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A linear ordinary differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e0364402.png" /> of the form
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<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/e036/e036440/e0364403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e0364404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e0364405.png" />, are constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e0364406.png" />. This equation was studied in detail by L. Euler, starting from 1740.
+
A linear ordinary differential equation of order  $  n $
 +
of the form
  
The change of the independent variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e0364407.png" /> transforms (1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e0364408.png" /> to the linear equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e0364409.png" /> with constant coefficients
+
$$ \tag{1 }
 +
\sum _ { i= } 0 ^ { n }  a _ {i} x  ^ {i}
  
<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/e036/e036440/e03644010.png" /></td> </tr></table>
+
\frac{d  ^ {i} y }{d x  ^ {i} }
 +
  = f ( x ) ,
 +
$$
  
The [[Characteristic equation|characteristic equation]] of the latter is called the indicial equation of the Euler equation (1). The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644011.png" /> is a [[Regular singular point|regular singular point]] of the homogeneous Euler equation. A fundamental system of (real) solutions of the real homogeneous equation (1) on the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644012.png" /> consists of functions of the form
+
where  $  a _ {i} $,
 +
$  i = 0 \dots n $,
 +
are constants and  $  a _ {n} \neq 0 $.  
 +
This equation was studied in detail by L. Euler, starting from 1740.
  
<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/e036/e036440/e03644013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
The change of the independent variable  $  x = e ^ {t} $
 +
transforms (1) for  $  x > 0 $
 +
to the linear equation of order  $  n $
 +
with constant coefficients
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644014.png" />, then (1) requires the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644015.png" />, and in (2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644016.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644017.png" />.
+
$$
 +
\sum _ { i= } 0 ^ { n }  a _ {i} D ( D - 1 ) \dots ( D- i+ 1) y  = f ( e  ^ {t} ) ,\ \
 +
D =  
 +
\frac{d}{dt}
 +
.
 +
$$
 +
 
 +
The [[Characteristic equation|characteristic equation]] of the latter is called the indicial equation of the Euler equation (1). The point  $  x = 0 $
 +
is a [[Regular singular point|regular singular point]] of the homogeneous Euler equation. A fundamental system of (real) solutions of the real homogeneous equation (1) on the semi-axis  $  x > 0 $
 +
consists of functions of the form
 +
 
 +
$$ \tag{2 }
 +
x  ^  \alpha  \cos ( \beta  \mathop{\rm ln}  x )  \mathop{\rm ln}  ^ {m}  x ,\ \
 +
x  ^  \alpha  \sin ( \beta  \mathop{\rm ln}  x )  \mathop{\rm ln}  ^ {m}  x .
 +
$$
 +
 
 +
If  $  x < 0 $,
 +
then (1) requires the substitution $  x = - e ^ {t} $,  
 +
and in (2) $  x $
 +
is replaced by $  | x | $.
  
 
A more general equation than (1) is the Lagrange equation
 
A more general equation than (1) is the Lagrange 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/e036/e036440/e03644018.png" /></td> </tr></table>
+
$$
 +
\sum _ { j= } 0 ^ { n }  a _ {j} ( \alpha x + \beta )  ^ {j} y  ^ {(} j)  = f ( x) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644021.png" /> are constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644023.png" />, which can also be reduced to a linear equation with constant coefficients by means of the substitution
+
where $  \alpha $,  
 +
$  \beta $
 +
and $  a _ {j} $
 +
are constants and $  \alpha \neq 0 $,  
 +
$  a _ {n} \neq 0 $,  
 +
which can also be reduced to a linear equation with constant coefficients by means of the substitution
  
<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/e036/e036440/e03644024.png" /></td> </tr></table>
+
$$
 +
\alpha x + \beta  = e  ^ {t} \ \
 +
\textrm{ or } \  \alpha x + \beta  = - e ^ {t} .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint  (1947)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint  (1947)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
Line 38: Line 82:
 
One of the problems of variational calculus consists in finding an extremum of the functional
 
One of the problems of variational calculus consists in finding an extremum of the functional
  
