#  A review on coisotropic reduction in Symplectic, Cosympletic, Contact and
Co-contact Hamiltonian systems

Rubén Izquierdo-López, Manuel de León

###### Abstract

In this paper we study coisotropic reduction in different types of dynamics
according to the geometry of the corresponding phase space. The relevance of
coisotropic reduction is motivated by the fact that these dynamics can always
be interpreted as Lagrangian or Legendrian submanifolds. Furthermore,
Lagrangian or Legendrian submanifolds can be reduced by a coisotropic one.

MSC2020 classification: 37J39, 37J55, 70H05, 70H33.

Key words: Coisotropic reduction; symplectic manifolds; cosymplectic
manifolds; contact manifolds; cocontact manifolds; Hamiltonian dynamics.

A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-
contact Hamiltonian systems

Manuel de León111E-mail<EMAIL_ADDRESS>
Instituto de Ciencias Matemáticas, Campus Cantoblanco

Consejo Superior de Investigaciones Científicas

C/ Nicolás Cabrera, 13–15, 28049, Madrid, Spain

and

Real Academia Española de las Ciencias.

C/ Valverde, 22, 28004 Madrid, Spain.

Rubén Izquierdo-López222E-mail<EMAIL_ADDRESS>
Instituto de Ciencias Matemáticas, Campus Cantoblanco

Consejo Superior de Investigaciones Científicas

C/ Nicolás Cabrera, 13–15, 28049, Madrid, Spain

###### Contents

  1. 1 Introduction
  2. 2 Symplectic vector spaces
  3. 3 Coisotropic reduction in symplectic geometry
    1. 3.1 Hamiltonian vector fields as Lagrangian submanifolds
    2. 3.2 Coisotropic reduction
    3. 3.3 Projection of Lagrangian submanifolds
    4. 3.4 An example
  4. 4 Poisson structures
  5. 5 Coisotropic reduction in cosymplectic geometry
    1. 5.1 Hamiltonian vector fields as Lagrangian submanifolds
    2. 5.2 Coisotropic reduction
    3. 5.3 Vertical coisotropic reduction
      1. 5.3.1 Projection of Lagrangian submanifolds
    4. 5.4 Horizontal coisotropic reduction
      1. 5.4.1 Projection of Lagrangian submanifolds
  6. 6 Jacobi structures
  7. 7 Coisotropic reduction in contact geometry
    1. 7.1 Hamiltonian vector fields as Lagrangian submanifolds
    2. 7.2 An overview of the use of contact geometry in thermodynamics
    3. 7.3 Coisotropic reduction
    4. 7.4 Vertical coisotropic reduction
      1. 7.4.1 Projection of Legendrian submanifolds
    5. 7.5 Horizontal coisotropic reduction
  8. 8 An interlude: Different geometries provide different dynamics
    1. 8.1 The Lagrangian picture
    2. 8.2 The Herglotz principle
    3. 8.3 The Legendre transformation and the Hamiltonian picture
  9. 9 Coisotropic reduction in cocontact geometry
    1. 9.1 Hamiltonian vector fields as Lagrangian submanifolds
    2. 9.2 Coisotropic reduction
    3. 9.3 $tz-$vertical reduction
      1. 9.3.1 Projection of Legendrian submanifolds
    4. 9.4 $t-$vertical, $z-$horizontal reduction
    5. 9.5 $z-$vertical, $t-$horizontal reduction
      1. 9.5.1 Projection of Legendrian submanifolds
    6. 9.6 $tz-$horizontal reduction
  10. 10  Coisotropic reduction in stable Hamiltonian structures
    1. 10.1 Hamiltonian vector fields as Lagrangian submanifolds
    2. 10.2 Vertical coisotropic reduction
      1. 10.2.1 Projection of Lagrangian submanifolds
    3. 10.3 Horizontal coisotropic reduction
      1. 10.3.1 Projection of Lagrangian submanifolds
  11. 11 Conclusions
  12. 12 Acknowledgements

##  1\. Introduction

The introduction of symplectic geometry in the study of Hamiltonian systems
was a tremendous breakthrough, both in quantitative and qualitative aspects.
For example, we have the results in the reduction of the original Hamiltonian
system when in the presence of symmetries, or the so-called coisotropic
reduction [abraham2008foundations, arnold, libermann2012symplectic,
de2011methods]. Another relevant example, in the quantitative aspects, is the
development of geometric integrators that respect geometric aspects and prove
to be more efficient than the traditional ones (see for instance [sanzserna,
marsdenbook]). It has also had a major influence on the study of completely
integrable systems and Hamilton-Jacobi theory [arnold, abraham2008foundations,
catalanes, vaquero]. In addition, the so-called geometric quantization relies
on symplectic geometry [kostant, SouriauDynamical].

Regarding the reduction in the presence of symmetries, the most relevant
result is the so-called Marsden-Weinstein symplectic reduction theorem
[marsden1974reduction] (a preliminary version can be found in Meyer [meyer])
using the momentum mapping, a natural extension of the classical linear and
angular momentum. The reduced manifold is obtained using a regular value of
the momentum mapping and the corresponding isotropy group, and the dynamics is
projected to this reduced manifold, gaining for integration a smaller number
of degrees of freedom. This theorem has been extended to many other contexts:
cosymplectic, contact, and more general settings (see [acakpo2022stable,
albert1989theoreme, de1993cosymplectic, de2019contact, garcia2022momentum,
Lainztesis, marsden1990reduction, willett] and the references therein). For a
recent review on reduction by symmetries in cosymplectic geometry we refer to
[delucas2023cosymplectic]. When the reduced space is not a manifold, we can
reduce the algebra of observables [Sniatycki1983ReductionAQ] (see also
[beppe]). This reduction recovers the Poisson algebra of the reduced space in
Marsden-Weinstein reduction.

Related to the geometric reduction is Noether’s theorem (in fact, this
reduction is a generalisation of it), which states that a symmetry of the
system produces a conserved quantity [binz]. The introduction of geometric
structures has revealed itself in a plethora of results relating symmetries
and conserved quantities [de2011methods].

Furthermore, Lagrangian submanifolds play a crucial role, since it is easy to
check that the image of a Hamiltonian vector field $X_{H}$ in a symplectic
manifold $(M,\omega)$ can be interpreted as a Lagrangian submanifold of the
symplectic manifold $(TM,\omega^{c})$, where $\omega^{c}$ is the complete or
tangent lift of $\omega$ to the tangent bundle $TM$. This result has its
equivalent in Lagrangian mechanics, and has led to the so-called Tulczyjew
triples, which elegantly relate the different Lagrangian submanifolds that
appear in Lagrangian and Hamiltonian descriptions of mechanics via the
Legendre transformation [tulczyjew1976hamiltonienne,
tulczyjew1976lagrangienne, de2011methods]. This interpretation of the dynamics
as a Lagrangian submanifold has been extended to other scenarios, including
the Tulczyjew triple [deleon2003tulczyjews, jcelisa, ogul, ogul2, grabowska,
ogul2021, silvia, willett]. Lagrangian submanifolds are also relevant to
develop the so-called Hamilton–Jacobi theory since they provide the geometric
setting for solutions of the Hamilton–Jacobi problem (see [ogulin] for a
recent topical review on the subject). In this sense, we follow Weinstein’s
creed: "Everything is a Lagrangian submanifold" [weinsteincreed].

Moreover, coisotropic submanifolds play a relevant role both in the theory of
constraints and in the theory of quantization. For instance, coisotropic
submanifolds are precisely the first class constraints considered by Dirac in
[Dirac_1950] (see also [BOJOWALD_2003]), where he developed the constraint
algorithm for singular Lagrangians in the Hamiltonian setting. The constraint
algorithm has been developed in geometrical terms in
[GotaySingularLagrangians1, GotaySinguarLagrangians2]. In this approach, the
phase space of a singular system is a presymplectic manifold and, in [got],
Gotay showed that every presymplectic manifold $(P,\Omega)$ may be imbedded in
a symplectic manifold $(M,\omega)$ as a closed coisotropic submanifold. More
precisely, there exists an imbedding $j:P\rightarrow M$ such that

  * •

$j(P)$ is closed in $M$;

  * •

$j^{\ast}\omega=\Omega$;

  * •

$TP^{\perp}\subseteq j_{\ast}(TP)$.

An alternative approach to the usual treatments of singular Lagrangians based
on a Hamiltonian regularization scheme inspired on the coisotropic embedding
of presymplectic systems was developed in [regular].

The ideas to develop the coisotropic reduction procedure came from Weinstein
[weinstein1977lectures] and were also partially inspired by Roels and
Weisntein [roels] and Marsden and Weinstein [marsden1974reduction].

Coisotropic reduction works when we give a coisotropic submanifold $N$ of a
symplectic manifold $(M,\omega)$ and we consider (if it is well defined) the
quotient manifold $N/(TN)^{\perp}$, where $(TN)^{\perp}$ is the symplectic
complement of $TN$. Being involutive, this distribution along $N$ defines a
foliation. The corresponding leaf space inherits a reduced symplectic form
from the symplectic structure given on $M$. If in addition we have a
Lagrangian submanifold $L$ with clean intersection with $N$, then $L\cap N$
projects into a Lagrangian submanifold of the quotient (see [weinsteincreed,
abraham2008foundations]). Coisotropic reduction can be also combined with
symplectic reduction to develop a reduction procedure for the Hamilton-Jacobi
equation in presence of symmetries (see [vaquero]).

Coisotropic reduction has been extended to the field of contact manifolds
(with the interest of being in a dissipative context) [de2019contact,
Tortorella2017], but it has not been studied in sufficient detail in the case
of cosymplectic manifolds nor in that of co-contact manifolds, the latter the
natural settings to study time-dependent Hamiltonian contact systems
[de2022time, de2022time2].

The objectives of this paper are twofold. On the one hand, to develop in
detail the coisotropic reduction in the case of cosymplectic manifolds and
those of co-contact, covering a gap in the literature. On the other hand, to
present a survey that brings together in one place the different cases that
appear in the study of Hamiltonian systems in classical mechanics.

The paper is structured as follows. Sections 2 and 3 are devoted to the main
ingredients concerning symplectic Hamiltonian systems and the classical
coisotropic reduction procedure. In order to go to the cosymplectic setting,
we recall some general notions in Poisson structures (Section 4) and then we
consider the case of coisotropic reduction in the cosymplectic setting in
Section 5 (remember that this is the scenario to develop time dependent
Hamiltonian systems). Contact manifolds require a more general notion than
Poisson structures; indeed, they are examples of Jacobi structures, so that we
give some fundamental notions in Section 6. The coisotropic reduction scheme
developed in contact manifolds is the subject of Section 7, which is very
different to the cosymplectic case since we are in presence of dissipative
systems. We emphasize these differences in Section 8, where we study the
corresponding Lagrangian settings. To combine dissipative systems with
Hamiltonians depending also on time, we consider cocontact manifolds in
Section 9, and develop there the corresponding coisotropic reduction
procedure. Finally, we discuss a recent generalization of contact and
cosymplectic systems called stable Hamiltonian systems in Section 10.

##  2\. Symplectic vector spaces

We refer to [abraham2008foundations, arnold, de2011methods,
godbillon1969geometrie, libermann2012symplectic, vaisman2012lectures] for the
main definitions and results.

######  Definition 2.1 (Symplectic vector space).

A symplectic vector space is a pair $(V,\omega)$ where $V$ is a finite
dimensional vector space and $\omega$ is a non-degenerate $2$-form, called the
symplectic form. Here, non-degeneracy means that the map

$\flat_{\omega}:V\rightarrow V^{*};\,\,v\mapsto i_{v}\omega$

is an isomorphism.

For every non-degenerate $2$-form on $V$ there exist a basis $(x_{i},y^{i})$
whith $i$ taking values from $1$ to $n$ such that, making use of the summation
convention, $\omega=x^{i}\wedge y_{i}$, where $(x^{i},y_{i})$ is the dual
basis. This implies that a symplectic vector space is necessarily of even
dimension $2n$.

######  Definition 2.2 ($\omega$-orthogonal).

Let $W\subseteq V$ be a subspace of $V$. We define its $\omega$-orthogonal
complement as

$W^{\perp_{\omega}}:=\\{v\in V\,\,|\,\,\omega(v,w)=0,\,\,\forall w\in W\\}.$

Note that $W^{\perp_{\omega}}=\operatorname{Ker}(i^{*}\flat_{\omega})$ where
$i:W\hookrightarrow V$ is the natural inclusion. Using the non-degeneracy of
$\omega$, this implies that
$\operatorname{dim}W^{\perp_{\omega}}=\operatorname{dim}V-\operatorname{dim}W$,
a result which will be useful throughout this paper.

The antisymmetry of $\omega$ gives rise to a wide variety of situations. In
particular, we say that $W\subseteq V$ is:

  * $i)$

Isotropic if $W\subseteq W^{\perp_{\omega}}$ (if $W$ is isotropic, necessarily
$\operatorname{dim}W\leq n$);

  * $ii)$

Coisotropic if $W^{\perp_{\omega}}\subseteq W$ (if $W$ is coisotropic,
necessarily $\operatorname{dim}W\geq n$);

  * $iii)$

Lagrangian if $W$ is isotropic and has an isotropic complement (if $W$ is
Lagrangian, necessarily $\operatorname{dim}W=n$);

  * $iv)$

Symplectic if $V=W\oplus W^{\perp_{\omega}}$.

A subspace $W$ is Lagrangian if and only if $W=W^{\perp_{\omega}}$. This
implies that Lagrangian subspaces are the isotropic subspaces of maximal
dimension and the coisotropic subspaces of minimal dimension.

It can be easily checked that the symplectic complement has the following
properties:

  * $i)$

$(W_{1}\cap
W_{2})^{\perp_{\omega}}=W_{1}^{\perp_{\omega}}+W_{2}^{\perp_{\omega}};$

  * $ii)$

$(W_{1}+W_{2})^{\perp_{\omega}}=W_{1}^{\perp_{\omega}}\cap
W_{2}^{\perp_{\omega}};$

  * $iii)$

$(W^{\perp_{\omega}})^{\perp_{\omega}}=W.$

##  3\. Coisotropic reduction in symplectic geometry

######  Definition 3.1 (Symplectic manifold).

A symplectic manifold is pair $(M,\omega)$ where $M$ is a manifold and
$\omega$ is a closed $2$-form such that $(T_{q}M,\omega_{q})$ is a symplectic
vector space, for every $q\in M$. As in the linear case, for the existence of
such a form, $M$ needs to have even dimension $2n$.

Every symplectic manifold is locally isomorphic, that is, there exists a set
of canonical coordinates around each point:

######  Theorem 3.1 (Darboux Theorem).

Let $(M,\omega)$ be a symplectic manifold and $q\in M$. There exist a
coordinate system $(q^{i},p_{i})$ around $q$ such that $\omega=dq^{i}\wedge
dp_{i}$. These coordinates are called Darboux coordinates.

This non-degenerate form induces a bundle isomorphism between the tangent and
cotangent bundles of $M$ point-wise, namely

$\flat_{\omega}:TM\rightarrow
T^{*}M;\,\,\,v_{q}\mapsto\flat_{\omega}(v_{q})=i_{v_{q}}\omega.$

######  Definition 3.2 (Hamiltonian vector field).

Given $H\in\mathcal{C}^{\infty}(M)$, we define the Hamiltonian vector field of
$H$ as

$X_{H}:=\sharp_{\omega}(dH),$

where $\sharp_{\omega}=\flat_{\omega}^{-1}.$ We say that a vector field $X$ is
Hamiltonian if $X=X_{H}$ for some function $H$ and say that $X$ is locally
Hamiltonian if $X=X_{H}$ for some local function defined in a neighborhood of
every point of the manifold.

######  Remark 3.1.

Notice that a vector field is locally Hamiltonian if and only if
$\flat_{\omega}(X)$ is closed, and Hamiltonian if and only if
$\flat_{\omega}(X)$ is exact.

Locally, Hamiltonian vector fields have the expression

$X_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{\partial}{\partial p_{i}}.$

Then, the integral curves of the Hamiltonian vector field
$X_{H},(q^{i}(t),p_{i}(t))$, satisfy the local differential equations

$\displaystyle\frac{dq^{i}}{dt}$ $\displaystyle=\frac{\partial H}{\partial
p_{i}},$ $\displaystyle\frac{dp_{i}}{dt}$ $\displaystyle=-\frac{\partial
H}{\partial q^{i}},$

which are the Hamilton’s equations of motion.

The definitions of the different cases of subspaces given in the linear case
can be extended point-wise to submanifolds $N\hookrightarrow M$. Consequently,
we say that $N\hookrightarrow M$ is:

  * i)

Isotropic if $T_{q}N\subseteq T_{q}M$ is for every $q\in N$;

  * ii)

Coisotropic if $T_{q}N\subseteq T_{q}M$ is for every $q\in N$;

  * iii)

Lagrangian if $N$ is isotropic and there is a isotropic subbundle (where we
understand isotropic point-wise) $E\subseteq TM|N$ such that $TM=TN\oplus E$
(here $\oplus$ denotes the Whitney sum). This is exactly the point-wise
definition of a Lagrangian subspace asking for the coisotropic complement to
vary smoothly;

  * iv)

Symplectic if $T_{q}N\subseteq T_{q}M$ is for every $q\in N$.

These definitions extend naturally to distributions.

Just like in the linear case, a submanifold $N\hookrightarrow M$ is Lagrangian
if and only if it is isotropic (or coisotropic) and has maximal (or minimal)
dimension. This is a useful characterization that we will use several times in
the rest of the paper.

######  Lemma 3.1.

Let $i:L\rightarrow M$ be a submanifold of dimension $n$. Then, $L$ is a
Lagrangian submanifold of $(M,\omega)$ if and only if $i^{*}\omega=0$.

###### Proof.

It is trivial, since Lagrangian submanifolds are the isotropic submanifolds of
maximal dimension, say $n$. ∎

###  3.1. Hamiltonian vector fields as Lagrangian submanifolds

######  Definition 3.3.

Let $(M,\omega)$ be a symplectic manifold. Define the tangent symplectic
structure on $TM$ as $\omega_{0}=-d\lambda_{0}$ where
$\lambda_{0}=\flat_{\omega}^{*}\lambda_{M}$, and $\lambda_{M}$ is the
Liouville $1$-form on the cotangent bundle.

Recall that $\lambda_{M}$ is defined as follows:

$\lambda_{M}(\alpha_{x})(X_{\alpha_{x}})=\alpha_{x}(d_{\alpha_{x}}\pi_{M}\cdot
X_{\alpha_{x}})$

where $X_{\alpha_{z}}\in T_{\alpha_{x}}(T^{*}M)$, $\alpha_{x}\in T^{*}_{x}M$,
and $\pi_{M}:T^{*}M\longrightarrow M$ is the canonical projection. The
Liouville $1$-form can be also defined as the unique $1$-form $\lambda_{M}$ on
$T^{*}M$ such that, for every $1$-form $\alpha:M\rightarrow T^{*}M$,

$\alpha^{*}(\lambda_{M})=\alpha.$

In coordinates $(q^{i},p_{i},\dot{q}^{i},\dot{p}_{i})$ the tangent symplectic
structure is

$\omega_{0}=-dq^{i}\wedge d\dot{p}_{i}-d\dot{q}^{i}\wedge dp_{i}.$

######  Proposition 3.1.

Let $X:M\rightarrow TM$ be a vector field. Then

$X^{*}(\lambda_{0})=\flat_{\omega}(X).$

###### Proof.

This is a straight-forward verification. Let $v\in T_{q}M$, then we have

$\displaystyle\langle X^{*}\lambda_{0},v\rangle$
$\displaystyle=\langle\lambda_{0},d_{q}X\cdot
v\rangle=\langle\lambda_{M},d_{X(q)}\flat_{\omega}\cdot d_{q}X\cdot v\rangle$
$\displaystyle=\langle\flat_{\omega}(X)^{*}(\lambda_{M}),v\rangle=\langle\flat_{\omega}(X),v\rangle.$

∎

######  Proposition 3.2.

Let $X:M\rightarrow TM$ be a vector field. Then $X$ is locally Hamiltonian if
and only if $X(M)$ is a Lagrangian submanifold of $(TM,\omega_{0})$.

