A Jacobi manifold [45, 47, 50, 70] is a triplet $(P,\Lambda^{J},\epsilon)$,
where $\Lambda^{J}$ is a bivector field (a skew-symmetric contravariant
2-tensor field) and $\epsilon$ is a vector field satisfying

$[\Lambda^{J},\Lambda^{J}]=2\epsilon\wedge\Lambda^{J},\qquad[\epsilon,\Lambda^{J}]=0,$
(A.1)

where the bracket is the Schouten–Nijenhuis bracket. We can define a Jacobi
bracket on the space $C^{\infty}(P)$ of smooth functions on $M$ by

$\\{F,H\\}^{J}=\Lambda^{J}(dF,dH)+F\epsilon(H)-H\epsilon(F).$ (A.2)

The bracket (A.2) is skew-symmetric and satisfies the Jacobi identity but not
the Leibniz identity. Instead, it is a first-order differential operator since

$\\{F,KH\\}^{J}=K\\{F,H\\}^{J}+H\\{F,K\\}^{J}+KH\epsilon(F),$ (A.3)

for all $F$, $K$ and $H$. This is equivalently written as the weak Leibniz
identity

$\operatorname{supp}(\\{H,F\\}^{J})\subseteq\operatorname{supp}(H)\cap\text{supp}(F).$
(A.4)

For a Hamiltonian function $H$ on a Jacobi manifold
$(P,\Lambda^{J},\epsilon)$, the Hamiltonian vector field is defined to be

$X_{H}^{J}=\sharp_{\Lambda^{J}}(dH)-H\epsilon.$ (A.5)

The relationship between the Hamiltonian vector field and the Jacobi bracket
is given by the equation

$X_{H}^{J}(F)=\\{F,H\\}^{J}-F\epsilon(H).$ (A.6)

The space of Hamiltonian vector fields is closed; moreover, we have that

$[X_{F},X_{H}]=-X_{\\{F,H\\}^{J}},$ (A.7)

where the bracket on the left-hand side is the Jacobi-Lie bracket of vector
fields. This result leads us to determine the Lie algebra

$\mathfrak{X}_{\mathrm{J-ham}}(P)=\big{\\{}X_{H}^{J}\in\mathfrak{X}(M):X_{H}^{J}=\sharp_{\Lambda^{J}}(dH)-H\epsilon\big{\\}}$
(A.8)

of Hamiltonian vector fields on the Jacobi manifold. Further, by taking the
Lie algebra bracket on the space $\mathfrak{X}_{\mathrm{J-ham}}(P)$ as the
minus of the Jacobi-Lie bracket of vector fields, the mapping

$\vartheta:C^{\infty}(P)\longrightarrow\mathfrak{X}_{\mathrm{J-ham}}(P),\qquad
H\mapsto X_{H}^{J}$ (A.9)

turns out to be a Lie algebra homomorphism.

###  A.2 Almost Poisson Manifolds and Dynamics

Consider a manifold $P$ equipped with a skew-symmetric $\mathbb{R}$-bilinear
bracket

$\\{\bullet,\bullet\\}:C^{\infty}(P)\times C^{\infty}(P)\longrightarrow
C^{\infty}(P)$ (A.10)

defined on the space $C^{\infty}(P)$ of smooth functions. The bracket is
called an almost Poisson bracket if it satisfies the Leibniz identity. In this
case, the pair $(P,\\{\bullet,\bullet\\})$ is called an almost Poisson
manifold [8, 25, 48, 69, 72, 73]. An almost Poison bracket is said to be
Poisson if it satisfies the Jacobi identity. In this case, the pair
$(P,\\{\bullet,\bullet\\})$ is called a Poisson manifold.

