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Euler–Lagrange equation

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Second-order partial differential equation describing motion of mechanical system

In thecalculus of variations andclassical mechanics, theEuler–Lagrange equations^[1] are a system of second-orderordinary differential equations whose solutions arestationary points of the givenaction functional. The equations were discovered in the 1750s by Swiss mathematicianLeonhard Euler and Italian mathematicianJoseph-Louis Lagrange.

Because a differentiable functional is stationary at its localextrema, the Euler–Lagrange equation is useful for solvingoptimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous toFermat's theorem incalculus, stating that at any point where a differentiable function attains a local extremum itsderivative is zero. InLagrangian mechanics, according toHamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for theaction of the system. In this context Euler equations are usually calledLagrange equations. Inclassical mechanics,^[2] it is equivalent toNewton'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 ofgeneralized coordinates, and it is better suited to generalizations. Inclassical field theory there is ananalogous equation to calculate the dynamics of afield.

History

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The Euler–Lagrange equation was developed in connection with their studies of thetautochrone problem.

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of thetautochrone 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 tomechanics, which led to the formulation ofLagrangian mechanics. Their correspondence ultimately led to thecalculus of variations, a term coined by Euler himself in 1766.^[3]

Statement

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Let $(X,L)$ be areal dynamical system with $n {\displaystyle n}$ degrees of freedom. Here $X {\displaystyle X}$ is theconfiguration space and $L=L(t,{\boldsymbol {q}}(t),{\boldsymbol {v}}(t))$ theLagrangian, i.e. a smoothreal-valued function such that ${\boldsymbol {q}}(t)\in X,$ and ${\boldsymbol {v}}(t)$ is an $n {\displaystyle n}$ -dimensional "vector of speed". (For those familiar withdifferential geometry, $X {\displaystyle X}$ is asmooth manifold, and $L\colon {\mathbb {R} }_{t}\times TX\to {\mathbb {R} },$ where $T X {\displaystyle TX}$ is thetangent bundle of $X).$ ^[4]

Let ${\cal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})$ be the set of smooth paths ${\boldsymbol {q}}:[a,b]\to X$ for which ${\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}$ and ${\boldsymbol {q}}(b)={\boldsymbol {x}}_{b}.$

Theaction functional $S\colon {\cal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})\to \mathbb {R}$ is defined via $S[{\boldsymbol {q}}]=\int _{a}^{b}L(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))\,dt.$

A path ${\boldsymbol {q}}\in {\cal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})$ is astationary point of $S {\displaystyle S}$ if and only if

${\frac {\partial L}{\partial q^{i}}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}^{i}}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))=0,\quad i=1,\dots ,n.$

Here, ${\dot {\boldsymbol {q}}}(t)$ is thetime derivative of ${\boldsymbol {q}}(t).$ When we say stationary point, we mean a stationary point of $S {\displaystyle S}$ with respect to any small perturbation in ${\boldsymbol {q}}$ . See proofs below for more rigorous detail.

Derivation of the one-dimensional Euler–Lagrange equation

The derivation of the one-dimensional Euler–Lagrange equation is one of the classic proofs inmathematics. It relies on thefundamental lemma of calculus of variations.

We wish to find a function $f {\displaystyle f}$ which satisfies the boundary conditions $f(a)=A$ , $f(b)=B$ , and which extremizes the functional $J[f]=\int _{a}^{b}L(x,f(x),f'(x))\,\mathrm {d} x\ .$

We assume that $L {\displaystyle L}$ is twice continuously differentiable.^[5] A weaker assumption can be used, but the proof becomes more difficult.^{[citation needed]}

If $f {\displaystyle f}$ extremizes the functional subject to the boundary conditions, then any slight perturbation of $f {\displaystyle f}$ that preserves the boundary values must either increase $J {\displaystyle J}$ (if $f {\displaystyle f}$ is a minimizer) or decrease $J {\displaystyle J}$ (if $f {\displaystyle f}$ is a maximizer).

Let $f+\varepsilon \eta$ be the result of such a perturbation $\varepsilon \eta$ of $f {\displaystyle f}$ , where $\varepsilon$ is small and $\eta$ is a differentiable function satisfying $\eta (a)=\eta (b)=0$ . Then define $\Phi (\varepsilon )=J[f+\varepsilon \eta ]=\int _{a}^{b}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\ .$

We now wish to calculate thetotal derivative of $\Phi$ with respect toε. ${\begin{aligned}{\frac {\mathrm {d} \Phi }{\mathrm {d} \varepsilon }}&={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\\&=\int _{a}^{b}{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\\&=\int _{a}^{b}\left[\eta (x){\frac {\partial L}{\partial {f}}}(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))+\eta '(x){\frac {\partial L}{\partial f'}}(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\right]\mathrm {d} x\ .\end{aligned}}$

