# Hamiltonian systems on almost cosymplectic manifolds

Stefan Berceanu  National Institute for Physics and Nuclear Engineering
Department of Theoretical Physics
PO BOX MG-6, Bucharest-Magurele, Romania<EMAIL_ADDRESS>
###### Abstract.

We determine the Hamiltonian vector field on an odd dimensional manifold
endowed with almost cosymplectic structure. This is a generalization of the
corresponding Hamiltonian vector field on manifolds with almost transitive
contact structures, which extends the contact Hamiltonian systems.
Applications are presented to the equations of motion on a particular five-
dimensional manifold, the extended Siegel-Jacobi upper-half plane
$\tilde{\mathcal{X}}^{J}_{1}$. The $\tilde{\mathcal{X}}^{J}_{1}$ manifold is
endowed with a generalized transitive almost cosymplectic structure, an almost
cosymplectic structure, more general than transitive almost contact structure
and cosymplectic structure. The equations of motion on
$\tilde{\mathcal{X}}^{J}_{1}$ extend the Riccati equations of motion on the
four-dimensional Siegel-Jacobi manifold $\mathcal{X}^{J}_{1}$ attached to a
linear Hamiltonian in the generators of the real Jacobi group
$G^{J}_{1}(\mathbb{R})$.

###### Key words and phrases:

Jacobi group, invariant metric, Siegel–Jacobi disk, Siegel–Jacobi upper half-
plane, extended Siegel–Jacobi upper half-plane, almost cosymplectic manifold,
cosymplectic manifold, contact Hamiltonian system, equations of motion

###### 1991 Mathematics Subject Classification:

37J55; 53D15; 53D22

###### Contents

  1. 1 Introduction
  2. 2 Preparation: Invariant metrics on $G^{J}_{1}(\mathbb{R})$-homogenous spaces
  3. 3 Equations of motion on almost cosymplectic manifolds
  4. 4 Equations of motion on $\tilde{{\mathcal{X}}}^{J}_{1}$
    1. 4.1 Generalized transitive almost cosymplectic Hamiltonian systems
    2. 4.2 Contact Hamiltonian systems
    3. 4.3 Linear Hamiltonian in the generators of the Jacobi group $G^{J}_{1}(\mathbb{R})$
  5. 5 Appendix – a breviar of terminology

##  1\. Introduction

The semi-direct product $G^{J}_{n}(\mathbb{R})$ of ${\rm Sp}(n,\mathbb{R})$
with the $(2n+1)$-dimensional Heisenberg group $\rm{H}_{n}(\mathbb{R})$ is
called real Jacobi group of degree $n$, while the isomorph group
${\mathrm{H}}_{n}\ltimes{\mathrm{Sp}}(n,\mathbb{R})_{\mathbb{C}}$ is denoted
$G^{J}_{n}$. Both Jacobi groups $G^{J}_{n}(\mathbb{R})$ and $G^{J}_{n}$ are
intensively studied in Mathematics, Mathematical Physics and Theoretical
Physics [7, 11, 12, 17, 18, 24, 35], [54]-[57].

The $G^{J}_{n}(\mathbb{R})$-homogeneous space
${\mathcal{X}}^{J}_{n}:=\frac{G^{J}_{n}(\mathbb{R})}{\rm{U}(n)\times\mathbb{R}}$
is called Siegel-Jacobi upper half space and
${\mathcal{X}}^{J}_{n}\approx{\mathcal{X}}_{n}\times\mathbb{R}^{2n}$, where
${\mathcal{X}}_{n}$ denotes the Siegel upper half space [11, 12, 54, 56].
Similarly, the Siegel-Jacobi ball is defined as
${\mathcal{D}}^{J}_{n}:=\frac{G^{J}_{n}}{\rm{U}(n)\times\mathbb{R}}\approx{\mathcal{D}}_{n}\times\mathbb{C}^{n}$
[8], where ${\mathcal{D}}_{n}$ denotes the Siegel (open) ball of degree $n$
[40].

Systems of coherent states (CS) based on the Siegel-Jacobi ball have
applications in quantum mechanics, geometric quantization, dequantization,
quantum optics, squeezed states, quantum teleportation, quantum tomography,
nuclear structure, signal processing, Vlasov kinetic equation, see references
in [3, 10, 12].

In order to construct invariant metric on homogeneous spaces attached to the
real Jacobi group of degree one, in [12] we gave up CS technique inspired by
Berezin [19]-[21] and we applied moving frame method [36], initiated by Cartan
[31, 32]. In [12] we got invariant metrics on ${\mathcal{X}}^{J}_{1}$ and
$\tilde{{\mathcal{X}}}^{J}_{1}$, while in [13] the results were generalized to
extended Siegel-Jacobi upper half space
$\tilde{{\mathcal{X}}}^{J}_{n}\approx{\mathcal{X}}^{J}_{n}\times\mathbb{R}$.

It raises the question: how can we distinguish between the different invariant
metrics obtained in [12] on ${\mathcal{X}}^{J}_{1}$ and
$\tilde{{\mathcal{X}}}^{J}_{1}$? In the present paper we give an answer to
this problem, underlining the differences between equations of motion on
${\mathcal{X}}^{J}_{1}$ and $\tilde{{\mathcal{X}}}^{J}_{1}$.

We developed a technique to calculate equations of classical and quantum
motion on a homogenous manifold $M=G/H$ attached to a linear Hamiltonian in
the generators of the group $G$, simultaneously with the determination of the
Berry phase [6]-[9]. The method works on the so called CS manifolds, i.e.
Kähler manifolds for which the generators of group $G$ admits a realisation as
first order holomorphic differential operators with polynomial coefficients
[6, 15, 16].

In this paper we are interested in a similar problem: find the equations of
motion on odd dimensional manifolds $M_{2n+1}$ with almost cosymplectic
structure in the meaning of [45]. For an almost cosymplectic manifold
$(M_{2n+1},\theta,\Omega)$, where $\theta$ ($\Omega$) is a one (respectively,
two)-form, we determine the equations of motion attached to a Hamiltonian
function $H$.

We determined the invariant metric to the action of the Jacobi group
$G^{J}_{1}(\mathbb{R})$ on $\tilde{{\mathcal{X}}}^{J}_{1}$ [12]. We pointed
out that the extended Siegel–Jacobi manifold does not admit a Sasaki structure
in the parametrization used in [12]. In the present paper we introduce on
$\tilde{{\mathcal{X}}}^{J}_{1}$ an almost cosymplectic structure in the sense
of Libermann [45], and moreover, we consider the particular case when
$\operatorname{d}\Omega=0$. Such a manifold is called generalized transitive
almost cosymplectic space (GTACOS). The place of the GTACOS structure in the
set of geometric structures with which a manifold of odd dimension can be
endowed is underlined in the Table in the Appendix: GTACOS is an almost
cosymplectic structure more general than the transive almost contact structure
and contact structure.

We also endow $\tilde{{\mathcal{X}}}^{J}_{1}$ with a contact structure in the
sens of [33].

The paper is laid out as follows. In Section 2 are collected several previous
results extracted from [3, 12, 14] used in this paper. Proposition 1 presents
the invariant metrics on five $G^{J}_{1}(\mathbb{R})$-homogeneous manifolds.
Proposition 2 expresses the Kähler two-form and the invariant metric on
${\mathcal{D}}^{J}_{1}$ and ${\mathcal{X}}^{J}_{1}$ in several sets of
variables. Proposition 3 recalls the invariant metric on
$\tilde{{\mathcal{X}}^{J}_{1}}$ in the S-variables $(x,y,p,q,\kappa)$ [24,
35]. In Section 3 are calculated the Hamiltonian vector field $X_{H}$ and
$\operatorname{grad}H$ associated with the Hamiltonian function $H$ and the
corresponding equations of motion on almost cosymplectic manifolds in the
variables $(q^{i},p_{i},\kappa)$ , $i=1,\dots,n$, which appear in the one-form
$\theta$, while $\Omega$ is expressed in Darboux coordinates (Theorem 1). The
new results on almost cosymplectic manifolds are compared with the
corresponding results for transitive almost contact manifolds [1] (Remark 1).
Section 4.1 particularizes the results of Section 3 to the extended Siegel-
Jacobi upper half-plane $\tilde{{\mathcal{X}}}^{J}_{1}$ organized as GTACOS
manifold. Proposition 4 displays the equations of motion on
$\tilde{{\mathcal{X}}}^{J}_{1}$ in the variables $(x,y,p,q,\kappa)$ organized
as a GTACOS manifold for a Hamiltonian function $H$ comparatively to the
equations of motion on ${\mathcal{X}}^{J}_{1}$ in the variables $(x,y,p,q)$.
Proposition 5 underlines that $\tilde{{\mathcal{X}}}^{J}_{1}$ does not admit
an almost contact structure with the invariant metric
$g_{\tilde{{\mathcal{X}}}^{J}_{1}}$ constructed in [12], but
$\tilde{{\mathcal{X}}}^{J}_{1}$ can be equipped with an almost contact metric
structure with a metric different of the metric
$g_{\tilde{{\mathcal{X}}}^{J}_{1}}$. We underline that the parametrization
introduced in [37] for Sasaki potential on Sasaki manifolds is not appropriate
for $\tilde{{\mathcal{X}}}^{J}_{1}$ (Remark 2). Proposition 6 of Section 4.2
shows the equations of motion on $\tilde{{\mathcal{X}}}^{J}_{1}$ organized as
contact Hamiltonian system. Proposition 7 in Section 4.3 presents the
equations of motion on $\tilde{{\mathcal{X}}}^{J}_{1}$ organized as GTACOS
manifold generated by a Hamiltonian linear in generators of the real Jacobi
group comparatively with the corresponding equations on
${{\mathcal{X}}}^{J}_{1}$. Finally, the Appendix summarizes the definitions of
different mathematical concepts used in the paper.

The new results of the paper are contained in Lemma 1, the six Remarks,
Propositions 4 – 7 and Theorem 1, which presents the equations of motion on
odd dimensional manifolds endowed with almost cosymplectic structure. The
Remark 3 emphasizes the differences between the equations of motion on
$\tilde{{\mathcal{X}}}^{J}_{1}$ endowed with almost cosymplectic structure
$(\tilde{{\mathcal{X}}}^{J}_{1},\theta,\omega)$ and contact structure
$(\tilde{{\mathcal{X}}}^{J}_{1},\eta_{0})$, which in principle could be
experimentally highlighted. Although we are not able to determine effectively
the almost contact structure associated with the concrete realization of the
contact structure $(\tilde{{\mathcal{X}}}^{J}_{1},\eta_{0})$, we report the
partial results obtained at Proposition 5 b) in order to emphasize the
difficulty of the problem.

Notation We denote by $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Z}$ and
$\mathbb{N}$ the field of real numbers, the field of complex numbers, the ring
of integers, and the set of non-negative integers, respectively. We denote the
imaginary unit $\sqrt{-1}$ by $\operatorname{i}$, the real and imaginary parts
of a complex number $z\in\mathbb{C}$ by $\operatorname{Re}z$ and
$\operatorname{Im}z$ respectively, and the complex conjugate of $z$ by
$\bar{z}$. We denote by ${\operatorname{d}}$ the differential. We use
Einstein’s summation convention, i.e. repeated indices are implicitly summed
over. The set of vector fields (1-forms) is denoted by ${\mathfrak{D}}^{1}$
(respectively ${\mathfrak{D}}_{1}$). If we denote with Roman capital letteres
the Lie groups, then their associated Lie algebras are denoted with the
corresponding lower-case letteres. The interior product $i_{X}\omega$
(interior multiplication or contraction) of the differential form $\omega$
with $X\in{\mathfrak{D}}^{1}$ is denoted $X\lrcorner\omega$. We denote by
$M(n,m,{\mathbb{F}})$ the set of $n\times m$ matrices with elements in the
field ${\mathbb{F}}$. If $X\in M(n,m,{\mathbb{F}})$, then $X^{t}$ denotes the
transpose of $X$.