###### Proof.

We only check that $X(M)$ is isotropic using Lemma 3.1, since
$\operatorname{dim}X(M)=\operatorname{dim}M=\frac{1}{2}\operatorname{dim}TM.$
In fact:

$\displaystyle
X^{*}\omega_{0}=-X^{*}(d\theta_{0})=-d(X^{*}\theta_{0})=-d(\flat_{\omega}(X)),$

which gives the characterization. ∎

We can also check this last proposition easily in coordinates. Indeed, let

$X=X^{i}\frac{\partial}{\partial q^{i}}+Y_{i}\frac{\partial}{\partial p_{i}}.$

We have

$-X^{*}\omega_{0}=\frac{\partial Y_{i}}{\partial q^{j}}dq^{i}\wedge
dq^{j}+\left(\frac{\partial Y_{i}}{\partial p_{j}}+\frac{\partial
X^{j}}{\partial q_{i}}\right)dq^{i}\wedge dp_{j}+\frac{\partial
X^{i}}{\partial p_{j}}dp_{j}\wedge dp_{i},$

and thus, $X$ defines a Lagrangian submanifold if and only if

$\displaystyle\frac{\partial Y_{i}}{\partial q^{j}}-\frac{\partial
Y_{j}}{\partial q^{i}}$ $\displaystyle=0,$ $\displaystyle\frac{\partial
Y_{i}}{\partial p_{j}}+\frac{\partial X^{j}}{\partial q_{i}}$
$\displaystyle=0,$ $\displaystyle\frac{\partial X^{i}}{\partial
p_{j}}-\frac{\partial X^{j}}{\partial p_{i}}$ $\displaystyle=0.$

Taking

$(G^{1},\dots,G^{n},G^{n+1},\dots,G^{2n}):=(X^{i},-Y_{i})$

and

$(x^{1},\dots,x^{n},x^{n+1},\dots,x^{2n}):=(q^{i},p_{i}),$

these conditions become

$\frac{\partial G^{i}}{\partial x^{j}}=\frac{\partial G^{j}}{\partial x^{i}}.$

This implies $G^{i}=\displaystyle\frac{\partial H}{\partial x^{i}},$ for some
local function $H$. It is clear that locally, we have $X=X_{H}$.

###  3.2. Coisotropic reduction

Now, given a coisotropic submanifold $N\hookrightarrow M$, we define the
distribution $(TN)^{\perp_{\omega}}$ on $N$ as the subbundle of $TM|_{N}$
consisting of all $\omega$-orthogonal spaces $(T_{q}N)^{\perp_{\omega}}$. Note
that this distribution is regular and its rank is
$\operatorname{dim}M-\operatorname{dim}N$.

######  Proposition 3.3.

Let $(M,\omega)$ be a symplectic manifold and $N\hookrightarrow M$ be a
coisotropic submanifold. The distribution $q\mapsto(T_{q}N)^{\perp_{\omega}}$
is involutive.

###### Proof.

Let $X,Y$ be vector fields along $N$ with values in $TN^{\perp_{\omega}}$ and
$Z$ be any other vector field tangent to $N$. Since $\omega$ is closed we have

$\displaystyle 0=$
$\displaystyle(d\omega)(X,Y,Z)=X(\omega(Y,Z))-Y(\omega(X,Z))+Z\omega(X,Y)$
$\displaystyle-\omega([X,Y],Z)+\omega([X,Z],Y)-\omega([Y,Z],X)=-\omega([X,Y],Z),$

since $X,Y$ belong to the orthogonal complement of $TN$. We conclude that
$\omega([X,Y],Z)=0$ for every field $Z$ tangent to $N$, that is,
$[X,Y]\in(TN)^{\perp_{\omega}}$. ∎

Since the distribution is involutive and regular, the Frobenius Theorem
guarantees the existence of a maximal regular foliation $\mathcal{F}$ of $N$,
that is, a decomposition of $N$ into maximal submanifolds tangent to the
distribution. In what follows, we suppose that $N/\mathcal{F}$ (the space of
all leaves) admits a manifold structure so that the projection

$\pi:N\rightarrow N/\mathcal{F}$

is a submersion. The main result is the Weinstein reduction theorem
[weinstein1977lectures]:

######  Theorem 3.2 (Coisotropic reduction in the symplectic setting).

Let $(M,\omega)$ be a symplectic manifold and $N\hookrightarrow M$ be a
coisotropic submanifold. If $N/\mathcal{F}$ (the spaces of all leaves under
the distribution $q\mapsto(T_{q}N)^{\perp_{\omega}}$) admits a manifold
structure such that $N\xrightarrow{\pi}N/\mathcal{F}$ is a submersion, there
exist an unique $2$-form $\omega_{N}$ on $N/\mathcal{F}$ that defines a
symplectic manifold structure such that, if $N\xrightarrow{i}M$ is the natural
inclusion, then $i^{*}\omega=\pi^{*}\omega_{N}$. The following diagram
summarizes the situation:

${N}$${M}$${N/\mathcal{F}}$$\scriptstyle{i}$$\scriptstyle{\pi}$

###### Proof.

Uniqueness is guaranteed from the imposed relation since it forces us to
define

$(\omega_{N})_{[q]}([u],[v]):=\omega(u,v),$

where $[u]:=T\pi(q)\cdot u$. We only need to check that this is a well-defined
closed form and that it is non-degenerate.

We begin showing that our definition does not depend on the representative of
the vector $[u]$. For this, it is sufficient to observe that
$(\omega_{N})_{[q]}([u],[v])=0$ whenever $u$ is a vector in the distribution.

Furthermore,

$\mathcal{L}_{X}\omega=di_{X}\omega+i_{X}d\omega=0$

for every vector field $X$ in $N$ with values in $(TN^{\perp_{\omega}})$, and
this implies the independece of the point (for every two points in the same
leaf of the foliation can be joined by a finite union of flows of such
fields).

It is clearly non-degenerate and it is closed, since
$d\pi^{*}\omega_{N}=i_{*}d\omega=0$ and $\pi$ is a submersion. ∎

###  3.3. Projection of Lagrangian submanifolds

######  Definition 3.4 (Clean intersection).

We say that two submanifolds $L,N\hookrightarrow M$ have clean intersection if
$L\cap N\hookrightarrow M$ is again a submanifold and $T_{q}(L\cap
N)=T_{q}L\cap T_{q}N$, for every $q\in L\cap N$.

######  Proposition 3.4.

Let $L\hookrightarrow M$ be a Lagrangian submanifold and $N\hookrightarrow M$
a coisotropic submanifold. If they have clean intersection and
$L_{N}:=\pi(L\cap N)$ is a submanifold of $N/\mathcal{F}$, $L_{N}$ is
Lagrangian.

###### Proof.

It is sufficient to see that is isotropic and that it has maximal dimension in
$N/\mathcal{F}$. It is isotropic since $[u]\in T_{q}(L_{N})$ implies
$\omega_{N}([u],[v])=\omega(u,v)=0$, for every $[v]\in T_{q}(L_{N})$. Now,
since $\operatorname{Ker}d_{q}\pi=(T_{q}N)^{\perp_{\omega}}$, the kernel-range
formula yields

$\operatorname{dim}L_{N}=\operatorname{dim}(L\cap
N)-\operatorname{dim}(T_{q}L\cap(T_{q}N)^{\perp_{\omega}}).$ (1)

Furthermore,

$\operatorname{dim}(L\cap
N)+\operatorname{dim}(T_{q}L+(T_{q}N)^{\perp_{\omega}})=\operatorname{dim}M,$
(2)

beacause $L$ is Lagrangian and $N$ coisotropic. Substituting (2)in (1) we
obtain

$\displaystyle\operatorname{dim}L_{N}$
$\displaystyle=\operatorname{dim}M-\operatorname{dim}(T_{q}L+(T_{q}N)^{\perp_{\omega}})-\operatorname{dim}(T_{q}L\cap(T_{q}N)^{\perp_{\omega}})$
$\displaystyle=\operatorname{dim}M-\operatorname{dim}L-\operatorname{dim}(T_{q}N)^{\perp_{\omega}}=\operatorname{dim}M-\operatorname{dim}L-(\operatorname{dim}M-\operatorname{dim}N)$
$\displaystyle=\operatorname{dim}N-\operatorname{dim}L=\operatorname{dim}N-\frac{1}{2}\operatorname{dim}M,$

which is exactly $\frac{1}{2}\operatorname{dim}N/\mathcal{F}$, as a direct
calculation shows. ∎

###  3.4. An example

As an example of coisotropic reduction, let us take

$M:=\mathbb{R}^{2(n+1)},$

and

$N:=\mathbb{S}^{2n+1}.$

Since it has codimension $1$, it defines a coisotropic submanifold if we endow
$M$ with the natural symplectic form $\omega:=dq^{i}\wedge dp_{i}.$ It is easy
to check that $(TN)^{\perp_{\omega}}$ is generated by

$\sum_{i}\left(p_{i}\frac{\partial}{\partial
q^{i}}-q^{i}\frac{\partial}{\partial p_{i}}\right),$

where $(q^{i},p_{i})$ are canonical coordinates in $\mathbb{R}^{2(n+1)}$.
Therefore, the leaves of the corresponding foliation are precisely the orbits
of the previous vector field in $\mathbb{S}^{2n+1}$. Making the identification

$\mathbb{R}^{2(n+1)}=\mathbb{C}^{n+1},$

two points $x,y\in\mathbb{S}^{2n+1}$ are in the same orbit if and only if
there exist some $\alpha\in\mathbb{C}$ ($|\alpha|=1$) such that

$\alpha x=y.$

This implies that

$\mathbb{S}^{2n+1}/\mathcal{F}$

is the complex projective space of complex dimension $n$,
$\mathbb{P}^{n}\mathbb{C}$. Therefore, we conclude that through coisotropic
reduction we can define a natural symplectic structure on
$\mathbb{P}^{n}\mathbb{C},$ for every $n$.

The above example is taken from Exercise 5.3B in [abraham2008foundations].

##  4\. Poisson structures

A symplectic structure $(M,\omega)$ induces a Lie algebra structure on the
ring of functions $\mathcal{C}^{\infty}(M)$.

######  Definition 4.1 (Poisson bracket).

Let $(M,\omega)$ be a symplectic manifold and $f,g\in\mathcal{C}^{\infty}(M)$.
We define the Poisson bracket of $f,g$ as the function

$\\{f,g\\}:=\omega(X_{f},X_{g}).$

It is easily checked that in Darboux coordinates the Poisson bracket is

$\\{f,g\\}=\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q^{i}}.$

######  Proposition 4.1.

The Poisson bracket satifies the following properties:

  * i)

It is bilinear with respect to $\mathbb{R}$;

  * ii)

$\\{f,g\cdot h\\}=g\cdot\\{f,h\\}+\\{f,g\\}\cdot h$ (the Leibniz rule);

  * iii)

$\\{f,\\{g,h\\}\\}+\\{h,\\{f,g\\}\\}+\\{g,\\{h,f\\}\\}=0$ (the Jacobi
identity).

Taking into consideration the previous definition, we can generalize the
notion of symplectic manifolds as follows:

######  Definition 4.2 (Poisson manifold).

A Poisson manifold is a pair $(P,\\{\cdot,\cdot\\})$ where $P$ is a manifold
and $\\{\cdot,\cdot\\}$ is an antisymmetric bracket in the ring of functions
$\mathcal{C}^{\infty}(P)$ satisfying the Leibniz rule and the Jacobi identity.

######  Definition 4.3 (Hamiltonian vector field, Characteristic
distribution).

Given $H\in\mathcal{C}^{\infty}(M),$ the Leibniz rule implies that
$\\{H,\cdot\\}$ defines a derivation on $C^{\infty}(M)$ an thus is associated
to a unique vector field $X_{H}$, which will be called the Hamiltonian vector
field of $H$. The collection of all Hamiltonian vector fields generates the
characteristic distribution, namely

$\mathcal{S}_{q}:=\langle v=X_{H}(q)\,\,\text{for
all}\,\,H\in\mathcal{C}^{\infty}(M)\rangle.$

The definition of Hamiltonian vector field implies that $\\{f,g\\}$ only
depends on the values of $df,dg$ and thus we can define a bivector field

$\Lambda(\alpha,\beta):=\\{f,g\\}$

where $df=\alpha,dg=\beta.$ We have $\\{f,g\\}=\Lambda(df,dg).$ $\Lambda$ also
satisfies the partial differential equation $[\Lambda,\Lambda]=0$, where
$[\cdot,\cdot]$ is the Schouten-Nijenhuis bracket [vaisman2012lectures]. This
last property is actually equivalent to the Jacobi identity, that is, given a
bivector field $\Lambda$, $\\{f,g\\}:=\Lambda(df,dg)$ defines a Poisson
structure if and only if $[\Lambda,\Lambda]=0.$

######  Definition 4.4.

Let $(P,\\{\cdot,\cdot\\})$ be a Poisson manifold. Then we define

$\sharp_{\Lambda}:T^{*}P\rightarrow TP;\,\,\alpha_{q}\mapsto
i_{\alpha_{q}}\Lambda.$

Notice that $\operatorname{Im}\sharp_{\Lambda}=\mathcal{S}$, the
characteristic distribution.

######  Remark 4.1.

In the case of symplectic manifolds $\sharp_{\omega}=\sharp_{\Lambda}$, and
the characteristic distribution is the whole tangent bundle; however, in the
general setting $\sharp_{\Lambda}$ need not be a bundle isomorphism. Actually,
if $\sharp_{\Lambda}$ is a bundle isomorphism, it arises form a symplectic
structure defined as
$\omega(v,w):=\Lambda(\sharp_{\Lambda}^{-1}(v),\sharp_{\Lambda}^{-1}(w))$
[de2011methods].

This type of distributions is in general not of constant rank, so we cannot
directly apply the Frobenius theorem. But there is an extension of the result,
due to Stefan [stefan] and Sussmann [sussmann1973orbits] (independently) that
works for generalised distributions, locally generated by vector fields that
leave the distribution invariant. This is the situation for characteristic
distributions in the case of Poisson manifolds (see
[libermann2012symplectic]).

So, the characteristic distribution is involutive and each leaf $S$ of the
foliation admits a symplectic structure defining for
$f,g\in\mathcal{C}^{\infty}(S)$ and $q\in S$,

$\\{f,g\\}(q):=\\{\widetilde{f},\widetilde{g}\\}(q)$

for arbitrary extensions
$\widetilde{f},\widetilde{g}\in\mathcal{C}^{\infty}(P)$ of $f,g$ respectively.
It can be easily checked that this definition does not depend on the chosen
functions and that it defines a non-degenerate Poisson structure and thus, $S$
is a symplectic manifold [vaisman2012lectures]. The symplectic form is given
by

$\omega_{S}(X_{f},X_{g})=\\{f,g\\},$

where $f,g\in C^{\infty}(S).$

######  Definition 4.5 ($\Lambda$-orthogonal).

Let $\Delta_{q}\subseteq T_{q}P$ be a subspace on a Poisson manifold
$(P,\\{\cdot,\cdot\\})$. We define the $\Lambda$-orthogonal complement
$\Delta_{q}^{\perp_{\Lambda}}=\sharp_{\Lambda}(\Delta_{q}^{0})$ where
$\Delta_{q}^{0}$ is the annihilator of $\Delta_{q}$, that is,
$\Delta_{q}^{0}:=\\{\alpha\in
T^{*}_{q}P\,\,|\,\,\alpha=0\,\,\text{in}\,\,\Delta_{q}\\}.$

Just as in the symplectic scenario, we say that a subspace
$\Delta_{q}\subseteq T_{q}P$ is

  * $i)$

Isotropic if $\Delta_{q}\subseteq\Delta_{q}^{\perp_{\Lambda}}$ for every $q\in
P$;

  * $ii)$

Coisotropic if $\Delta_{q}^{\perp_{\Lambda}}\subseteq\Delta_{q}$ for every
$q\in P$;

  * $iii)$

Lagrangian if $\Delta_{q}=\Delta_{q}^{\perp_{\Lambda}}\cap\mathcal{S}_{q}$ for
every $q\in P$. Notice that this is equivalent to
$\Delta_{q}\cap\mathcal{S}_{q}$ being Lagrangian in each symplectic vector
space $\mathcal{S}_{q}.$

The $\Lambda$-orthogonal complement satisfies the following properties:

  * $i)$

$(W_{1}\cap
W_{2})^{\perp_{\Lambda}}=W_{1}^{\perp_{\Lambda}}+W_{2}^{\perp_{\Lambda}};$

  * $ii)$

$(W_{1}+W_{2})^{\perp_{\Lambda}}\subseteq W_{1}^{\perp_{\Lambda}}\cap
W_{2}^{\perp_{\Lambda}}.$

######  Remark 4.2.

For symplectic manifolds, the above definitions coincide with the ones
previously given.

##  5\. Coisotropic reduction in cosymplectic geometry

Cosymplectic structures are relevant precisely because they are the natural
arena to develop time-dependent Lagrangian and Hamiltonian mechanics
[de2011methods].

######  Definition 5.1 (Cosymplectic manifold).

A cosymplectic manifold is a triple $(M,\Omega,\theta)$ where $M$ is a
$(2n+1)$-manifold, $\theta$ is a closed $1$-form and $\Omega$ is a closed
$2$-form such that $\theta\wedge\Omega^{n}\neq 0$.

Similar to the symplectic setting, there exist canonical coordinates, which
will be called Darboux coordinates $(q^{i},p_{i},t)$ such that
$\Omega=dq^{i}\wedge dp_{i}$ and $\theta=dt$. The existence of such coordinate
charts is proven in [godbillon1969geometrie].

There are two natural distributions defined on $M$:

  * i)

The horizontal distribution $\mathcal{H}:=\operatorname{Ker}\theta$;

  * ii)

The vertical distribution $\mathcal{V}:=\operatorname{Ker}\Omega$.

These distributions induce the following types of tangent vectors in each
tangent space. A vector $v\in T_{q}M$ will be called:

  * i)

Horizontal if $v\in\mathcal{H}_{q}$;

  * ii)

Vertical if $v\in\mathcal{V}_{q}$.

In Darboux coordinates, these distributions are locally generated as follows:

$\mathcal{H}=\langle\frac{\partial}{\partial q^{i}},\frac{\partial}{\partial
p_{i}}\rangle;\,\,\mathcal{V}=\langle\frac{\partial}{\partial t}\rangle.$

Just as before, we can define a bundle isomorphism between the tangent and
cotangent bundles:

$\flat_{\theta,\Omega}:TM\rightarrow
T^{*}M;\,\,\,v_{q}\mapsto\flat_{\theta,\Omega}(v_{q})=i_{v_{q}}\Omega+\theta(v_{q})\cdot\theta.$

Its inverse is denoted by $\sharp_{\theta,\Omega}$.

The vector field defined as $\mathcal{R}:=\sharp_{\theta,\Omega}(\theta)$ is
called the Reeb vector field. The Reeb vector field is locally given by

$\mathcal{R}=\frac{\partial}{\partial t}.$

Let $H$ be a differentiable function on $M$. We define the following vector
fields

  * i)

The gradient vector field $\operatorname{grad}H:=\sharp_{\theta,\Omega}(dH)$;

  * ii)

The Hamiltonian vector field
$X_{H}:=\operatorname{grad}H-\mathcal{R}(H)\mathcal{R}$;

  * iii)

The evolution vector field $\mathcal{E}_{H}:=X_{H}+\mathcal{R}.$

These vector fields have the local expressions:

$\displaystyle\operatorname{grad}H$ $\displaystyle=\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial
q^{i}}\frac{\partial}{\partial p_{i}}+\frac{\partial H}{\partial
t}\frac{\partial}{\partial t},$ $\displaystyle X_{H}$
$\displaystyle=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{\partial}{\partial p_{i}},$
$\displaystyle\mathcal{E}_{H}$ $\displaystyle=\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial
q^{i}}\frac{\partial}{\partial p_{i}}+\frac{\partial}{\partial t}.$

Notice that an integral curve $(q^{i}(\lambda),p_{i}(\lambda),t(\lambda)$ of
the evolution vector field satisfies the following differential equations

