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''Liouville formula''
 
''Liouville formula''
  
 
A relation that connects the [[Wronskian|Wronskian]] of a system of solutions and the coefficients of an ordinary linear differential equation.
 
A relation that connects the [[Wronskian|Wronskian]] of a system of solutions and the coefficients of an ordinary linear differential equation.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l0596601.png" /> be an arbitrary system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l0596602.png" /> solutions of a homogeneous system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l0596603.png" /> linear first-order equations
+
Let $  x _ {1} ( t) \dots x _ {n} ( t) $
 +
be an arbitrary system of $  n $
 +
solutions of a homogeneous system of $  n $
 +
linear first-order 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/l/l059/l059660/l0596604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
x  ^  \prime  = A ( t) x ,\ \
 +
x \in \mathbf R  ^ {n} ,
 +
$$
  
with an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l0596605.png" /> that is continuous on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l0596606.png" />, and let
+
with an operator $  A ( t) $
 +
that is continuous on an interval $  I $,  
 +
and let
  
<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/l/l059/l059660/l0596607.png" /></td> </tr></table>
+
$$
 +
W ( x _ {1} ( t) \dots x _ {n} ( t) )  = W ( t)
 +
$$
  
 
be the Wronskian of this system of solutions. The Liouville–Ostrogradski formula has the form
 
be the Wronskian of this system of solutions. The Liouville–Ostrogradski formula has 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/l/l059/l059660/l0596608.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
 
 +
\frac{d}{dt}
 +
W ( t)  = W ( t) \cdot
 +
\mathop{\rm Tr}  A ( t) ,\  t \in I ,
 +
$$
  
 
or, equivalently,
 
or, equivalently,
  
<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/l/l059/l059660/l0596609.png" /></td> </tr></table>
+
$$
 +
W ( x _ {1} ( t) \dots x _ {n} ( t) ) =
 +
$$
 +
 
 +
$$
 +
= \
 +
W ( x _ {1} ( t _ {0} ) \dots x _ {n} ( t _ {0} ) ) \cdot  \mathop{\rm exp}  \int\limits _ {t _ {0} } ^ { t }
 +
\mathop{\rm Tr}  A ( s)  d s ,\  t , t _ {0} \in I .
 +
$$
  
<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/l/l059/l059660/l05966010.png" /></td> </tr></table>
+
Here  $  \mathop{\rm Tr}  A ( t) $
 +
is the [[Trace|trace]] of the operator  $  A ( t) $.
 +
The Liouville–Ostrogradski formula can be written by means of the [[Cauchy operator|Cauchy operator]]  $  X ( t , t _ {0} ) $
 +
of the system (1) as follows:
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966011.png" /> is the [[Trace|trace]] of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966012.png" />. The Liouville–Ostrogradski formula can be written by means of the [[Cauchy operator|Cauchy operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966013.png" /> of the system (1) as follows:
+
$$ \tag{4 }
 +
\mathop{\rm det}  X ( t , t _ {0} )  = \
 +
\mathop{\rm exp}  \int\limits _ {t _ {0} } ^ { t }
 +
\mathop{\rm Tr}  A ( s)  d s ,\ \
 +
t , t _ {0} \in I .
 +
$$
  
<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/l/l059/l059660/l05966014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
The geometrical meaning of (4) (or ) is that as a result of the transformation  $  X ( t , t _ {0} ) : \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $
 +
the oriented volume of any body is increased by a factor  $  \mathop{\rm exp}  \int _ {t _ {0}  }  ^ {t}  \mathop{\rm Tr}  A ( s) d s $.
  
The geometrical meaning of (4) (or ) is that as a result of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966015.png" /> the oriented volume of any body is increased by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966016.png" />.
+
If one considers a linear homogeneous  $  n $-
 +
th order equation
  
If one considers a linear homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966017.png" />-th order equation
+
$$ \tag{5 }
 +
p _ {0} ( t) y  ^ {(} n) + \dots + p _ {n} ( t) y  = 0
 +
$$
  
<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/l/l059/l059660/l05966018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
with continuous coefficients on an interval  $  I $,
 +
and if  $  p _ {0} ( t) \neq 0 $
 +
for  $  t \in I $,
 +
then the Liouville–Ostrogradski formula is the equality
  
with continuous coefficients on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966019.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966020.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966021.png" />, then the Liouville–Ostrogradski formula is the equality
+
$$ \tag{6 }
 +
W ( y _ {1} ( t) \dots y _ {n} ( t) ) =
 +
$$
  
