Editing Euler–Lagrange equation

{{Short description|Second-order partial differential equation describing motion of mechanical system}}
In the [[calculus of variations]] and [[classical mechanics]], the '''Euler–Lagrange equations'''<ref>{{cite book|first=Charles|last=Fox|title=An introduction to the calculus of variations|publisher=Courier Dover Publications|year=1987|isbn=978-0-486-65499-7}}</ref> are a system of second-order [[ordinary differential equation]]s whose solutions are [[stationary point]]s of the given [[action (physics)|action functional]]. The equations were discovered in the 1750s by Swiss mathematician [[Leonhard Euler]] and Italian mathematician [[Joseph-Louis Lagrange]].

Because a differentiable functional is stationary at its local [[maxima and minima|extrema]], the Euler–Lagrange equation is useful for solving [[optimization (mathematics)|optimization]] problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to [[Fermat's theorem (stationary points)|Fermat's theorem]] in [[calculus]], stating that at any point where a differentiable function attains a local extremum its [[derivative (mathematics)|derivative]] is zero. 
In [[Lagrangian mechanics]], according to [[Hamilton's principle]] of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the [[action (physics)#Action (functional)|action]] of the system. In this context Euler equations are usually called '''Lagrange equations'''. In [[classical mechanics]],<ref name="Goldstein"> {{cite book|author1-link=Herbert Goldstein|author2-link=Charles P. Poole|first1=H.|last1=Goldstein|first2=C.P.|last2=Poole|first3=J.|last3=Safko|title=Classical Mechanics|publisher=Addison Wesley|year=2014|edition=3rd}}</ref> it is equivalent to [[Newton's laws of motion]]; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws.  This is particularly useful when analyzing systems whose force vectors are particularly complicated. It has the advantage that it takes the same form in any system of [[generalized coordinate]]s, and it is better suited to generalizations. In [[classical field theory]] there is an [[classical field theory#Lagrangian dynamics|analogous equation]] to calculate the dynamics of a [[field (physics)|field]].

==History==
[[File:Tautochrone curve.gif|thumb|The Euler–Lagrange equation was developed in connection with their studies of the [[tautochrone]] problem. ]]
The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.

Lagrange solved this problem in 1755  and sent the solution to Euler. Both further developed Lagrange's method and applied it to [[mechanics]], which led to the formulation of [[Lagrangian mechanics]]. Their correspondence ultimately led to the [[calculus of variations]], a term coined by Euler himself in 1766.<ref>[http://numericalmethods.eng.usf.edu/anecdotes/lagrange.pdf A short biography of Lagrange] {{webarchive|url=https://web.archive.org/web/20070714022022/http://numericalmethods.eng.usf.edu/anecdotes/lagrange.pdf |date=2007-07-14 }}</ref>

==Statement==
Let <math>(X,L)</math> be a [[real dynamical system]] with <math>n</math> degrees of freedom. Here <math>X</math> is the [[configuration space (physics)|configuration space]] and <math>L=L(t,{\boldsymbol q}(t), {\boldsymbol v}(t))</math> the ''[[Lagrangian mechanics#Lagrangian|Lagrangian]]'', i.e. a smooth [[real-valued function]] such that <math>{\boldsymbol q}(t) \in X,</math> and <math>{\boldsymbol v}(t)</math> is an <math>n</math>-dimensional "vector of speed". (For those familiar with [[differential geometry]], <math>X</math> is a [[smooth manifold]], and <math>L : {\mathbb R}_t \times X \times TX \to {\mathbb R},</math> where <math>TX</math> is the [[tangent bundle]] of <math>X).</math>

Let <math>{\cal P}(a,b,\boldsymbol x_a,\boldsymbol x_b)</math> be the set of smooth paths <math>\boldsymbol q: [a,b] \to X</math> for which <math>\boldsymbol q(a) = \boldsymbol x_a</math> and <math>\boldsymbol q(b) = \boldsymbol x_b. </math>  

The [[action (physics)|action functional]] <math>S : {\cal P}(a,b,\boldsymbol x_a,\boldsymbol x_b) \to \mathbb{R}</math> is defined via
<math display="block"> S[\boldsymbol q] = \int_a^b L(t,\boldsymbol q(t),\dot{\boldsymbol q}(t))\, dt.</math>

A path <math>\boldsymbol q \in {\cal P}(a,b,\boldsymbol x_a,\boldsymbol x_b)</math> is a [[stationary point]] of <math>S</math> if and only if

{{Equation box 1
|indent =:
|equation = <math>\frac{\partial L}{\partial q^i}(t,\boldsymbol q(t),\dot{\boldsymbol q}(t))-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot q^i}(t,\boldsymbol q(t),\dot{\boldsymbol q}(t)) = 0,
\quad i = 1, \dots, n.</math>
|border colour = #50C878
|background colour = #ECFCF4}} 
Here, <math>\dot{\boldsymbol q}(t) </math> is the [[time derivative]] of <math>\boldsymbol q(t).</math> When we say stationary point, we mean a stationary point of <math>S</math> with respect to any small perturbation in <math>\boldsymbol q</math>. See proofs below for more rigorous detail.