$\displaystyle\frac{dq^{i}}{d\lambda}$ $\displaystyle=\frac{\partial
H}{\partial p_{i}},$ $\displaystyle\frac{dp_{i}}{d\lambda}$
$\displaystyle=-\frac{\partial H}{\partial q^{i}},$
$\displaystyle\frac{dt}{d\lambda}$ $\displaystyle=1,$

which immediately give time-dependent Hamilton’s equations

$\displaystyle\frac{dq^{i}}{dt}$ $\displaystyle=\frac{\partial H}{\partial
p_{i}},$ $\displaystyle\frac{dp_{i}}{dt}$ $\displaystyle=-\frac{\partial
H}{\partial q^{i}}$

since we have $t=\lambda+const.$

Notice that the horizontal distribution $\mathcal{H}$ is the distribution
generated by all Hamiltonian vector fields. Just as in the symplectic case, we
can define a Poisson bracket:

######  Definition 5.2 (Poisson bracket).

Let $\\{\cdot,\cdot\\}$ be the bracket in the ring $\mathcal{C}^{\infty}(M)$
given by

$\\{f,g\\}:=\Omega(X_{f},X_{g}).$

We can easily check that this is indeed a Poisson structure by observing that
in coordinates it is given by

$\\{f,g\\}=\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q^{i}}.$

Thus, the coordinate expression of the Poisson tensor is

$\Lambda=\frac{\partial}{\partial q^{i}}\wedge\frac{\partial}{\partial
p_{i}}.$

So, the Hamiltonian vector fields coincide with the ones provided by this
induced Poisson structure, following the notions and results given in Section
4. In particular, given $\Delta_{q}\subseteq T_{q}M$, we have

$\Delta_{q}^{\perp_{\Lambda}}=\sharp_{\Lambda}(\Delta_{q}^{0}).$

Note that $\operatorname{Ker}\sharp_{\Lambda}=\langle\theta\rangle$ and that
$\operatorname{Im}\sharp_{\Lambda}=\mathcal{H},$ that is, $\mathcal{H}$ is the
characteristic distribution of the Poisson structure induced by
$(\theta,\Omega)$. This implies the following result:

######  Proposition 5.1.

$i:L\rightarrow M$ is a Lagrangian submanifold if and only if

$T_{q}L^{\perp_{\Lambda}}=T_{q}L\cap\mathcal{H}_{q}$

for every $q\in L.$

###### Proof.

It follows from the definition of Lagrangian submanifold (Section 4) and the
fact that $\mathcal{H}$ is the characteristic distribution on $M$. ∎

It is also easy to see that

$\Lambda(\alpha,\beta)=\Omega(\sharp_{\theta,\Omega}(\alpha),\sharp_{\theta,\Omega}(\beta))$

observing that we have
$\Omega(X_{f},X_{g})=\Omega(\operatorname{grad}f,\operatorname{grad}g)$.

###  5.1. Gradient, Hamiltonian and evolution vector fields as Lagrangian
submanifolds

######  Definition 5.3.

Given a cosymplectic manifold $(M,\Omega,\theta)$, we define the symplectic
structure on $TM$ as $\Omega_{0}:=-d\lambda_{0}$, where
$\lambda_{0}=\flat_{\theta,\Omega}^{*}\lambda_{M}$, $\lambda_{M}$ being the
Liouville $1$-form in the cotangent bundle $T^{*}M$.

There is another expression of $\Omega_{0}$, namely

$\Omega_{0}=-\Omega^{c}-\theta^{c}\wedge\theta^{v}$

as one can verify [cantrijn1992gradient]. Here, $\alpha^{v},\alpha^{c}$ denote
the complete and vertical lifts of a form $\alpha$ on $M$ to its tangent
bundle $TM$ [de2011methods]. This implies that in the induced coordinates in
$TM$, $(q^{i},p_{i},t,\dot{q}^{i},\dot{p}_{i},\dot{t})$,

$\Omega_{0}=-dq^{i}\wedge d\dot{p}_{i}-d\dot{q}^{i}\wedge
dp_{i}-d\dot{t}\wedge dt.$

######  Proposition 5.2.

Let $(M,\Omega,\theta)$ be a cosymplectic manifold and $X:M\rightarrow TM$ a
vector field. Then $X(M)$ is a Lagrangian submanifold of $(TM,\Omega_{0})$ if
and only if $X$ is locally a gradient vector field.

###### Proof.

It is easily checked that $X^{*}\lambda_{0}=\flat_{\theta,\Omega}(X)$ (just
like in Proposition 3.1) and then, $X(M)$ is Lagrangian if and only if

$0=X^{*}\Omega_{0}=-X^{*}d\lambda_{0}=-d\flat_{\theta,\Omega}(X),$

that is, $X$ is locally a gradient vector field. ∎

We can also check this in coordinates. Indeed, let

$X=X^{i}\frac{\partial}{\partial q^{i}}+Y_{i}\frac{\partial}{\partial
p_{i}}+Z\frac{\partial}{\partial t}$

be a vector field on $M$. $X:M\hookrightarrow TM$ defines a Lagrangian
submanifold if and only if $X^{*}\Omega_{0}=0.$ An easy calculation gives

$\displaystyle-X^{*}\Omega_{0}=$ $\displaystyle\left(\frac{\partial
X^{i}}{\partial q^{j}}+\frac{\partial Y_{j}}{\partial
p_{i}}\right)dq^{j}\wedge dp_{i}+\left(\frac{\partial X^{i}}{\partial
t}-\frac{\partial Z}{\partial p_{i}}\right)dt\wedge
dp_{i}+\left(\frac{\partial Y_{i}}{\partial t}+\frac{\partial Z}{\partial
q^{i}}\right)dq^{i}\wedge dt$ $\displaystyle\frac{\partial X^{i}}{\partial
p_{j}}dp_{j}\wedge dp_{i}+\frac{\partial Y_{i}}{\partial q^{j}}dq^{j}\wedge
dq^{i}.$

Therefore, $X$ defines a Lagrangian submanifold of $(TM,\Omega_{0})$ if and
only if

$\displaystyle\frac{\partial X^{i}}{\partial q^{j}}+\frac{\partial
Y_{j}}{\partial p_{i}}=0,$ (3) $\displaystyle\frac{\partial X^{i}}{\partial
t}-\frac{\partial Z}{\partial p_{i}}=0,$ (4) $\displaystyle\frac{\partial
Y_{i}}{\partial t}+\frac{\partial Z}{\partial q^{i}}=0,$ (5)
$\displaystyle\frac{\partial X^{i}}{\partial p_{j}}-\frac{\partial
X^{j}}{\partial p_{i}}=0,$ (6) $\displaystyle\frac{\partial Y_{i}}{\partial
q^{j}}-\frac{\partial Y_{j}}{\partial q^{i}}=0.$ (7)

The equations above can be summarized by taking

$(G^{1},\dots,G^{n},G^{n+1},\dots,G^{2n},G^{2n+1}):=(X^{i},-Y_{i},Z),$
$(x^{1},\dots,x^{n},x^{n+1},\dots,x^{2n},x^{2n+1}):=(q^{i},p_{i},t),$

since they translate to

$\frac{\partial G^{i}}{\partial x^{j}}=\frac{\partial G^{j}}{\partial x_{i}}.$

We conclude that $G^{i}=\displaystyle\frac{\partial H}{\partial x^{i}}$, for
some local function $H$, that is, locally, $X=\operatorname{grad}H$.

In general, the Hamiltonian and evolution vector field do not define
Lagrangian submanifolds in $(TM,\Omega_{0})$. However, modifying the form we
can achieve this. First, let us study the Hamiltonian vector field $X_{H}$. We
have

$X_{H}^{*}\Omega_{0}=(\operatorname{grad}H-\mathcal{R}(H)\mathcal{R})^{*}\Omega_{0}=-d(\mathcal{R}(H)\theta)=-d(\mathcal{R}(H))\wedge\theta.$

The form defined as

$\Omega_{H}:=\Omega_{0}+\left(d\mathcal{R}(H)\wedge\theta\right)^{v}$

is a symplectic form and has the local expression

$\Omega_{H}=-dq\wedge d\dot{p}_{i}-d\dot{q}^{i}\wedge dp_{i}-d\dot{t}\wedge
dt+d\left(\frac{\partial H}{\partial t}\right)\wedge dt.$

Also,

$X_{H}^{*}\Omega_{H}=-d\mathcal{R}(H)\wedge\theta+d(\mathcal{R}(H))\wedge\theta=0.$

We have proved that $X_{H}$ defines a Lagrangian submanifold of
$(TM,\Omega_{H}).$ Furthermore, since

$\mathcal{R}^{*}\Omega_{0}=0,$

it follows that the evolution vector field $\mathcal{E}_{H}$ also defines a
Lagrangian submanifold of $(TM,\Omega_{H})$.

This also gives a way of interpreting both vector fields as Lagrangian
submanifolds of a the cosymplectic submanifold
$(TM\times\mathbb{R},\Omega_{H},ds),$ taking the coordinate in $\mathbb{R}$ to
be constant.

###  5.2. Coisotropic reduction

We can interpret the orthogonal complement defined by the Poisson structure
using the cosymplectic structure. We note that $\Omega|_{\mathcal{H}}$ defined
as $\Omega$ restricted to the distribution $\mathcal{H}$ induces a symplectic
vector space in each $\mathcal{H}_{q}$ and thus we have a symplectic vector
bunde $\mathcal{H}\rightarrow M$. If $\Delta_{q}\subseteq\mathcal{H}_{q}$, we
have the $\Omega|_{\mathcal{H}}$-orthogonal complement

$(\Delta_{q})^{\perp_{\Omega|_{\mathcal{H}}}}=\\{v\in\mathcal{H}\,\,|\,\,\Omega(v,w)=0,\,\,\forall
w\in\Delta_{q}\\}.$

######  Proposition 5.3.

Let $\Delta_{q}\subseteq T_{q}M$. Then
$\Delta_{q}^{\perp_{\Lambda}}=(\Delta_{q}\cap\mathcal{H})^{\perp_{\Omega|_{\mathcal{H}}}}.$

###### Proof.

Let $v\in\Delta_{q}^{\perp_{\Lambda}}$, that is, $v=\sharp_{\Lambda}(\alpha)$
with $\alpha\in\Delta_{q}^{0}$. This implies that $v$ is horizontal. We only
need to check that $\Omega(v,w)=0$ for every
$w\in\Delta_{q}\cap\mathcal{H}_{q}$. Indeed, since $\theta(w)=0$,

