Liouville-Ostrogradski formula
Liouville formula
A relation that connects the Wronskian of a system of solutions and the coefficients of an ordinary linear differential equation.
Let 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 |
Ostrogradski-Liouville formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ostrogradski-Liouville_formula&oldid=43063