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''Euler–Lagrange operator''
 
''Euler–Lagrange operator''
  
A fundamental object, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202301.png" />, in the calculus of variations (cf. also [[Variational calculus|Variational calculus]]), used to formulate the system of partial differential equations, called the Euler–Lagrange equations or the variational equations, that the extremals for variational problems must satisfy (cf. also [[Euler–Lagrange equation|Euler–Lagrange equation]]).
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A fundamental object, $\cal E$, in the calculus of variations (cf. also [[Variational calculus|Variational calculus]]), used to formulate the system of partial differential equations, called the Euler–Lagrange equations or the variational equations, that the extremals for variational problems must satisfy (cf. also [[Euler–Lagrange equation|Euler–Lagrange equation]]).
  
In essence, to each [[Lagrangian|Lagrangian]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202302.png" />, the Euler operator assigns a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202303.png" /> whose components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202305.png" />, are the expressions for the Euler–Lagrange equations.
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In essence, to each [[Lagrangian|Lagrangian]] $L$, the Euler operator assigns a geometric object $\mathcal{E} ( L )$ whose components $\mathcal{E} ^ { a } ( L )$, $a = 1 , \dots , m$, are the expressions for the Euler–Lagrange equations.
  
For trivial fibre bundles (or locally on appropriate charts) and for first-order Lagrangians, the Euler operator is easy to describe. Thus, suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202307.png" /> are open sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202308.png" /> is compact, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e1202309.png" /> is the trivial fibre bundle (cf. also [[Fibre space|Fibre space]]) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023010.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023011.png" /> and projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023012.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023013.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023015.png" />. Then the first-order jet bundle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023016.png" /> is the set
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For trivial fibre bundles (or locally on appropriate charts) and for first-order Lagrangians, the Euler operator is easy to describe. Thus, suppose $U \subseteq \mathbf{R} ^ { n }$, $F \subseteq {\bf R} ^ { m }$ are open sets, $M = \overline { U }$ is compact, and $E = M \times F$ is the trivial fibre bundle (cf. also [[Fibre space|Fibre space]]) over $M$ with fibre $F$ and projection $\pi : E \rightarrow M$ given by $\pi ( x , y ) = x$. Here $x = ( x _ { 1 } , \ldots , x _ { n } )$ and $y = ( y ^ { 1 } , \dots , y ^ { m } )$. Then the first-order jet bundle for $E$ is the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023017.png" /></td> </tr></table>
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\begin{equation*} E ^ { 1 } = J ^ { 1 } ( E ) = M \times F \times \mathbf{R} ^ { n m }, \end{equation*}
  
whose points are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023018.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023019.png" />. A first-order Lagrangian is a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023020.png" /> that has continuous partial derivatives up to the second order and determines a variational problem as follows.
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whose points are $( x , y , y ^ { \prime } )$ where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023019.png"/>. A first-order Lagrangian is a real-valued function $L : E ^ { 1 } \rightarrow \mathbf{R}$ that has continuous partial derivatives up to the second order and determines a variational problem as follows.
  
The set of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023021.png" /> consists of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023022.png" /> of the form
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The set of sections $\Gamma ( E )$ consists of functions $\sigma : M \rightarrow E$ of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023023.png" /></td> </tr></table>
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\begin{equation*} \sigma ( x ) = ( x , y ( x ) ) \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023024.png" /> is twice continuously differentiable. Each section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023025.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023026.png" />-jet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023027.png" />, which is the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023028.png" /> given by
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where $y : M \rightarrow F$ is twice continuously differentiable. Each section $\sigma$ has a $1$-jet $\sigma ^ { 1 }$, which is the section $\sigma ^ { 1 } : M \rightarrow E ^ { 1 }$ given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023029.png" /></td> </tr></table>
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\begin{equation*} \sigma ^ { 1 } ( x ) = ( x , y ( x ) , y ^ { \prime } ( x ) ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023030.png" />. With this notation, the variational problem associated with the Lagrangian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023031.png" /> is to determine the extreme values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023032.png" />, which is the action (or action integral) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023033.png" />:
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023030.png"/>. With this notation, the variational problem associated with the Lagrangian $L$ is to determine the extreme values of the function $\mathcal{A} : \Gamma ( E ) \rightarrow \mathbf{R}$, which is the action (or action integral) for $L$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023034.png" /></td> </tr></table>
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\begin{equation*} \mathcal{A} ( \sigma ) = \int _ { M } L ( \sigma ^ { 1 } ( x ) ) d x = \int _ { M } L ( x , y ( x ) , y ^ { \prime } ( x ) ) d x. \end{equation*}
  
In the trivial bundle setting, it is an easy exercise to derive the partial differential equations, called the Euler–Lagrange equations (cf. [[Euler–Lagrange equation|Euler–Lagrange equation]]), that any extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023036.png" /> must satisfy. This derivation is given here since it will clarify the difficulties in obtaining the global, or intrinsic, version of these equations when the fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023037.png" /> is not trivial.
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In the trivial bundle setting, it is an easy exercise to derive the partial differential equations, called the Euler–Lagrange equations (cf. [[Euler–Lagrange equation|Euler–Lagrange equation]]), that any extremal $\sigma$ of $\mathcal{A}$ must satisfy. This derivation is given here since it will clarify the difficulties in obtaining the global, or intrinsic, version of these equations when the fibre bundle $E$ is not trivial.
  
