Liouville-Ostrogradski formula

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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 solutions of a homogeneous system of linear first-order equations


with an operator that is continuous on an interval , and let

be the Wronskian of this system of solutions. The Liouville–Ostrogradski formula has the form


or, equivalently,

Here is the trace of the operator . The Liouville–Ostrogradski formula can be written by means of the Cauchy operator of the system (1) as follows:


The geometrical meaning of (4) (or ) is that as a result of the transformation the oriented volume of any body is increased by a factor .

If one considers a linear homogeneous -th order equation


with continuous coefficients on an interval , and if for , then the Liouville–Ostrogradski formula is the equality


where is the Wronskian of the system of solutions 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 was found by N.H. Abel in 1827 (see [1]), and for arbitrary 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


under the assumption that the vector-valued function

and the matrix are continuous. If is a set of finite measure and the image of this set under the linear mapping , where is the Cauchy operator of the system (7), has measure , then


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

does not change the volume of any body in the phase space if and only if for all ; in particular, the phase volume is conserved by the flow of a Hamiltonian system.


[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
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