##  2\. Preparation: Invariant metrics on $G^{J}_{1}(\mathbb{R})$-homogenous
spaces

In [12, (4.10), (5.15), (5.17)] we have introduced 6 invariant one-forms
$\lambda_{1},\dots,\lambda_{6}$ in the S-coordinates $(x,y,\theta,p,q,\kappa)$
[24] associated with the real Jacobi group $G^{J}_{1}(\mathbb{R})$

(2.1a) $\displaystyle\lambda_{1}$ $\displaystyle=\frac{\sqrt{\alpha}}{y}(\cos
2\theta\operatorname{d}x+\sin 2\theta\operatorname{d}y),\quad\alpha>0,$ (2.1b)
$\displaystyle\lambda_{2}$ $\displaystyle=\frac{\sqrt{\alpha}}{y}(-\sin
2\theta\operatorname{d}x+\cos 2\theta\operatorname{d}y),$ (2.1c)
$\displaystyle\lambda_{3}$
$\displaystyle=\sqrt{\beta}(\frac{\operatorname{d}x}{y}+2\operatorname{d}\theta),\quad\beta>0,$
(2.1d) $\displaystyle\lambda_{4}$
$\displaystyle=\sqrt{\gamma}[-y^{-\frac{1}{2}}\sin\theta\operatorname{d}q+\big{(}y^{\frac{1}{2}}\cos\theta-
xy^{-\frac{1}{2}}\sin\theta\big{)}\operatorname{d}p],\quad\gamma>0,$ (2.1e)
$\displaystyle\lambda_{5}$
$\displaystyle=\sqrt{\gamma}[y^{-\frac{1}{2}}\cos\theta\operatorname{d}q+\big{(}y^{\frac{1}{2}}\sin\theta+xy^{-\frac{1}{2}}\cos\theta\big{)}\operatorname{d}p],$
(2.1f) $\displaystyle\lambda_{6}$
$\displaystyle=\sqrt{\delta}(\operatorname{d}\kappa-p\operatorname{d}q+q\operatorname{d}p),\qquad\delta>0,$

and we have expressed the invariant metric on several homogenous spaces
associated with $G^{J}_{1}(\mathbb{R})$ [12, Theorem 5.7]:

######  Proposition 1.

The four-parameter left-invariant metric on the real Jacobi group
$G^{J}_{1}(\mathbb{R})$ in the S-coordinates $(x,y,\theta,$ $p,q,\kappa)$ is

(2.2) $\begin{split}{\rm
d}s^{2}_{G^{J}_{1}(\mathbb{R})}&=\sum_{i=1}^{6}\lambda_{i}^{2}=\alpha\frac{{\rm
d}x^{2}+{\rm d}y^{2}}{y^{2}}+\beta\left(\frac{{\rm d}x}{y}+2{\rm
d}\theta\right)^{2},\\\ &+\frac{\gamma}{y}\big{(}{\rm d}q^{2}+S{\rm
d}p^{2}+2x{\rm d}p{\rm d}q\big{)}+\delta({\rm d}\kappa-p{\rm d}q+q{\rm
d}p)^{2},~{}~{}S:=x^{2}+y^{2}.\end{split}$

Depending on the parameter values $\alpha,\beta,\gamma,\delta\geq 0$ in the
metric (2.2), we have invariant metric on the following
$G^{J}_{1}\mathbb{R})$-homogeneous manifolds$:$

  1. $1)$

the Siegel upper half-plane ${\mathcal{X}}_{1}$ if $\beta,\gamma,\delta=0$,

  2. $2)$

the group ${\mbox{\rm{SL}}(2,\mathbb{R})}$ if $\gamma,\delta=0$,
$\alpha\beta\not=0$,

  3. $3)$

the Siegel–Jacobi half-plane ${\mathcal{X}}^{J}_{1}$ if $\beta,\delta=0$,

  4. $4)$

the extended Siegel–Jacobi half-plane $\tilde{{\mathcal{X}}}^{J}_{1}$ if
$\beta=0$,

  5. $5)$

the Jacobi group $G^{J}_{1}$ if $\alpha\beta\gamma\delta\not=0$.

The action of Jacobi group on several homogeneous spaces associated with it is
presented in [3, Lemma 1], [12, Lemma 5.1]. We have studied the geometry of
the Siegel–Jacobi upper half-plane ${\mathcal{X}}^{J}_{1}$ in [3, 7, 9, 12,
14].

The parts a), b), c) of Proposition 2 below are extracted from [12,
Proposition 2.1 in], [3, Proposition 1]. Below
$(w,z)\in({\mathcal{D}}_{1},\mathbb{C})$,
$(v,u)\in({\mathcal{X}}_{1},\mathbb{C})$. and the parameters $k$ and $\nu$
come from representation theory of the Jacobi group: $k$ indexes the positive
discrete series of ${\rm SU}(1,1)$, $2k\in\mathbb{N}$, while $\nu>0$ indexes
the representations of the Heisenberg group ${\mathrm{H}}_{1}$. See also [7],
[11]. By the two-form of Berndt-Kähler we mean the two-parameter invariant
Kähler two-form on ${\mathcal{X}}^{J}_{1}$ determined in [22, 23, 41, 42], see
also [3, 12]. Part d) is taken from [14, Proposition 3]. Part e) was proved in
[12, Proposition 5.4, see (5.21b)] and appears also in [14, Proposition 7, see
(4.18), (4.23)].

######  Proposition 2.

a) The Kähler two-form on ${\mathcal{D}}^{J}_{1}$, invariant to the action of
$G^{J}_{0}=\rm{SU}(1,1)\ltimes\mathbb{C}$, is

(2.3)
$-\operatorname{i}\omega_{{\mathcal{D}}^{J}_{1}}(w,z)\\!=\\!\frac{2k}{P^{2}}\operatorname{d}w\wedge\operatorname{d}\bar{w}+\nu\frac{{\mathcal{A}}\wedge\bar{{\mathcal{A}}}}{P},~{}P:=1-|w|^{2},~{}{\mathcal{A}}={\mathcal{A}}(w,z):=\operatorname{d}z+\bar{\eta}\operatorname{d}w.$

We have the change of variables $FC:(w,z)\rightarrow(w,\eta)$

$\displaystyle{\rm FC}\colon\ z=\eta-w\bar{\eta},\qquad{\rm FC}^{-1}\colon\
\eta=\frac{z+\bar{z}w}{P},$

and

${\rm FC}\colon\
{\mathcal{A}}(w,z)\rightarrow\operatorname{d}\eta-w\operatorname{d}\bar{\eta}.$

b) Using the partial Cayley transform $\Phi~{}:(w,z)\rightarrow(v,u)$ and its
inverse

(2.4a) $\displaystyle\Phi:w$
$\displaystyle=\frac{v-\operatorname{i}}{v+\operatorname{i}},~{}~{}z=2\operatorname{i}\frac{u}{v+\operatorname{i}},~{}~{}v,u\in\mathbb{C},~{}\operatorname{Im}v>0,$
(2.4b) $\displaystyle\Phi^{-1}:v$
$\displaystyle=\operatorname{i}\frac{1+w}{1-w},~{}~{}u=\frac{z}{1-w},~{}~{}w,z\in\mathbb{C},~{}|w|<1,$

we obtain

${\mathcal{A}}\left(\frac{v-\operatorname{i}}{v+\operatorname{i}},\frac{2\operatorname{i}u}{v+\operatorname{i}}\right)=\frac{2\operatorname{i}}{v+\operatorname{i}}{\mathcal{B}}(v,u),$

where

(2.5) ${\mathcal{B}}(v,u):={\rm d}u-r{\rm
d}v,~{}r:=\frac{u-\bar{u}}{v-\bar{v}}.$

The Berndt–Kähler’s two-form, invariant to the action of
$G^{J}(\mathbb{R})_{0}=\rm{SL}(2,\mathbb{R})\ltimes\mathbb{C}$, is

(2.6)
$-\operatorname{i}\omega_{{\mathcal{X}}^{J}_{1}}(v,u)=-\frac{2k}{(\bar{v}-v)^{2}}\operatorname{d}v\wedge\operatorname{d}\bar{v}+\frac{2\nu}{\operatorname{i}(\bar{v}-v)}{\mathcal{B}}\wedge\bar{{\mathcal{B}}}.$

We have the change of variables ${\rm FC}_{1}\colon\ (v,u)\rightarrow(v,\eta)$

${\rm FC}_{1}\colon\
2\operatorname{i}u=(v+\operatorname{i})\eta-(v-\operatorname{i})\bar{\eta},\qquad{\rm
FC}^{-1}_{1}\colon\eta=\frac{u\bar{v}-\bar{u}v}{\bar{v}-v}+\operatorname{i}r.$

c) If we apply the change of coordinates
${\mathcal{D}}^{J}_{1}\ni(v,u)\rightarrow(x,y,p,q)\in{\mathcal{X}}^{J}_{1}$

$\mathbb{C}\ni u:=pv+q,~{}~{}p,q\in\mathbb{R},\quad\mathbb{C}\ni
v:=x+\operatorname{i}y,~{}x,y\in\mathbb{R},~{}y>0,$

then

$r=p,\quad{\mathcal{B}}(v,u)=\operatorname{d}u-p\operatorname{d}v,$ (2.7)
${\mathcal{B}}(v,u)={\mathcal{B}}(x,y,p,q):=F\operatorname{d}t=v\operatorname{d}p+\operatorname{d}q=(x+\operatorname{i}y)\operatorname{d}p+\operatorname{d}q,\quad
F:=\dot{p}v+\dot{q}.$

d) The second partial Cayley transform
$\Phi_{1}:{\mathcal{D}}^{J}_{1}\rightarrow{\mathcal{X}}^{J}_{1}$

$\Phi_{1}:=FC_{1}\circ\Phi:(w,z)\rightarrow(v=x+\operatorname{i}y,\eta=q+\operatorname{i}p)$

is given by

(2.8a) $\displaystyle\Phi_{1}:$
$\displaystyle~{}w=\frac{v-\operatorname{i}}{v+\operatorname{i}},\quad
z=2\operatorname{i}\frac{pv+q}{v+\operatorname{i}},$ (2.8b)
$\displaystyle\Phi_{1}^{-1}:$
$\displaystyle~{}v=\operatorname{i}\frac{1+w}{1-w},\quad\eta=\frac{(1+\operatorname{i}\bar{v})(z-\bar{z})+v(\bar{v}-\operatorname{i})(z+\bar{z})}{2\operatorname{i}(\bar{v}-v)}=\frac{z+\bar{z}w}{P}.$

e) The two-parameter balanced metric on the Siegel–Jacobi upper half-plane
${\mathcal{X}}^{J}_{1}$, the particular case of (2.2) corresponding to item
_3)_ in _Proposition 1_, associated to the Kähler two-form (2.6), (2.7), is

(2.9)
$\operatorname{d}s^{2}_{{\mathcal{X}}^{J}_{1}}(x,y,p,q)\\!=\\!\alpha\frac{\operatorname{d}x^{2}\\!+\\!\operatorname{d}y^{2}}{y^{2}}+\frac{\gamma}{y}(S\operatorname{d}p^{2}+\operatorname{d}q^{2}+2x\operatorname{d}p\operatorname{d}q),$

where $S$ was defined in (2.2) and

(2.10) $\alpha:=k/2,\quad\gamma:=\nu.$

The metric matrix associated with (2.9) is

$g_{{{\mathcal{X}}}^{J}_{1}}\\!=\\!\left(\begin{array}[]{cccc}g_{xx}&0&0&0\\\
0&g_{yy}&0&0\\\ 0&0&g_{pp}&g_{pq}\\\
0&0&g_{qp}&g_{qq}\end{array}\right),\\!\begin{array}[]{cc}\qquad
g_{xx}\\!=\frac{\alpha}{y^{2}},&\\!g_{yy}\\!=\\!\frac{\alpha}{y^{2}};\\\
g_{pq}\\!=\\!\gamma\frac{x}{y},&g_{pp}\\!=\\!\gamma\frac{S}{y},\quad
g_{qq}\\!=\\!\frac{\gamma}{y}.\end{array}$

We have also obtained invariant metric to the action of the Jacobi group
$G^{J}_{1}(\mathbb{R})$ on the extended Siegel-Jacobi upper half-plane
$\tilde{{\mathcal{X}}}^{J}_{1}$ [12, Proposition 5.6, (5.25), (5.26)], see
also [14, Theorem 1, (5.1) (5.5)]

######  Proposition 3.

The three-parameter metric of the extended Siegel-Jacobi upper half-plane
$\tilde{{\mathcal{X}}}^{J}_{1}$ expressed in the S-coordinates
$(x,y,p,q,\kappa)$, left-invariant with respect to the action of the Jacobi
group $G^{J}_{1}(\mathbb{R})$, is given by item _4)_ in _Proposition 1_ as

(2.11) $\begin{split}{\rm
d}s^{2}_{\tilde{{\mathcal{X}}}^{J}_{1}}(x,y,p,q,\kappa)&\\!=\\!{\rm
d}s^{2}_{{\mathcal{X}}^{J}_{1}}(x,y,p,q)+\lambda^{2}_{6}(p,q,\kappa)\\\
&\\!=\\!\frac{\alpha}{y^{2}}\big{(}{\rm d}x^{2}+{\rm
d}y^{2}\big{)}+\left(\frac{\gamma}{y}S+\delta q^{2}\right){\rm
d}p^{2}+\left(\frac{\gamma}{y}+\delta p^{2}\right){\rm d}q^{2}+\delta{\rm
d}\kappa^{2}\\\ &\\!+2\left(\gamma\frac{x}{y}-\delta pq\right){\rm d}p{\rm
d}q+2\delta(q{\rm d}p{\rm d}\kappa-p{\rm d}q{\rm d}\kappa).\end{split}$

The metric matrix associated to the metric (2.11) is

(2.12)
$g_{\tilde{{\mathcal{X}}}^{J}_{1}}\\!=\\!\left(\begin{array}[]{ccccc}g_{xx}&0&0&0&0\\\
0&g_{yy}&0&0&0\\\ 0&0&g_{pp}&g_{pq}&g_{p\kappa}\\\
0&0&g_{qp}&g_{qq}&g_{q\kappa}\\\ 0&0&g_{\kappa p}&g_{\kappa
q}&g_{\kappa\kappa}\end{array}\right),\\!\begin{array}[]{cc}g_{xx}\\!=\frac{\alpha}{y^{2}},&\\!g_{yy}\\!=\\!\frac{\alpha}{y^{2}},\\\
g_{pq}\\!=\\!\gamma\frac{x}{y}-\delta pq,&~{}g_{p\kappa}\\!=\\!\delta
q,g_{q\kappa}\\!=\\!-\delta p,\\\ g_{pp}\\!=\\!\gamma\frac{S}{y}+\delta
q^{2},&g_{qq}\\!=\\!\frac{\gamma}{y}+\delta
p^{2},g_{\kappa\kappa}\\!=\\!\delta.\end{array}$

The extended Siegel–Jacobi upper half-plane $\tilde{{\mathcal{X}}}^{J}_{1}$
does not admit an almost contact structure $(\Phi,\xi,\eta)$ in the sense of
_Definition 1_ with a contact form $\eta=\lambda_{6}$ and Reeb vector
$\xi=\operatorname{Ker}(\eta)$.