For simplicity, assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023038.png" /> is a bounded closed interval in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023040.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023041.png" /> has a maximum or minimum value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023042.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023043.png" /> be the section
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For simplicity, assume $M = [ a , b ]$ is a bounded closed interval in $\mathbf{R}$ and $m = 1$. Suppose that $\mathcal{A}$ has a maximum or minimum value at $\sigma$. Let $\sigma _ { t }$ be the section
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023044.png" /></td> </tr></table>
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\begin{equation*} \sigma _ { t } ( x ) = ( x , y ( x ) + t z ( x ) ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023045.png" /> is any twice continuously differentiable function with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023046.png" /> (so, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023047.png" />). Then for a suitably chosen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023048.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023049.png" /> defined by
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where $z: M \rightarrow F$ is any twice continuously differentiable function with compact support in $M$ (so, in particular, $z ( a ) = 0 = z ( b )$). Then for a suitably chosen $\epsilon &gt; 0$, the function $f : ( - \epsilon , \epsilon ) \rightarrow \mathbf{R}$ defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023050.png" /></td> </tr></table>
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\begin{equation*} f ( t ) = A ( \sigma _ { t } ) = \int _ { a } ^ { b } L ( x , y ( x ) + t z ( x ) , y ^ { \prime } ( x ) + t z ^ { \prime } ( x ) ) d x \end{equation*}
  
has a maximum or minimum value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023051.png" />. Consequently,
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has a maximum or minimum value at $t = 0$. Consequently,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023052.png" /></td> </tr></table>
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\begin{equation*} 0 = f ^ { \prime } ( 0 ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023053.png" /></td> </tr></table>
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\begin{equation*} = \int _ { a } ^ { b } \left[ \frac { \partial L } { \partial y } ( \sigma ^ { 1 } ( x ) ) z ( x ) + \frac { \partial L } { \partial y ^ { \prime } } ( \sigma ^ { 1 } ( x ) ) z ^ { \prime } ( x ) \right] d x = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023054.png" /></td> </tr></table>
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\begin{equation*} = \int _ { a } ^ { b } \left[ \frac { \partial L } { \partial y } ( \sigma ^ { 1 } ( x ) ) - \frac { d } { d x } \left( \frac { \partial L } { \partial y ^ { \prime } } ( \sigma ^ { 1 } ( x ) ) \right) \right] z ( x ) d x = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023055.png" /></td> </tr></table>
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\begin{equation*} = \int _ { a } ^ { b } {\cal E} ( L ) ( \sigma ^ { 2 } ( x ) ) z ( x ) d x. \end{equation*}
  
In the last equation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023056.png" /> denotes the function on the second-order jet bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023057.png" /> defined by
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In the last equation, $\mathcal{E} ( L )$ denotes the function on the second-order jet bundle $E ^ { 2 }$ defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023058.png" /></td> </tr></table>
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\begin{equation*} \mathcal{E} ( L ) = \frac { \partial L } { \partial y } - D \left( \frac { \partial L } { \partial y ^ { \prime } } \right), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023059.png" /> is the differential operator
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where $D$ is the differential operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023060.png" /></td> </tr></table>
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\begin{equation*} D = \frac { \partial } { \partial x } + y ^ { \prime } \frac { \partial } { \partial y } + y ^ { \prime \prime } \frac { \partial } { \partial y ^ { \prime } }. \end{equation*}
  
In this setting, then, the Euler operator is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023061.png" />. The differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023062.png" /> is called the total derivative operator.
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In this setting, then, the Euler operator is $L \mapsto \mathcal{E} ( L )$. The differential operator $D$ is called the total derivative operator.
  
It is important to note that the next to the last equation above comes from integrating by parts and uses the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023063.png" /> vanishes on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023064.png" />.
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It is important to note that the next to the last equation above comes from integrating by parts and uses the assumption that $z$ vanishes on the boundary of $[ a , b ]$.
  
From the arbitrariness of the variation function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023065.png" /> (up to the stated conditions), the above shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023066.png" /> must satisfy the second-order partial differential equation
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From the arbitrariness of the variation function $z$ (up to the stated conditions), the above shows that $\sigma$ must satisfy the second-order partial differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023067.png" /></td> </tr></table>
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\begin{equation*} \mathcal{E} ( L ) ( \sigma ^ { 2 } ( x ) ) = 0, \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023068.png" />. This is the Euler–Lagrange equation for this special case.
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for all $x \in ( a , b )$. This is the Euler–Lagrange equation for this special case.
  
For the higher-dimensional cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023070.png" /> (but still first-order Lagrangians), the above variational argument is entirely similar and one can show than each extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023071.png" /> must satisfy the system of partial differential equations
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For the higher-dimensional cases $n &gt; 1$, $m &gt; 1$ (but still first-order Lagrangians), the above variational argument is entirely similar and one can show than each extremal $\sigma$ must satisfy the system of partial differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023072.png" /></td> </tr></table>
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\begin{equation*} {\cal E} ^ { a } ( L ) ( \sigma ^ { 2 } ( x ) ) = 0, \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023073.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023074.png" />. Here,
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$a = 1 , \dots , m$, for all $x \in M$. Here,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023075.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{E} ^ { a  } ( L ) = \frac { \partial L } { \partial y ^ { a } } - D _ { i } \left( \frac { \partial L } { \partial y ^ { a _ { i } } } \right), \end{equation*}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023076.png" /> is the differential operator
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and $D_i$ is the differential operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023077.png" /></td> </tr></table>
+
\begin{equation*} D _ { i } = \frac { \partial } { \partial x _ { i } } + y ^ { b _ { i } } \frac { \partial } { \partial y ^ { b } } + y ^ { b _ { i j } } \frac { \partial } { \partial y ^ { b _ { j } } }. \end{equation*}
  
These expressions involve (Einstein) summation on repeated indices, as is customary (cf. also [[Einstein rule|Einstein rule]]). Again, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023078.png" /> is called the total derivative operator and the Euler operator for this setting is the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023079.png" />, assigning to each first-order Lagrangian a function on the second-order jet bundle.
+
These expressions involve (Einstein) summation on repeated indices, as is customary (cf. also [[Einstein rule|Einstein rule]]). Again, the operator $D_i$ is called the total derivative operator and the Euler operator for this setting is the mapping $\mathcal E ( L ) = ( \mathcal E ^ { 1 } ( L ) , \ldots , \mathcal E ^ { m } ( L ) )$, assigning to each first-order Lagrangian a function on the second-order jet bundle.
  