<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/e036/e036440/e03644025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
J ( x)  = \int\limits _ { t _ {1} } ^ { {t _ 2 } } F
 +
( t , x , \dot{x} ) dt
 +
$$
  
 
for prescribed conditions at the end points:
 
for prescribed conditions at the end points:
  
<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/e036/e036440/e03644026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
x ( t _ {1} )  = x _ {1} ,\ \
 +
x ( t _ {2} ) =  x _ {2} .
 +
$$
  
If a continuously-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644028.png" />, is a solution of (1) and (2), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644029.png" /> satisfies the Euler equation
+
If a continuously-differentiable function $  x ( t) $,  
 +
$  t _ {1} \leq  t \leq  t _ {2} $,  
 +
is a solution of (1) and (2), then $  x ( t) $
 +
satisfies the Euler 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/e036/e036440/e03644030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
F _ {x} -  
 +
\frac{d}{dt}
 +
F _ {\dot{x} }  = 0 ,
 +
$$
  
 
or, in expanded form,
 
or, in expanded 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/e036/e036440/e03644031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
F _ {x} - F _ {t \dot{x} }  - F _ {x \dot{x} }  \dot{x} -
 +
F _ {\dot{x} \dot{x} }  \dot{x} dot  = 0 .
 +
$$
  
A smooth solution of (3) or (4) is called an extremal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644032.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644033.png" /> on an extremal, then at this point the extremal has a continuous second derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644034.png" />. An extremal such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644035.png" /> at all its points is called non-singular. For a non-singular extremal the Euler equation can be written in a form that is solvable with respect to the second derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644036.png" />.
+
A smooth solution of (3) or (4) is called an extremal. If $  F _ {\dot{x} \dot{x} }  \neq 0 $
 +
at a point $  ( t , x ) $
 +
on an extremal, then at this point the extremal has a continuous second derivative $  \dot{x} dot $.  
 +
An extremal such that $  F _ {\dot{x} \dot{x} }  \neq 0 $
 +
at all its points is called non-singular. For a non-singular extremal the Euler equation can be written in a form that is solvable with respect to the second derivative $  \dot{x} dot $.
  
The solution of the variational problem (1), (2) need not be continuously differentiable. In general, the optimal solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644037.png" /> may be a piecewise-differentiable function. Then at the corner points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644038.png" /> the [[Weierstrass–Erdmann corner conditions|Weierstrass–Erdmann corner conditions]] must be satisfied, which ensure the continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644040.png" /> at the passage through a corner point, while on the segments between consecutive corner points the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644041.png" /> must satisfy the Euler equation. The piecewise-smooth curves consisting of pieces of extremals and satisfying the Weierstrass–Erdmann corner conditions are called polygonal (broken) extremals. In general, the differential Euler equation is an equation of the second order. Hence, its general solution depends on two arbitrary constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644043.png" />:
+
The solution of the variational problem (1), (2) need not be continuously differentiable. In general, the optimal solution $  x ( t) $
 +
may be a piecewise-differentiable function. Then at the corner points of $  x ( t) $
 +
the [[Weierstrass–Erdmann corner conditions|Weierstrass–Erdmann corner conditions]] must be satisfied, which ensure the continuity of $  F _ {\dot{x} }  $
 +
and $  F - \dot{x} F _ {\dot{x} }  $
 +
at the passage through a corner point, while on the segments between consecutive corner points the function $  x ( t) $
 +
must satisfy the Euler equation. The piecewise-smooth curves consisting of pieces of extremals and satisfying the Weierstrass–Erdmann corner conditions are called polygonal (broken) extremals. In general, the differential Euler equation is an equation of the second order. Hence, its general solution depends on two arbitrary constants $  c _ {1} $
 +
and $  c _ {2} $:
  
<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/e036/e036440/e03644044.png" /></td> </tr></table>
+
$$
 +
= f ( t , c _ {1} , c _ {2} ) .
 +
$$
  
 
These arbitrary constants can be determined from the boundary conditions (2):
 
These arbitrary constants can be determined from the boundary conditions (2):
  