##  3\. Equations of motion on almost cosymplectic manifolds

In the present paper we adopt the terminology of Libermann [45] recalled at
ACOS in Appendix. We consider the almost cosymplectic manifold
$(M_{2n+1},\theta,\Omega)$ such that condition (5.5) is satisfied for the real
one-form $\theta$

(3.1)
$\theta=a_{i}\operatorname{d}q^{i}+b_{i}\operatorname{d}p_{i}+c\kappa,\quad
a_{i},b_{i}\in\mathbb{R},~{}i=1,\dots,n,\quad c\neq 0,$

while for the two-form $\Omega$ we use the Darboux parametrization given in
(5.2).

We determine the Hamiltonian vector field $X_{H}$ verifying a relation similar
to (5.15) for the contact manifold $(M,\eta)$ for the one-form $\theta$ (3.1)
and two-form $\Omega$ (5.2):

######  Theorem 1.

Let $(M_{2n+1},\theta,\Omega)$ be an almost cosymplectic manifold and let
$H\in C^{\infty}(M)$ be a smooth Hamiltonian. Then the coefficients that
define the Hamiltonian vector field $X_{H}$ attached the Hamiltonian $H$

$X_{H}=A_{i}\frac{\partial}{\partial
q^{i}}+B_{i}\frac{\partial}{p_{i}}+C\frac{\partial}{\partial\kappa},$

are solution of the equation

$\flat(X_{H})=\operatorname{d}H-(R\lrcorner H+H)\theta,$

where the Reeb vector which is the solution of the equation (5.6) is

(3.2) $R=\frac{1}{c}\frac{\partial}{\partial\kappa},$

and the isomorphism $\flat$ is defined in (5.7).

It is obtained

(3.3) $A_{i}=\frac{\partial H}{\partial
p^{i}}-b_{i}R(H),~{}B_{i}=-\frac{\partial H}{\partial
q_{i}}+a_{i}R(H),~{}C=\frac{1}{c}(-a_{i}\frac{\partial H}{\partial
p^{i}}+b_{i}\frac{\partial H}{\partial q^{i}}-H),$

and then

(3.4) $X_{H}=(\frac{\partial H}{\partial
p^{i}}-b_{i}R(H))\frac{\partial}{\partial q^{i}}+(-\frac{\partial H}{\partial
q_{i}}+a_{i}R(H))\frac{\partial}{\partial p_{i}}+(-a_{i}\frac{\partial
H}{\partial p^{i}}+b_{i}\frac{\partial H}{\partial q^{i}}-H)R.$

The vector field $\operatorname{grad}H$, i.e. the solution of the equation

$(\operatorname{grad}H)^{\flat}=\operatorname{d}H$

has the expression

(3.5) $\operatorname{grad}H=(\frac{\partial H}{\partial
p_{i}}-b_{i}R(H))\frac{\partial}{\partial q^{i}}+(-\frac{\partial H}{\partial
p_{i}}+a_{i}R(H))\frac{\partial}{\partial p_{i}}+(-a_{i}\frac{\partial
H}{\partial p_{j}}+b_{i}\frac{\partial H}{\partial q^{j}}+R(H))R.$

The vector fields $X_{H}$ (3.4) and $\operatorname{grad}H$ (3.5) are related
by the relation

(3.6) $X_{H}=\operatorname{grad}{H}-(H+R(H))R,$

which is similar with (5.15) but with Reeb vector (3.2) instead of (5.14).

The equations of motion on the almost cosymplectic manifold
$(M,\theta,\Omega)$ are

(3.7)
$\frac{\operatorname{d}q^{i}}{\operatorname{d}t}-A_{i}=0,\quad\frac{\operatorname{d}p_{i}}{\operatorname{d}t}-B_{i}=0,\quad\frac{\operatorname{d}\kappa}{\operatorname{d}t}-C=0,\quad
i=1,\dots,n,$

where $(A_{i},B_{i},C)$ have the expressions given in (3.3).

In particular, we find

(3.8a) $\displaystyle\operatorname{grad}q^{i}$
$\displaystyle=-\frac{\partial}{\partial p_{i}}+b_{i}R,$ (3.8b)
$\displaystyle\operatorname{grad}p_{i}$
$\displaystyle=\frac{\partial}{\partial q^{i}}-a_{i}R,$ (3.8c)
$\displaystyle\operatorname{grad}\kappa$
$\displaystyle=\frac{1}{c}(-b_{i}\frac{\partial}{\partial
q^{i}}+a_{i}\frac{\partial}{\partial p_{i}}+R),$ (3.8d) $\displaystyle
X_{q^{i}}$ $\displaystyle=-\frac{\partial}{\partial p_{i}}+(b_{i}-q^{i})R,$
(3.8e) $\displaystyle X_{p_{i}}$ $\displaystyle=\frac{\partial}{\partial
q^{i}}-(a_{i}+p_{i})R,$ (3.8f) $\displaystyle X_{\kappa}$
$\displaystyle=\frac{1}{c}(-b_{i}\frac{\partial}{\partial
q^{i}}+a_{i}\frac{\partial}{\partial
p_{i}}-\kappa\frac{\partial}{\partial\kappa}).$

###### Proof.

The proof is elementary. We indicate the main steps.

In order to apply (5.7) we calculate

$X_{H}\lrcorner\theta=A_{i}a_{i}+B_{i}b_{i}+C,\quad
X_{H}\lrcorner\Omega=-B_{i}\operatorname{d}q^{i}+A_{i}\operatorname{d}p_{i},$

and we find

(3.9)
$X^{\flat}_{H}\\!=\\![-B_{i}\\!+\\!a_{i}(A_{j}a_{j}\\!+\\!B_{j}b_{j}\\!+\\!C)]\\!\operatorname{d}\\!q^{i}\\!+\\![A_{i}\\!+\\!b_{i}(A_{j}a_{j}\\!+\\!B_{j}b_{j}\\!+\\!C)]\\!\operatorname{d}\\!p_{i}\\!+\\!(A_{j}a_{j}\\!+\\!B_{j}b_{j}+\\!C)c\operatorname{d}\\!\kappa.$

We calculate

(3.10) $\operatorname{d}H-(R\lrcorner H+H)\theta=[\frac{\partial H}{\partial
q^{i}}-a_{i}(R(H)+H)]\operatorname{d}q^{i}+[\frac{\partial H}{\partial
p_{i}}-b_{i}(R(H)+H)]\operatorname{d}p_{i}-cH\operatorname{d}\kappa$

and (3.3) follows from the identification of (3.9) with (3.10).

The calculation of $\operatorname{grad}H$ is done similarly and from (3.4) and
(3.5) it follows (3.6). ∎

Now we compare our results in Theorem 1 with the results of Albert [1].

######  Remark 1.

The transitive almost contact structure considered by Albert [1] at TACS in
the _Appendix_ corresponds in our case in (3.1) to

$a_{i}=\epsilon p_{i},\quad b_{i}=0,\quad c=1,\quad i=1,\dots,n,$

and the Reeb vector is

$R(H)=\frac{\partial H}{\partial\kappa}.$

(3.4) becomes

$X_{H}=\frac{\partial H}{\partial p^{i}}\frac{\partial}{\partial
q^{i}}+(-\frac{\partial H}{\partial q_{i}}+\epsilon p^{i}\frac{\partial
H}{\partial\kappa})\frac{\partial}{\partial p_{i}}-(\epsilon
p^{i}\frac{\partial H}{\partial p^{i}}+H)\frac{\partial}{\partial\kappa},$

and $X_{H}$ verifies the relations

(3.11) $X_{H}\lrcorner\theta=-H,\quad
X_{H}\lrcorner\Omega=\operatorname{d}H-R(H)\theta.$

Equation of $\operatorname{grad}f$ becomes

$\operatorname{grad}f=\frac{\partial f}{\partial
p_{i}}\frac{\partial}{\partial q^{i}}+(\epsilon p_{i}\frac{\partial
f}{\partial\kappa}-\frac{\partial f}{\partial q^{i}})\frac{\partial}{\partial
p_{i}}+(\frac{\partial f}{\partial\kappa}-\epsilon p_{i}\frac{\partial
f}{\partial p_{i}})\frac{\partial}{\partial\kappa}.$

Equations (3.8) become

(3.12a) $\displaystyle\operatorname{grad}q^{i}$
$\displaystyle=-\frac{\partial}{\partial p_{i}},$ (3.12b)
$\displaystyle\operatorname{grad}p_{i}$
$\displaystyle=\frac{\partial}{\partial q^{i}}-\epsilon p_{i}R,$ (3.12c)
$\displaystyle\operatorname{grad}\kappa$ $\displaystyle=\epsilon
p_{i}\frac{\partial}{\partial p_{i}}+\frac{\partial}{\partial\kappa},$ (3.12d)
$\displaystyle X_{q^{i}}$ $\displaystyle=-\frac{\partial}{\partial
p_{i}}-q^{i}\frac{\partial}{\partial\kappa},$ (3.12e) $\displaystyle
X_{p_{i}}$ $\displaystyle=\frac{\partial}{\partial
q_{i}}-(\epsilon+1)p_{i}\frac{\partial}{\partial\kappa},$ (3.12f)
$\displaystyle X_{\kappa}$ $\displaystyle=+\epsilon
p_{i}\frac{\partial}{\partial p_{i}}-\kappa\frac{\partial}{\partial\kappa}.$

Equations of the vector field $X_{f}$ at [1, p 636] are in agreement with our
(3.3), except the following two differences:

  * •

the coefficient

$\epsilon(f-p_{i}\frac{\partial}{\partial t})$

of $\frac{\partial}{\partial t}$ should be replaced with

$-\epsilon p^{i}\frac{\partial f}{\partial p_{i}}-f.$

This comes from the fact that:

  * •

in formulas [1, (3)] instead of

$X_{f}\lrcorner\theta=\epsilon f,$

we should have

$X_{f}\lrcorner\theta=-f$

as in (3.11). With these correction, Albert’s formulas for $X_{f}$ on _p 636_
are particular cases of our formula (3.4) and, if $\epsilon=-1$, corrected
equation _(2.12)_ in [33] is find again. Similarly, if instead of (3.1) we
consider (5.12), equations of motion [1, (2.13), (2.14), (2.15)] in [33] are
particular cases of ours equations (3.7).

  * •

in formula (3.12) Albert gives the different values for $X_{q^{i}}$,
$X_{p_{i}}$ and $X_{\kappa}$ in (3.12d), (3.12e), respectively (3.12f)

$X_{q^{i}}=-\frac{\partial}{\partial p_{i}}+\epsilon
q^{i}\frac{\partial}{\partial t},\quad X_{p_{i}}=\frac{\partial}{\partial
q^{i}},\quad X_{t}=\epsilon(t\frac{\partial}{\partial
t}+p_{i}\frac{\partial}{\partial p_{i}}).$

##  4\. Equations of motion on $\tilde{{\mathcal{X}}}^{J}_{1}$

###  4.1. Generalized transitive almost cosymplectic Hamiltonian systems

In the standard approach of CS, the $G$-invariant Kähler two-form on a
$2n$-dimensional homogenous manifold $M=G/H$ is obtained from the Kähler
potential $f$ via the recipe

$\displaystyle-\operatorname{i}\omega_{M}$
$\displaystyle=\partial\bar{\partial}f,~{}f(z,\bar{z})=\log{\mathcal{K}}(z,\bar{z}),~{}{\mathcal{K}}(z,\bar{z}):=(e_{{z}},e_{{z}}),$
$\displaystyle\omega_{M}(z,\bar{z})$
$\displaystyle=\operatorname{i}\sum_{\alpha,\beta}h_{\alpha\bar{\beta}}\operatorname{d}z_{\alpha}\wedge\operatorname{d}\bar{z}_{\beta},~{}h_{\alpha\bar{\beta}}=\frac{\partial^{2}f}{\partial
z_{\alpha}\partial\bar{z}_{\beta}},~{}h_{\alpha\bar{\beta}}=\bar{h}_{\beta\bar{\alpha}},~{}\alpha,\beta=1,\dots,n,$

where ${\mathcal{K}}(z,\bar{z})$ is the scalar product of two un-normalized
Perelomov’s CS-vectors $e_{{z}}$ at $z\in M$ [8, 11, 48].

In accord with [4, p 42 ], [39, p 28], [12, Appendix B], the Riemannian metric
associated with the Hermitian metric on the manifold $M$ in local coordinates
is

(4.2)
$\operatorname{d}s^{2}_{M}(z,\bar{z})=\sum_{\alpha,\beta=1}^{n}h_{\alpha\bar{\beta}}\operatorname{d}z_{\alpha}\otimes\operatorname{d}\bar{z}_{\beta}.$

From the Kähler two-form (2.3) obtained via CS in [8] we get with (4.2) the
balanced metric on the Siegel-Jacobi disk ${\mathcal{D}}^{J}_{1}$. If in
$\operatorname{d}s^{2}(w,z)_{{\mathcal{D}}^{J}_{1}}$ we apply the second
partial Cayley transform (2.8), then we obtain the invariant metric (2.9) on
${\mathcal{X}}^{J}_{1}$.

We endow $\tilde{{\mathcal{X}}}^{J}_{1}$ with a generalized transitive almost
cosymplectic structure, i.e. an almost cosymplectic structure
$(M,\theta,\Omega)$ such that

$\operatorname{d}\Omega=0.$

######  Lemma 1.

If we introduce into the Kähler two-form $\omega_{{\mathcal{D}}^{J}_{1}}(w,z)$
(2.3) the second partial Cayley transform (2.8a)
$(w,z)\rightarrow(x,y,q,p),y>0$ we get the symplectic two-form

(4.3)
$\omega_{{\mathcal{X}}^{J}_{1}}(x,y,q,p)=\frac{k}{y^{2}}\operatorname{d}x\wedge\operatorname{d}y+2\nu\operatorname{d}q\wedge\operatorname{d}p,\quad
y>0.$

In the notation of [12], we introduce on the extended 5-dimensional Siegel-
Jacobi half-plane $\tilde{{\mathfrak{X}}}^{J}_{1}$ parametrized in
$(x,y,p,q,\kappa)$ the almost cosymplectic structure
$(\tilde{{\mathcal{X}}}^{J}_{1},\theta,\omega)$, where $\theta=\lambda_{6}$
and $\omega$ is (4.3), i.e.

(4.4a) $\displaystyle\theta$
$\displaystyle=\sqrt{\delta}(\operatorname{d}\kappa-p\operatorname{d}q+q\operatorname{d}p),\quad\delta>0,$
(4.4b) $\displaystyle\omega$
$\displaystyle=\frac{k}{y^{2}}\operatorname{d}x\wedge\operatorname{d}y+2\nu\operatorname{d}q\wedge\operatorname{d}p,\quad
y>0.$

We have

(4.5)
$\operatorname{d}\omega=0,\quad\theta\wedge\omega^{2}=2\frac{k\nu\sqrt{\delta}}{y^{2}}\operatorname{d}x\wedge\operatorname{d}y\wedge\operatorname{d}q\wedge\operatorname{d}p\wedge\operatorname{d}\kappa,$

and $(\tilde{{\mathfrak{X}}}^{J}_{1},\theta,\omega)$ verifies the condition
(5.5) of an almost cosymplectic manifold.

(4.4a) corresponds in (3.1) to the choice:

(4.6) $a_{1}=b_{1}=0,\quad a_{2}=-\frac{\sqrt{\delta}}{2\nu}p,\quad
b_{2}=\sqrt{\delta}q,\quad c=\sqrt{\delta},\quad n=2,$

and (4.4b) corresponds in (5.2) to the choice:

(4.7) $q^{1}=kx,\quad p_{1}=-\frac{1}{y},\quad q^{2}=2\nu q,\quad
p_{2}=p,\quad n=2.$

In Darboux coordinates we have a particular almost cosymplectic manifold
$(\tilde{{\mathfrak{X}}}^{J}_{1},\theta,\omega)$ verifying (5.5) and in
addition the condition

$\operatorname{d}\omega=0.$

$(\tilde{{\mathfrak{X}}}^{J}_{1},\theta,\omega)$ is a generalized transitive
almost cosymplectic manifold.

Introducing (3.3) in Theorem 1, we get

######  Proposition 4.

The coefficients (3.3) of vector field $X_{H}$ (3.4) on the generalized
transitive almost cosymplectic manifold
$(\tilde{{\mathcal{X}}}^{J}_{1},\theta,\omega)$ have the expressions

(4.8a) $\displaystyle A_{1}$ $\displaystyle=y^{2}\frac{\partial}{\partial
y},\quad\quad A_{2}=\frac{\partial H}{\partial
p}\color[rgb]{1,0,0}-q\frac{\partial H}{\partial\kappa}\color[rgb]{0,0,0},$
(4.8b) $\displaystyle B_{1}$ $\displaystyle=-\frac{1}{k}\frac{\partial
H}{\partial x},\quad B_{2}=-\frac{1}{2\nu}(\frac{\partial H}{\partial
q}\color[rgb]{1,0,0}+p\frac{\partial H}{\partial\kappa})\color[rgb]{0,0,0},$
(4.8c) $\displaystyle\color[rgb]{1,0,0}C$
$\displaystyle\color[rgb]{1,0,0}=\frac{1}{2\nu}(p\frac{\partial H}{\partial
p}+q\frac{\partial H}{\partial q})-H.\color[rgb]{0,0,0}$

Equations of motion (3.7) on the 5-dimensional extended Siegel-Jacobi half-
plane $\tilde{{\mathcal{X}}}^{J}_{1}$ are

(4.9a) $\displaystyle\dot{x}$ $\displaystyle=\frac{y^{2}}{k}\frac{\partial
H}{\partial y},\quad\qquad\qquad\dot{y}=-\frac{y^{2}}{k}\frac{\partial
H}{\partial x},$ (4.9b) $\displaystyle\dot{q}$
$\displaystyle=\frac{1}{2\nu}(\frac{\partial H}{\partial
p}\color[rgb]{1,0,0}-q\frac{\partial
H}{\partial\kappa}\color[rgb]{0,0,0}),\quad\dot{p}=-\frac{1}{2\nu}(\frac{\partial
H}{\partial q}\color[rgb]{1,0,0}+p\frac{\partial
H}{\partial\kappa}\color[rgb]{0,0,0}),$ (4.9c)
$\displaystyle\color[rgb]{1,0,0}\dot{\kappa}$
$\displaystyle\color[rgb]{1,0,0}=\frac{1}{2\nu}(p\frac{\partial H}{\partial
p}+q\frac{\partial H}{\partial q})-H.$

###  4.2. Contact Hamiltonian systems

We recall that in [12] we noticed that for $\eta=\lambda_{6}$ we have
$\operatorname{d}\eta^{2}=0$ and if we want to introduce a contact structure
on $\tilde{{\mathcal{X}}}^{J}_{1}$, we have to choose $\eta\not=\lambda_{6}$.

Now we organize $\tilde{{\mathcal{X}}}^{J}_{1}$ as a contact Hamiltonian
system in the sense of [33, § 2.3]. Then we apply Proposition 9 to determine
the associated almost contact metric structure:

######  Proposition 5.

Let $\eta_{0}$ be such that
$\operatorname{d}\eta_{0}=\omega_{{\mathcal{X}}^{J}_{1}}$, where
$\omega_{{\mathcal{X}}^{J}_{1}}$ is given by (4.3). We chose

(4.10)
$\eta_{o}=\sqrt{\delta}\operatorname{d}\kappa+\frac{k}{y}\operatorname{d}x+\nu(-p\operatorname{d}q+q\operatorname{d}p).$

$\eta_{0}\wedge\operatorname{d}\eta_{0}^{2}$ has the same nonzero value as the
volume form given by the second equation (4.5), (5.11) is verified, and
$(\tilde{{\mathcal{X}}}^{J}_{1},\eta_{0})$ is a contact manifold.

a) For (4.10) chosen as $\eta_{0}$ and $g_{\tilde{{\mathcal{X}}}^{J}_{1}}$
given by (2.12), there is no $\Phi$ so that
$(\tilde{{\mathcal{X}}}^{J}_{1},\Phi,\xi,\eta_{0},$
$g_{\tilde{{\mathcal{X}}}^{J}_{1}})$ is an almost contact metric structure,
where $\xi$ is given by (4.15), (4.19).

b) The contact manifold $(\tilde{{\mathcal{X}}}^{J}_{1},\eta_{0})$ can be
endowed with an almost contact metric structure
$(\tilde{{\mathcal{X}}}^{J}_{1},\Phi,\xi,\eta_{0},g^{\prime}_{\tilde{{\mathcal{X}}}^{J}_{1}})$.
The six components
$\Phi_{xx},\Phi_{xy},\Phi_{xq},\Phi_{xp},\Phi_{yx},\Phi_{qq}$ of $\Phi$ in
(4.25) can be expressed as function of the four remaining independent
variables $\Phi_{yq},\Phi_{yp},\Phi_{qp},\Phi_{pq}$, while the rest of the
components of $\Phi$ are obtained with (4.20), (4.24), (4.26). The fundamental
quadratic form associated to the metric tensor $g^{\prime}$ (4.27) must be
positive definite.

###### Proof.

a) We use (5.29) for $\hat{\Phi}$ corresponding to $\eta_{0}$ and (2.12)
written as

(4.11) $g_{\tilde{{\mathcal{X}}}^{J}_{1}}=\left(\begin{array}[]{cc}A&0\\\
0&B\end{array}\right),~{}~{}A=\left(\begin{array}[]{cc}g_{xx}&0\\\
0&g_{yy}\end{array}\right),~{}B=\left(\begin{array}[]{ccc}g_{qq}&g_{qp}&g_{q\kappa}\\\
g_{pq}&g_{pp}&g_{p\kappa}\\\ g_{\kappa q}&g_{\kappa
p}&g_{\kappa\kappa}\end{array}\right).$

We write down $\Phi$ as

(4.12) $\Phi\\!=\\!\left(\begin{array}[]{cc}\Phi_{1}&\Phi_{2}\\\
\Phi_{3}&\Phi_{4}\end{array}\right),\text{~{}~{}~{}}\Phi_{1}\in
M(2,2,\mathbb{R}),~{}\Phi_{4}\in
M(3,3,\mathbb{R}),~{}\Phi_{2},~{}\Phi_{3}^{t}\in M(2,3,\mathbb{R}).$

With (4.11) and (4.12) we get

$g_{\tilde{{\mathcal{X}}}^{J}_{1}}\Phi=\left(\begin{array}[]{cc}A\Phi_{1}&A\Phi_{2}\\\
B\Phi_{3}&B\Phi_{4}\end{array}\right).$

We write down $\hat{\Phi}$ as

(4.13) $\hat{\Phi}=\left(\begin{array}[]{cc}\hat{\Phi}_{1}&0\\\
0&\hat{\Phi}_{4}\end{array}\right),~{}\hat{\Phi}_{1}=\left(\begin{array}[]{cc}0&\tau\\\
-\tau&0\end{array}\right),~{}\hat{\Phi}_{4}=\left(\begin{array}[]{ccc}0&\sigma&0\\\
-\sigma&0&0\\\
0&0&0\end{array}\right),~{}\tau=\frac{k}{y^{2}},~{}\sigma=2\nu.$

With (5.29) we find the relations

(4.14) $\Phi_{1}=\left(\begin{array}[]{cc}0&\frac{\tau}{g_{xx}}\\\
-\frac{\tau}{g_{yy}}&0\end{array}\right),~{}\Phi_{2}=0,~{}\Phi_{3}=0,~{}\Phi_{4}=\left(\begin{array}[]{ccc}g_{qq}&g_{qp}&0\\\
g_{pq}&g_{pp}&0\\\ g_{\kappa q}&g_{\kappa p}&0\end{array}\right).\\\ $

With (4.14), the first condition (5.27) implies

$\xi^{t}=(0,0,0,0,\xi_{\kappa}),$

while the second condition (5.27) gives

(4.15) $\xi=\frac{1}{\sqrt{\delta}}\frac{\partial}{\partial\kappa}.$

The second condition (5.27) implies

(4.16a) $\displaystyle 0=$
$\displaystyle\frac{k}{y}\Phi^{x}_{x}-\nu\Phi^{q}_{x}+\nu\Phi^{p}_{x}+\sqrt{\delta}\Phi^{\kappa}_{x},$
(4.16b) $\displaystyle 0=$ $\displaystyle\frac{k\tau}{yg_{xx}},$ (4.16c)
$\displaystyle 0=$ $\displaystyle\dots.$

But (4.16b) can’t be satisfied and a) is proved.

b) With (5.26), (5.27), (5.28), for that the contact manifold
$(M_{2n+1},\eta)$ to have an almost contact metric structure
$(M,\Phi,\xi,\eta,g)$, the following equations must be satisfied

(4.17a) $\displaystyle\text{rank~{}}\Phi=2n,$ (4.17b)
$\displaystyle\eta\lrcorner\xi=1,\quad\xi^{t},\eta\in M(1,n,\mathbb{R}),$
(4.17c) $\displaystyle\Phi\xi=0,$ (4.17d) $\displaystyle\eta\Phi=0,$ (4.17e)
$\displaystyle\Phi^{2}=-{\mbox{\boldmath{$1$}}}_{n}+\xi\otimes\eta,$ (4.17f)
$\displaystyle\eta=g\xi,$ (4.17g) $\displaystyle
g=\eta\otimes\eta-\hat{\Phi}\Phi,$ (4.17h) $\displaystyle\hat{\Phi}\xi=0.$

We recall that $\hat{\Phi}$ is defined in (5.29) and is connected with
$\operatorname{d}\eta$ as in (5.31). Equations (4.17a)-(4.17g) appear in [49]
(and also in [51]), where it is underlined that the first 5 relations are not
independent. Equation (4.17h) is a consequence of (4.17d), (5.29). See also
Theorem 3.1 in [50].

Below we denote the components $\Phi^{i}_{j}$ of the (1,1)-tensor field $\Phi$
by $\Phi_{ij}$, i.e.