Within the trivial bundle setting (or on local charts, for non-trivial bundles), the Euler operator for higher-order Lagrangians <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023081.png" />, is also easy to describe. This requires the multi-index notation.
+
Within the trivial bundle setting (or on local charts, for non-trivial bundles), the Euler operator for higher-order Lagrangians $L : E ^ { k } \rightarrow  \bf R$, $k &gt; 1$, is also easy to describe. This requires the multi-index notation.
  
A multi-index is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023082.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023083.png" /> of non-negative integers and the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023084.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023085.png" />. Also,
+
A multi-index is an $n$-tuple $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } )$ of non-negative integers and the order of $\alpha$ is $| \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }$. Also,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023086.png" /></td> </tr></table>
+
\begin{equation*} ( \frac { \partial } { \partial x } ) ^ { \alpha } = ( \frac { \partial } { \partial x _ { 1 } } ) ^ { \alpha _ { 1 } } \dots ( \frac { \partial } { \partial x _ { n } } ) ^ { \alpha _ { n } }. \end{equation*}
  
With this notation, a point in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023088.png" />th-order jet bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023089.png" /> is denoted by
+
With this notation, a point in the $k$th-order jet bundle $E ^ { k } = M \times F \times F ^ { ( 1 ) } \times \ldots F ^ { ( k ) }$ is denoted by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023090.png" /></td> </tr></table>
+
\begin{equation*} ( x , y , y ^ { \prime } , \dots , y ^ { ( k ) } ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023091.png" />. For a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023092.png" />, its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023093.png" />-jet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023094.png" /> is given by
+
where $y ^ { ( r ) } = \{ y _ { \alpha } ^ { a } \} _ { | \alpha | = r } ^ { a = 1 , \ldots , m }$. For a section $\sigma ( x ) = ( x , y ( x ) )$, its $k$-jet $\sigma ^ { k } : M \rightarrow E ^ { k }$ is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023095.png" /></td> </tr></table>
+
\begin{equation*} \sigma ^ { k } ( x ) = ( x , y ( x ) , y ^ { \prime } ( x ) , \ldots , y ^ { ( k ) } ( x ) ), \end{equation*}
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023096.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023096.png"/></td> </tr></table>
  
Using a variational argument similar to that above, but now integrating by parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023097.png" /> times, one can show that if the action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023098.png" /> has a local maximum or minimum value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e12023099.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230100.png" /> must satisfy the system of partial differential equations
+
Using a variational argument similar to that above, but now integrating by parts $k$ times, one can show that if the action $\mathcal{A} ( \sigma ) = \int _ { M } L ( \sigma ^ { k } ( x ) ) d x$ has a local maximum or minimum value at $\sigma$, then $\sigma$ must satisfy the system of partial differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230101.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{E} ^ { a } ( L ) ( \sigma ^ { 2 k } ( x ) ) = 0, \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230102.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230103.png" />. Here,
+
$a = 1 , \dots , m$, for all $x \in M$. Here,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230104.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{E} ^ { a } ( L ) = \sum _ { | \alpha | = 0 } ^ { k } ( - 1 ) ^ { | \alpha | } D ^ { \alpha } \left( \frac { \partial L } { \partial y _ { \alpha } ^ { a } } \right), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230105.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230106.png" /> is the total derivative operator:
+
where $D ^ { \alpha } = D _ { 1 } ^ { \alpha _ { 1 } } \ldots D _ { n } ^ { \alpha _ { n } }$, and $D_i$ is the total derivative operator:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230107.png" /></td> </tr></table>
+
\begin{equation*} D _ { i } = \frac { \partial } { \partial x _ { i } } + \sum _ { | \alpha | = 0 } ^ { 2 k } y _ { \alpha + e _ { i } } ^ { b } \frac { \partial } { \partial y _ { \alpha } ^ { b } }. \end{equation*}
  
Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230108.png" /> is the multi-index of all zeros except for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230109.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230110.png" />th position.
+
Note that $e _ { i }$ is the multi-index of all zeros except for a $1$ in the $i$th position.
  
In the general setting, the intrinsic construction of the Euler operator is more complicated and many different approaches occur in the literature. See [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]. One approach realizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230111.png" /> as a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230112.png" />-form-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230113.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230114.png" /> which is globally defined and has, in any chart, the local expression
+
In the general setting, the intrinsic construction of the Euler operator is more complicated and many different approaches occur in the literature. See [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]. One approach realizes $\mathcal{E} ( L )$ as a certain $n$-form-valued $1$-form on $E ^ { 2 k }$ which is globally defined and has, in any chart, the local expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230115.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{E} ( L ) = \mathcal{E} ^ { a } ( L ) \omega ^ { a } \bigotimes \Delta, \end{equation*}
  
using Einstein summation, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230116.png" />-s are the local contact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230117.png" />-forms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230118.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230119.png" /> is a volume form on the base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230120.png" />. Because of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230121.png" /> in the local expression for the volume form, the components in the local expression of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230122.png" /> are slightly modified from above to
+
using Einstein summation, the $\omega ^ { a}$-s are the local contact $1$-forms, $\omega ^ { a } = d y ^ { a } - y _ { e _ { i } } ^ { a } d x _ { i }$, and $\Delta = \gamma d x _ { 1 } \wedge \ldots \wedge d x _ { n }$ is a volume form on the base space $M$. Because of the function $\gamma : M \rightarrow {\bf R}$ in the local expression for the volume form, the components in the local expression of $\mathcal{E} ( L )$ are slightly modified from above to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230123.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{E} ^ { a } ( L ) = \sum _ { | \alpha | = 0 } ^ { k } ( - 1 ) ^ { | \alpha | } \gamma ^ { - 1 } D ^ { \alpha } \left( \gamma \frac { \partial L } { \partial y _ { \alpha } ^ { a } } \right). \end{equation*}
  
 
This approach to the Euler operator is briefly described as follows.
 
