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 $ 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