One may identify an almost Poisson bracket $\\{\bullet,\bullet\\}$ with a
bivector field $\Lambda$ on $P$ according to the following definition

$\Lambda(dF,dH):=\\{F,H\\}$ (A.11)

for all $F$ and $H$ in $C^{\infty}(P)$. One can extend this definition to all
one-forms by linearity. So, we may represent an almost Poisson manifold by a
pair $(P,\Lambda)$ as well. This geometry enables us to define a musical
mapping $\sharp_{\Lambda}$ induced by the bivector field $\Lambda$ as

$\sharp_{\Lambda}:\Gamma^{1}(P)\longrightarrow\mathfrak{X}(P),\qquad\sharp_{\Lambda}(\alpha)(\beta):=\Lambda(\beta,\alpha),$
(A.12)

where the pairing on the left-hand side is the one between the space
$\Gamma^{1}(P)$ of one-form sections and the space $\mathfrak{X}(P)$ of vector
fields. The bracket $\\{\bullet,\bullet\\}$ satisfies the Jacobi identity if
and only if the bivector field $\Lambda$ commutes with itself under the
Schouten-Nijenhuis bracket, that is,

$[\Lambda,\Lambda]=0.$ (A.13)

A bivector field is a Poisson bivector field if the condition (A.13) holds.
So, we can denote a Poisson manifold by a pair $(P,\Lambda)$ with $\Lambda$
satisfying the Jacobi identity (A.13). Accordingly, a Poisson manifold is a
particular instance of a Jacobi manifold where the vector field vanishes.

Consider an (almost) Poisson manifold $(P,\\{\bullet,\bullet\\})$. Given a
Hamiltonian function $H$ (any real-valued function on $P$), the Hamiltonian
vector field $X_{H}$ is defined to be, for all $F$ in $C^{\infty}(P)$,

$X_{H}(F):=\\{F,H\\},$ (A.14)

where the notation $X_{H}(F)$ stands for the directional derivative of $F$ in
the direction of $X_{H}$. In terms of $\sharp_{\Lambda}$, the Hamiltonian
vector field $X_{H}$ can be defined as

$\sharp_{\Lambda}(dH)=X_{H}.$ (A.15)

The Hamiltonian dynamics generated by a Hamiltonian function $H$ is then

$\dot{x}=\\{x,H\\},$ (A.16)

for $x$ in $P$. The skew-symmetry of the bracket implies that the Hamiltonian
function $H$ is conserved all along the motion that is $\dot{H}=0$. This
physically corresponds to the conservation of energy.

###  A.3 Lie-Poisson Dynamics of Kinetic Theories

Let $G$ be a Lie group, $\mathfrak{g}$ be its Lie algebra with skew-symmetric
bilinear bracket

$[\bullet,\bullet]:\mathfrak{g}\times\mathfrak{g}\mapsto\mathfrak{g}$ (A.17)

satisfying the Jacobi identity [37, 71] . Referring to this Lie bracket, one
can define (left) adjoint and (right) coadjoint actions of the Lie algebra
$\mathfrak{g}$ on itself and the dual space as

$\begin{split}ad&:\mathfrak{g}\times\mathfrak{g}\mapsto\mathfrak{g},\qquad
ad_{\xi}\eta:=[\xi,\eta],\\\
ad^{*}&:\mathfrak{g}\times\mathfrak{g}^{*}\mapsto\mathfrak{g}^{*},\qquad\langle
ad^{\ast}_{\xi}\rho,\eta\rangle=\langle\rho,ad_{\xi}\eta\rangle,\end{split}$
(A.18)

respectively. The dual space $\mathfrak{g}^{*}$ is equipped with a Lie Poisson
bracket [54, 43, 44], defined as follows: for two functionals $\mathcal{F}$
and $\mathcal{H}$ defined on $\mathfrak{g}^{*}$, we put

$\left\\{\mathcal{F},\mathcal{H}\right\\}_{\mathfrak{g}^{\ast}}(\rho)=\Big{\langle}\rho,\left[\frac{\delta\mathcal{F}}{\delta\rho},\frac{\delta\mathcal{H}}{\delta\rho}\right]\Big{\rangle}.$
(A.19)