(4.18) $\Phi=(\Phi)_{ij},\quad i,j=x,y,q,p,\kappa.$

1) With $\eta_{0}$ given by (4.10) and (4.17b), we get for
$\xi=(\xi_{x},\xi_{y},\xi_{q},\xi_{p},\xi_{\kappa})$

$\frac{k}{y}\xi_{x}-\nu p\xi_{q}+\nu q\xi_{p}+\sqrt{\delta}\xi_{k}=1.$

With (4.17h) for $\hat{\Phi}$ given by (4.13) we get again for $\xi$ the form
(4.15)

(4.19) $\xi^{t}=(0,0,0,0,\frac{1}{\sqrt{\delta}}).$

2) With (4.17c) and (4.19) we get the relations

(4.20) $\Phi_{z\kappa}=0,\quad z=x,y,q,p,\kappa.$

3) With (4.17d), we get

(4.21) $\Phi_{\kappa z}=-\frac{1}{\sqrt{\delta}}(\frac{k}{y}\Phi_{xz}-\nu
p\Phi_{qz}+\nu q\Phi_{pz}),\quad z=x,y,q,p.$

4) With the values (4.10) for $\eta$ and (4.19) for $\xi$ introduced in
(4.17f), we get

(4.22)
$(g_{x\kappa},g_{y\kappa},g_{q\kappa},g_{p\kappa},g_{\kappa\kappa})=(\frac{k\sqrt{\delta}}{y},0,-\nu\sqrt{\delta}p,\nu\sqrt{\delta}q,\delta).$

5) Introducing (4.13) in (4.17g), with (4.20), (4.22) we get

(4.23) $\displaystyle g$
$\displaystyle=\left(\begin{array}[]{ccccc}g_{xx}&g_{xy}&g_{xq}&g_{xp}&g_{x\kappa}\\\
&g_{yy}&g_{yq}&g_{yp}&g_{y\kappa}\\\ &&g_{qq}&g_{qp}&g_{q\kappa}\\\
&&&g_{pp}&g_{p\kappa}\\\ &&&&g_{\kappa\kappa}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccccc}\frac{k^{2}}{y^{2}}-\tau\Phi_{xx}&\tau\Phi_{xx}&-\frac{\nu
kp}{y}-\tau\Phi_{yq}&\frac{\nu
kq}{y}-\tau\Phi_{yp}&\frac{k\sqrt{\delta}}{y}\\\
&\tau\Phi_{xy}&\tau\Phi_{xq}&\tau\Phi_{xp}&0\\\
&&\nu^{2}p^{2}-\sigma\Phi_{pq}&-\nu^{2}pq+\sigma\Phi_{pp}&-\nu\sqrt{\delta}p\\\
&&&\nu^{2}q^{2}+\sigma\Phi_{qp}&\nu\sqrt{\delta}q\\\
&&&&\delta\end{array}\right),$

where we have not written down the matrix elements under the diagonal of the
symmetric matrix $g$.

6) But the symmetry of the matrix $g$ (4.23) imposes the following
restrictions on components of the $(1,1)$-tensor $\Phi$:

(4.24a) $\displaystyle\Phi_{yy}=$ $\displaystyle-\Phi_{xx},$ (4.24b)
$\displaystyle\Phi_{qx}=$
$\displaystyle-\zeta\Phi_{yp},\quad\zeta:=\frac{\tau}{\sigma},$ (4.24c)
$\displaystyle\Phi_{qy}=$ $\displaystyle\zeta\Phi_{xp},$ (4.24d)
$\displaystyle\Phi_{px}=$ $\displaystyle\zeta\Phi_{yq},$ (4.24e)
$\displaystyle\Phi_{pp}=$ $\displaystyle-\Phi_{qq},$ (4.24f)
$\displaystyle\Phi_{py}=$ $\displaystyle-\zeta\Phi_{xq}.$

We choose as independent components of tensor field $\Phi$ the components
$\Phi$ minus the l.h.s. components of (4.24), i.e. the submatrix of $\Phi$
with the following 10 components

(4.25)
$\left(\begin{array}[]{ccccc}\Phi_{xx}&\Phi_{xy}&\Phi_{xq}&\Phi_{xp}&\\\
\Phi_{yx}&&\Phi_{yq}&\Phi_{yp}&\\\ &&\Phi_{qq}&\Phi_{qp}&\\\ &&\Phi_{pq}&&\\\
&&&&\end{array}\right).$

With (4.24), the relations (4.21) became

(4.26a) $\displaystyle\Phi_{\kappa x}$
$\displaystyle=-\frac{1}{\sqrt{\delta}}(\frac{k}{y}\Phi_{xx}+\nu\zeta
p\Phi_{yp}+\nu\zeta q\Phi_{yq}),$ (4.26b) $\displaystyle\Phi_{\kappa y}$
$\displaystyle=-\frac{1}{\sqrt{\delta}}(\frac{k}{y}\Phi_{xy}-\nu\zeta
p\Phi_{xp}-\nu\zeta q\Phi_{xq}),$ (4.26c) $\displaystyle\Phi_{\kappa q}$
$\displaystyle=-\frac{1}{\sqrt{\delta}}(\frac{k}{y}\Phi_{xq}-\nu
p\Phi_{qq}+\nu q\Phi_{pq}),$ (4.26d) $\displaystyle\Phi_{\kappa p}$
$\displaystyle=-\frac{1}{\sqrt{\delta}}(\frac{k}{y}\Phi_{xp}-\nu
p\Phi_{qp}-\nu q\Phi_{qq}).$

With (4.24), equation (4.23) becomes

(4.27)
$g^{\prime}=\left(\begin{array}[]{ccccc}\frac{k^{2}}{y^{2}}-\tau\Phi_{xx}&-\tau\Phi_{yy}&-\frac{\nu
kp}{y}-\tau\Phi_{yq}&\frac{\nu
kq}{y}-\tau\Phi_{yp}&\frac{k\sqrt{\delta}}{y}\\\
&\tau\Phi_{xy}&\tau\Phi_{xq}&\tau\Phi_{xp}&0\\\
&&\nu^{2}p^{2}-\sigma\Phi_{pq}&-\nu^{2}pq-\sigma\Phi_{qq}&-\nu\sqrt{\delta}p\\\
&&&\nu^{2}q^{2}+\sigma\Phi_{qp}&\nu\sqrt{\delta}q\\\
&&&&\delta\end{array}\right).$

Taking into account (4.20), (4.21), (4.24), (4.26), the 25 equations (4.17e)
for the tensor $\Phi$ in the convention (4.18) expressed as function of the 10
independent components (4.25) are reduced only to the following 6 independent
equations

(4.28a)
$\displaystyle\Phi^{2}_{xx}+\Phi_{xy}\Phi_{yx}+\zeta(\Phi_{xp}\Phi_{yq}-\Phi_{xq}\Phi_{yp})=-1,$
(4.28b)
$\displaystyle\Phi_{xq}(\Phi_{xx}+\Phi_{qq})+\Phi_{xy}\Phi_{yq}+\Phi_{xp}\Phi_{pq}=0,$
(4.28c)
$\displaystyle\Phi_{xp}(\Phi_{xx}-\Phi_{qq})+\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp}=0,$
(4.28d)
$\displaystyle\Phi_{yq}(\Phi_{qq}-\Phi_{xx})+\Phi_{yx}\Phi_{xq}+\Phi_{yp}\Phi_{pq}=0,$
(4.28e)
$\displaystyle-\Phi_{yp}(\Phi_{xx}+\Phi_{qq})+\Phi_{yx}\Phi_{xp}+\Phi_{yq}\Phi_{qp}=0,$
(4.28f)
$\displaystyle\zeta(\Phi_{xp}\Phi_{yq}-\Phi_{yp}\Phi_{xq})+\Phi^{2}_{qq}+\Phi_{qp}\Phi_{pq}=-1.$

In (4.28) the first 3 (respectively, next 2, last, no, no) equations are the
only independent equations of the product of the first (respectively, second,
third, fourth, fifth) line of $\Phi$ with $\Phi$ in (4.17e).

7) The ten independent variables $\Phi$ (4.25) verify the 6 independent
equations (4.28). We shall choose 6 independent variables in the set (4.25)
and express the four remaining variables using six independent equations
(4.28).

From equations (4.28b) and (4.28c), we get if $\Phi_{xq}\Phi_{xp}\not=0$

(4.29a) $\displaystyle\Phi_{xx}+\Phi_{qq}$
$\displaystyle=-\frac{1}{\Phi_{xq}}(\Phi_{xy}\Phi_{yq}+\Phi_{xp}\Phi_{pq}),$
(4.29b) $\displaystyle\Phi_{xx}-\Phi_{qq}$
$\displaystyle=-\frac{1}{\Phi_{xp}}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp}).$

Introducing (4.29b) into (4.28d), we get if $\Phi_{xq}\Phi_{xp}\not=0$

(4.30)
$\Phi_{yx}=-\frac{1}{\Phi_{xq}\Phi_{xp}}[\Phi_{yq}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp})+\Phi_{yp}\Phi_{pq}\Phi_{xq}].$

From (4.29), we get if $\Phi_{xq}\Phi_{xp}\not=0$

(4.31a) $\displaystyle 2\Phi_{xx}$
$\displaystyle=-\frac{1}{\Phi_{xq}}(\Phi_{xy}\Phi_{yq}+\Phi_{pq}\Phi_{xp})-\frac{1}{\Phi_{xp}}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp}),$
(4.31b) $\displaystyle 2\Phi_{qq}$
$\displaystyle=-\frac{1}{\Phi_{xq}}(\Phi_{xy}\Phi_{yq}+\Phi_{pq}\Phi_{xp})+\frac{1}{\Phi_{xp}}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp}).$

But from (4.28a), (4.28f) we get

(4.32)
$\zeta(\Phi_{xp}\Phi_{yq}-\Phi_{xq}\Phi_{yp})+1=-\Phi^{2}_{xx}-\Phi_{xy}\Phi_{yx}=-\Phi^{2}_{qq}-\Phi_{qp}\Phi_{pq}.$

Introducing (4.30) into (4.28e), we get

$\Phi_{xy}[\Phi_{xq}(\Phi_{yx}\Phi_{xp}+\Phi_{qp}\Phi_{yq})+\Phi_{yp}(\Phi_{xp}\Phi_{pq}+\Phi_{yq}\Phi_{xy})]=0.$

If $\Phi_{xy}\not=0$, we get

(4.33)
$\Phi_{yx}=-\frac{1}{\Phi_{xq}\Phi_{xp}}[\Phi_{yq}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp})+\Phi_{yp}\Phi_{pq}\Phi_{xp}].$

But comparing (4.30) with (4.33), we get

(4.34) $\Phi_{xp}=\Phi_{xq}.$

With (4.31), (4.34), equations (4.28a), (4.28f) became

(4.35a) $\displaystyle 2\Phi_{xx}$
$\displaystyle=-\frac{1}{\Phi_{xq}}[\Phi_{xy}(\Phi_{yq}+\Phi_{yp})+\Phi_{xq}(\Phi_{pq}+\Phi_{qp})],$
(4.35b) $\displaystyle 2\Phi_{qq}$
$\displaystyle=\frac{1}{\Phi_{xq}}[\Phi_{xy}(-\Phi_{yq}+\Phi_{yp})+\Phi_{xq}(-\Phi_{pq}+\Phi_{qp})].$

With (4.34), equations (4.32) became

(4.36a) $\displaystyle\Phi_{xx}^{2}+\zeta\Phi_{xq}(\Phi_{yq}-\Phi_{yp})+1$
$\displaystyle=-\Phi_{xy}\Phi_{yx},$ (4.36b)
$\displaystyle\Phi_{qq}^{2}+\zeta\Phi_{xq}(\Phi_{yq}-\Phi_{yp})+1$
$\displaystyle=-\Phi_{qp}\Phi_{pq}.$

From (4.36a) ((4.36b)) we get with (4.35a), (4.33) ((4.35b)) the value
$\Phi_{xy}$ (respectively $\Phi_{xq}$).

We have shown that the six components
$\Phi_{xx},\Phi_{xy},\Phi_{xq},\Phi_{xp},\Phi_{yx},\Phi_{qq}$ of $\Phi$ in
(4.25) can be expressed as function of the remaining independent variables
$\Phi_{yq},\Phi_{yp},\Phi_{qp},\Phi_{pq}$, while the rest of the components of
$\Phi$ are obtained with (4.24).

Once the (1,1)-tensor $\Phi$ is known, it should be verified that the
condition (4.17a) is fulfilled. Then the metric tensor $g^{\prime}$ is
determined with (4.27) and we have to impose the condition that the
fundamental quadratic form associated to the tensor $g^{\prime}$ must be
positive definite.

∎

######  Remark 2.

In order to compare the metric matrices (2.12) and (4.27) on
${\tilde{{\mathcal{X}}}^{J}_{1}}$ we have to make the replacement
$k,\nu\rightarrow\sqrt{k},\sqrt{\nu}$ in (2.12) due to the parametrization
(2.1) of the one-forms in [12] and we get instead of (2.12)

(4.37) $\begin{split}{\rm
d}s^{2}_{\tilde{{\mathcal{X}}}^{J}_{1}}(x,y,p,q,\kappa)&=\nu(p\operatorname{d}q^{2}+q\operatorname{d}p^{2}-2pq\operatorname{d}q\operatorname{d}p)+\delta\operatorname{d}\kappa^{2}+2\frac{\sqrt{k\delta}}{y}\operatorname{d}x\operatorname{d}\kappa\\\
&+2\frac{\sqrt{k\nu}}{y}(-p\operatorname{d}p+q\operatorname{d}q)\operatorname{d}x-2\sqrt{\nu\delta}(p\operatorname{d}q+q\operatorname{d}p)\operatorname{d}\kappa\end{split}$

in the convention (2.10).

Also it would be interesting to check that the metric
$g_{\tilde{{\mathcal{X}}}^{J}_{1}}(x,y,p,q,\kappa)$ (4.27) can not be obtained
from a potential type (5.36).

######  Proposition 6.

$(\tilde{{\mathcal{X}}}^{J}_{1},\eta_{0})$ is a contact Hamiltonian system as
in _Appendix_ , C.