{{math proof|title=Derivation of the one-dimensional Euler–Lagrange equation|proof=
The derivation of the one-dimensional Euler–Lagrange equation is one of the classic proofs in [[mathematics]]. It relies on the [[fundamental lemma of calculus of variations]].

We wish to find a function <math>f</math> which satisfies the boundary conditions <math>f(a) = A</math>, <math>f(b) = B</math>, and which extremizes the functional
<math display="block"> J[f] = \int_a^b L(x,f(x),f'(x))\, \mathrm{d}x\ . </math>

We assume that <math>L</math> is twice continuously differentiable.<ref name='CourantP184'>{{harvnb|Courant|Hilbert|1953|p=184}}</ref> A weaker assumption can be used, but the proof becomes more difficult.{{Citation needed|date=September 2013}}

If <math>f</math> extremizes the functional subject to the boundary conditions, then any slight perturbation of <math>f</math> that preserves the boundary values must either increase <math>J</math> (if <math>f</math> is a minimizer) or decrease <math>J</math> (if <math>f</math> is a maximizer).

Let <math>f + \varepsilon \eta</math> be the result of such a perturbation <math>\varepsilon \eta</math> of <math>f</math>, where <math>\varepsilon</math> is small and <math>\eta</math> is a differentiable function satisfying <math>\eta (a) = \eta (b) = 0</math>. Then define
<math display="block"> \Phi(\varepsilon) =  J[f+\varepsilon\eta] = \int_a^b L(x,f(x)+\varepsilon\eta(x), f'(x)+\varepsilon\eta'(x))\, \mathrm{d}x \ .</math>

We now wish to calculate the [[total derivative]] of <math> \Phi</math> with respect to ''ε''.
<math display="block">\begin{align}\frac{\mathrm{d} \Phi}{\mathrm{d} \varepsilon} &= \frac{\mathrm d}{\mathrm d\varepsilon}\int_a^b L(x,f(x)+\varepsilon\eta(x), f'(x)+\varepsilon\eta'(x)) \, \mathrm{d}x \\
&= \int_a^b \frac{\mathrm d}{\mathrm d\varepsilon} L(x,f(x)+\varepsilon\eta(x), f'(x)+\varepsilon\eta'(x)) \, \mathrm{d}x
\\
&= \int_a^b \left[\eta(x)\frac{\partial L}{\partial {f} }(x,f(x)+\varepsilon\eta(x),f'(x)+\varepsilon\eta'(x)) + \eta'(x)\frac{\partial L}{\partial f'}(x,f(x)+\varepsilon\eta(x),f'(x)+\varepsilon\eta'(x))\right] \mathrm{d}x \ . \end{align}</math>

The third line follows from the fact that <math> x </math> does not depend on <math> \varepsilon </math>, i.e. <math> \frac{\mathrm{d} x}{\mathrm{d} \varepsilon} = 0</math>.

When <math>\varepsilon = 0</math>, <math>\Phi</math> has an [[extremum]] value, so that
<math display="block"> \left.\frac{\mathrm d \Phi}{\mathrm d\varepsilon}\right|_{\varepsilon=0}  = \int_a^b \left[ \eta(x) \frac{\partial L}{\partial f}(x,f(x),f'(x)) + \eta'(x) \frac{\partial L}{\partial f'}(x,f(x),f'(x)) \,\right]\,\mathrm{d}x = 0 \ .</math>

The next step is to use [[integration by parts]] on the second term of the integrand, yielding
<math display="block"> \int_a^b \left[ \frac{\partial L}{\partial f}(x,f(x),f'(x)) - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial L}{\partial f'}(x,f(x),f'(x)) \right] \eta(x)\,\mathrm{d}x + \left[ \eta(x) \frac{\partial L}{\partial f'}(x,f(x),f'(x)) \right]_a^b = 0 \ . </math>

Using the boundary conditions <math>\eta (a) = \eta (b) = 0</math>,
<math display="block"> \int_a^b \left[ \frac{\partial L}{\partial f}(x,f(x),f'(x)) - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial L}{\partial f'}(x,f(x),f'(x)) \right] \eta(x)\,\mathrm{d}x = 0 \, . </math>

Applying the [[fundamental lemma of calculus of variations]] now yields the Euler–Lagrange equation
<math display="block"> \frac{\partial L}{\partial f}(x,f(x),f'(x)) - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial L}{\partial f'}(x,f(x),f'(x)) = 0 \, . </math>
}}
{{math proof|title=Alternative derivation of the one-dimensional Euler–Lagrange equation|proof=
Given a functional
<math display="block">J = \int^b_a L(t, y(t), y'(t))\,\mathrm{d}t</math>
on <math>C^1([a, b])</math> with the boundary conditions <math>y(a) = A</math> and <math>y(b) = B</math>, we proceed by approximating the extremal curve by a polygonal line with <math>n</math> segments and passing to the limit as the number of segments grows arbitrarily large.