One may consider the minus Lie-Poisson bracket as well. In this work, we
prefer the positive one. For a Hamiltonian functional $\mathcal{H}$, the
dynamics is governed by the Lie-Poisson equations computed in terms of the
coadjoint action as

$\dot{\rho}=-ad_{\delta\mathcal{H}/\delta\rho}^{\ast}\rho.$ (A.20)

Lie-Poisson Dynamics of Kinetic Theories. Consider a bunch of particles rest
in a manifold $M\subset\mathbb{R}^{3}$. $\mathrm{Diff}(M)$ is the group of all
diffeomorphisms on $M$, [3, 6, 26, 51]. The motion of the particles is
determined by the left action of $\mathrm{Diff}(M)$ on the particle space $M$.
The right action commutes with the particle motion and constitutes an infinite
dimensional symmetry group called the particle relabelling symmetry. The Lie
algebra of $\mathrm{Diff}(M)$ is the space $\mathfrak{X}(M)$ of vector fields.
We define the dual space $\mathfrak{X}^{\ast}(M)$ of the Lie algebra as space
of one-form densities $\Lambda^{1}(M)\otimes\mathrm{Den}(M)$ on $M$. Here, the
pairing between a vector field $X$ and a dual element $\Pi\otimes\mu$ is
defined to be $L_{2}$ (simply multiply-and-integrate form) pairing

$\langle\bullet,\bullet\rangle_{L_{2}}:\Gamma^{1}(M)\otimes\mathrm{Den}(M)\times\mathfrak{X}(M)\longrightarrow\mathbb{R},\qquad(\Pi\otimes\mu,X)\mapsto\int_{M}\langle\Pi,X\rangle\mu.$
(A.21)

Here, the pairing inside the integral is between the one-form $\Pi$ and the
vector field $X$, $\mu$ is a density, that is, a volume form, on $M$. The
adjoint and coadjoint actions are

$\begin{split}ad_{X}Y&=[X,Y]_{\mathfrak{X}\left(M\right)}=-[X,Y]_{JL},\\\
ad_{X}^{\ast}(\Pi\otimes\mu)&=(\mathcal{L}_{X}\Pi+\mathrm{div}(X)\Pi)\otimes\mu,\end{split}$
(A.22)

respectively. Here, $\mathrm{div}(X)$ is the divergence of the vector field
with respect to the volume form $\mu$.

If a particle in a medium is flowing, following the dynamics generated by the
vector field $X$ defined on the manifold $M$, then the kinetic dynamics of the
medium can be understood as the lift of this motion. To describe it, we
introduce a linear Hamiltonian functional defined as follows

$\mathcal{H}(\Pi\otimes\mu)=\int_{M}\langle\Pi,X\rangle\mu,$ (A.23)

This functional operates on the space of one-form densities
$\Gamma^{1}(M)\otimes\mathrm{Den}(M)$. We can prove that $\delta
H/\delta\Pi=X$. When the volume is fixed, the Lie-Poisson equation takes the
form

$\dot{\Pi}=-\mathcal{L}_{X}\Pi-\mathrm{div}(X)\Pi.$ (A.24)

However, if the dynamics is generated by a divergence-free vector field (such
as in the case of incompressible fluid flow or Vlasov flow), then the second
term on the right-hand side of (A.24) vanishes, and we obtain

$\dot{\Pi}=-\mathcal{L}_{X}\Pi.$ (A.25)

This intermediate level represents the momentum formulation of kinetic
dynamics. To express the dynamics in terms of density functions, one must
establish a Poisson mapping from the space of one-forms to the space of
densities. In the present work, this has been accomplished for Hamiltonian
dynamics through the dualization of the Lie algebra homomorphism $H\mapsto
X_{H}.$

## Acknowledgements

We acknowledge the financial support of Grant PID2019-106715GBC21, and the
Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S).

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