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m (using eqref)
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linear first-order equations
 
linear first-order equations
  
$$ \tag{1 }
+
$$ \tag{1}
 
x  ^  \prime  =  A ( t) x ,\ \  
 
x  ^  \prime  =  A ( t) x ,\ \  
 
x \in \mathbf R  ^ {n} ,
 
x \in \mathbf R  ^ {n} ,
Line 35: Line 35:
 
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
  
$$ \tag{2 }
+
$$ \tag{2}
  
 
\frac{d}{dt}
 
\frac{d}{dt}
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is the [[Trace|trace]] of the operator  $  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} ) $
 
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:
+
of the system \eqref{1} as follows:
  
$$ \tag{4 }
+
$$ \tag{4}
 
  \mathop{\rm det}  X ( t , t _ {0} )  = \  
 
  \mathop{\rm det}  X ( t , t _ {0} )  = \  
 
  \mathop{\rm exp}  \int\limits _ {t _ {0} } ^ { t }  
 
  \mathop{\rm exp}  \int\limits _ {t _ {0} } ^ { t }  
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$$
 
$$
  
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 geometrical meaning of \eqref{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 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 $-
+
If one considers a linear homogeneous  $ n$-th order equation
th order equation
 
  
$$ \tag{5 }
+
$$ \tag{5}
 
p _ {0} ( t) y  ^ {(} n) + \dots + p _ {n} ( t) y  =  0
 
p _ {0} ( t) y  ^ {(} n) + \dots + p _ {n} ( t) y  =  0
 
$$
 
$$
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then the Liouville–Ostrogradski formula is the equality
 
then the Liouville–Ostrogradski formula is the equality
  
$$ \tag{6 }
+
$$ \tag{6}
 
W ( y _ {1} ( t) \dots y _ {n} ( t) ) =
 
W ( y _ {1} ( t) \dots y _ {n} ( t) ) =
 
$$
 
$$
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is the Wronskian of the system of  $  n $
 
is the Wronskian of the system of  $  n $
 
solutions  $  y _ {1} ( t) \dots y _ {n} ( t) $
 
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.
+
of \eqref{5}. The Liouville–Ostrogradski formulas , \eqref{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 \eqref{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 $
+
The relation \eqref{6} for equation \eqref{5} with $n = 2$
was found by N.H. Abel in 1827 (see [[#References|[1]]]), and for arbitrary $ n $
+
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
 
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,
 
 
is sometimes called the Jacobi formula).
 
is sometimes called the Jacobi formula).
  
 
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
 
+
$$ \tag{7}
$$ \tag{7 }
 
 
x  ^  \prime  =  f ( t , x ) ,\  x \in \mathbf R  ^ {n} ,
 
x  ^  \prime  =  f ( t , x ) ,\  x \in \mathbf R  ^ {n} ,
 
$$
 
$$
Line 155: Line 151:
  
 
====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>
====Comments====
+
<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>
====References====
+
<TR><TD valign="top">[4]</TD> <TD valign="top"> C.G.J. Jacobi,  "Gesammelte Werke" , '''4''' , Chelsea, reprint  (1969)</TD></TR>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>
+
<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>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR>
 +
</table>

Revision as of 12:16, 19 March 2023


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 \eqref{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 \eqref{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 \eqref{5}. The Liouville–Ostrogradski formulas , \eqref{6} are ordinarily used in the case when the system of solutions in question is fundamental (cf. Fundamental system of solutions). For example, formula \eqref{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 \eqref{6} for equation \eqref{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)
[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=52971
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article