Using the same parametrization (4.6) and (4.7) in _Section 4.1_, we get for
$X_{H}$ (5.16) the expression

(4.38) $\begin{split}X_{H}&=\frac{y^{2}}{k}\frac{\partial H}{\partial
y}\frac{\partial}{\partial x}+\frac{1}{2\nu}\frac{\partial H}{\partial
p}\frac{\partial}{\partial q}+(-\frac{y^{2}}{k}\frac{\partial H}{\partial
x}+\color[rgb]{0,1,0}y\frac{\partial
H}{\partial\kappa}\color[rgb]{0,0,0})\frac{\partial}{\partial y}\\\
&-(\frac{1}{2\nu}\frac{\partial H}{\partial
q}+\color[rgb]{0,1,0}p\frac{\partial
H}{\partial\kappa}\color[rgb]{0,0,0})\frac{\partial}{\partial
p}+\color[rgb]{0,1,0}(-y\frac{\partial H}{\partial y}+p\frac{\partial
H}{\partial
p}-H)\frac{\partial}{\partial\kappa}\color[rgb]{0,0,0}.\end{split}$

The equations of motion associated to the vector field $X_{H}$ (4.38) on the
extended Sigel–Jacobi upper half-plane organized as the contact manifold
$(\tilde{{\mathcal{X}}}^{J}_{1},\eta_{0})$ are

(4.39a) $\displaystyle\dot{x}$ $\displaystyle=\frac{y^{2}}{k}\frac{\partial
H}{\partial y},\quad\dot{y}=-\frac{y^{2}}{k}\frac{\partial H}{\partial
x}+\color[rgb]{0,1,0}y\frac{\partial H}{\partial\kappa}\color[rgb]{0,0,0},$
(4.39b) $\displaystyle\dot{q}$ $\displaystyle=\frac{1}{2\nu}\frac{\partial
H}{\partial p},\quad\dot{p}=-(\frac{1}{2\nu}\frac{\partial H}{\partial
q}+\color[rgb]{0,1,0}p\frac{\partial H}{\partial\kappa}\color[rgb]{0,0,0}),$
(4.39c) $\displaystyle\color[rgb]{0,1,0}\dot{\kappa}$
$\displaystyle=\color[rgb]{0,1,0}(-y\frac{\partial H}{\partial
y}+p\frac{\partial H}{\partial p}-H)\frac{\partial
H}{\partial\kappa}\color[rgb]{0,0,0}.$

The Jacobi bracket (5.25) on $\tilde{{\mathcal{X}}}^{J}_{1}$ is

$\\{f,g\\}=\\{f,g\\}_{P}+f_{e}\frac{\partial
g}{\partial\kappa}-g_{e}\frac{\partial f}{\partial\kappa},$

where the Poisson bracket (5.4) with the convention (4.7) reads

$\\{f,g\\}_{P}=\frac{1}{k}\frac{y^{2}+1}{y^{2}}(\frac{\partial f}{\partial
x}\frac{\partial g}{\partial y}-\frac{\partial g}{\partial x}\frac{\partial
f}{\partial y})+\frac{1}{2\nu}[\frac{\partial f}{\partial q}\frac{\partial
g}{\partial p}-\frac{\partial g}{\partial q}\frac{\partial f}{\partial
p}+\frac{1}{y^{2}}(\frac{\partial f}{\partial q}\frac{\partial g}{\partial
y}-\frac{\partial g}{\partial q}\frac{\partial f}{\partial y})].$

while the Euler operator (5.23) is

$f_{e}=f+\frac{1}{y^{3}}\frac{\partial f}{\partial y}-p\frac{\partial
f}{\partial p}.$

######  Remark 3.

If in the equations of motion expressed in the S-variables $(x,y,p,q,\kappa)$
in (4.9) (respectively (4.39)) on the generalized transitive almost
cosymplectic manifold $(\tilde{{\mathcal{X}}}^{J}_{1},\theta,\omega)$
(respectively, on the contact manifold
$(\tilde{{\mathcal{X}}}^{J}_{1},\eta_{0})$) we ignore the “red” (respectively
“green”) parts we get the equations of motion on ${\mathcal{X}}^{J}_{1}$ in
$(x,y,p,q)$.

###  4.3. Linear Hamiltonian in the generators of the Jacobi group
$G^{J}_{1}(\mathbb{R})$

In [9, (4.7)] we have considered a linear Hermitian Hamiltonian $H$ in the
generators of the Jacobi group $G^{J}_{1}$

(4.40)
${\mbox{\boldmath{$H$}}}=\epsilon_{a}{\mbox{\boldmath{$a$}}}+\bar{\epsilon}_{a}{\mbox{\boldmath{$a$}}}^{\dagger}+\epsilon_{0}{\mbox{\boldmath{$K$}}}_{0}+\epsilon_{+}{\mbox{\boldmath{$K$}}}_{+}+\epsilon_{-}{\mbox{\boldmath{$K$}}}_{-},\quad\bar{\epsilon}_{+}=\epsilon_{-},\quad\bar{\epsilon}_{0}=\epsilon_{0}.$

We use the notation introduced in [9, § 4.1 3.]

$\epsilon_{a}:=a+\operatorname{i}b,~{}\epsilon_{+}:=m-\operatorname{i}n,~{}\epsilon_{0}:=2c,\quad
a,b,c,m,n\in\mathbb{R}.$

We have proved in [9, (4.29) in § 4.3] that the energy function
${\mathcal{H}}$ associated to the linear Hamiltonian (4.40) expressed in the
variables $(\eta,v)$ splits into the sum of two independent functions

(4.41) ${\mathcal{H}}(\eta,v)={\mathcal{H}}(\eta)+{\mathcal{H}}(v),\quad
v=x+\operatorname{i}y,~{}y>0,~{}\eta=q+\operatorname{i}p,$

where

(4.42a) $\displaystyle{\mathcal{H}}(q,p)$
$\displaystyle=\nu[(m+c)q^{2}+(c-m)p^{2}+2nqp+2(aq+bp)],$ (4.42b)
$\displaystyle{\mathcal{H}}(x,y)$
$\displaystyle=k\\{\frac{1}{y}[(m+c)(x^{2}+y^{2})-2(nx+cy)]+3c-m\\}.$

We have underlined in [3, Remark 1], [14, Proposition 3] the connection of
parameter $\eta=q+\operatorname{i}p$ which appear in the FC-transform with the
S-variables $(p,q)$ which parametrize the Jacobi group
$G^{J}_{1}(\mathbb{R})$.

Now we particularize equations (4.9) to the linear Hamiltonian (4.41) to which
we add a term ${h}(\kappa)$

(4.43) ${\mathcal{H}}={\mathcal{H}}(p,q)+{\mathcal{H}}(x,y)+{h}(\kappa),$

and we get

######  Proposition 7.

The equations of motion (3.7) on the extended Siegel-Jacobi upper half-plane
organized as generalized transitive almost cosymplectic manifold
$(\tilde{{\mathcal{X}}}^{J}_{1},\theta,\omega)$ corresponding to the energy
function (4.43) are

(4.44a) $\displaystyle\dot{x}$
$\displaystyle=(c+m)(-x^{2}+y^{2})+mx-c+m,\quad\dot{y}=-2(c+m)y^{2}+2ny,$
(4.44b) $\displaystyle\dot{q}$
$\displaystyle=\\!-\\!(m+c)q-\\!np-\\!a\color[rgb]{1,0,0}\\!-\\!\frac{q}{2\nu}\frac{\partial{h}}{\partial\kappa}\color[rgb]{0,0,0},\quad\dot{p}=qn+(c-m)p+b\color[rgb]{1,0,0}-\frac{p}{2\nu}\frac{\partial
h}{\partial\kappa}\color[rgb]{0,0,0},$ (4.44c)
$\displaystyle\color[rgb]{1,0,0}\dot{\kappa}$
$\displaystyle\color[rgb]{1,0,0}=(c+m)qa^{2}+(-c+m)p^{2}+(m-n)pq+nq+bp\\!-\\!\frac{1}{\sqrt{\delta}}h.\color[rgb]{0,0,0}$

######  Remark 4.

If in the equations of motion (4.44) on $\tilde{{\mathcal{X}}}^{J}_{1}$
generated by the linear Hamiltonian (4.43) we ignore the “red parts”, the
well-known matrix Riccati equation (4.44a) in $(x,y)$ and the linear system of
differential equations in $(p,q)$ generated by the linear Hamiltonian (4.41),
(4.42) on ${\mathcal{X}}^{J}_{1}$ found in [9] are obtained.

##  5\. Appendix – a breviar of terminology

In this section are collected definitions of the main geometric structures
used in our paper. We use the following abbreviations:
SH: symplectic Hamiltonian; COS: cosymplectic; ACOS: almost cosymplectic,
GTACOS: generalized transitive almost cosymplectic, TACS: transitive almost
contact structure, CH: contact Hamiltonian, C: contact, ACM: almost contact
metric, SAC: (strict) almost contact, N: normal, SAS: Sasakian, ACOK: almost
coKähler.

Table: Geometric structures on odd dimensional manifolds

$\boxed{\begin{array}[]{cc cc cc cc
cc}\text{COS}&\subset&\color[rgb]{1,0,0}\text{GTACOS}\color[rgb]{0,0,0}&\supset&\text{TACS}&&&\\\
&&\cap&&&&&\\\ &&\text{ACOS}&&&&&\\\ &&\cup&&&&&\\\
\text{CH}&\subset&\text{C}&\subset&\text{ACM}&\subset&\text{SAC}&\supset\text{N=SAS}\\\
&&&&\cap&&&\\\ &&&&\text{ACOK}&&&\end{array}}$

SH

In [46, Chapter III, Symplectic manifolds and Poisson manifolds]: a symplectic
Hamiltonian system – is the pair $(M,\Omega)$, where $\dim M=2n$,
$n\in\mathbb{N}$, $\Omega$ is a closed non-degenerate two-form, and

$\Omega^{n}\neq 0.$

If $H:M\rightarrow\mathbb{R}$ is a Hamiltonian function, then the Hamiltonian
vector field $X_{H}$, $\operatorname{grad}H$, is the solution of the equation

(5.1)
$\flat(X_{H})=\operatorname{d}H,\quad{\text{where~{}~{}~{}}}\flat:TM\rightarrow
T^{*}M,\quad\flat(X)=X\lrcorner\Omega.$

In canonical Darboux coordinates $(q^{i},p_{i}),~{}i=1,\dots,n$ , we have

(5.2) $\Omega=\operatorname{d}q^{i}\wedge\operatorname{d}p_{i},$ (5.3)
$X_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{\partial}{\partial p_{i}},$

and Hamilton equations of motion are

$\dot{q}^{i}=\frac{\partial H}{\partial
p_{i}}=\\{q^{i},H\\}_{P},\quad\dot{p}_{i}=-\frac{\partial H}{\partial
q^{i}}=\\{p_{i},H\\}_{P},$

where the Poisson bracket $\\{f,g\\}_{P}$ in Darboux coordinates is

(5.4) $\\{f,g\\}_{P}=\Omega(X_{f},X_{g})=\frac{\partial f}{\partial
q^{i}}\frac{\partial g}{\partial p_{i}}-\frac{\partial g}{\partial
q^{i}}\frac{\partial f}{\partial p_{i}},\quad f,g\in C^{\infty}(M).$

Note that if in (5.4) we make the change of coordinates
$(q,p)\rightarrow(p,q)$, then $\\{f,g\\}_{P}\rightarrow-\\{f,g\\}_{P}$.

ACOS

In [45]: almost cosymplectic manifold – is the triplet $(M,\theta,\Omega)$,
where $M$ is a $(2n+1)$-dimensional manifold, $\theta\in{\mathfrak{D}}^{1}$,
$\Omega$ is a 2-form with $\text{rank}(\Omega)=2n$, and

(5.5) $\theta\wedge\Omega^{n}\neq 0.$

The Reeb vector $R\in{\mathfrak{D}}^{1}$ is defined by the equations

(5.6) $R\lrcorner\Omega=0,\quad R\lrcorner\theta=1.$

In [1]: the almost cosymplectic manifold $(M,\theta,\Omega)$ of [45] is called
almost contact manifold.
It is proved in [1, Proposition 1] that the application $\flat:TM\rightarrow
TM^{*}$

(5.7) $X\rightarrow X^{\flat}=X\lrcorner\Omega+(X\lrcorner\theta)\theta$

is a vector bundle isomorphism.

COS

In [45]: cosymplectic manifold – is the triplet $(M,\theta,\Omega)$, defined
in ACOS, where

$\operatorname{d}\theta=0,\quad\operatorname{d}\Omega=0.$

In [33]: the isomorphism $\flat$ for the cosymplectic manifold
$(M,\eta,\Omega)$ is defined as in (5.7).

The Darboux coordinates are $(z,q^{i},p_{i},)$, $i=1,\dots,n$, $\Omega$ is
defined in (5.2), and

$\eta=\operatorname{d}z.$

The gradient vector field $\operatorname{grad}H$, the Hamiltonian vector field
$X_{H}$ and the evolution vector field ${\mathcal{E}}_{H}$ attached to the
function $H$ are defined as solutions of the equations:

(5.8a) $\displaystyle\flat(\operatorname{grad}H)$
$\displaystyle=\operatorname{d}H,$ (5.8b) $\displaystyle X_{H}$
$\displaystyle=\operatorname{grad}H-R(H)R,$ (5.8c)
$\displaystyle{\mathcal{E}}_{H}$ $\displaystyle=X_{H}+R.$

TACS

In [1, 29]: transitive almost contact structure \- is an almost contact
manifold $(M,\theta,\Omega)$ defined at ACOS with

$\operatorname{d}\Omega=0,$

and around every point of $M$ there is a neighbourhood where there are local
Darboux coordinates $(\kappa,q^{1},\dots,q^{n},p_{1},\dots,p_{n})$ such that

(5.9) $\theta=\operatorname{d}\kappa+\epsilon p_{i}\operatorname{d}q^{i},\quad
i=1,\dots,n,~{}~{}\epsilon\in\mathbb{R}.$

To a function $f\in C^{\infty}(M)$ it is associated the Hamiltonian vector
field $X_{f}$ defined by [1, (3)] as solution of the equations

$\displaystyle X_{f}\lrcorner\theta$ $\displaystyle=\epsilon f,$
$\displaystyle X_{f}\lrcorner\Omega$
$\displaystyle=\operatorname{d}f-(Rf)\theta.$

CH

In [33, 34]: contact Hamiltonian system $(M,\eta)$– is defined as the almost
cosymplectic manifold $(M,\eta,\operatorname{d}\eta)$ and (5.5) verified

(5.11) $\eta\wedge\operatorname{d}\eta^{n}\neq 0.$

Apparently the denomination contact structure for a manifold $(M,\eta)$
verifying the condition (5.11) was used firstly by Gray [38].

