, \mathrm{d}\mathbf{x}\,\! ~;~~
      f_{j} := \cfrac{\partial f}{\partial x_j}
</math>

is extremized only if ''f'' satisfies the [[partial differential equation]]

: <math> \frac{\partial \mathcal{L}}{\partial f} - \sum_{j=1}^{n} \frac{\partial}{\partial x_j}\left(\frac{\partial \mathcal{L}}{\partial f_{j}}\right) = 0. </math>

When ''n'' = 2 and functional <math>\mathcal I</math> is the [[energy functional]], this leads to the soap-film [[minimal surface]] problem.

===Several functions of several variables with single derivative===
If there are several unknown functions to be determined and several variables such that
: <math> 
   I[f_1,f_2,\dots,f_m] = \int_{\Omega} \mathcal{L}(x_1, \dots , x_n, f_1, \dots, f_m, f_{1,1}, \dots , f_{1,n},  \dots, f_{m,1}, \dots, f_{m,n}) \, \mathrm{d}\mathbf{x}\,\! ~;~~
      f_{i,j} := \cfrac{\partial f_i}{\partial x_j}
</math>
the system of Euler–Lagrange equations is<ref name=Courant/>
: <math> 
  \begin{align}
    \frac{\partial \mathcal{L}}{\partial f_1} - \sum_{j=1}^{n} \frac{\partial}{\partial x_j}\left(\frac{\partial \mathcal{L}}{\partial f_{1,j}}\right) &= 0_1 \\
    \frac{\partial \mathcal{L}}{\partial f_2} - \sum_{j=1}^{n} \frac{\partial}{\partial x_j}\left(\frac{\partial \mathcal{L}}{\partial f_{2,j}}\right) &= 0_2 \\
    \vdots \qquad \vdots \qquad &\quad \vdots  \\
    \frac{\partial \mathcal{L}}{\partial f_m} - \sum_{j=1}^{n} \frac{\partial}{\partial x_j}\left(\frac{\partial \mathcal{L}}{\partial f_{m,j}}\right) &= 0_m.
  \end{align}
 </math>

===Single function of two variables with higher derivatives===
If there is a single unknown function ''f'' to be determined that is dependent on two variables ''x''<sub>1</sub> and ''x''<sub>2</sub> and if the functional depends on higher derivatives of ''f'' up to ''n''-th order such that
: <math>
   \begin{align}
     I[f] & = \int_{\Omega} \mathcal{L}(x_1, x_2, f, f_{1}, f_{2}, f_{11}, f_{12}, f_{22},
                                        \dots, f_{22\dots 2})\, \mathrm{d}\mathbf{x} \\
     & \qquad \quad
        f_{i} := \cfrac{\partial f}{\partial x_i} \; , \quad
        f_{ij} := \cfrac{\partial^2 f}{\partial x_i\partial x_j} \; , \;\; \dots
   \end{align} 
</math>
then the Euler–Lagrange equation is<ref name=Courant/>
:<math>
  \begin{align}
    \frac{\partial \mathcal{L}}{\partial f}
    & - \frac{\partial}{\partial x_1}\left(\frac{\partial \mathcal{L}}{\partial f_{1}}\right)
      - \frac{\partial}{\partial x_2}\left(\frac{\partial \mathcal{L}}{\partial f_{2}}\right) 
      + \frac{\partial^2}{\partial x_1^2}\left(\frac{\partial \mathcal{L}}{\partial f_{11}}\right)
      + \frac{\partial^2}{\partial x_1\partial x_2}\left(\frac{\partial \mathcal{L}}{\partial f_{12}}\right)
      + \frac{\partial^2}{\partial x_2^2}\left(\frac{\partial \mathcal{L}}{\partial f_{22}}\right) \\
    & - \dots
      + (-1)^n \frac{\partial^n}{\partial x_2^n}\left(\frac{\partial \mathcal{L}}{\partial f_{22\dots 2}}\right) = 0
  \end{align}
 </math>
which can be represented shortly as:
:<math>
    \frac{\partial \mathcal{L}}{\partial f} +\sum_{j=1}^n \sum_{\mu_1 \leq \ldots \leq \mu_j} (-1)^j \frac{\partial^j}{\partial x_{\mu_{1}}\dots \partial x_{\mu_{j}}} \left( \frac{\partial \mathcal{L} }{\partial f_{\mu_1\dots\mu_j}}\right)=0
 </math>
wherein <math>\mu_1 \dots \mu_j</math> are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the <math>\mu_1 \dots \mu_j</math> indices is only over <math>\mu_1 \leq \mu_2 \leq \ldots \leq \mu_j</math> in order to avoid counting the same [[partial derivative]] multiple times, for example <math>f_{12} = f_{21}</math> appears only once in the previous equation.

===Several functions of several variables with higher derivatives===
If there are ''p'' unknown functions ''f''<sub>i</sub> to be determined that are dependent on ''m'' variables ''x''<sub>1</sub> ... ''x''<sub>m</sub> and if the functional depends on higher derivatives of the ''f''<sub>i</sub> up to ''n''-th order such that
:<math>
   \begin{align}
     I[f_1,\ldots,f_p] & = \int_{\Omega} \mathcal{L}(x_1, \ldots, x_m; f_1,\ldots,f_p; f_{1,1},\ldots,
     f_{p,m}; f_{1,11},\ldots, f_{p,mm};\ldots; f_{p,1\ldots 1}, \ldots, f_{p,m\ldots m})\, \mathrm{d}\mathbf{x} \\
     & \qquad \quad
        f_{i,\mu} := \cfrac{\partial f_i}{\partial x_\mu} \; , \quad
        f_{i,\mu_1\mu_2} := \cfrac{\partial^2 f_i}{\partial x_{\mu_1}\partial x_{\mu_2}} \; , \;\; \dots
   \end{align}
</math>

where <math>\mu_1 \dots \mu_j</math> are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is

:<math>
    \frac{\partial \mathcal{L}}{\partial f_i} +\sum_{j=1}^n \sum_{\mu_1 \leq \ldots \leq \mu_j} (-1)^j \frac{\partial^j}{\partial x_{\mu_{1}}\dots \partial x_{\mu_{j}}} \left( \frac{\partial \mathcal{L} }{\partial f_{i,\mu_1\dots\mu_j}}\right)=0
 </math>

where the summation over the <math>\mu_1 \dots \mu_j</math> is avoiding counting the same derivative <math> f_{i,\mu_1\mu_2} = f_{i,\mu_2\mu_1}</math> several times, just as in the previous subsection. This can be expressed more compactly as

:<math>
\sum_{j=0}^n \sum_{\mu_1 \leq \ldots \leq \mu_j} (-1)^j \partial_{ \mu_{1}\ldots \mu_{j} }^j \left( \frac{\partial \mathcal{L} }{\partial f_{i,\mu_1\dots\mu_j}}\right)=0
 </math>

===Field theories===
{{Main|Lagrangian (field theory)}}