(A related notion is that of a inverse Jacobi multiplier).
Given a vector field $X \in \mathfrak{X}(M)$ on a $n$-dimensional manifold $M$ with a distinguished volume form $\Omega$, a Jacobi last multiplier (JLM) is a smooth function $f$ such that $fX$ has null divergence, i.e. , $\mathcal{L}_{fX} \Omega=0$.
In the particular case of $\mathbb{R}^n$ with the standard volume form and $X=\sum_k X_k \partial_{x_k}$ we get the expression:
$$ \sum_k \dfrac{\partial (f X_k)}{\partial x_k}=0 \tag{1} $$or equivalently
$$ \sum_k X_k \dfrac{\partial f}{\partial x_k}=-f \mbox{div}(X) $$or
$$ X(f)+f\mbox{div}(X)=0 $$From here, if $X$ is a divergence free vector field, $f$ is a Jacobi multiplier if and only if is a first integral of $X$. Of course, $f=1$ would be one of them.
If $f_1,f_2$ are two Jacobi multipliers, then $f_1/f_2$ is a first integral of $X$:
$$ X\left(\frac{f_1}{f_2}\right)=\frac{X(f_1)f_2-f_1X(f_2)}{f_2^2}=0 $$Alternative definition:
Since for any volume form we have
$$ \mathcal{L}_X (f\Omega)=\mathcal{L}_{fX} (\Omega) $$(see here) we can alternatively define a Jacobi last multiplier like a coefficient for the volume form such that with this new volume form the vector field is divergence-free. Of course, this is clearly equivalent to saying that the modified volume form $f\Omega$ is invariant with respect to $X$, since $\mathcal{L}_X (f\Omega)=0$.
Another equivalent characterization is what follows: $f$ is a JLM for $X$ with respect to $\Omega$ if there exists a $(n-2)$-form $\sigma$ such that
$$ d\sigma=X\,\lrcorner\,f\Omega=fX\,\lrcorner\,\Omega, $$which is equivalente, in a local sense, to requiring $fX\,\lrcorner\,\Omega$ to be closed (by Poincare lemma). This is because of the identity (for $n$-forms)
$$ \mathcal{L}_X \Omega=d(X\,\lrcorner\,\Omega). $$If we fix a function $f$ on the manifold $(M,\Omega)$, the vector fields $X$ admitting $f$ as a JLM constitute a real Lie algebra. This follows from the formula
$$ \mathcal{L}_{X} \circ \mathcal{L}_{Y}-\mathcal{L}_{Y} \circ \mathcal{L}_{X}=\mathcal{L}_{[X, Y]} $$When $X$ is the vector field of a system of ODEs of first order $\dot{x}_i=X_i$ then:
Theorem (Jacobi) (see Berrone_2003)
Given $n-2$ first integrals $\Phi_1, \Phi_2, \ldots, \Phi_{n-2}$ of the vector field $X$ reduced in such a way that $x_1$ does not appear in $\Phi_2$, $x_1,x_2$ do not appear in $\Phi_3$ and so on; then the integrating factor of the final reduced system of ODEs with $x_{n-1}$ and $x_n$ will be given by
$$ {\mu}=\frac{M}{\frac{\partial \Phi_{1}}{\partial x_{1}} \frac{\partial \Phi_{2}}{\partial x_{2}} \cdots \frac{\partial \Phi_{n-2}}{\partial x_{n-2}}} $$in which $M$ is a Jacobi last multiplier in the sense of $(1)$.
$\blacksquare$
Theorem (general version):
There is a version in Muriel_2014 that says that if the $n-2$ first integral are "not so good" then the integrating factor is
$$ {\mu}=\frac{M}{\frac{\partial (y_1,\ldots,y_n)}{\partial (x_1,\ldots,x_n)}} $$where $\frac{\partial (y_1,\ldots,y_n)}{\partial (x_1,\ldots,x_n)}$ is the Jacobian determinant of the change of variables
$$ \left\{ \begin{array}{l} y_i=\Phi_i, 1\leq i \leq n-2\\ y_{n-1}=x_{n-1}\\ y_{n}=x_{n} \end{array}\right. $$$\blacksquare$
Proof: I think the proof has to do with this lemma.
Also, I have a version of the statement and the proof in xournal_147.
$\blacksquare$
In the particular case of the vector field $A$ the associated to the $m$th order ODE
$$ u_m=\phi(x,u,\ldots,u_{m-1}) $$a Jacobi last multiplier must satisfy
$$ \dfrac{\partial M}{\partial x}+\sum \dfrac{\partial (M u_{k+1})}{\partial x_k}+\dfrac{\partial (M\phi)}{\partial u_{m-1}}=0 $$My personal researchs on JLM are in ideas about JLM.
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Author of the notes: Antonio J. Pan-Collantes
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