This approach to the Euler operator is briefly described as follows.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230124.png" /> is a fibre bundle with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230125.png" />-dimensional fibre and base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230126.png" /> which is a smooth, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230127.png" />-dimensional manifold with volume form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230128.png" />. For simplicity of exposition, assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230129.png" /> is compact. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230130.png" />th-order jet bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230131.png" /> consists of equivalence classes of local sections at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230132.png" />, all of whose partial derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230133.png" /> are the same at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230134.png" />. There are naturally defined projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230135.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230136.png" /> and it is common, to simplify the notation, to identify a differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230137.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230138.png" /> with its pullback <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230139.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230140.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230141.png" /> and, for a Lagrangian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230142.png" />, the action integral can be written as
+
Suppose that $\pi : E \rightarrow M$ is a fibre bundle with $m$-dimensional fibre and base space $M$ which is a smooth, $n$-dimensional manifold with volume form $\Delta$. For simplicity of exposition, assume that $M$ is compact. The $k$th-order jet bundle $E ^ { k } = \left\{ [ \sigma ] _ { x } ^ { k } : x \in M , \sigma \in \Gamma _ { x } ( E ) \right\}$ consists of equivalence classes of local sections at $x \in M$, all of whose partial derivatives up to order $k$ are the same at $x$. There are naturally defined projections $\pi ^ { k } : E ^ { k } \rightarrow M$ and $\pi _ { r } ^ { k } : E ^ { k } \rightarrow E ^ { r }$ and it is common, to simplify the notation, to identify a differential form $\theta$ on $E ^ { r }$ with its pullback $\pi _ { r } ^ { k * } ( \theta )$ to $E ^ { k }$. Thus, $\Delta = \pi ^ { k ^ { * } } ( \Delta )$ and, for a Lagrangian $L : E ^ { k } \rightarrow  \bf R$, the action integral can be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230143.png" /></td> </tr></table>
+
\begin{equation*} {\cal A} ( \sigma ) = \int _ { M } L \circ \sigma ^ { k } \Delta = \int _ { M } \sigma ^ { k ^ { * } } ( L \Delta ). \end{equation*}
  
To make a variation in the action, as was done above in the trivial case, suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230144.png" /> is a vertical vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230145.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230146.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230147.png" />) and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230148.png" /> is its corresponding flow. Then the prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230149.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230150.png" /> to a vertical vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230151.png" /> has flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230152.png" /> (cf. also [[Prolongation of solutions of differential equations|Prolongation of solutions of differential equations]]). Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230153.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230154.png" />, and consequently
+
To make a variation in the action, as was done above in the trivial case, suppose $Z$ is a vertical vector field on $E$ (i.e. $d \pi _ { e } Z _ { e } = 0$ for all $e \in E$) and that $\phi _ { t }$ is its corresponding flow. Then the prolongation $Z^k$ of $Z$ to a vertical vector field on $E ^ { k }$ has flow $\phi _ { t } ^ { k }$ (cf. also [[Prolongation of solutions of differential equations|Prolongation of solutions of differential equations]]). Letting $\sigma _ { t } = \phi _ { t } \circ \sigma$, one has $\sigma _ { t } ^ { k } = \phi _ { t } ^ { k } \circ \sigma ^ { k }$, and consequently
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230155.png" /></td> </tr></table>
+
\begin{equation*} \left. \frac { d } { d t } {\cal A} ( \sigma _ { t } ) \right| _ { t = 0 } = \left. \frac { d } { d t } \int _ { M } \sigma ^ { k ^ { * } } \phi _ { t } ^ { k ^ { * } } ( L \Delta ) \right| _ { t = 0 } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230156.png" /></td> </tr></table>
+
\begin{equation*} = \int _ { M } \sigma ^ { k ^ { * } } \mathcal{L} _ { Z ^ { k } } ( L \Delta ). \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230157.png" /> is the Lie derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230158.png" />. Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230159.png" /> has compact support contained in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230160.png" />. Use this together with the [[Stokes theorem|Stokes theorem]] to reduce the variation of the action to
+
Here, $\mathcal{L} _ { \text{Z} ^ { k } } ( L , \Delta ) = Z ^ { k }\lrcorner d L \Delta + d ( Z ^ { k } \lrcorner L  \Delta )$ is the Lie derivative of $L \Delta$. Suppose now that $Z$ has compact support contained in the interior of $\sigma ( M )$. Use this together with the [[Stokes theorem|Stokes theorem]] to reduce the variation of the action to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230161.png" /></td> </tr></table>
+
\begin{equation*} \frac { d } { d t } {\cal A} ( \sigma _ { t } ) | _ { t = 0 } = \int _ { M } \sigma ^ { k ^ { * } } ( Z ^ { k }  \lrcorner d L \Delta ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230162.png" /></td> </tr></table>
+
\begin{equation*} = \int _ { M } \sigma ^ { k + 1* } [ \Omega ( d L \Delta ) ( Z ^ { k + 1 } ) ]. \end{equation*}
  
The latter equation results from using the variational operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230163.png" />, which maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230164.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230165.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230166.png" />-form-valued contact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230167.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230168.png" />. For the case under consideration here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230169.png" /> has, on each chart, a component expression:
+
The latter equation results from using the variational operator $\Omega$, which maps $n + 1$-forms on $E ^ { r }$ into $n$-form-valued contact $1$-forms on $E ^{r+1} $. For the case under consideration here, $\Omega ( d L \Delta )$ has, on each chart, a component expression:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230170.png" /></td> </tr></table>
+
\begin{equation*} \Omega ( d L \Delta ) = \sum _ { | \alpha | = 0 } ^ { k } \frac { \partial L } { \partial y _ { \alpha } ^ { a } } \omega _ { \alpha } ^ { a } \bigotimes \Delta . \end{equation*}
  
 
Consequently, the component expression for the integrand of the first variation is
 
Consequently, the component expression for the integrand of the first variation is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230171.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { | \alpha | = 0 } ^ { k } \left( \frac { \partial L } { \partial y _ { \alpha } ^ { a }}  \circ \sigma ^ { k }  \right) ( \frac { \partial } { \partial x } ) ^ { \alpha } ( Z ^ { a } \circ \sigma ) \Delta. \end{equation*}
  