<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/e036/e036440/e03644045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
f ( t _ {1} , c _ {1} , c _ {2} )  = x _ {1} ,\ \
 +
f ( t _ {2} , c _ {1} , c _ {2} )  = x _ {2} .
 +
$$
  
 
If the functional depends on several functions, that is,
 
If the functional depends on several functions, that 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/e036/e036440/e03644046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
J ( x  ^ {1} \dots x  ^ {n} )  = \int\limits _ { t _ {1} } ^ { {t _ 2 } } F ( t , x  ^ {1} \dots x  ^ {n} ,\
 +
\dot{x}  ^ {1} \dots \dot{x}  ^ {n} ) dt ,
 +
$$
  
then one obtains a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644047.png" /> Euler equations instead of one:
+
then one obtains a system of $  n $
 +
Euler equations instead of one:
  
<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/e036/e036440/e03644048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
F _ {x  ^ {i}  } -  
 +
\frac{d}{dt}
 +
F _ {\dot{x}  ^ {i}  }  = 0 ,\ \
 +
i = 1 \dots n .
 +
$$
  
The general solution of (7) depends on 2n arbitrary constants, which are determined from given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644049.png" /> boundary conditions (in a problem with fixed end points).
+
The general solution of (7) depends on 2n arbitrary constants, which are determined from given $  2n $
 +
boundary conditions (in a problem with fixed end points).
  
 
In variational problems with variable end points, where the left-hand and right-hand end points of the extremal can move on given hypersurfaces, the missing boundary conditions, which make it possible to obtain a closed system of relations of the type (5), are determined by means of the necessary [[Transversality condition|transversality condition]].
 
In variational problems with variable end points, where the left-hand and right-hand end points of the extremal can move on given hypersurfaces, the missing boundary conditions, which make it possible to obtain a closed system of relations of the type (5), are determined by means of the necessary [[Transversality condition|transversality condition]].
Line 78: Line 163:
 
For variational problems concerning the extremum of functionals that depend on functions of several variables, a necessary condition analogous to the Euler equation is written in the form of the Euler–Ostrogradski equation, which is a partial differential equation (see [[#References|[2]]]).
 
For variational problems concerning the extremum of functionals that depend on functions of several variables, a necessary condition analogous to the Euler equation is written in the form of the Euler–Ostrogradski equation, which is a partial differential equation (see [[#References|[2]]]).
  
In the case of variational problems for a conditional extremum the system of Euler equations is obtained by means of [[Lagrange multipliers|Lagrange multipliers]]. For example, for the [[Bolza problem|Bolza problem]], which requires one to find the extremum of a functional depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644050.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644051.png" />,
+
In the case of variational problems for a conditional extremum the system of Euler equations is obtained by means of [[Lagrange multipliers|Lagrange multipliers]]. For example, for the [[Bolza problem|Bolza problem]], which requires one to find the extremum of a functional depending on $  n $
 +
functions $  x = ( x  ^ {1} \dots x  ^ {n} ) $,
  
<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/e036/e036440/e03644052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
J ( x)  = \int\limits _ { t _ {1} } ^ { {t _ 2 } } f
 +
( t , x , \dot{x} )  dt + g ( t _ {1} , x ( t _ {1} ) ,\
 +
t _ {2} , x ( t _ {2} ) )
 +
$$
  
 
under differential constraints of the form
 
under differential constraints 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/e036/e036440/e03644053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
$$ \tag{9 }
 +
\phi _ {i} ( t , x , \dot{x} )  = 0 ,\ \
 +
i = 1 \dots m ,\  m < n ,
 +
$$
  
 
and boundary conditions
 
and boundary conditions
  
<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/e036/e036440/e03644054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>
+
$$ \tag{10 }
 +
\psi _  \mu  ( t _ {1} , x ( t _ {1} ) , t _ {2} , x
 +
( t _ {2} ) )  = 0 ,\ \
 +
\mu = 1 \dots p ,\  p \leq  2n + 2 ,
 +
$$
  
after using Lagrange multipliers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644057.png" />, to construct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644059.png" /> the function
+
after using Lagrange multipliers $  \lambda _ {0} $
 +
and $  \lambda _ {i} ( t) $,  
 +
$  i = 1 \dots m $,  
 +
to construct from $  f $
 +
and $  \phi _ {i} $
 +
the function
  
