View source for Liouville-Ostrogradski formula

''Liouville formula''

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

<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>

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

<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>

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>

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>

<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 <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:

<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 <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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059660/l05966017.png" />-th order 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/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 <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

<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>

<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>

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.

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

was obtained by Liouville [[#References|[2]]] and C.G.J. Jacobi [[#References|[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

<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>

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>

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>

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>

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>

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]].

====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>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>