The problem now is to construct a (horizontal) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230172.png" />-form-valued, contact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230173.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230174.png" /> on a higher-order jet bundle (as suggested by using integration by parts) so that
+
The problem now is to construct a (horizontal) $n$-form-valued, contact $1$-form $\mathcal{E} ( L )$ on a higher-order jet bundle (as suggested by using integration by parts) so that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230175.png" /></td> </tr></table>
+
\begin{equation*} \sigma ^ { 2 k *  } [ {\cal E} ( L ) ( Z ^ { 2 k } ) ] = \sigma ^ { k + 1  *  } [ \Omega ( d L \Delta ) ( Z ^ { k + 1 } ) ], \end{equation*}
  
and so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230176.png" />, locally on each chart. Consequently, the component expression for the integrand of the first variation is now
+
and so that $\mathcal{E} ( L ) = \mathcal{E} ^ { a } ( L ) \omega ^ { a } \otimes \Delta$, locally on each chart. Consequently, the component expression for the integrand of the first variation is now
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230177.png" /></td> </tr></table>
+
\begin{equation*} ( \mathcal{E} ^ { a } ( L ) \circ \sigma ^ { 2 k } ) ( Z ^ { a } \circ \sigma ) \Delta. \end{equation*}
  
Thus, it follows that if the first variation vanishes identically for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230178.png" /> of the stated form, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230179.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230180.png" />, which is the global version of the Euler–Lagrange equations. This problem can be solved by using a shift operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230181.png" />.
+
Thus, it follows that if the first variation vanishes identically for all $Z$ of the stated form, then $\sigma$ satisfies $\sigma ^ { 2 k  *  } \mathcal{E} ( L ) = 0$, which is the global version of the Euler–Lagrange equations. This problem can be solved by using a shift operator $S$.
  
It is shown in [[#References|[a1]]] that there is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230182.png" />, called a shift operator, which maps contact-horizontal forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230183.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230184.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230185.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230186.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230187.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230188.png" />) and which has, on each local chart, the form
+
It is shown in [[#References|[a1]]] that there is an operator $S$, called a shift operator, which maps contact-horizontal forms $\phi$ on $E ^ { k + 1 }$ into $n$-forms $S ( \phi )$ on $E ^ { k + 1 }$ (for $k = 0,1 , \ldots$) and which has, on each local chart, the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230189.png" /></td> </tr></table>
+
\begin{equation*} S ( \phi ) = \sum _ { | \alpha | = 0 } ^ { k - 1 } S _ { \alpha i } ^ { a } ( \phi ) \omega _ { \alpha } ^ { a } \bigwedge \left( \frac { \partial } { \partial x _ { i } } \lrcorner ( d x _ { 1 } \bigwedge \ldots \bigwedge d x _ { n } ) \right). \end{equation*}
  
By repeated application of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230190.png" /> in conjunction with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230191.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230192.png" />, one gets the Euler operator defined in a global way by
+
By repeated application of $S$ in conjunction with $\Omega$ and $d$, one gets the Euler operator defined in a global way by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120230/e120230193.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{E} ( L ) \equiv ( 1 + \Omega d S ) ^ { k } \Omega d ( L \Delta ). \end{equation*}
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Betounes,  "Global shift operators and the higher order calculus of variations"  ''J. Geom. Phys.'' , '''10'''  (1993)  pp. 185–201</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Kolar,  "A geometric version of the higher ordered Hamilton formalism in fibered manifolds"  ''J. Geom. Phys.'' , '''1'''  (1984)  pp. 127–137</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Krupka,  "Lepagean forms in the higher order variational calculus" , ''Geometrical Dynamics. Proc. IUTAM–ISIMM Symp. Modern Developments in Analytic. Mech. (Turin, 1982)'' , '''I''' , Technoprint, Bologna  (1983)  pp. 197–238</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Munoz Masque,  "Poincare–Cartan forms in higher order variational calculus on fibered manifolds"  ''Rev. Mat. Iberoamercana'' , '''1'''  (1985)  pp. 85–126</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.J. Sanders,  "The geometry of jet bundles" , ''London Math. Soc. Lecture Notes'' , '''142''' , Cambridge Univ. Press  (1989)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  D. Betounes,  "Global shift operators and the higher order calculus of variations"  ''J. Geom. Phys.'' , '''10'''  (1993)  pp. 185–201</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Kolar,  "A geometric version of the higher ordered Hamilton formalism in fibered manifolds"  ''J. Geom. Phys.'' , '''1'''  (1984)  pp. 127–137</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Krupka,  "Lepagean forms in the higher order variational calculus" , ''Geometrical Dynamics. Proc. IUTAM–ISIMM Symp. Modern Developments in Analytic. Mech. (Turin, 1982)'' , '''I''' , Technoprint, Bologna  (1983)  pp. 197–238</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Munoz Masque,  "Poincare–Cartan forms in higher order variational calculus on fibered manifolds"  ''Rev. Mat. Iberoamercana'' , '''1'''  (1985)  pp. 85–126</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  D.J. Sanders,  "The geometry of jet bundles" , ''London Math. Soc. Lecture Notes'' , '''142''' , Cambridge Univ. Press  (1989)</td></tr></table>

Latest revision as of 17:45, 1 July 2020

Euler–Lagrange operator

A fundamental object, $\cal E$, in the calculus of variations (cf. also Variational calculus), used to formulate the system of partial differential equations, called the Euler–Lagrange equations or the variational equations, that the extremals for variational problems must satisfy (cf. also Euler–Lagrange equation).

In essence, to each Lagrangian $L$, the Euler operator assigns a geometric object $\mathcal{E} ( L )$ whose components $\mathcal{E} ^ { a } ( L )$, $a = 1 , \dots , m$, are the expressions for the Euler–Lagrange equations.