<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/e036/e036440/e03644060.png" /></td> </tr></table>
+
$$
 +
F ( t , x , \dot{x} , \lambda )  = \lambda _ {0} f
 +
( t , x , \dot{x} ) + \sum _ { i= } 1 ^ { m }  \lambda _ {i} \phi _ {i} ( t , x , \dot{x} ) ,
 +
$$
  
 
the Euler equations can be written in the from
 
the Euler equations can be written in the from
  
<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/e036/e036440/e03644061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>
+
$$ \tag{11 }
 +
\left .
 +
\begin{array}{c}
 +
 
 +
F _ {\lambda _ {i}  } -
 +
\frac{d}{dt}
 +
F _ {\dot \lambda  _ {i}  }
 +
\equiv  \phi _ {i} ( t , x , \dot{x} )  = 0 ,\  i = 1 \dots m ,
 +
\\
 +
 
 +
F _ {x  ^ {i}  } -  
 +
\frac{d}{dt}
 +
F _ {\dot{x}  ^ {i}  }  = 0 ,\ \
 +
i = 1 \dots n.  
 +
\end{array}
 +
\right \}
 +
$$
  
In this way, an optimal solution of the variational problem (8)–(10) must satisfy the system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644062.png" /> Euler differential equations (11) the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644063.png" /> of which coincide with the given constraints (9). The additional use of the necessary transversality condition leads to a closed boundary value problem for determining the solution of (8)–(10).
+
In this way, an optimal solution of the variational problem (8)–(10) must satisfy the system of $  m + n $
 +
Euler differential equations (11) the first $  m $
 +
of which coincide with the given constraints (9). The additional use of the necessary transversality condition leads to a closed boundary value problem for determining the solution of (8)–(10).
  
 
Besides the Euler equation and the transversality condition, the solution of a variational problem must also satisfy the remaining necessary conditions, that is, those of Clebsch (Legendre), Weierstrass and Jacobi.
 
Besides the Euler equation and the transversality condition, the solution of a variational problem must also satisfy the remaining necessary conditions, that is, those of Clebsch (Legendre), Weierstrass and Jacobi.
Line 108: Line 231:
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
Line 115: Line 237:
 
The Euler equation is a differential equation of the form
 
The Euler equation is a differential equation 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/e036/e036440/e03644064.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{dx}{\sqrt X }
 +
+
 +
\frac{dy}{\sqrt Y }
 +
  = 0 ,
 +
$$
  
 
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/e036/e036440/e03644065.png" /></td> </tr></table>
+
$$
 +
X ( x)  = a _ {0} x  ^ {4} + a _ {1} x  ^ {3}
 +
+ a _ {2} x  ^ {2} + a _ {3} x + a _ {4} ,
 +
$$
  
<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/e036/e036440/e03644066.png" /></td> </tr></table>
+
$$
 +
Y ( x)  = a _ {0} y  ^ {4} + a _ {1} y  ^ {3} + a _ {2} y  ^ {2} + a _ {3} y + a _ {4} .
 +
$$
  
L. Euler considered this equation in a number of papers, starting from 1753. He showed that its general solution has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644068.png" /> is a symmetric polynomial of degree 4 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644070.png" />.
+
L. Euler considered this equation in a number of papers, starting from 1753. He showed that its general solution has the form $  F ( x , y ) = 0 $,  
 +
where $  F ( x , y ) $
 +
is a symmetric polynomial of degree 4 in $  x $
 +
and $  y $.
  