$\displaystyle\Omega(\sharp_{\Lambda}\alpha,w)$
$\displaystyle=-\Omega(w,\sharp_{\Lambda}\alpha)-\theta(w)\theta(\sharp_{\Lambda}\alpha)=-(\flat_{\theta,\Omega}w)(\sharp_{\Lambda}\alpha)=-\Lambda(\alpha,\flat_{\theta,\Omega}w)$
$\displaystyle=-\Omega(\sharp_{\theta,\Omega}\alpha,w)=-\Omega(\sharp_{\theta,\Omega}\alpha,w)-\theta(\sharp_{\theta,\Omega}\alpha)\theta(w)=-\alpha(w)=0.$

Now we compare dimensions. We distinguish two cases, if
$\theta\in\Delta_{q}^{0}$, we have

$\operatorname{dim}\Delta_{q}^{\perp_{\Lambda}}=\operatorname{dim}\Delta_{q}^{0}-1=2n-\operatorname{dim}\Delta_{q}$

which is exactly
$\operatorname{dim}(\Delta_{q}\cap\mathcal{H}_{q})^{\perp_{\Omega|_{\mathcal{H}}}}$,
for $\Delta_{q}\subseteq\mathcal{H}_{q}$ and
$(\mathcal{H}_{q},\Omega|_{\mathcal{H}})$ is symplectic. Now, if
$\theta\not\in\Delta_{q}^{0}$, then

$\operatorname{dim}\Delta_{q}^{\perp_{\Lambda}}=2n+1-\operatorname{dim}\Delta_{q}$

and, since $\Delta_{q}\not\subseteq\mathcal{H}_{q}$, we have
$\operatorname{dim}(\Delta_{q}\cap\mathcal{H}_{q})=\operatorname{dim}\Delta_{q}-1$
which implies that
$\operatorname{dim}(\Delta_{q}\cap\mathcal{H}_{q})^{\perp_{\Omega|_{\mathcal{H}}}}=2n+1-\operatorname{dim}\Delta.$
∎

This last proposition clarifies the situation. The $\Lambda$-orthogonal of a
subspace $\Delta$ is just the symplectic orthogonal of the intersection with
the symplectic leaf. This means that coisotropic reduction in cosymplectic
geometry will be performed in each leaf of the characteristic distribution
$\mathcal{H}$. Also, because the $\Lambda$-orthogonal complement is just the
symplectic complement of the intersection with $\mathcal{H}$, we have the
following properties:

  * $i)$

$(\Delta_{1}\cap\Delta_{2})^{\perp_{\Lambda}}=\Delta_{1}^{\perp_{\Lambda}}+\Delta_{2}^{\perp_{\Lambda}}.$

  * $ii)$

$(\Delta_{1}+\Delta_{2})^{\perp_{\Lambda}}=\Delta_{1}^{\perp_{\Lambda}}\cap\Delta_{2}^{\perp_{\Lambda}}.$

  * $iii)$

$(\Delta^{\perp_{\Lambda}})^{\perp_{\Lambda}}=\Delta\cap\mathcal{H}.$

It will also be important to distinguish submanifolds $N\hookrightarrow M$
acording to the position relative to the distributions
$\mathcal{H},\mathcal{V}$.

######  Definition 5.4 (Horizontal, non-horizontal and vertical submanifolds).

Let $i:N\hookrightarrow M$ be a submanifold. $N$ will be called a:

  * $i)$

Horizontal submanifold if $T_{q}N\subseteq\mathcal{H}_{q}$ for every $q\in N$;

  * $ii)$

Non-horizontal submanifold if $T_{q}N\not\subseteq\mathcal{H}_{q}$ for every
$q\in N$;

  * $iii)$

Vertical submanifold if the Reeb vector field is tangent to $N$, that is,
$\mathcal{R}(q)\in T_{q}N$ for every $q\in N$.

######  Remark 5.1.

Note that if $N\hookrightarrow M$ is a vertical submanifold, then $N$ is non-
horizontal.

Lagrangian submanifolds are characterized as follows:

######  Lemma 5.1.

Let $L\hookrightarrow M$ be a Lagrangian submanifold and $q\in L$. Then

  * i)

If $T_{q}L\subseteq\mathcal{H}_{q}$,
$\operatorname{dim}T_{q}L^{\perp_{\Lambda}}=\operatorname{dim}M-\operatorname{dim}L-1$.

  * ii)

If $T_{q}L\not\subseteq\mathcal{H}_{q}$,
$\operatorname{dim}T_{q}L^{\perp_{\Lambda}}=\operatorname{dim}M-\operatorname{dim}L$.

and, in either case, $\operatorname{dim}T_{q}L^{\perp_{\Lambda}}=n$, where
$\operatorname{dim}M=2n+1$.

###### Proof.

Since $\theta\in T_{q}L^{0}$ we have

$\operatorname{dim}T_{q}L^{\perp_{\Lambda}}=\operatorname{dim}\sharp_{\Lambda}(T_{q}L^{0})=\operatorname{dim}M-\operatorname{dim}L-\operatorname{dim}(\operatorname{Ker}\sharp_{\Lambda}\cap
T_{q}L)=\operatorname{dim}M-\operatorname{dim}L-1.$

It follows from the previous calculation using that $\theta\not\in T_{q}L^{0}$
because

$\operatorname{dim}(\operatorname{Ker}\sharp_{\Lambda}\cap T_{q}L)=0.$

The proof of the equality $\operatorname{dim}T_{q}L^{\perp_{\Lambda}}=n$ is
straightforward using that $T_{q}L\cap\mathcal{H}_{q}$ is a Lagrangian
subspace of $(\mathcal{H}_{q},\Omega|_{\mathcal{H}})$. ∎

Lemma 5.1 guarantees that either $\operatorname{dim}L=n$, in which case $L$ is
horizontal, or $\operatorname{dim}L=n+1$, in which case $L$ is non-horizontal.
We have the following useful characterization of Larangian submanifolds:

######  Lemma 5.2.

Let $L\hookrightarrow M$ be a submanifold. We have

  * i)

If $\operatorname{dim}L=n$, then $L$ is Lagrangian if and only if
$i^{*}\theta=0,i^{*}\Omega=0$.

  * ii)

If $L$ is non-horizontal and $\operatorname{dim}L=n+1$, then $L$ is Lagrangian
if and only if $i^{*}\Omega=0$.

###### Proof.

Both assertions are proved by a comparison of dimensions. ∎

######  Proposition 5.4.

Let $i:N\hookrightarrow M$ be a coisotropic submanifold. Then the distribution
$(TN)^{\perp_{\Lambda}}$ is involutive.

###### Proof.

We start proving that $\mathcal{H}$ is an involutive distribution. Let $X,Y$
be vector fields tangent to $\mathcal{H}$. Since $\theta$ is closed we have

$\displaystyle 0$
$\displaystyle=(d\theta)(X,Y)=X(\theta(Y))-Y(\theta(X))-\theta([X,Y])=-\theta([X,Y]),$

that is, $[X,Y]$ is tangent to $\mathcal{H}.$

Denote $\Omega_{0}:=i^{*}\Omega$. Let $X,Y$ be vector fields in $N$ tangent to
$(TN)^{\perp_{\Lambda}}$. Using Proposition 5.3,
$[X,Y]\in(TN)^{\perp_{\Lambda}}$ if and only if
$[X,Y]\in(TN\cap\mathcal{H})^{\perp_{\Omega|_{\mathcal{H}}}}$. In order to see
this, we take an arbitrary vector field $Z$ on $N$ tangent to $\mathcal{H}$
and check that $\Omega_{0}([X,Y],Z)=0$. Because $\Omega$ is closed, we have

$\displaystyle 0=$ $\displaystyle
i^{*}(d\Omega)=(d\Omega_{0})(X,Y,Z)=X(\Omega_{0}(Y,Z))-Y(\Omega_{0}(X,Z))+Z(\Omega_{0}(X,Y))$
$\displaystyle-\Omega_{0}([X,Y],Z)+\Omega_{0}([X,Z],Y)-\Omega_{0}([Y,Z],X)=-\Omega_{0}([X,Y],Z)$

where we have used that $X,Y,Z,[Y,Z],[X,Z]$ are horizontal (since
$\mathcal{H}$ is involutive) and that
$X,Y\in(TN\cap\mathcal{H})^{\perp_{\Omega|_{\mathcal{H}}}}$. ∎

###  5.3. Vertical coisotropic reduction

We shall now study coisotropic reduction of a vertical submanifold
$N\hookrightarrow M$. Let $q\in N$. We have
$\operatorname{dim}(T_{q}N)^{0}=\operatorname{dim}M-\operatorname{dim}N$.
Since $N$ is vertical, $\theta\not\in(T_{q}N)^{0}$ and we have

$\operatorname{dim}(T_{q}N)^{\perp_{\Lambda}}=\operatorname{dim}\sharp_{\Lambda}(T_{q}N)^{0}=\operatorname{dim}M-\operatorname{dim}N-\operatorname{dim}(\operatorname{Ker}\sharp_{\Lambda}\cap(T_{q}N)^{0})=\operatorname{dim}M-\operatorname{dim}N.$

In particular, $(TN)^{\perp_{\Lambda}}$ is a regular distribution.

######  Theorem 5.1 (Vertical coisotropic reduction in the cosymplectic
setting).

Let $(M,\Omega,\theta)$ be a cosymplectic manifold and $i:N\hookrightarrow M$
be an coisotropic vertical submanifold. Denote by $\mathcal{F}$ the maximal
foliation of the involutive regular distribution $(TN)^{\perp_{\Lambda}}$. If
the space of all leaves $N/\mathcal{F}$ admits a manifold structure such that
the projection $\pi:N\rightarrow N/\mathcal{F}$ is a submersion, then there
exist unique $\theta_{N}$, $\omega_{N}$ such that

$\displaystyle i^{*}\omega$ $\displaystyle=\pi^{*}\omega_{N},$ $\displaystyle
i^{*}\theta$ $\displaystyle=\pi^{*}\theta_{N},$

and they define a cosymplectic structre on $N/\mathcal{F}.$ The following
diagram summarizes the situation:

${N}$${M}$${N/\mathcal{F}}$$\scriptstyle{i}$$\scriptstyle{\pi}$

###### Proof.

Uniqueness is clear from the imposed relation. Denote
$\Omega_{0}:=i^{*}\Omega$, $\theta_{0}:=i^{*}\theta$. We only need to verify
that the following forms are closed, well defined and define a cosymplectic
structure:

$\displaystyle\Omega_{N}([u],[v]):=\Omega_{0}(u,v),$
$\displaystyle\theta_{N}([u]):=\theta_{0}(u),$

where $[u]:=T\pi(q)\cdot u\in T_{[q]}N/\mathcal{F}$. If they were well
defined, it is clear that they are smooth and closed since
$\pi^{*}d\theta_{N}=d\theta_{0}=0$, $\pi^{*}\Omega_{N}=d\Omega_{0}=0$ and
$\pi$ is a submersion.