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

When $\varepsilon =0$ , $\Phi$ has anextremum value, so that $\left.{\frac {\mathrm {d} \Phi }{\mathrm {d} \varepsilon }}\right|_{\varepsilon =0}=\int _{a}^{b}\left[\eta (x){\frac {\partial L}{\partial f}}(x,f(x),f'(x))+\eta '(x){\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\,\right]\,\mathrm {d} x=0\ .$

The next step is to useintegration by parts on the second term of the integrand, yielding $\int _{a}^{b}\left[{\frac {\partial L}{\partial f}}(x,f(x),f'(x))-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\right]\eta (x)\,\mathrm {d} x+\left[\eta (x){\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\right]_{a}^{b}=0\ .$

Using the boundary conditions $\eta (a)=\eta (b)=0$ , $\int _{a}^{b}\left[{\frac {\partial L}{\partial f}}(x,f(x),f'(x))-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial L}{\partial f'}}(x,f(x),f'(x))\right]\eta (x)\,\mathrm {d} x=0\,.$

Applying thefundamental lemma of calculus of variations now yields the Euler–Lagrange equation ${\frac {\partial L}{\partial f}}(x,f(x),f'(x))-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial L}{\partial f'}}(x,f(x),f'(x))=0\,.$

Alternative derivation of the one-dimensional Euler–Lagrange equation

Given a functional $J=\int _{a}^{b}L(t,y(t),y'(t))\,\mathrm {d} t$ on $C^{1}([a,b])$ with the boundary conditions $y(a)=A$ and $y(b)=B$ , we proceed by approximating the extremal curve by a polygonal line with $n {\displaystyle n}$ segments and passing to the limit as the number of segments grows arbitrarily large.

Divide the interval $[a,b]$ into $n {\displaystyle n}$ equal segments with endpoints $t_{0}=a,t_{1},t_{2},\ldots ,t_{n}=b$ and let $\Delta t=t_{k}-t_{k-1}$ . Rather than a smooth function $y(t)$ we consider the polygonal line with vertices $(t_{0},y_{0}),\ldots ,(t_{n},y_{n})$ , where $y_{0}=A$ and $y_{n}=B$ . Accordingly, our functional becomes a real function of $n-1$ variables given by $J(y_{1},\ldots ,y_{n-1})\approx \sum _{k=0}^{n-1}L\left(t_{k},y_{k},{\frac {y_{k+1}-y_{k}}{\Delta t}}\right)\Delta t.$

Extremals of this new functional defined on the discrete points $t_{0},\ldots ,t_{n}$ correspond to points where ${\frac {\partial J(y_{1},\ldots ,y_{n})}{\partial y_{m}}}=0.$

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

Evaluating the partial derivative gives ${\frac {\partial J}{\partial y_{m}}}=L_{y}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right)\Delta t+L_{y'}\left(t_{m-1},y_{m-1},{\frac {y_{m}-y_{m-1}}{\Delta t}}\right)-L_{y'}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right).$

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

The left hand side of the previous equation is thefunctional derivative $\delta J/\delta y$ of the functional $J {\displaystyle J}$ . 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

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A standard example^{[citation needed]} is finding the real-valued functiony(x) on the interval [a,b], such thaty(a) =c andy(b) =d, for which thepath length along thecurve traced byy is as short as possible.

{\text{s}}=\int _{a}^{b}{\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}}}=\int _{a}^{b}{\sqrt {1+y'^{2}}}\,\mathrm {d} x,

the integrand function being ${\textstyle L(x,y,y')={\sqrt {1+y'^{2}}}}$ .

The partial derivatives ofL are:

{\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.

By substituting these into the Euler–Lagrange equation, we obtain

{\begin{aligned}{\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{aligned}}

that is, the function must have a constant first derivative, and thus itsgraph is astraight line.

Generalizations

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Single function of single variable with higher derivatives

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The stationary values of the functional

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} ^{2}f}{\mathrm {d} x^{2}}},~f^{(k)}:={\cfrac {\mathrm {d} ^{k}f}{\mathrm {d} x^{k}}}

can be obtained from the Euler–Lagrange equation^[6]

{\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

under fixed boundary conditions for the function itself as well as for the first $k-1$ derivatives (i.e. for all $f^{(i)},i\in \{0,...,k-1\}$ ). The endpoint values of the highest derivative $f^{(k)}$ remain flexible.

Several functions of single variable with single derivative

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If the problem involves finding several functions ( $f_{1},f_{2},\dots ,f_{m}$ ) of a single independent variable ( $x {\displaystyle x}$ ) that define an extremum of the functional

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}}

then the corresponding Euler–Lagrange equations are^[7]

{\begin{aligned}{\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{aligned}}

Single function of several variables with single derivative

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A multi-dimensional generalization comes from considering a function on n variables. If $\Omega$ is some surface, then

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}}}

is extremized only iff satisfies thepartial differential equation

{\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.

Whenn = 2 and functional ${\mathcal {I}}$ is theenergy functional, this leads to the soap-filmminimal surface problem.

