##### Actions

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 ), and for arbitrary $n$ in 1838 by J. Liouville  and M.V. Ostrogradski ; the equality was obtained by Liouville  and C.G.J. Jacobi  (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.

How to Cite This Entry: