Jacobi last multiplier

(A related notion is that of a inverse Jacobi multiplier).
Given a vector field $X \in X (M)$ on a $n$ -dimensional manifold $M$ with a distinguished volume form $Ω$ , a Jacobi last multiplier (JLM) is a smooth function $f$ such that $f X$ has null divergence, i.e. , $L_{f X} Ω = 0$ .
In the particular case of $R^{n}$ with the standard volume form and $X = \sum_{k} X_{k} \partial_{x_{k}}$ we get the expression:

\begin{matrix} (1) & \sum_{k} \frac{\partial (f X_{k})}{\partial x_{k}} = 0 \end{matrix}

or equivalently

\sum_{k} X_{k} \frac{\partial f}{\partial x_{k}} = - f div (X)

X (f) + f 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 (\frac{f_{1}}{f_{2}}) = \frac{X (f_{1}) f_{2} - f_{1} X (f_{2})}{f_{2}^{2}} = 0

Alternative definition:
Since for any volume form we have

L_{X} (f Ω) = L_{f X} (Ω)

(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 $f Ω$ is invariant with respect to $X$ , since $L_{X} (f Ω) = 0$ .

Another equivalent characterization is what follows: $f$ is a JLM for $X$ with respect to $Ω$ if there exists a $(n - 2)$ -form $σ$ such that

d σ = X ⌟ f Ω = f X ⌟ Ω,

which is equivalente, in a local sense, to requiring $f X ⌟ Ω$ to be closed (by Poincare lemma). This is because of the identity (for $n$ -forms)

L_{X} Ω = d (X ⌟ Ω) .

If we fix a function $f$ on the manifold $(M, Ω)$ , the vector fields $X$ admitting $f$ as a JLM constitute a real Lie algebra. This follows from the formula

L_{X} \circ L_{Y} - L_{Y} \circ L_{X} = L_{[X, Y]}

Jacobi's theorem

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 $Φ_{1}, Φ_{2}, \dots, Φ_{n - 2}$ of the vector field $X$ reduced in such a way that $x_{1}$ does not appear in $Φ_{2}$ , $x_{1}, x_{2}$ do not appear in $Φ_{3}$ 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

μ = \frac{M}{\frac{\partial Φ_{1}}{\partial x_{1}} \frac{\partial Φ_{2}}{\partial x_{2}} \dots \frac{\partial Φ_{n - 2}}{\partial x_{n - 2}}}

in which $M$ is a Jacobi last multiplier in the sense of $(1)$ .
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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

μ = \frac{M}{\frac{\partial (y_{1}, \dots, y_{n})}{\partial (x_{1}, \dots, x_{n})}}

where $\frac{\partial (y_{1}, \dots, y_{n})}{\partial (x_{1}, \dots, x_{n})}$ is the Jacobian determinant of the change of variables

{\begin{cases} y_{i} = Φ_{i}, 1 \leq i \leq n - 2 \\ y_{n - 1} = x_{n - 1} \\ y_{n} = x_{n} \end{cases}

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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.
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In the particular case of the vector field $A$ the associated to the $m$ th order ODE

u_{m} = ϕ (x, u, \dots, u_{m - 1})

a Jacobi last multiplier must satisfy

\frac{\partial M}{\partial x} + \sum \frac{\partial (M u_{k + 1})}{\partial x_{k}} + \frac{\partial (M ϕ)}{\partial u_{m - 1}} = 0

My personal researchs on JLM are in ideas about JLM.