<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/l/l059/l059660/l05966022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$
 +
= \
 +
W ( y _ {1} ( t _ {0} ) \dots y _ {n} ( t _ {0} ) ) \cdot  \mathop{\rm exp}  \left [ - \int\limits _ {t _ {0} } ^ { t }
  
<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/l/l059/l059660/l05966023.png" /></td> </tr></table>
+
\frac{p _ {1} ( s) }{p _ {0} ( s) }
 +
  ds \right ] ,\  t , t _ {0} \in I ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966024.png" /> is the Wronskian of the system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966025.png" /> solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966026.png" /> of (5). The Liouville–Ostrogradski formulas , (6) are ordinarily used in the case when the system of solutions in question is fundamental (cf. [[Fundamental system of solutions|Fundamental system of solutions]]). For example, formula (6) makes it possible to find by quadratures the general solution of a linear homogeneous equation of the second order if one knows one particular non-trivial solution of it.
+
where $  W ( y _ {1} ( t) \dots y _ {n} ( t) ) $
 +
is the Wronskian of the system of $  n $
 +
solutions $  y _ {1} ( t) \dots y _ {n} ( t) $
 +
of (5). The Liouville–Ostrogradski formulas , (6) are ordinarily used in the case when the system of solutions in question is fundamental (cf. [[Fundamental system of solutions|Fundamental system of solutions]]). For example, formula (6) makes it possible to find by quadratures the general solution of a linear homogeneous equation of the second order if one knows one particular non-trivial solution of it.
  
The relation (6) for equation (5) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966027.png" /> was found by N.H. Abel in 1827 (see [[#References|[1]]]), and for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966028.png" /> in 1838 by J. Liouville [[#References|[2]]] and M.V. Ostrogradski [[#References|[3]]]; the equality
+
The relation (6) for equation (5) with $  n = 2 $
 +
was found by N.H. Abel in 1827 (see [[#References|[1]]]), and for arbitrary $  n $
 +
in 1838 by J. Liouville [[#References|[2]]] and M.V. Ostrogradski [[#References|[3]]]; the equality
  
 
was obtained by Liouville [[#References|[2]]] and C.G.J. Jacobi [[#References|[4]]] (as a consequence of this,
 
was obtained by Liouville [[#References|[2]]] and C.G.J. Jacobi [[#References|[4]]] (as a consequence of this,
Line 47: Line 108:
 
The Liouville–Ostrogradski formula (2) can be generalized to a non-linear system
 
The Liouville–Ostrogradski formula (2) can be generalized to a non-linear system
  
<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/l/l059/l059660/l05966029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
x  ^  \prime  = f ( t , x ) ,\  x \in \mathbf R  ^ {n} ,
 +
$$
  
 
under the assumption that the vector-valued function
 
under the assumption that the vector-valued 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/l/l059/l059660/l05966030.png" /></td> </tr></table>
+
$$
 +
f ( t , x )  = ( f _ {1} ( t , x _ {1} \dots x _ {n} )
 +
\dots f _ {n} ( t , x _ {1} \dots x _ {n} ))
 +
$$
 +
 
 +
and the matrix  $  \partial  f / \partial  x $
 +
are continuous. If  $  \Omega _ {t _ {0}  } \subset  \mathbf R  ^ {n} $
 +
is a set of finite measure  $  \mu ( t _ {0} ) $
 +
and the image  $  \Omega _ {t} $
 +
of this set under the linear mapping  $  X ( t , t _ {0} ) : \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $,
 +
where  $  X ( t , t _ {0} ) $
 +
is the Cauchy operator of the system (7), has measure  $  \mu ( t) $,
 +
then
  
and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966031.png" /> are continuous. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966032.png" /> is a set of finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966033.png" /> and the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966034.png" /> of this set under the linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966036.png" /> is the Cauchy operator of the system (7), has measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966037.png" />, then
+
$$
  
<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/l/l059/l059660/l05966038.png" /></td> </tr></table>
+
\frac{d \mu }{dt}
 +
  = \int\limits _ {\Omega _ {t} }  \mathop{\rm div} _ {x}  f ( t , x )  dx ;
 +
$$
  
 
here
 
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/l/l059/l059660/l05966039.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm div} _ {x}  f ( t , x )  = \sum _ { i= } 1 ^ { n } 
 +
\frac{\partial
 +
f _ {i} ( t , x _ {1} \dots x _ {n} ) }{\partial  x _ {i} }
 +
.
 +
$$
  