Divide the interval <math>[a, b]</math> into <math>n</math> equal segments with endpoints <math>t_0 = a, t_1, t_2, \ldots, t_n = b</math> and let <math>\Delta t = t_k - t_{k - 1}</math>. Rather than a smooth function <math>y(t)</math> we consider the polygonal line with vertices <math>(t_0, y_0),\ldots,(t_n, y_n)</math>, where <math>y_0 = A</math> and <math>y_n = B</math>. Accordingly, our functional becomes a real function of <math>n - 1</math> variables given by
<math display="block">J(y_1, \ldots, y_{n - 1}) \approx \sum^{n - 1}_{k = 0}L\left(t_k, y_k, \frac{y_{k + 1} - y_k}{\Delta t}\right)\Delta t.</math>

Extremals of this new functional defined on the discrete points <math>t_0,\ldots,t_n</math> correspond to points where
<math display="block">\frac{\partial J(y_1,\ldots,y_n)}{\partial y_m} = 0.</math>

Note that change of <math> y_m </math> affects L not only at m but also at m-1 for the derivative of the 3rd argument. 
<math display="block"> L(\text{3rd argument}) \left( \frac{y_{m+1} - (y_{m} + \Delta y_{m})}{\Delta t} \right) = L \left(\frac{y_{m+1} - y_{m}}{\Delta t}\right) - \frac{\partial L}{\partial y'} \frac{\Delta y_m}{\Delta t} 
</math><math display="block">L \left( \frac{(y_{m} + \Delta y_{m}) - y_{m-1}}{\Delta t} \right) 
= L \left(\frac{y_{m} - y_{m-1}}{\Delta t}\right) + \frac{\partial L}{\partial y'} \frac{\Delta y_m}{\Delta t}</math>

Evaluating the partial derivative gives
<math display="block">\frac{\partial J}{\partial y_m} = L_y\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right)\Delta t + L_{y'}\left(t_{m - 1}, y_{m - 1}, \frac{y_m - y_{m - 1}}{\Delta t}\right) - L_{y'}\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right).</math>

Dividing the above equation by <math>\Delta t</math> gives
<math display="block">\frac{\partial J}{\partial y_m \Delta t} = L_y\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right) - \frac{1}{\Delta t}\left[L_{y'}\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right) - L_{y'}\left(t_{m - 1}, y_{m - 1}, \frac{y_m - y_{m - 1}}{\Delta t}\right)\right],</math>
and taking the limit as <math>\Delta t \to 0</math> of the right-hand side of this expression yields
<math display="block">L_y - \frac{\mathrm{d}}{\mathrm{d}t}L_{y'} = 0.</math>

The left hand side of the previous equation is the [[functional derivative]] <math>\delta J/\delta y</math> of the functional <math>J</math>. A necessary condition for a differentiable functional to have an extremum on some function is that its functional derivative at that function vanishes, which is granted by the last equation.
}}

==Example==
A standard example{{Citation needed|reason=this example seems like a poor one as it is not even presented as a dynamical system|date=July 2023}} is finding the real-valued function ''y''(''x'') on the interval [''a'', ''b''], such that ''y''(''a'') = ''c'' and ''y''(''b'') = ''d'', for which the [[path (topology)|path]]  [[arc length|length]] along the [[Curve#length of a curve|curve]] traced by ''y'' is as short as possible. 
:<math> \text{s} = \int_{a}^{b} \sqrt{\mathrm{d}x^2+\mathrm{d}y^2} = \int_{a}^{b} \sqrt{1+y'^2}\,\mathrm{d}x,</math>
the integrand function being <math display="inline"> L(x,y, y') = \sqrt{1+y'^2} </math>.