 
''BSE-3''
 
''BSE-3''
Line 132: Line 268:
 
For instance, for an inviscuous fluid the Euler equations of motion are
 
For instance, for an inviscuous fluid the Euler equations of motion are
  
<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/e036/e036440/e03644071.png" /></td> </tr></table>
+
$$
 +
\rho \left (
 +
 
 +
\frac{\partial  u _ {i} }{\partial  t }
 +
+
 +
u _  \alpha 
 +
\frac{\partial  u _ {i} }{\partial  x _  \alpha  }
 +
\right )  = \
 +
-  
 +
\frac{\partial  p }{\partial  x _ {i} }
 +
+ \rho X _ {i} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644072.png" /> is time, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644073.png" /> is the density of the fluid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644074.png" /> is the pressure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644075.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644076.png" />-th component of the body force per unit mass, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644077.png" /> is the velocity component in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644079.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644080.png" /> are Cartesian coordinates, and repeated indices in the equation above indicate summation.
+
where $  t $
 +
is time, $  \rho $
 +
is the density of the fluid, $  p $
 +
is the pressure, $  X _ {i} $
 +
is the $  i $-
 +
th component of the body force per unit mass, and $  u _ {i} $
 +
is the velocity component in the direction of $  x _ {i} $,
 +
$  i = 1 , 2 , 3 $.  
 +
Here $  x _ {1} , x _ {2} , x _ {3} $
 +
are Cartesian coordinates, and repeated indices in the equation above indicate summation.
  
 
Finally, partial differential equations of the type
 
Finally, partial differential equations of the type
  
<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/e036/e036440/e03644081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
 
 +
\frac{\partial  ^ {2} F }{\partial  t _ {1} \partial  t _ {2} }
 +
 
 +
+
 +
\frac{1}{t _ {1} - t _ {2} }
 +
 
 +
\left (
 +
a
 +
\frac \partial {\partial  t _ {1} }
 +
- b
 +
 
 +
\frac \partial {\partial  t _ {2} }
 +
\right ) F  = 0 ,
 +
$$
 +
 
 +
where  $  a $,
 +
$  b $
 +
are constants, are called Euler partial differential equations. Certain solutions can be expressed as integrals
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644083.png" /> are constants, are called Euler partial differential equations. Certain solutions can be expressed as integrals
+
$$
 +
\int\limits
 +
\frac{dx}{( Q ( x) )  ^ {1/n} }
  
<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/e036/e036440/e03644084.png" /></td> </tr></table>
+
$$
  
on Riemann surfaces given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644085.png" /> depending on two parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644087.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644088.png" /> is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644089.png" />.)
+
on Riemann surfaces given by $  Y  ^ {n} = Q ( X , t _ {1} , t _ {2} ) $
 +
depending on two parameters $  t _ {1} $,  
 +
$  t _ {2} $.  
 +
(Here $  Q ( X , t _ {1} , t _ {2} ) $
 +
is a polynomial in $  X $.)
  
There are connections between the monodromy of these solutions and automorphic functions on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036440/e03644090.png" />-dimensional unit ball. Cf. [[#References|[a3]]] for a great deal of material on this topic. The Euler partial differential equation
+
There are connections between the monodromy of these solutions and automorphic functions on the $  2 $-
 +
dimensional unit ball. Cf. [[#References|[a3]]] for a great deal of material on this topic. The Euler partial differential equation
  
 
is also called Euler–Darboux–Poisson equation (cf. [[Mixed and boundary value problems for hyperbolic equations and systems|Mixed and boundary value problems for hyperbolic equations and systems]]) and Euler–Poisson–Darboux equation (cf. [[Differential equation, partial, with singular coefficients|Differential equation, partial, with singular coefficients]]).
 
is also called Euler–Darboux–Poisson equation (cf. [[Mixed and boundary value problems for hyperbolic equations and systems|Mixed and boundary value problems for hyperbolic equations and systems]]) and Euler–Poisson–Darboux equation (cf. [[Differential equation, partial, with singular coefficients|Differential equation, partial, with singular coefficients]]).

Revision as of 19:38, 5 June 2020


A linear ordinary differential equation of order $ n $ of the form

$$ \tag{1 } \sum _ { i= } 0 ^ { n } a _ {i} x ^ {i} \frac{d ^ {i} y }{d x ^ {i} } = f ( x ) , $$

where $ a _ {i} $, $ i = 0 \dots n $, are constants and $ a _ {n} \neq 0 $. This equation was studied in detail by L. Euler, starting from 1740.