Let us first check that these definitions do not depend on the chosen
representatives of the vectors. It suffices to observe that for vectors in the
distribution, say $v\in(T_{q}N)^{\perp_{\Lambda}}$, we have
$i_{v}\Omega_{0}=0$ and $i_{v}\theta_{0}=0$. This easily follows from
Proposition 5.3 using that the horizontal proyection of every vector $u\in
T_{q}N$ is tangent to $N$ (here we use the condition $\mathcal{R}(q)\in
T_{q}N$). To see the independence of the point in the leaf chosen, it is
enough to observe that

$\mathcal{L}_{X}\Omega_{0}=0;\,\,\,\mathcal{L}_{X}\theta_{0}=0$

for every vector field on $N$ tangent to the distribution
$(T_{q}N)^{\perp_{\Lambda}}$ (since every two points in the same leaf of the
foliation can be joined by a finite union of flows of such fields). Indeed, we
have

$\displaystyle\mathcal{L}_{X}\Omega_{0}$
$\displaystyle=i_{X}d\Omega_{0}+di_{X}\Omega_{0}=0,$
$\displaystyle\mathcal{L}_{X}\theta_{0}$
$\displaystyle=i_{X}d\theta_{0}+di_{X}\theta_{0}=0.$

Now we check that they define a cosymplectic structure. Assuming
$k=\operatorname{dim}N$ and $2n+1=\operatorname{dim}M$, from the remark above
we have

$\operatorname{dim}N/\mathcal{F}=\operatorname{dim}N-\operatorname{rank}\,(TN)^{\perp_{\Lambda}}=2k-2n-1=2(k-n-1)+1$

and hence, $(N/\mathcal{F},\Omega_{N},\theta_{N})$ is a cosymplectic manifold
if and only if

$\theta_{N}\wedge\Omega_{N}^{k-n-1}\neq 0,$

which is equivalent to $\theta_{0}\wedge\Omega_{0}^{k-n-1}\neq 0$, because
$\pi$ is a submersion. For every point $q\in N$, $TqN$ can be decomposed in

$TqN=TqN_{\mathcal{H}}\oplus\mathcal{V}$

where $T_{q}N_{\mathcal{H}}=(T_{q}N)\cap\mathcal{H}_{q}$. It is easy to see
that $(T_{q}N)_{\mathcal{H}}$ is a coisotropic subspace of
$(\mathcal{H}_{q},\Omega|_{\mathcal{H}})$. This implies (using symplectic
reduction) that there are
$\operatorname{dim}(T_{q}N)_{\mathcal{H}}-\operatorname{dim}(T_{q}N)_{\mathcal{H}}^{\perp_{\widetilde{\Omega}}}=k-1-(2n+1-k)=2k-2n-2$
horizontal vectors, say $u_{1},\dots,u_{2k-2n-2}$ such that
$\Omega_{0}^{k-n-1}(u_{1},\dots,u_{2k-2n-2})\neq 0$. Taking the last vector to
be $\mathcal{R}(q)$, it is clear that

$(\theta_{0}\wedge\Omega_{0}^{k-n-1})(\mathcal{R}(q),u_{1},\dots,u_{2k-2n-2})\neq
0.$

∎

####  5.3.1. Projection of Lagrangian submanifolds

Now we will prove that Lagrangian submanifolds $L\hookrightarrow M$ project to
Lagrangian submanifolds in $N/\mathcal{F}.$

######  Proposition 5.5 (Projection of horizontal Lagrangian submanifolds is
Lagrangian).

Under the hypotheses of Theorem 5.1, let $L\hookrightarrow M$ be an horizontal
Lagrangian submanifold such that $L$ and $N$ have clean intersection. If
$L_{N}:=\pi(L\cap N)$ is a submanifold of $N/\mathcal{F}$, then $L_{N}$ is
Lagrangian.

###### Proof.

Let $\mathcal{H}_{N}$ be the horizontal distribution in $N/\mathcal{F}$. It is
clear that $L_{N}$ is horizontal, because $T_{[q]}L_{N}=T\pi(q)(T_{q}L\cap
T_{q}N)$. Using Proposition 5.3 we have

$T_{[q]}L_{N}^{\perp_{\Lambda_{N}}}=(T_{[q]}L_{N}\cap\mathcal{H}_{N})^{\perp_{\Omega_{N}|_{\mathcal{H}_{N}}}}=T_{[q]}L_{N}^{\perp_{\Omega_{N}|_{\mathcal{H}_{N}}}}.$

We will check that

$T_{[q]}L_{N}\subseteq T_{[q]}L_{N}^{\perp_{\Lambda_{N}}}$

and prove that $\operatorname{dim}L_{N}=\operatorname{dim}N-n-1$, which with
Lemma 5.2 together with the calculation of the dimension of $N/\mathcal{F}$
done in Theorem 5.1, yields the result.

Let $[v],[w]\in T_{[q]}L_{N}$. Then

$\Omega_{N}([v],[w])=\Omega(v,w)=0,$

since $L$ is Lagrangian. Because $[w]$ is arbitrary, this last calculation
implies that $[v]\in
T_{[q]}L_{N}^{\perp_{\Omega_{N}|_{\mathcal{H}_{N}}}}\subseteq
T_{[q]}L_{N}^{\perp_{\Lambda_{N}}}$. Now,

$\operatorname{dim}L_{N}=\operatorname{dim}(L\cap
N)-\operatorname{dim}(T_{q}L\cap(T_{q}N)^{\perp_{\Lambda}}).$ (8)

Furthermore, since $L$ is Lagrangian and horizontal,
$(T_{q}L\cap(T_{q}N)^{\perp_{\Lambda}})^{\perp_{\Lambda}}=T_{q}L\cap\mathcal{H}_{q}+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}=T_{q}L+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}$
and thus (using that $T_{q}L\cap(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}$
is necessarily horizontal),

$\operatorname{dim}(T_{q}L\cap(T_{q}N)^{\perp_{\Lambda}})=\operatorname{dim}M-\operatorname{dim}(T_{q}L+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}})-1.$

Since
$\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}})=\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+T_{q}N)-1$
(which comes from the fact that
$\operatorname{dim}(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}=\operatorname{dim}T_{q}N-1$)
we have

$\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+T_{q}N^{\perp_{\Lambda}})=\operatorname{dim}M-\operatorname{dim}(T_{q}L\cap
T_{q}N^{\perp_{\Lambda}}).$ (9)

Substituting (9) in (8) and using $\operatorname{dim}L=n$, we conclude

$\displaystyle\operatorname{dim}L_{N}$ $\displaystyle=\operatorname{dim}(L\cap
N)-\left(\operatorname{dim}M-\operatorname{dim}(T_{q}L+T_{q}N)\right)$
$\displaystyle=\operatorname{dim}(L\cap
N)-\operatorname{dim}M+\operatorname{dim}L+\operatorname{dim}N-\operatorname{dim}(L\cap
N)$ $\displaystyle=-2n-1+n+\operatorname{dim}N=\operatorname{dim}N-n-1.$

∎

######  Proposition 5.6 (Projection of non-horizontal Lagrangian submanifold
is Lagrangian).

Under the hypotheses of Theorem 5.1, let $L\hookrightarrow M$ be a non-
horizontal Lagragian submanifold. If $L$ and $N$ have clean intersection and
$L_{N}:=\pi(L\cap N)\hookrightarrow N/\mathcal{F}$ is a submanifold, then
$L_{N}$ is Lagrangian.

###### Proof.

The proof follows the same lines as that of Proposition 5.5. That $L_{N}$ is
isotropic follows easily from Proposition 5.3. However, in order to calculate
$\operatorname{dim}L_{N}$, we need to distinguish whether $L\cap N$ is
horizontal or not.

If $L\cap N$ is horizontal, we need to check that
$\operatorname{dim}L_{N}=\operatorname{dim}N-n-1$, since $L_{N}$ is
horizontal. Because $(T_{q}L\cap
T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}=T_{q}L\cap\mathcal{H}_{q}+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}$,
we have

$\displaystyle\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}})$
$\displaystyle=\operatorname{dim}M-\operatorname{dim}(T_{q}L\cap
T_{q}N^{\perp_{\Lambda}})-1.$

It is easy to check that
$\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}})=\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+T_{q}N)-1$
and thus,

$\operatorname{dim}(T_{q}L\cap
T_{q}N^{\perp_{\Lambda}})=\operatorname{dim}M-\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+T_{q}N).$

We conclude that

$\displaystyle\operatorname{dim}L_{N}$ $\displaystyle=\operatorname{dim}(L\cap
N)-\operatorname{dim}(T_{q}L\cap T_{q}N^{\perp_{\Lambda}})$
$\displaystyle=\operatorname{dim}(L\cap
N)-\left(\operatorname{dim}M-\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}+T_{q}N)\right)$
$\displaystyle=\operatorname{dim}(L\cap
N)-\left(\operatorname{dim}M-\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q})-\operatorname{dim}N+\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}\cap
T_{q}N)\right)$ $\displaystyle=\operatorname{dim}(L\cap
N)-\operatorname{dim}M+\operatorname{dim}L-1+\operatorname{dim}N-\operatorname{dim}(T_{q}L\cap
T_{q}N)$
$\displaystyle=\operatorname{dim}N-\operatorname{dim}M+\operatorname{dim}L-1=\operatorname{dim}N-2n-1+n+1-1=\operatorname{dim}N-n-1,$

where we have used that $T_{q}L\cap\mathcal{H}_{q}\cap T_{q}N=T_{q}L\cap
T_{q}N$, since $N\cap L$ is horizontal, and
$\operatorname{dim}T_{q}L\cap\mathcal{H}_{q}=\operatorname{dim}L-1$, because
$T_{q}L\not\subseteq\mathcal{H}_{q}$.