For trivial fibre bundles (or locally on appropriate charts) and for first-order Lagrangians, the Euler operator is easy to describe. Thus, suppose $U \subseteq \mathbf{R} ^ { n }$, $F \subseteq {\bf R} ^ { m }$ are open sets, $M = \overline { U }$ is compact, and $E = M \times F$ is the trivial fibre bundle (cf. also Fibre space) over $M$ with fibre $F$ and projection $\pi : E \rightarrow M$ given by $\pi ( x , y ) = x$. Here $x = ( x _ { 1 } , \ldots , x _ { n } )$ and $y = ( y ^ { 1 } , \dots , y ^ { m } )$. Then the first-order jet bundle for $E$ is the set

\begin{equation*} E ^ { 1 } = J ^ { 1 } ( E ) = M \times F \times \mathbf{R} ^ { n m }, \end{equation*}

whose points are $( x , y , y ^ { \prime } )$ where . A first-order Lagrangian is a real-valued function $L : E ^ { 1 } \rightarrow \mathbf{R}$ that has continuous partial derivatives up to the second order and determines a variational problem as follows.

The set of sections $\Gamma ( E )$ consists of functions $\sigma : M \rightarrow E$ of the form

\begin{equation*} \sigma ( x ) = ( x , y ( x ) ) \end{equation*}

where $y : M \rightarrow F$ is twice continuously differentiable. Each section $\sigma$ has a $1$-jet $\sigma ^ { 1 }$, which is the section $\sigma ^ { 1 } : M \rightarrow E ^ { 1 }$ given by

\begin{equation*} \sigma ^ { 1 } ( x ) = ( x , y ( x ) , y ^ { \prime } ( x ) ), \end{equation*}

where . With this notation, the variational problem associated with the Lagrangian $L$ is to determine the extreme values of the function $\mathcal{A} : \Gamma ( E ) \rightarrow \mathbf{R}$, which is the action (or action integral) for $L$:

\begin{equation*} \mathcal{A} ( \sigma ) = \int _ { M } L ( \sigma ^ { 1 } ( x ) ) d x = \int _ { M } L ( x , y ( x ) , y ^ { \prime } ( x ) ) d x. \end{equation*}

In the trivial bundle setting, it is an easy exercise to derive the partial differential equations, called the Euler–Lagrange equations (cf. Euler–Lagrange equation), that any extremal $\sigma$ of $\mathcal{A}$ must satisfy. This derivation is given here since it will clarify the difficulties in obtaining the global, or intrinsic, version of these equations when the fibre bundle $E$ is not trivial.

For simplicity, assume $M = [ a , b ]$ is a bounded closed interval in $\mathbf{R}$ and $m = 1$. Suppose that $\mathcal{A}$ has a maximum or minimum value at $\sigma$. Let $\sigma _ { t }$ be the section

\begin{equation*} \sigma _ { t } ( x ) = ( x , y ( x ) + t z ( x ) ), \end{equation*}

where $z: M \rightarrow F$ is any twice continuously differentiable function with compact support in $M$ (so, in particular, $z ( a ) = 0 = z ( b )$). Then for a suitably chosen $\epsilon > 0$, the function $f : ( - \epsilon , \epsilon ) \rightarrow \mathbf{R}$ defined by

\begin{equation*} f ( t ) = A ( \sigma _ { t } ) = \int _ { a } ^ { b } L ( x , y ( x ) + t z ( x ) , y ^ { \prime } ( x ) + t z ^ { \prime } ( x ) ) d x \end{equation*}

has a maximum or minimum value at $t = 0$. Consequently,

\begin{equation*} 0 = f ^ { \prime } ( 0 ) = \end{equation*}

\begin{equation*} = \int _ { a } ^ { b } \left[ \frac { \partial L } { \partial y } ( \sigma ^ { 1 } ( x ) ) z ( x ) + \frac { \partial L } { \partial y ^ { \prime } } ( \sigma ^ { 1 } ( x ) ) z ^ { \prime } ( x ) \right] d x = \end{equation*}

\begin{equation*} = \int _ { a } ^ { b } \left[ \frac { \partial L } { \partial y } ( \sigma ^ { 1 } ( x ) ) - \frac { d } { d x } \left( \frac { \partial L } { \partial y ^ { \prime } } ( \sigma ^ { 1 } ( x ) ) \right) \right] z ( x ) d x = \end{equation*}

\begin{equation*} = \int _ { a } ^ { b } {\cal E} ( L ) ( \sigma ^ { 2 } ( x ) ) z ( x ) d x. \end{equation*}

In the last equation, $\mathcal{E} ( L )$ denotes the function on the second-order jet bundle $E ^ { 2 }$ defined by

\begin{equation*} \mathcal{E} ( L ) = \frac { \partial L } { \partial y } - D \left( \frac { \partial L } { \partial y ^ { \prime } } \right), \end{equation*}

where $D$ is the differential operator

\begin{equation*} D = \frac { \partial } { \partial x } + y ^ { \prime } \frac { \partial } { \partial y } + y ^ { \prime \prime } \frac { \partial } { \partial y ^ { \prime } }. \end{equation*}

In this setting, then, the Euler operator is $L \mapsto \mathcal{E} ( L )$. The differential operator $D$ is called the total derivative operator.

It is important to note that the next to the last equation above comes from integrating by parts and uses the assumption that $z$ vanishes on the boundary of $[ a , b ]$.

From the arbitrariness of the variation function $z$ (up to the stated conditions), the above shows that $\sigma$ must satisfy the second-order partial differential equation

\begin{equation*} \mathcal{E} ( L ) ( \sigma ^ { 2 } ( x ) ) = 0, \end{equation*}

for all $x \in ( a , b )$. This is the Euler–Lagrange equation for this special case.