$\theta$ defined in (5.9) with $\epsilon=-1$ corresponds to Darboux
coordinates and in accord with [25, Theorem, page 1] $\eta$ can be taken

(5.12) $\eta=\operatorname{d}\kappa-
p_{i}\operatorname{d}q^{i},\quad\operatorname{d}\eta=\operatorname{d}q^{i}\wedge\operatorname{d}p_{i}.$

The conditions (5.6) defining the Reeb vector became

(5.13) $R\lrcorner\operatorname{d}\eta=0,\quad R\lrcorner\eta=1,$

where

(5.14) $R=\frac{\partial}{\partial\kappa}.$

The Hamiltonian vector field $X_{H}$ attached to the real function $H$ is
defined by [33, (2.11)]

(5.15) $\flat(X_{H})=\operatorname{d}H-(R\lrcorner H+H)\eta,$

where the Reeb vector $R$ is defined in (5.14) and the application $\flat$ is
defined in (5.7).

The vector field $X_{H}$ in Darboux coordinates reads [33, (2.12)]

(5.16) $X_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-(\frac{\partial H}{\partial
q^{i}}+\color[rgb]{0,1,0}p_{i}\frac{\partial
H}{\partial\kappa})\color[rgb]{0,0,0}\frac{\partial}{\partial
p_{i}}+\color[rgb]{0,1,0}(p_{i}\frac{\partial H}{\partial
p_{i}}-H)\frac{\partial}{\partial\kappa}\color[rgb]{0,0,0}.$

We determine the vector field $\operatorname{grad}H$ solution of equation
(5.8a)

(5.17) $\operatorname{grad}H=\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial q^{i}}-(\frac{\partial H}{\partial
q^{i}}+\color[rgb]{0,1,0}p_{i}\frac{\partial
H}{\partial\kappa}\color[rgb]{0,0,0})\frac{\partial}{\partial
p_{i}}+\color[rgb]{0,1,0}(\frac{\partial
H}{\partial\kappa}+p_{i}\frac{\partial H}{\partial
p_{i}})R\color[rgb]{0,0,0},$

but instead of (5.8b), we have

(5.18) $X_{H}=\operatorname{grad}H-(H+R(H))R.$

If in equations (5.16) and (5.17) we neglect the “green parts”, we get $X_{H}$
(5.3) on the symplectic manifold $(M,\Omega)$.

In order to recall the notion of Jacobi bracket [43, 47, 52], we follow [33,
§4, Contact manifolds as Jacobi structures].

The isomorphism (5.7) becomes

$X\rightarrow X^{\flat}=X\lrcorner\operatorname{d}\eta+(X\lrcorner\eta)\eta.$

If

(5.19) $X=A_{i}\frac{\partial}{\partial q^{i}}+B_{i}\frac{\partial}{\partial
p^{i}}+C\frac{\partial}{\partial\kappa},\quad\text{then}\quad
X^{\flat}=\alpha_{i}\operatorname{d}q^{i}+\beta_{i}\operatorname{d}p_{i}+\gamma\operatorname{d}\kappa,\quad
i=1,\dots,n,$

where

(5.20a)
$\displaystyle\flat:\quad\alpha_{i}=-B_{i}+p_{i}(p_{j}A_{j}-C),\quad\beta_{i}=A_{i},\quad\gamma=C-p_{j}A_{j},$
(5.20b) $\displaystyle\sharp=\flat^{-1}:\quad A_{i}=\beta_{i},\quad
B_{i}=-\alpha_{i}-\gamma p_{i},\quad C=p_{i}\beta_{i}+\gamma.$

We have to calculate the bracket $\\{f,g\\}_{J}$ [33, p. 11]

(5.21)
$\\{f,g\\}_{J}=-\operatorname{d}\eta(\sharp\operatorname{d}f,\sharp\operatorname{d}g)-R(g)f+gR(f).$

With (5.17), (5.19), (5.20), we find

######  Remark 5.

The Jacobi bracket (5.21) has the expression

(5.22) $\\{f,g\\}_{J}=\\{g,f\\}_{P}+\frac{\partial
f}{\partial\kappa}g_{e}-\frac{\partial g}{\partial\kappa}f_{e},$

where $f_{e}$ denotes Euler’s operator [2]

(5.23) $f_{e}:=f-p_{i}\frac{\partial f}{\partial p_{i}}.$

In [46, Chapter V, Contact manifolds] the conventions expressed in (5.11), …,
(5.14) are used.

Instead of (5.1) it is used [46, Chapter V, Proposition 6.13 page 293]

(5.24) $\eta^{\flat}:X\rightarrow-X\lrcorner\operatorname{d}\eta,$

which associates to vector fields semi-basic differential forms [46, pages 56,
68]. Also it is used the notation

$\eta^{\sharp}(\gamma)=^{\sharp}\gamma,\quad\gamma\in{\mathfrak{D}}_{1}.$

Note that in [45, Lemma p 44] instead of (5.24) it is used

$\eta^{\flat}:X\rightarrow X\lrcorner\operatorname{d}\eta.$

According to [46, Proposition 6.11 p 292]: We have decomposition of the
tangent space $TM$ into the direct sum

$TM=\text{Ker}\operatorname{d}\eta\oplus\text{Ker}~{}\eta,$

of the vertical bundle of rank 1 generated by the Reeb vector (5.14) and the
horizontal bundle of rank $2n$

$X=(X\lrcorner\eta)R+(X-(X\lrcorner\eta)R).$

According to [46, Proposition 13.1 p 318]: The vector field $X$ on the contact
manifold $(M,\eta)$ is an infinitesimal contact automorphism if and only if
there exists a differentiable function $\rho$ such that

${\mathcal{L}}_{X}\eta=\rho\eta.$

According to [46, Theorem 13.3 p 319]: The choice of a contact form $\eta$ on
the strictly contact manifold $(M,\eta)$ defines an isomorphism
$\Phi:~{}L\rightarrow L^{\prime}$

$\Phi(X)=X\lrcorner\eta,\quad
X_{f}=\Phi^{-1}(f)=fR+\eta^{\sharp}(\operatorname{d}f-(R\lrcorner\operatorname{d}f)\eta),\quad\rho=R\lrcorner\operatorname{d}f,$

where $L$ ($L^{\prime}$) is the space of the infinitesimal automorphisms of
$\eta$ (respectively differentiable real-valued functions on $M$).

According to [46, Proposition 14.2 p 325]: The Lie algebra structure of
$L^{\prime}$ is defined by the bracket

(5.25a) $\displaystyle\\{f,g\\}$ $\displaystyle=[X_{f},X_{g}]\lrcorner\eta,$
(5.25b)
$\displaystyle=\operatorname{d}\eta(X_{f},X_{g})+f(R\lrcorner\operatorname{d}g)-g(R\lrcorner\operatorname{d}f),$
(5.25c) $\displaystyle=\\{f,g\\}_{P}+f_{e}\frac{\partial
g}{\partial\kappa}-g_{e}\frac{\partial f}{\partial\kappa}.$

######  Remark 6.

Note that the expresion of the Jacobi bracket (5.25) is minus the expression
(5.22)

$\\{f,g\\}=-\\{f,g\\}_{J},$

and coincides with the expresion [1, {f,g} on p 636] for $\epsilon=-1$.

SAC

In [12]: following Sasaki [50] and [28, Definition 6.2.5], we used the

######  Definition 1.

The manifold $M_{2n+1}$ has a (strict) almost contact structure
$(\Phi,\xi,\eta)$ if there exists a $(1,1)$-tensor field $\Phi$, the
contravariant vector field $\xi\in{\mathcal{D}}^{1}$ (Reeb vector field, or
characteristic vector field), and the covariant vector field
$\eta\in{\mathfrak{D}}_{1}$ verifying the relations

(5.26) $\eta\lrcorner\xi=1,\quad\Phi^{2}X=-X+\eta(X)\xi.$

C

Following [27], in [12] we used the definition of a contact structure
($M_{2n+1},\eta)$ when $\eta\in{\mathcal{D}}_{1}$ satisfies (5.11).

The contact structure can be given by a codimension one subbundle
${\mathcal{D}}$ of the tangent bundle $TM$ which is as far from being
integrable as possible, and ${\mathcal{D}}:=\text{Ker}(\eta)$.

ACM

Sasaki has proved [50, Theorem 1.1] and [51, (5.16)]:

######  Proposition 8.

For an almost contact structure $(\Phi,\xi,\eta)$, the following relations
hold

(5.27)
$\Phi\xi=0,\quad\eta\Phi=0,\quad\text{{\emph{Rank~{}}}}(\Phi^{i}_{j})=2n,\quad\xi,\eta^{t}\in
M(n,1,\mathbb{R}).$

There exists a positive Riemannian metric $g$ such that

(5.28) $\eta=g\xi,\quad\Phi^{t}g\Phi=g-\eta^{t}\otimes\eta.$

If we put

(5.29) $\hat{\Phi}:=g\Phi,$

the two-form $\hat{\Phi}$ is antisymmetric.

Sasaki has proved [50, Theorem 3.1], [51], see also [12, (9.12 ), Theorem 14]:

######  Proposition 9.

Let $(M,\eta)$ be a contact manifold. Then we can find an almost contact
metric structure $(M,\Phi,\xi,\eta,g)$ such that (5.29) is satisfied,

(5.30) $\operatorname{d}\eta(X,Y)=g(X,\Phi(Y)),$

and

(5.31)
$\operatorname{d}\eta\\!\\!=\\!\\!\frac{1}{2}\sum_{i,j=1}^{2n+1}\\!\hat{\Phi}_{ij}\operatorname{d}x^{i}\\!\wedge\\!\operatorname{d}x^{j}\\!=\\!\sum_{1\leq
i<j\leq
2n+1}\\!\hat{\Phi}_{ij}\operatorname{d}x^{i}\\!\wedge\\!\operatorname{d}x^{j},\quad\hat{\Phi}_{ij}\\!\\!=\\!\\!\partial_{i}\eta_{j}\\!-\\!\partial_{j}\eta_{i}.$

N=SAS

Following [25, p. 47], let us introduce

######  Definition 2.

Let $h$ be a $(1,1)$-tensor field. Then the Nijenhuis torsion $[h,h]$ of $h$
is the tensor field of type $(1,2)$ given by

$[h,h](X,Y)=h^{2}[X,Y]+[hX,hY]-h[hX,Y]-h[X,hY],\quad
X,Y\in{\mathfrak{D}}^{1}.$

Let us define the $(1,2)$-tensor

(5.32) $N^{1}:=[\Phi,\Phi]+2\operatorname{d}\eta\otimes\xi.$

According to [28, Theorem 6.5.9]

######  Proposition 10.

An almost contact structure $(\xi,\eta,\Phi)$ on $M$ is normal if and only if
$N^{1}=0$.

A normal contact metric structure $(M,\xi,\eta,\Phi,g)$ is called a Sasakian
structure, see more details [28, Definitions 6.4.7, 6.5.7, 6.5.13].

Under the conditions (5.26), (5.28), but instead of (5.30) and (5.32) with

$\operatorname{d}\eta(X,Y)=g(\Phi(X),Y),$ (5.33)
$N^{1}:=[\Phi,\Phi]+\operatorname{d}\eta\otimes\xi,$

in local coordinates $(\kappa,z^{1},\dots,z^{n})$ on a small neighbourhood
$\mathbb{R}\times\mathbb{C}^{k}$ of a Sasaki manifold
$(M_{2n+1},\xi,\eta,\Phi,g)$ according with [37, Theorem 1], see also [53], we
have

(5.34a) $\displaystyle\xi$ $\displaystyle=\frac{\partial}{\partial\kappa},$
(5.34b) $\displaystyle\eta$
$\displaystyle=\operatorname{d}\kappa+\operatorname{i}\sum_{j=1}^{n}(K_{j}\operatorname{d}z^{j}-K_{\bar{j}}\operatorname{d}\bar{z}_{j}),$
(5.34c) $\displaystyle\operatorname{d}\eta$
$\displaystyle=-2\operatorname{i}\sum_{j,\bar{k}=1}^{n}K_{j\bar{k}}\operatorname{d}z^{j}\wedge\operatorname{d}\bar{z}^{k},$
(5.34d) $\displaystyle g$
$\displaystyle=\eta\otimes\eta+2\sum_{j,\bar{k}=1}^{n}K_{j\bar{k}}\operatorname{d}z^{j}\operatorname{d}\bar{z}^{k},$
(5.34e) $\displaystyle\Phi$
$\displaystyle=-\operatorname{i}\sum_{j=1}^{n}(\partial_{j}-K_{j}\partial_{\kappa})\otimes\operatorname{d}z^{j}+\operatorname{i}\sum_{j=1}^{n}(\partial_{\bar{j}}-K_{\bar{j}}\partial_{\kappa})\otimes\operatorname{d}\bar{z}^{j},$

where the Sasaki potential $K$ does not depend on $\kappa$.