The partial derivatives of ''L'' are:
:<math>\frac{\partial L(x, y, y')}{\partial y'} = \frac{y'}{\sqrt{1 + y'^2}} \quad \text{and} \quad
\frac{\partial L(x, y, y')}{\partial y} = 0.</math>
By substituting these into the Euler–Lagrange equation, we obtain
:<math> 
\begin{align}
\frac{\mathrm{d}}{\mathrm{d}x} \frac{y'(x)}{\sqrt{1 + (y'(x))^2}} &= 0 \\ 
\frac{y'(x)}{\sqrt{1 + (y'(x))^2}} &= C = \text{constant} \\
\Rightarrow y'(x)&= \frac{C}{\sqrt{1-C^2}} =: A \\
\Rightarrow y(x) &= Ax + B
\end{align}
</math>
that is, the function must have a constant first derivative, and thus its [[graph of a function|graph]] is a [[straight line]].

==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)}}

==Generalization to manifolds==
Let <math>M</math> be a [[smooth manifold]], and let <math>C^\infty([a,b])</math> denote the space of [[smooth functions]] <math>f\colon [a,b]\to M</math>. Then, for functionals <math>S\colon C^\infty ([a,b])\to \mathbb{R}</math> of the form
:<math>
S[f]=\int_a^b (L\circ\dot{f})(t)\,\mathrm{d} t
</math>
where <math>L\colon TM\to\mathbb{R}</math> is the Lagrangian, the statement <math>\mathrm{d} S_f=0</math> is equivalent to the statement that, for all <math>t\in [a,b]</math>, each coordinate frame [[fiber bundle|trivialization]] <math>(x^i,X^i)</math> of a neighborhood of <math>\dot{f}(t)</math> yields the following <math>\dim M</math> equations:
:<math>
\forall i:\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial X^i}\bigg|_{\dot{f}(t)}=\frac{\partial L}{\partial x^i}\bigg|_{\dot{f}(t)}.
</math>
Euler-Lagrange equations can also be written in a coordinate-free form as <ref>{{Cite book |last1=José |last2=Saletan |year=1998 |title=Classical Dynamics: A contemporary approach |url=https://www.cambridge.org/in/academic/subjects/physics/general-and-classical-physics/classical-dynamics-contemporary-approach,%20https://www.cambridge.org/in/academic/subjects/physics/general-and-classical-physics |access-date=2023-09-12 |publisher=Cambridge University Press |language=en |isbn=9780521636360}}</ref>

:<math>
\mathcal{L}_\Delta \theta_L=dL
</math>
where <math>\theta_L</math> is the canonical momenta [[One-form (differential geometry)|1-form]] corresponding to the Lagrangian <math>L</math>. The vector field generating time translations is denoted by <math>\Delta</math> and the [[Lie derivative]] is denoted by <math>\mathcal{L}</math>. One can use local charts <math>(q^\alpha,\dot{q}^\alpha)</math> in which <math>\theta_L=\frac{\partial L}{\partial \dot{q}^\alpha}dq^\alpha</math> and <math>\Delta:=\frac{d}{dt}=\dot{q}^\alpha\frac{\partial}{\partial q^\alpha}+\ddot{q}^\alpha\frac{\partial}{\partial \dot{q}^\alpha}</math> and use coordinate expressions for the Lie derivative to see equivalence with coordinate expressions of the Euler Lagrange equation. The coordinate free form is particularly suitable for geometrical interpretation of the Euler Lagrange equations.

==See also==
{{Wiktionary|Euler–Lagrange equation}}
*[[Lagrangian mechanics]]
*[[Hamiltonian mechanics]]
*[[Analytical mechanics]]
*[[Beltrami identity]]
*[[Functional derivative]]

==Notes==
{{Reflist}}

==References==
* {{springer|title=Lagrange equations (in mechanics)|id=p/l057150}}
* {{MathWorld|urlname=Euler-LagrangeDifferentialEquation|title=Euler-Lagrange Differential Equation}}
* {{planetmath| urlname=CalculusOfVariations| title=Calculus of Variations}}
* {{cite book |last=Gelfand |first=Izrail Moiseevich |author-link=Israel Gelfand |title=Calculus of Variations |publisher=Dover |year=1963 |isbn=0-486-41448-5}}
* Roubicek, T.: ''[https://web.archive.org/web/20150510023928/http://www.wiley-vch.de/books/sample/3527411887_c17.pdf Calculus of variations]. Chap.17 in: [https://web.archive.org/web/20150510021514/http://www.wiley-vch.de/publish/en/books/forthcomingTitles/MA00/3-527-41188-7/?sID=nrgsqk516u2v9ffab8u7io1dq4 Mathematical Tools for Physicists]. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, {{ISBN|978-3-527-41188-7}}, pp.&nbsp;551–588.

{{Leonhard Euler}}
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{{DEFAULTSORT:Euler-Lagrange Equation}}
[[Category:Eponymous equations of mathematics]]
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[[Category:Ordinary differential equations]]
[[Category:Partial differential equations]]
[[Category:Calculus of variations]]
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[[Category:Leonhard Euler]]