 
This implies Liouville's theorem on the conservation of phase volume, which has important applications in the theory of dynamical systems and in [[Statistical mechanics, mathematical problems in|statistical mechanics, mathematical problems in]]: The flow of a smooth [[Autonomous system|autonomous system]]
 
This implies Liouville's theorem on the conservation of phase volume, which has important applications in the theory of dynamical systems and in [[Statistical mechanics, mathematical problems in|statistical mechanics, mathematical problems in]]: The flow of a smooth [[Autonomous system|autonomous system]]
  
<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/l/l059/l059660/l05966040.png" /></td> </tr></table>
+
$$
 +
x  ^  \prime  = f ( x) ,\  x \in \mathbf R  ^ {n} ,
 +
$$
  
does not change the volume of any body in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966041.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966043.png" />; in particular, the phase volume is conserved by the flow of a [[Hamiltonian system|Hamiltonian system]].
+
does not change the volume of any body in the phase space $  \mathbf R  ^ {n} $
 +
if and only if $  \mathop{\rm div}  f ( x) = 0 $
 +
for all $  x $;  
 +
in particular, the phase volume is conserved by the flow of a [[Hamiltonian system|Hamiltonian system]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.H. Abel,  "Ueber einige bestimmte Integrale"  ''J. Reine Angew. Math.'' , '''2'''  (1827)  pp. 22–30</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Liouville,  ''J. Math. Pures Appl.'' , '''3'''  (1838)  pp. 342–349</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.V. Ostrogradskii,  , ''Collected works'' , '''3''' , Kiev  (1961)  pp. 124–126  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C.G.J. Jacobi,  "Gesammelte Werke" , '''4''' , Chelsea, reprint  (1969)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.H. Abel,  "Ueber einige bestimmte Integrale"  ''J. Reine Angew. Math.'' , '''2'''  (1827)  pp. 22–30</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Liouville,  ''J. Math. Pures Appl.'' , '''3'''  (1838)  pp. 342–349</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.V. Ostrogradskii,  , ''Collected works'' , '''3''' , Kiev  (1961)  pp. 124–126  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C.G.J. Jacobi,  "Gesammelte Werke" , '''4''' , Chelsea, reprint  (1969)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>

Latest revision as of 22:17, 5 June 2020


Liouville formula

A relation that connects the Wronskian of a system of solutions and the coefficients of an ordinary linear differential equation.

Let $ x _ {1} ( t) \dots x _ {n} ( t) $ be an arbitrary system of $ n $ solutions of a homogeneous system of $ n $ linear first-order equations

$$ \tag{1 } x ^ \prime = A ( t) x ,\ \ x \in \mathbf R ^ {n} , $$

with an operator $ A ( t) $ that is continuous on an interval $ I $, and let

$$ W ( x _ {1} ( t) \dots x _ {n} ( t) ) = W ( t) $$

be the Wronskian of this system of solutions. The Liouville–Ostrogradski formula has the form

$$ \tag{2 } \frac{d}{dt} W ( t) = W ( t) \cdot \mathop{\rm Tr} A ( t) ,\ t \in I , $$

or, equivalently,

$$ W ( x _ {1} ( t) \dots x _ {n} ( t) ) = $$

$$ = \ W ( x _ {1} ( t _ {0} ) \dots x _ {n} ( t _ {0} ) ) \cdot \mathop{\rm exp} \int\limits _ {t _ {0} } ^ { t } \mathop{\rm Tr} A ( s) d s ,\ t , t _ {0} \in I . $$

Here $ \mathop{\rm Tr} A ( t) $ is the trace of the operator $ A ( t) $. The Liouville–Ostrogradski formula can be written by means of the Cauchy operator $ X ( t , t _ {0} ) $ of the system (1) as follows:

$$ \tag{4 } \mathop{\rm det} X ( t , t _ {0} ) = \ \mathop{\rm exp} \int\limits _ {t _ {0} } ^ { t } \mathop{\rm Tr} A ( s) d s ,\ \ t , t _ {0} \in I . $$

The geometrical meaning of (4) (or ) is that as a result of the transformation $ X ( t , t _ {0} ) : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $ the oriented volume of any body is increased by a factor $ \mathop{\rm exp} \int _ {t _ {0} } ^ {t} \mathop{\rm Tr} A ( s) d s $.