For the higher-dimensional cases $n > 1$, $m > 1$ (but still first-order Lagrangians), the above variational argument is entirely similar and one can show than each extremal $\sigma$ must satisfy the system of partial differential equations

\begin{equation*} {\cal E} ^ { a } ( L ) ( \sigma ^ { 2 } ( x ) ) = 0, \end{equation*}

$a = 1 , \dots , m$, for all $x \in M$. Here,

\begin{equation*} \mathcal{E} ^ { a } ( L ) = \frac { \partial L } { \partial y ^ { a } } - D _ { i } \left( \frac { \partial L } { \partial y ^ { a _ { i } } } \right), \end{equation*}

and $D_i$ is the differential operator

\begin{equation*} D _ { i } = \frac { \partial } { \partial x _ { i } } + y ^ { b _ { i } } \frac { \partial } { \partial y ^ { b } } + y ^ { b _ { i j } } \frac { \partial } { \partial y ^ { b _ { j } } }. \end{equation*}

These expressions involve (Einstein) summation on repeated indices, as is customary (cf. also Einstein rule). Again, the operator $D_i$ is called the total derivative operator and the Euler operator for this setting is the mapping $\mathcal E ( L ) = ( \mathcal E ^ { 1 } ( L ) , \ldots , \mathcal E ^ { m } ( L ) )$, assigning to each first-order Lagrangian a function on the second-order jet bundle.

Within the trivial bundle setting (or on local charts, for non-trivial bundles), the Euler operator for higher-order Lagrangians $L : E ^ { k } \rightarrow \bf R$, $k > 1$, is also easy to describe. This requires the multi-index notation.

A multi-index is an $n$-tuple $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } )$ of non-negative integers and the order of $\alpha$ is $| \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }$. Also,

\begin{equation*} ( \frac { \partial } { \partial x } ) ^ { \alpha } = ( \frac { \partial } { \partial x _ { 1 } } ) ^ { \alpha _ { 1 } } \dots ( \frac { \partial } { \partial x _ { n } } ) ^ { \alpha _ { n } }. \end{equation*}

With this notation, a point in the $k$th-order jet bundle $E ^ { k } = M \times F \times F ^ { ( 1 ) } \times \ldots F ^ { ( k ) }$ is denoted by

\begin{equation*} ( x , y , y ^ { \prime } , \dots , y ^ { ( k ) } ), \end{equation*}

where $y ^ { ( r ) } = \{ y _ { \alpha } ^ { a } \} _ { | \alpha | = r } ^ { a = 1 , \ldots , m }$. For a section $\sigma ( x ) = ( x , y ( x ) )$, its $k$-jet $\sigma ^ { k } : M \rightarrow E ^ { k }$ is given by

\begin{equation*} \sigma ^ { k } ( x ) = ( x , y ( x ) , y ^ { \prime } ( x ) , \ldots , y ^ { ( k ) } ( x ) ), \end{equation*}

where

Using a variational argument similar to that above, but now integrating by parts $k$ times, one can show that if the action $\mathcal{A} ( \sigma ) = \int _ { M } L ( \sigma ^ { k } ( x ) ) d x$ has a local maximum or minimum value at $\sigma$, then $\sigma$ must satisfy the system of partial differential equations

\begin{equation*} \mathcal{E} ^ { a } ( L ) ( \sigma ^ { 2 k } ( x ) ) = 0, \end{equation*}

$a = 1 , \dots , m$, for all $x \in M$. Here,

\begin{equation*} \mathcal{E} ^ { a } ( L ) = \sum _ { | \alpha | = 0 } ^ { k } ( - 1 ) ^ { | \alpha | } D ^ { \alpha } \left( \frac { \partial L } { \partial y _ { \alpha } ^ { a } } \right), \end{equation*}

where $D ^ { \alpha } = D _ { 1 } ^ { \alpha _ { 1 } } \ldots D _ { n } ^ { \alpha _ { n } }$, and $D_i$ is the total derivative operator:

\begin{equation*} D _ { i } = \frac { \partial } { \partial x _ { i } } + \sum _ { | \alpha | = 0 } ^ { 2 k } y _ { \alpha + e _ { i } } ^ { b } \frac { \partial } { \partial y _ { \alpha } ^ { b } }. \end{equation*}

Note that $e _ { i }$ is the multi-index of all zeros except for a $1$ in the $i$th position.

In the general setting, the intrinsic construction of the Euler operator is more complicated and many different approaches occur in the literature. See [a1], [a2], [a3], [a4], [a5]. One approach realizes $\mathcal{E} ( L )$ as a certain $n$-form-valued $1$-form on $E ^ { 2 k }$ which is globally defined and has, in any chart, the local expression

\begin{equation*} \mathcal{E} ( L ) = \mathcal{E} ^ { a } ( L ) \omega ^ { a } \bigotimes \Delta, \end{equation*}

using Einstein summation, the $\omega ^ { a}$-s are the local contact $1$-forms, $\omega ^ { a } = d y ^ { a } - y _ { e _ { i } } ^ { a } d x _ { i }$, and $\Delta = \gamma d x _ { 1 } \wedge \ldots \wedge d x _ { n }$ is a volume form on the base space $M$. Because of the function $\gamma : M \rightarrow {\bf R}$ in the local expression for the volume form, the components in the local expression of $\mathcal{E} ( L )$ are slightly modified from above to

\begin{equation*} \mathcal{E} ^ { a } ( L ) = \sum _ { | \alpha | = 0 } ^ { k } ( - 1 ) ^ { | \alpha | } \gamma ^ { - 1 } D ^ { \alpha } \left( \gamma \frac { \partial L } { \partial y _ { \alpha } ^ { a } } \right). \end{equation*}

This approach to the Euler operator is briefly described as follows.