Several functions of several variables with single derivative

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If there are several unknown functions to be determined and several variables such that

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}}}

the system of Euler–Lagrange equations is^[6]

{\begin{aligned}{\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{aligned}}

Single function of two variables with higher derivatives

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If there is a single unknown functionf to be determined that is dependent on two variablesx₁ andx₂ and if the functional depends on higher derivatives off up ton-th order such that

{\begin{aligned}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{aligned}}

then the Euler–Lagrange equation is^[6]

{\begin{aligned}{\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{aligned}}

which can be represented shortly as:

{\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

wherein $\mu _{1}\dots \mu _{j}$ are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the $\mu _{1}\dots \mu _{j}$ indices is only over $\mu _{1}\leq \mu _{2}\leq \ldots \leq \mu _{j}$ in order to avoid counting the samepartial derivative multiple times, for example $f_{12}=f_{21}$ appears only once in the previous equation.

Several functions of several variables with higher derivatives

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If there arep unknown functionsf_i to be determined that are dependent onm variablesx₁ ...x_m and if the functional depends on higher derivatives of thef_i up ton-th order such that

{\begin{aligned}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{aligned}}

where $\mu _{1}\dots \mu _{j}$ are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is

{\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

where the summation over the $\mu _{1}\dots \mu _{j}$ is avoiding counting the same derivative $f_{i,\mu _{1}\mu _{2}}=f_{i,\mu _{2}\mu _{1}}$ several times, just as in the previous subsection. This can be expressed more compactly as

\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

Field theories

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Main article:Lagrangian (field theory)

Generalization to manifolds

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Let $M {\displaystyle M}$ be asmooth manifold, and let $C^{\infty }([a,b])$ denote the space ofsmooth functions $f\colon [a,b]\to M$ . Then, for functionals $S\colon C^{\infty }([a,b])\to \mathbb {R}$ of the form

S[f]=\int _{a}^{b}(L\circ {\dot {f}})(t)\,\mathrm {d} t

where $L\colon TM\to \mathbb {R}$ is the Lagrangian, the statement $\mathrm {d} S_{f}=0$ is equivalent to the statement that, for all $t\in [a,b]$ , each coordinate frametrivialization $(x^{i},X^{i})$ of a neighborhood of ${\dot {f}}(t)$ yields the following $\dim M$ equations:

\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)}.

Euler-Lagrange equations can also be written in a coordinate-free form as^[8]

{\mathcal {L}}_{\Delta }\theta _{L}=dL

where $\theta _{L}$ is the canonical momenta1-form corresponding to the Lagrangian $L {\displaystyle L}$ . The vector field generating time translations is denoted by $\Delta$ and theLie derivative is denoted by ${\mathcal {L}}$ . One can use local charts $(q^{\alpha },{\dot {q}}^{\alpha })$ in which $\theta _{L}={\frac {\partial L}{\partial {\dot {q}}^{\alpha }}}dq^{\alpha }$ and $\Delta :={\frac {d}{dt}}={\dot {q}}^{\alpha }{\frac {\partial }{\partial q^{\alpha }}}+{\ddot {q}}^{\alpha }{\frac {\partial }{\partial {\dot {q}}^{\alpha }}}$ 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.

Notes

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^Fox, Charles (1987).An introduction to the calculus of variations. Courier Dover Publications.ISBN 978-0-486-65499-7.
^Goldstein, H.;Poole, C.P.; Safko, J. (2014).Classical Mechanics (3rd ed.). Addison Wesley.
^A short biography of Lagrange Archived 2007-07-14 at theWayback Machine
^"Elements of the Calculus of Variations",Lectures on the Geometry of Manifolds (3 ed.), WORLD SCIENTIFIC, pp. 201–226, November 2020,doi:10.1142/9789811214820_0005,ISBN 978-981-12-1481-3, retrieved2025-10-31
^Courant & Hilbert 1953, p. 184
^^a ^b ^cCourant, R;Hilbert, D (1953).Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc.ISBN 978-0471504474.{{cite book}}:ISBN / Date incompatibility (help)
^Weinstock, R. (1952).Calculus of Variations with Applications to Physics and Engineering. New York: McGraw-Hill.
^José; Saletan (1998).Classical Dynamics: A contemporary approach. Cambridge University Press.ISBN 9780521636360. Retrieved2023-09-12.

References

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"Lagrange equations (in mechanics)",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Weisstein, Eric W."Euler-Lagrange Differential Equation".MathWorld.
Calculus of Variations atPlanetMath.
Gelfand, Izrail Moiseevich (1963).Calculus of Variations. Dover.ISBN 0-486-41448-5.{{cite book}}:ISBN / Date incompatibility (help)
Roubicek, T.:Calculus of variations. Chap.17 in:Mathematical Tools for Physicists. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014,ISBN 978-3-527-41188-7, pp. 551–588.

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