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 (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)

References

[a1]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) pp. 220–227