If one considers a linear homogeneous $ n $- th order equation

$$ \tag{5 } p _ {0} ( t) y ^ {(} n) + \dots + p _ {n} ( t) y = 0 $$

with continuous coefficients on an interval $ I $, and if $ p _ {0} ( t) \neq 0 $ for $ t \in I $, then the Liouville–Ostrogradski formula is the equality

$$ \tag{6 } W ( y _ {1} ( t) \dots y _ {n} ( t) ) = $$

$$ = \ W ( y _ {1} ( t _ {0} ) \dots y _ {n} ( t _ {0} ) ) \cdot \mathop{\rm exp} \left [ - \int\limits _ {t _ {0} } ^ { t } \frac{p _ {1} ( s) }{p _ {0} ( s) } ds \right ] ,\ t , t _ {0} \in I , $$

where $ W ( y _ {1} ( t) \dots y _ {n} ( t) ) $ is the Wronskian of the system of $ n $ solutions $ y _ {1} ( t) \dots y _ {n} ( t) $ of (5). The Liouville–Ostrogradski formulas , (6) are ordinarily used in the case when the system of solutions in question is fundamental (cf. Fundamental system of solutions). For example, formula (6) makes it possible to find by quadratures the general solution of a linear homogeneous equation of the second order if one knows one particular non-trivial solution of it.

The relation (6) for equation (5) with $ n = 2 $ was found by N.H. Abel in 1827 (see [1]), and for arbitrary $ n $ in 1838 by J. Liouville [2] and M.V. Ostrogradski [3]; the equality

was obtained by Liouville [2] and C.G.J. Jacobi [4] (as a consequence of this,

is sometimes called the Jacobi formula).

The Liouville–Ostrogradski formula (2) can be generalized to a non-linear system

$$ \tag{7 } x ^ \prime = f ( t , x ) ,\ x \in \mathbf R ^ {n} , $$

under the assumption that the vector-valued function

$$ f ( t , x ) = ( f _ {1} ( t , x _ {1} \dots x _ {n} ) \dots f _ {n} ( t , x _ {1} \dots x _ {n} )) $$

and the matrix $ \partial f / \partial x $ are continuous. If $ \Omega _ {t _ {0} } \subset \mathbf R ^ {n} $ is a set of finite measure $ \mu ( t _ {0} ) $ and the image $ \Omega _ {t} $ of this set under the linear mapping $ X ( t , t _ {0} ) : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $, where $ X ( t , t _ {0} ) $ is the Cauchy operator of the system (7), has measure $ \mu ( t) $, then

$$ \frac{d \mu }{dt} = \int\limits _ {\Omega _ {t} } \mathop{\rm div} _ {x} f ( t , x ) dx ; $$

here

$$ \mathop{\rm div} _ {x} f ( t , x ) = \sum _ { i= } 1 ^ { n } \frac{\partial f _ {i} ( t , x _ {1} \dots x _ {n} ) }{\partial x _ {i} } . $$

This implies Liouville's theorem on the conservation of phase volume, which has important applications in the theory of dynamical systems and in statistical mechanics, mathematical problems in: The flow of a smooth autonomous system

$$ x ^ \prime = f ( x) ,\ x \in \mathbf R ^ {n} , $$

does not change the volume of any body in the phase space $ \mathbf R ^ {n} $ if and only if $ \mathop{\rm div} f ( x) = 0 $ for all $ x $; in particular, the phase volume is conserved by the flow of a Hamiltonian system.

References

[1] N.H. Abel, "Ueber einige bestimmte Integrale" J. Reine Angew. Math. , 2 (1827) pp. 22–30
[2] J. Liouville, J. Math. Pures Appl. , 3 (1838) pp. 342–349
[3] M.V. Ostrogradskii, , Collected works , 3 , Kiev (1961) pp. 124–126 (In Russian)
[4] C.G.J. Jacobi, "Gesammelte Werke" , 4 , Chelsea, reprint (1969)
[5] L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)
[6] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)

Comments

References

[a1] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) pp. 220–227
How to Cite This Entry:
Liouville-Ostrogradski formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville-Ostrogradski_formula&oldid=47669
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article