Suppose that $\pi : E \rightarrow M$ is a fibre bundle with $m$-dimensional fibre and base space $M$ which is a smooth, $n$-dimensional manifold with volume form $\Delta$. For simplicity of exposition, assume that $M$ is compact. The $k$th-order jet bundle $E ^ { k } = \left\{ [ \sigma ] _ { x } ^ { k } : x \in M , \sigma \in \Gamma _ { x } ( E ) \right\}$ consists of equivalence classes of local sections at $x \in M$, all of whose partial derivatives up to order $k$ are the same at $x$. There are naturally defined projections $\pi ^ { k } : E ^ { k } \rightarrow M$ and $\pi _ { r } ^ { k } : E ^ { k } \rightarrow E ^ { r }$ and it is common, to simplify the notation, to identify a differential form $\theta$ on $E ^ { r }$ with its pullback $\pi _ { r } ^ { k * } ( \theta )$ to $E ^ { k }$. Thus, $\Delta = \pi ^ { k ^ { * } } ( \Delta )$ and, for a Lagrangian $L : E ^ { k } \rightarrow \bf R$, the action integral can be written as

\begin{equation*} {\cal A} ( \sigma ) = \int _ { M } L \circ \sigma ^ { k } \Delta = \int _ { M } \sigma ^ { k ^ { * } } ( L \Delta ). \end{equation*}

To make a variation in the action, as was done above in the trivial case, suppose $Z$ is a vertical vector field on $E$ (i.e. $d \pi _ { e } Z _ { e } = 0$ for all $e \in E$) and that $\phi _ { t }$ is its corresponding flow. Then the prolongation $Z^k$ of $Z$ to a vertical vector field on $E ^ { k }$ has flow $\phi _ { t } ^ { k }$ (cf. also Prolongation of solutions of differential equations). Letting $\sigma _ { t } = \phi _ { t } \circ \sigma$, one has $\sigma _ { t } ^ { k } = \phi _ { t } ^ { k } \circ \sigma ^ { k }$, and consequently

\begin{equation*} \left. \frac { d } { d t } {\cal A} ( \sigma _ { t } ) \right| _ { t = 0 } = \left. \frac { d } { d t } \int _ { M } \sigma ^ { k ^ { * } } \phi _ { t } ^ { k ^ { * } } ( L \Delta ) \right| _ { t = 0 } = \end{equation*}

\begin{equation*} = \int _ { M } \sigma ^ { k ^ { * } } \mathcal{L} _ { Z ^ { k } } ( L \Delta ). \end{equation*}

Here, $\mathcal{L} _ { \text{Z} ^ { k } } ( L , \Delta ) = Z ^ { k }\lrcorner d L \Delta + d ( Z ^ { k } \lrcorner L \Delta )$ is the Lie derivative of $L \Delta$. Suppose now that $Z$ has compact support contained in the interior of $\sigma ( M )$. Use this together with the Stokes theorem to reduce the variation of the action to

\begin{equation*} \frac { d } { d t } {\cal A} ( \sigma _ { t } ) | _ { t = 0 } = \int _ { M } \sigma ^ { k ^ { * } } ( Z ^ { k } \lrcorner d L \Delta ) = \end{equation*}

\begin{equation*} = \int _ { M } \sigma ^ { k + 1* } [ \Omega ( d L \Delta ) ( Z ^ { k + 1 } ) ]. \end{equation*}

The latter equation results from using the variational operator $\Omega$, which maps $n + 1$-forms on $E ^ { r }$ into $n$-form-valued contact $1$-forms on $E ^{r+1} $. For the case under consideration here, $\Omega ( d L \Delta )$ has, on each chart, a component expression:

\begin{equation*} \Omega ( d L \Delta ) = \sum _ { | \alpha | = 0 } ^ { k } \frac { \partial L } { \partial y _ { \alpha } ^ { a } } \omega _ { \alpha } ^ { a } \bigotimes \Delta . \end{equation*}

Consequently, the component expression for the integrand of the first variation is

\begin{equation*} \sum _ { | \alpha | = 0 } ^ { k } \left( \frac { \partial L } { \partial y _ { \alpha } ^ { a }} \circ \sigma ^ { k } \right) ( \frac { \partial } { \partial x } ) ^ { \alpha } ( Z ^ { a } \circ \sigma ) \Delta. \end{equation*}

The problem now is to construct a (horizontal) $n$-form-valued, contact $1$-form $\mathcal{E} ( L )$ on a higher-order jet bundle (as suggested by using integration by parts) so that

\begin{equation*} \sigma ^ { 2 k * } [ {\cal E} ( L ) ( Z ^ { 2 k } ) ] = \sigma ^ { k + 1 * } [ \Omega ( d L \Delta ) ( Z ^ { k + 1 } ) ], \end{equation*}

and so that $\mathcal{E} ( L ) = \mathcal{E} ^ { a } ( L ) \omega ^ { a } \otimes \Delta$, locally on each chart. Consequently, the component expression for the integrand of the first variation is now

\begin{equation*} ( \mathcal{E} ^ { a } ( L ) \circ \sigma ^ { 2 k } ) ( Z ^ { a } \circ \sigma ) \Delta. \end{equation*}

Thus, it follows that if the first variation vanishes identically for all $Z$ of the stated form, then $\sigma$ satisfies $\sigma ^ { 2 k * } \mathcal{E} ( L ) = 0$, which is the global version of the Euler–Lagrange equations. This problem can be solved by using a shift operator $S$.

It is shown in [a1] that there is an operator $S$, called a shift operator, which maps contact-horizontal forms $\phi$ on $E ^ { k + 1 }$ into $n$-forms $S ( \phi )$ on $E ^ { k + 1 }$ (for $k = 0,1 , \ldots$) and which has, on each local chart, the form

\begin{equation*} S ( \phi ) = \sum _ { | \alpha | = 0 } ^ { k - 1 } S _ { \alpha i } ^ { a } ( \phi ) \omega _ { \alpha } ^ { a } \bigwedge \left( \frac { \partial } { \partial x _ { i } } \lrcorner ( d x _ { 1 } \bigwedge \ldots \bigwedge d x _ { n } ) \right). \end{equation*}

By repeated application of $S$ in conjunction with $\Omega$ and $d$, one gets the Euler operator defined in a global way by

\begin{equation*} \mathcal{E} ( L ) \equiv ( 1 + \Omega d S ) ^ { k } \Omega d ( L \Delta ). \end{equation*}

References

[a1] D. Betounes, "Global shift operators and the higher order calculus of variations" J. Geom. Phys. , 10 (1993) pp. 185–201
[a2] I. Kolar, "A geometric version of the higher ordered Hamilton formalism in fibered manifolds" J. Geom. Phys. , 1 (1984) pp. 127–137
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How to Cite This Entry:
Euler operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_operator&oldid=16951
This article was adapted from an original article by D. Betounes (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article