The change of the independent variable $ x = e ^ {t} $ transforms (1) for $ x > 0 $ to the linear equation of order $ n $ with constant coefficients

$$ \sum _ { i= } 0 ^ { n } a _ {i} D ( D - 1 ) \dots ( D- i+ 1) y = f ( e ^ {t} ) ,\ \ D = \frac{d}{dt} . $$

The characteristic equation of the latter is called the indicial equation of the Euler equation (1). The point $ x = 0 $ is a regular singular point of the homogeneous Euler equation. A fundamental system of (real) solutions of the real homogeneous equation (1) on the semi-axis $ x > 0 $ consists of functions of the form

$$ \tag{2 } x ^ \alpha \cos ( \beta \mathop{\rm ln} x ) \mathop{\rm ln} ^ {m} x ,\ \ x ^ \alpha \sin ( \beta \mathop{\rm ln} x ) \mathop{\rm ln} ^ {m} x . $$

If $ x < 0 $, then (1) requires the substitution $ x = - e ^ {t} $, and in (2) $ x $ is replaced by $ | x | $.

A more general equation than (1) is the Lagrange equation

$$ \sum _ { j= } 0 ^ { n } a _ {j} ( \alpha x + \beta ) ^ {j} y ^ {(} j) = f ( x) , $$

where $ \alpha $, $ \beta $ and $ a _ {j} $ are constants and $ \alpha \neq 0 $, $ a _ {n} \neq 0 $, which can also be reduced to a linear equation with constant coefficients by means of the substitution

$$ \alpha x + \beta = e ^ {t} \ \ \textrm{ or } \ \alpha x + \beta = - e ^ {t} . $$

References

[1] E. Kamke, "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint (1947)

Comments

References

[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) MR1570308 MR0010757 MR1524980 MR0000325 MR1522581 Zbl 0612.34002 Zbl 0191.09801 Zbl 0063.02971 Zbl 0022.13601 Zbl 65.1253.02 Zbl 53.0399.07

The Euler equation is a necessary condition for an extremum in problems of variational calculus; it was obtained by L. Euler (1744). Later J.L. Lagrange (1759) derived it by a different method. For this reason it is sometimes called the Euler–Lagrange equation. The Euler equation is a necessary condition for the vanishing of the first variation of a functional.

One of the problems of variational calculus consists in finding an extremum of the functional

$$ \tag{1 } J ( x) = \int\limits _ { t _ {1} } ^ { {t _ 2 } } F ( t , x , \dot{x} ) dt $$

for prescribed conditions at the end points:

$$ \tag{2 } x ( t _ {1} ) = x _ {1} ,\ \ x ( t _ {2} ) = x _ {2} . $$

If a continuously-differentiable function $ x ( t) $, $ t _ {1} \leq t \leq t _ {2} $, is a solution of (1) and (2), then $ x ( t) $ satisfies the Euler equation

$$ \tag{3 } F _ {x} - \frac{d}{dt} F _ {\dot{x} } = 0 , $$

or, in expanded form,

$$ \tag{4 } F _ {x} - F _ {t \dot{x} } - F _ {x \dot{x} } \dot{x} - F _ {\dot{x} \dot{x} } \dot{x} dot = 0 . $$

A smooth solution of (3) or (4) is called an extremal. If $ F _ {\dot{x} \dot{x} } \neq 0 $ at a point $ ( t , x ) $ on an extremal, then at this point the extremal has a continuous second derivative $ \dot{x} dot $. An extremal such that $ F _ {\dot{x} \dot{x} } \neq 0 $ at all its points is called non-singular. For a non-singular extremal the Euler equation can be written in a form that is solvable with respect to the second derivative $ \dot{x} dot $.