However, using (5.32) and (5.30), equation (5.34d) should be replaced with

(5.35)
$g=\eta\otimes\eta-4\sum_{j,\bar{k}=1}^{n}K_{j\bar{k}}\operatorname{d}z^{j}\operatorname{d}\bar{z}^{k}.$

In the holonomic cobasis
$(\operatorname{d}y^{\mu})=(\operatorname{d}\kappa,\operatorname{d}z^{j},\operatorname{d}\bar{z}^{j})$
the covariant components of the Sasakian metric are [37, (15)]

(5.36)
$g_{\mu\nu}=\left(\begin{array}[]{ccc}1&\operatorname{i}K_{j}&-\operatorname{i}K_{\bar{j}}\\\
\operatorname{i}K_{i}&-K_{i}K_{j}&K_{i\bar{j}}+K_{i}K_{\bar{j}}\\\
-\operatorname{i}K_{\bar{i}}&K_{\bar{i}j}+K_{\bar{i}}K_{j}&-K_{\bar{i}}K_{\bar{j}}\end{array}\right).$

In the conventions used in [12, § 9.3] for the Heisenberg group
${\mathrm{H}}_{1}$, we make the

######  Remark 7.

The Sasaki potential for the Heisenberg group
$({\mathrm{H}}_{1},\xi,\eta,\Phi,g_{{\mathrm{H}}_{1}})$ has the expression

(5.37)
$K_{{\mathrm{H}}_{1}}(z,\bar{z})=\frac{1}{8}(z-\bar{z})^{2}=-\frac{y^{2}}{2},\quad\mathbb{C}\ni
z=x+\operatorname{i}y,~{}x,y\in\mathbb{R},$

corresponding to the left invariant metric

(5.38)
$g_{{\mathrm{H}}_{1}}=\operatorname{d}x^{2}+\operatorname{d}y^{2}+\eta^{2},\quad\eta=\operatorname{d}\kappa-y\operatorname{d}x,$

and

$\Phi=\left(\begin{array}[]{ccc}0&1&0\\\ -1&0&0\\\ 0&y&0\end{array}\right).$

The Kähler potential for the manifold
${\mathrm{H}}_{1}/\mathbb{R}\approx\mathbb{C}$ corresponding to the scalar
product of two coherent state vectors $e_{z}$ is

(5.39)
${\mathcal{K}}_{{\mathrm{H}}_{1}}(z,\bar{z}):=(e_{z},e_{z})=\exp(z\bar{z}).$

###### Proof.

We recall also [12, Remark 18, Propositions 25, 26].

We apply (5.36) for the Heisenberg group ${\mathrm{H}}_{1}$ with the left
invariant metric (5.38) [12, Section 9.3].

We apply formulas (5.34) with (5.34d) replaced with (5.35) to the Sasaki
potential (5.37).

Formula (5.39) is extracted from [12, (7.75)]. ∎

ACOK

In [30]: the notion of almost cosymplectic manifold is adopted as in ACOS, the
definition of cosymplectic manifold is adopted as in COS, the notion of almost
contact structure is introduced as in Definition 1.

An almost coKähler manifold \- is an almost contact metric manifold
$(M,\Phi,\xi,\eta,g)$ such that the two-form $\hat{\Phi}$ (5.29) and
$\eta\in{\mathcal{D}}_{1}$ are both closed. If in addition the almost contact
structure is normal, $M$ is called a coKähler manifold.

According to [30, Theorem 3.9]

######  Proposition 11.

The almost contact manifold $(M,\Phi,\xi,\eta,g)$ is coKähler if and only if
$\nabla^{g}\Phi=0$ or equivalently $\nabla^{g}\hat{\Phi}=0$.

The same definition of co-symplectic/co-Kähler manifold used by [30] is
adopted in the review paper [44].

Apparently the denomination of co-Kähler manifold was used firstly in [5].

In [26]: Blair used the term cosymplectic manifold denoting a cosymplectic
manifold in the sense of Libermann [45] endowed with a compatible almost
contact metric structure as in N=SAS satisfying the normality condition. i.e.
the coKähler manifold of [30].

### Acknowledgements

This research was conducted in the framework of the ANCS project program PN 19
06 01 01/2019. I am grateful to Mihai Visinescu for references on cosymplectic
structures and the remarks on the drafts of the paper. I would like to thank
Mrs. Popescu Elena from IMAR for facilitating my access to several references.

## References

  * [1] C. Albert, Le théoreme de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact, J. Geom. Phys. 6 (1989) 627–649
  * [2] V.I. Arnold, Les méthodes mathématiques de la mécanique classique, Traduit du russe par D. Embarek, Éditions Mir, Moscou, 1976
  * [3] E.M. Babalic, S. Berceanu, Remarks on the geometry of the extended Siegel–Jacobi upper half-plane, Romanian J. Phys. 65 (2020) 113, 27 pages; arXiv:2002.04452
  * [4] W. Ballmann, Lectures on Kähler Manifolds, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2006
  * [5] A. Bejancu, D. Smaranda, Semi-invariant submanifolds of a coKähler manifold, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 29 (1983), no. 2, suppl., 27–32
  * [6] S. Berceanu, Realization of coherent state algebras by differential operators, in Advances in Operator Algebras and Mathematical Physics, Eds. F. Boca, O. Bratteli, R. Longo and H. Siedentop, The Theta Foundation, Bucharest, 1–24, 2005, arXiv: math/0504053v1 [math.DG]
  * [7] S. Berceanu, A holomorphic representation of the Jacobi algebra, Rev. Math. Phys. 18 (2006) 163–199; Errata, Rev. Math. Phys. 24 (2012) 1292001, 2 pages; arXiv:math.DG/0408219
  * [8] S. Berceanu, A holomorphic representation of Jacobi algebra in several dimensions, in Perspectives in Operator Algebra and Mathematical Physics, Editors F.-P. Boca, R. Purice, S. Stratila, Theta Ser. Adv. Math., Vol. 8, Theta, Bucharest, 2008, 1–25, arXiv:math.DG/0604381
  * [9] S. Berceanu, Consequences of the fundamental conjecture for the motion on the Siegel–Jacobi disk, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250076, 18 pages; arXiv:1110.5469
  * [10] S. Berceanu, Geodesics associated to the balanced metric on the Siegel-Jacobi ball, Romanian J. Phys. 61 (2016) 1137–1160; arXiv:math.DG/1605.02962v1
  * [11] S. Berceanu, Balanced metric and Berezin quantization on the Siegel–Jacobi ball, SIGMA 12 (2016) 064, 24 pages; arXiv:1512.00601
  * [12] S. Berceanu, The real Jacobi group revisited, SIGMA 15 (2019) 096, 50 pages; arXiv:1903.1072 [math.DG], v1, 93 pages; v2, 54 pages
  * [13] S. Berceanu, Invariant metric on the extended Siegel-Jacobi upper half space, J. Geom. Phys. 162 (2021) 104049, 20 pages; arXiv:math.DG/2006.03319
  * [14] S. Berceanu, Geodesics on the extended Siegel-Jacobi upper half-plane, Romanian J. Phys. 66 (2021) 107, 28 pages; arXiv:2101.08015 [math.DG]
  * [15] S. Berceanu, A. Gheorghe, Linear Hamiltonians on homogeneous Kähler manifolds of coherent states, An. Univ. Timisoara Ser. Mat.-Inform. XXXIX (2001) 31–56, Special Issue: Mathematics; arXiv:math.DG/0408254
  * [16] S. Berceanu, A. Gheorghe, Differential operators on orbits of coherent states, Romanian J. Phys. 48 (2003), 545–556; arXiv: math.DG/0211054
  * [17] S. Berceanu, A. Gheorghe, Applications of the Jacobi group to Quantum Mechanics, Romanian J. Phys. 53 (2008) 1013–1021; arXiv:0812.0448
  * [18] S. Berceanu, A. Gheorghe, On the geometry of Siegel–Jacobi domains, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1783–1798; arXiv:1011.3317
  * [19] F.A. Berezin, Quantization, Math. USSR-Izv. 38 (1974) 1116–1175
  * [20] F.A. Berezin, Quantization in complex symmetric spaces, Math. USSR-Izv. 39 (1975) 363–402
  * [21] F.A. Berezin, The general concept of quantization, Commun. Math. Phys. 40 (1975) 153–174
  * [22] R. Berndt, Some differential operators in the theory of Jacobi forms, preprint IHES/M/84/10, 1984, 31 pages
  * [23] R. Berndt, Sur l’arithmétique du corps des fonctions elliptiques de niveau $N$, in Seminar on number theory, Paris 1982–83, Progr. Math., Vol. 51, Birkhäuser Boston, Boston, MA, 1984, 21–32
  * [24] R. Berndt, R. Schmidt, Elements of the representation theory of the Jacobi group, Progress in Mathematics, Vol. 163, Birkhäuser Verlag, Basel, 1998
  * [25] D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics 509, Spriger-Verlag Berrlin - Heidelberg - New York, 1976
  * [26] D.E. Blair, Riemannian geometry of contact and symplectic manifolds, second ed., Progr. Math. 203, Birkäuser Boston Inc., Boston, MA, 2010
  * [27] C.P. Boyer, The Sasakian Geometry of the Heisenberg Group, Bull. Math. Soc. Sci. Math. Roumanie 52 (100) (2009) 251–262
  * [28] C.P. Boyer, K. Galicki, Sasakian geometry, Oxford mathematical monographs, University Press, Oxford, 2008
  * [29] F. Cantrijn, M. de León, E.A. Lacomba, Gradient vector fields on cosymplectic manifolds, J. Phys. A: Math. Gen. 25 (1992) 175–188
  * [30] B. Cappelletti-Montano, A. De Nicola, I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys. 25 (2013) 1343002, 55 pages; arXiv:1305.3704 [math.DG]
  * [31] É. Cartan, La méthode du repére mobile, la théorie des groupes continus et les espaces généralisés, Actualités scientifiques et industrielles, Vol. 194, Hermann & Cie., Paris, 1935
  * [32] É. Cartan, Les espaces à connexion projective, Abh. Sem. Vektor \- Tensor analysis, Moskau 4 (1937) 147–173
  * [33] M. de León, M. Lainz, A review on contact Hamiltonian and Lagrangian systems, arXiv:2011.05579 [math-ph]
  * [34] M. de León, M. L. Valcźar, Singular Lagrangians and precontact Hamiltonian Systems, Int. J. Geom. Methods Mod. Phys. 16 (2019) 1950158, 39 pages; arXiv:1904.11429 [math-ph]
  * [35] M. Eichler, D. Zagier, The theory of Jacobi forms, Progress in Mathematics, Vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985
  * [36] E.L. Evtushik (originator) Moving-frame method, in Encyclopedia of Mathematics.
http://www.encyclopediaofmath.org/index.php?title=Moving-frame-
method&oldid=17828

  * [37] M. Godliński, W. Kopczyński, P. Nurowski, Locally Sasakian manifolds, Class. Quantum Grav. 17 (2000) L105–L115
  * [38] W. Gray, Some global properties of contact structures, Ann. of Math. 69 (1959) 421–450
  * [39] R. Greene, Complex differential geometry, in Differential geometry, Lecture Notes in Mathematics, Vol. 1263, Editor V. L. Hansen, Springer – Verlag, Berlin Heidelberg, 1987, 228–288
  * [40] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York – London, 1978
  * [41] E. Kähler, Raum-Zeit-Individuum, Rend. Accad. Naz. Sci. XL Mem. Mat. 16 (1992) 115–177
  * [42] Erich Kähler: Mathematische Werke/Mathematical Works, Editors R. Berndt, O. Riemenschneider, Walter de Gruyter & Co., Berlin, 2003
  * [43] A.A. Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976) 55–75
  * [44] H. Li, Topology of co-symplectic/co-Kähler manifolds; Asian J. Math. (2008) 527–544
  * [45] P. Libermann, Sur les automorphismes infinitésimaux des structures symplecliques et des structures de contact, in Colloque Géom. Diff. Globale, Bruxelles 1958, Gauthier Villars, Paris, 1959, 37–59
  * [46] P. Libermann, C.-H. Marle, Symplectic Geometry and Analytical Mechanics, Translated from the French by B.E. Schwarzbach, Mathematics and its Applications 35, D. Reidel Publishing Co., Dordrecht, 1987
  * [47] A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures et App., 57 (1978) 453–488
  * [48] A.M. Perelomov, Generalized Coherent States and their Applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986
  * [49] S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure, I, Tohoku Math. J. 12 (1960) 459–476
  * [50] S. Sasaki, Almost contact manifolds, Lecture Notes Part 1, Math. Inst. Tôhoku Univ., 1965
  * [51] S. Sasaki, Y. Hatakeyama, On differentiable manifolds with contact metric structures, J. Math. Soc. Japan 14 (1962) 249 – 271
  * [52] I. Vaisman, Jacobi manifolds, in Selected topics in Geometry and Mathematical Physics, Vol. 1, Universita degli Studi della Basilicata 1959, 81-100
  * [53] M. Visinescu, Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces $T^{1,1}$ and $Y^{p,q}$, Symmetry, 13 (2021) 591, 10 pages, https://doi.org/10.3390/sym13040591
  * [54] J.-H. Yang, The method of orbits for real Lie groups, Kyungpook Math. J. 42 (2002) 199–272; arXiv:math.RT/060205
  * [55] J.-H. Yang, Invariant metrics and Laplacians on the Siegel–Jacobi spaces, J. Number Theory 127 (2007) 83–102; arXiv:math.NT/0507215
  * [56] J.-H. Yang, A partial Cayley transform for Siegel–Jacobi disk, J. Korean Math. Soc. 45 (2008) 781–794; arXiv:math.NT/0507216
  * [57] J.-H. Yang, Invariant metrics and Laplacians on the Siegel–Jacobi disk, Chin. Ann. Math. 31B (2010) 85–100; arXiv:math.NT/0507217