The solution of the variational problem (1), (2) need not be continuously differentiable. In general, the optimal solution $ x ( t) $ may be a piecewise-differentiable function. Then at the corner points of $ x ( t) $ the Weierstrass–Erdmann corner conditions must be satisfied, which ensure the continuity of $ F _ {\dot{x} } $ and $ F - \dot{x} F _ {\dot{x} } $ at the passage through a corner point, while on the segments between consecutive corner points the function $ x ( t) $ must satisfy the Euler equation. The piecewise-smooth curves consisting of pieces of extremals and satisfying the Weierstrass–Erdmann corner conditions are called polygonal (broken) extremals. In general, the differential Euler equation is an equation of the second order. Hence, its general solution depends on two arbitrary constants $ c _ {1} $ and $ c _ {2} $:

$$ x = f ( t , c _ {1} , c _ {2} ) . $$

These arbitrary constants can be determined from the boundary conditions (2):

$$ \tag{5 } f ( t _ {1} , c _ {1} , c _ {2} ) = x _ {1} ,\ \ f ( t _ {2} , c _ {1} , c _ {2} ) = x _ {2} . $$

If the functional depends on several functions, that is,

$$ \tag{6 } J ( x ^ {1} \dots x ^ {n} ) = \int\limits _ { t _ {1} } ^ { {t _ 2 } } F ( t , x ^ {1} \dots x ^ {n} ,\ \dot{x} ^ {1} \dots \dot{x} ^ {n} ) dt , $$

then one obtains a system of $ n $ Euler equations instead of one:

$$ \tag{7 } F _ {x ^ {i} } - \frac{d}{dt} F _ {\dot{x} ^ {i} } = 0 ,\ \ i = 1 \dots n . $$

The general solution of (7) depends on 2n arbitrary constants, which are determined from given $ 2n $ boundary conditions (in a problem with fixed end points).

In variational problems with variable end points, where the left-hand and right-hand end points of the extremal can move on given hypersurfaces, the missing boundary conditions, which make it possible to obtain a closed system of relations of the type (5), are determined by means of the necessary transversality condition.

For functionals containing higher-order derivatives (not just the first one, as in (1) and (6)), a necessary condition analogous to the Euler equation can be written in the form of the Euler–Poisson differential equation (see [1]).

For variational problems concerning the extremum of functionals that depend on functions of several variables, a necessary condition analogous to the Euler equation is written in the form of the Euler–Ostrogradski equation, which is a partial differential equation (see [2]).

In the case of variational problems for a conditional extremum the system of Euler equations is obtained by means of Lagrange multipliers. For example, for the Bolza problem, which requires one to find the extremum of a functional depending on $ n $ functions $ x = ( x ^ {1} \dots x ^ {n} ) $,

$$ \tag{8 } J ( x) = \int\limits _ { t _ {1} } ^ { {t _ 2 } } f ( t , x , \dot{x} ) dt + g ( t _ {1} , x ( t _ {1} ) ,\ t _ {2} , x ( t _ {2} ) ) $$

under differential constraints of the form

$$ \tag{9 } \phi _ {i} ( t , x , \dot{x} ) = 0 ,\ \ i = 1 \dots m ,\ m < n , $$

and boundary conditions

$$ \tag{10 } \psi _ \mu ( t _ {1} , x ( t _ {1} ) , t _ {2} , x ( t _ {2} ) ) = 0 ,\ \ \mu = 1 \dots p ,\ p \leq 2n + 2 , $$

after using Lagrange multipliers $ \lambda _ {0} $ and $ \lambda _ {i} ( t) $, $ i = 1 \dots m $, to construct from $ f $ and $ \phi _ {i} $ the function

$$ F ( t , x , \dot{x} , \lambda ) = \lambda _ {0} f ( t , x , \dot{x} ) + \sum _ { i= } 1 ^ { m } \lambda _ {i} \phi _ {i} ( t , x , \dot{x} ) , $$

the Euler equations can be written in the from

$$ \tag{11 } \left . \begin{array}{c} F _ {\lambda _ {i} } - \frac{d}{dt} F _ {\dot \lambda _ {i} } \equiv \phi _ {i} ( t , x , \dot{x} ) = 0 ,\ i = 1 \dots m , \\ F _ {x ^ {i} } - \frac{d}{dt} F _ {\dot{x} ^ {i} } = 0 ,\ \ i = 1 \dots n. \end{array} \right \} $$

In this way, an optimal solution of the variational problem (8)–(10) must satisfy the system of $ m + n $ Euler differential equations (11) the first $ m $ of which coincide with the given constraints (9). The additional use of the necessary transversality condition leads to a closed boundary value problem for determining the solution of (8)–(10).

Besides the Euler equation and the transversality condition, the solution of a variational problem must also satisfy the remaining necessary conditions, that is, those of Clebsch (Legendre), Weierstrass and Jacobi.

References

[1] N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian) MR0142019 Zbl 0718.49001
[2] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)

I.B. Vapnyarskii

Comments

References

[a1] W.H. Fleming, R.W. Rishel, "Deterministic and stochastic optimal control" , Springer (1975) MR0454768 Zbl 0323.49001
[a2] I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian) Zbl 0534.58024

The Euler equation is a differential equation of the form

$$ \frac{dx}{\sqrt X } + \frac{dy}{\sqrt Y } = 0 , $$

where

$$ X ( x) = a _ {0} x ^ {4} + a _ {1} x ^ {3} + a _ {2} x ^ {2} + a _ {3} x + a _ {4} , $$

$$ Y ( x) = a _ {0} y ^ {4} + a _ {1} y ^ {3} + a _ {2} y ^ {2} + a _ {3} y + a _ {4} . $$

L. Euler considered this equation in a number of papers, starting from 1753. He showed that its general solution has the form $ F ( x , y ) = 0 $, where $ F ( x , y ) $ is a symmetric polynomial of degree 4 in $ x $ and $ y $.

BSE-3

Comments

In fluid mechanics, the system of equations of motion are also called the Euler equations.

For instance, for an inviscuous fluid the Euler equations of motion are

$$ \rho \left ( \frac{\partial u _ {i} }{\partial t } + u _ \alpha \frac{\partial u _ {i} }{\partial x _ \alpha } \right ) = \ - \frac{\partial p }{\partial x _ {i} } + \rho X _ {i} , $$

where $ t $ is time, $ \rho $ is the density of the fluid, $ p $ is the pressure, $ X _ {i} $ is the $ i $- th component of the body force per unit mass, and $ u _ {i} $ is the velocity component in the direction of $ x _ {i} $, $ i = 1 , 2 , 3 $. Here $ x _ {1} , x _ {2} , x _ {3} $ are Cartesian coordinates, and repeated indices in the equation above indicate summation.

Finally, partial differential equations of the type

$$ \tag{a1 } \frac{\partial ^ {2} F }{\partial t _ {1} \partial t _ {2} } + \frac{1}{t _ {1} - t _ {2} } \left ( a \frac \partial {\partial t _ {1} } - b \frac \partial {\partial t _ {2} } \right ) F = 0 , $$

where $ a $, $ b $ are constants, are called Euler partial differential equations. Certain solutions can be expressed as integrals

$$ \int\limits \frac{dx}{( Q ( x) ) ^ {1/n} } $$

on Riemann surfaces given by $ Y ^ {n} = Q ( X , t _ {1} , t _ {2} ) $ depending on two parameters $ t _ {1} $, $ t _ {2} $. (Here $ Q ( X , t _ {1} , t _ {2} ) $ is a polynomial in $ X $.)

There are connections between the monodromy of these solutions and automorphic functions on the $ 2 $- dimensional unit ball. Cf. [a3] for a great deal of material on this topic. The Euler partial differential equation

is also called Euler–Darboux–Poisson equation (cf. Mixed and boundary value problems for hyperbolic equations and systems) and Euler–Poisson–Darboux equation (cf. Differential equation, partial, with singular coefficients).

References

[a1] A.J. Chorin, J.E. Marsden, "A mathematical introduction to fluid dynamics" , Springer (1979)
[a2] C.-S. Yih, "Stratified flows" , Acad. Press (1980) MR0569474 Zbl 0458.76095
[a3] R.-P. Holzapfel, "Geometry and arithmetic around Euler partial differential equations" , Reidel (1986) MR0867406 MR0849778 Zbl 0595.14017 Zbl 0595.14016
How to Cite This Entry:
Euler equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_equation&oldid=46858
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article