If $L\cap N$ is not horizontal, we need to check that
$\operatorname{dim}L_{N}=\operatorname{dim}N-n$. This follows from the same
calculation done in i), using that

$\operatorname{dim}(T_{q}L\cap\mathcal{H}_{q}\cap
T_{q}N)=\operatorname{dim}(T_{q}L\cap T_{q}N)-1.$

∎

###  5.4. Horizontal coisotropic reduction

We will restrict the study to horizontal coisotropic submanifolds
$N\hookrightarrow M$, that is, manifolds satisfying
$T_{q}N\subseteq\mathcal{H}_{q}$ for every $q\in N$. Note that in this case
the distribution $(TN)^{\perp_{\Lambda}}$ is also regular, since

$\operatorname{dim}(T_{q}N)^{\perp_{\Lambda}}=\operatorname{dim}M-\operatorname{dim}N-\operatorname{dim}(\operatorname{Ker}\sharp_{\Lambda}\cap(T_{q}N)^{0})=\operatorname{dim}M-\operatorname{dim}N-1.$

######  Theorem 5.2 (Horizontal coisotropic reduction in the cosymplectic
setting).

Let $(M,\Omega,\theta)$ be a cosymplectic manifold and $i:N\hookrightarrow M$
be an horizontal coisotropic submanifold. Denote by $\mathcal{F}$ the space of
leaves determined by the regular and involutive distribution
$(TN)^{\perp_{\Lambda}}.$ If $N/\mathcal{F}$ admits a manifold structure such
that $\pi:N\rightarrow N/\mathcal{F}$ is a submersion, then there exists a
unique $2$\- form $\Omega_{N}$ in $N/\mathcal{F}$ such that

$\pi^{*}\Omega_{N}=i^{*}\Omega$

and $(N/\mathcal{F},\Omega_{N})$ is a symplectic manifold.

###### Proof.

Since $N$ is horizontal and the horizontal distribution is integrable, $N$
will be contained in an unique symplectic leaf and thus, we are performing
symplectic reduction. The proof is just repeating what has been done in
Theorem 3.2. ∎

We can generalize this process to arbitrary submanifolds. Let
$N\hookrightarrow M$ be a coisotropic submanifold. Since in general we cannot
guarantee the well-definedness of the $2$-form in the quotient, we will reduce
the intersection of $N$ with each one of the symplectic leaves. It is clear
that $TN\cap\mathcal{H}$ is an involutive distribution, since $TN$ and
$\mathcal{H}$ are. If this distribution was regular, for every $q\in N$ there
would exist an unique maximal leaf of the distribution, say $S_{q}$. Notice
that $S_{q}\hookrightarrow M$ is an horizontal submanifold. We can perform
coisotropic reduction on each of these submanifolds.

####  5.4.1. Projection of Lagrangian submanifolds

######  Proposition 5.7.

Let $L\hookrightarrow M$ be a Lagrangian submanifold. If $L$ and $N$ have
clean intersection and $L_{N}:=\pi(L\cap N)$ is a submanifold, then $L_{N}$ is
Lagrangian.

###### Proof.

Let $q\in N\cap L$, we have to prove that

$T_{[q]}L_{N}=d_{q}\pi\cdot\left(T_{q}L\cap T_{q}N\right)$

is a Lagrangian subspace of
$T_{[q]}N/\mathcal{F}=T_{[q]}N/\left(T_{q}N\right)^{\perp_{\Lambda}}.$ Since
$N$ is horizontal, $T_{q}N\subseteq\mathcal{H}_{q}$. Now, from Proposition
5.3, we know that

$(T_{q}N)^{\perp_{\Omega|_{\mathcal{H}}}}=(T_{q}N)^{\perp_{\Lambda}}\subseteq
T_{q}N,$

that is, $T_{q}N$ is a coisotropic subspace of $\mathcal{H}_{q}$, with its
natural symplectic structure. A similar argument shows that $T_{q}L$ is a
Lagrangian subspace of $\mathcal{H}_{q}.$ Now, from symplectic reduction
(linear symplectic reduction) we conclude that

$T_{[q]}L_{N}=d_{q}\pi\cdot\left(T_{q}L\cap T_{q}N\right)$

is a Lagrangian submanifold of
$T_{q}N/\left(T_{q}N\right)^{\perp_{\Lambda}}=T_{[q]}N/\mathcal{F}$, with the
symplectic structure induced by $\Omega_{q}|_{\mathcal{H}}$, which coincides
sith the symplectic structure induced by $\Omega_{q}$, as a quick check shows.
∎

##  6\. Jacobi structures

Contact and cocontact manifolds are not Poisson manifolds. However, there is
still a Lie bracket defined in the algebra of functions, as we will see. This
bracket induces what is called a Jacobi manifold. In this section we define
and study such structures (see [libermann2012symplectic, raul] for more
details).

######  Definition 6.1 (Jacobi Manifold).

A Jacobi structure on a manifold $M$ is a Lie bracket defined in the algebra
of functions $(\mathcal{C}^{\infty}(M),\\{\cdot,\cdot\\})$ that satisfies the
weak Leibniz rule, that is,

$\operatorname{supp}\\{f,g\\}\subseteq\operatorname{supp}f\cap\operatorname{supp}g.$

Every Jacobi bracket can be uniquely expressed as

$\\{f,g\\}=\Lambda(df,dg)+fE(g)-gE(f),$

where $\Lambda$ is a bivector field (called the Jacobi tensor) and $E$ is a
vector field. $\Lambda$ and $E$ satisfy the equalities

$[E,\Lambda]=0,\,\,[\Lambda,\Lambda]=2E\wedge\Lambda;$

where $[\cdot,\cdot]$ is the Schouten-Nijenhuis bracket. Conversely, given a
bivector field $\Lambda$ and a vector field $E$,

$\\{f,g\\}:=\Lambda(df,dg)+fE(g)-gE(f)$

defines a Jacobi bracket if and only if both equalities above hold.

######  Remark 6.1.

It is clear that Poisson manifolds are Jacobi manifolds, taking $E=0$.

The Jacobi tensor allows us to define the morphism

$\sharp_{\Lambda}:T^{*}M\rightarrow TM;\,\,\alpha\mapsto i_{\alpha}\Lambda.$

Define the $\Lambda$-orthogonal of distributions $\Delta$ as

$\Delta^{\perp_{\Lambda}}:=\sharp_{\Lambda}(\Delta^{0}).$

We can define the Hamiltonian vector field defined by a function $H$ as

$X_{H}=\sharp_{\Lambda}(dH)+HE$

Just like in the Poisson case, we say that a distribution $\Delta$ is:

  * $i)$

Isotropic if $\Delta\subseteq\Delta^{\perp_{\Lambda}}$;

  * $ii)$

Coisotropic if $\Delta^{\perp_{\Lambda}}\subseteq\Delta$;

  * $iii)$

Legendrian if $\Delta^{\perp_{\Lambda}}=\Delta$.

These definitions extend naturally to submanifolds.

######  Remark 6.2.

As in the case of Poisson manifolds, a Jacobi structure on a manifold $M$
defines a characteristic distribution ${\cal S}$ as follows: ${\cal S}_{x}$ is
the vector subspace of $T_{x}M$ generated by the values of all Hamiltonian
vector fields at $x$ and the vector field $E$ evaluated at $x$. This is again
an involutive distribution in the sense of Stefan and Sussmann [stefan,
sussmann1973orbits], and the leaves of the corresponding foliation are contact
manifolds if the leaf has odd dimension, and locally conformal symplectic
manifolds, if the leaf has even dimension. The definition of the Jacobi
bracket on the leaves follows the same path that in the case of Poisson
manifolds.

##  7\. Coisotropic reduction in contact geometry

Contact manifolds are the natural setting for Hamiltonian systems with
dissipation, instead of symplectic Hamiltonian systems where the antisymmetry
of the symplectic form provides conservative properties [bravetti0,
de2019contact]. In the Lagrangian picture, contact Lagrangian systems
correspond to the so-called Lagrangians depending on the action, and instead
of Hamilton’s principle, one has to use the so-called Herglotz principle to
obtain the dynamics [miguel].

######  Definition 7.1 (Contact manifold).

A contact manifold is a couple $(M,\eta)$ where $M$ is a $(2n+1)$-dimensional
manifold, $\eta$ is a $1$-form and $\eta\wedge(d\eta)^{n}\neq 0$.

In this case we also have Darboux coordinates $(q^{i},p_{i},z)$ in $M$
[godbillon1969geometrie] such that

$\eta=dz-p_{i}dq_{i};$

We have also have a bundle isomorphism defined as in the cosymplectic case

$\flat_{\eta}:TM\rightarrow T^{*}M;\,\,v_{q}\mapsto
i_{v_{q}}d\eta+\eta(v_{q})\cdot\eta,$

its inverse $\sharp_{\eta}=\flat_{\eta}^{-1}$, and a couple of natural
distributions:

  * $i)$

The horizontal distribution $\mathcal{H}:=\operatorname{Ker}\eta$;

  * $ii)$

The vertical distribution $\mathcal{V}:=\operatorname{Ker}d\eta$.

We can find different types of tangent vectors at a point $q\in M$. Indeed, a
tangent vector $v\in T_{q}M$ will be called

  * $i)$

Horizontal if $v\in\mathcal{H}_{q}$;

  * $ii)$

Vertical if $v\in\mathcal{V}_{q}$.

This time, however, we cannot define a canonical Poisson structure since the
bivector field

$\Lambda(\alpha,\beta):=-d\eta(\sharp_{\eta}(\alpha),\sharp_{\eta}(\beta))$

is not a Poisson tensor. In fact,

$[\Lambda,\Lambda]=-2\mathcal{R}\wedge\Lambda;\,\,[E,\Lambda]=0$

where $\mathcal{R}$ is the Reeb vector field defined as
$\mathcal{R}:=\sharp_{\eta}(\eta)$ (locally
$\mathcal{R}=\frac{\partial}{\partial z}$). This is easily seen performing a
direct calculation in Darboux coordinates using the local expresion

$\Lambda=\frac{\partial}{\partial p_{i}}\wedge\frac{\partial}{\partial
q^{i}}+p_{i}\frac{\partial}{\partial p_{i}}\wedge\frac{\partial}{\partial z}.$

This defines a Jacobi structure in $M$ taking $\Lambda$ as above and
$E=-\mathcal{R}$ (see Section 6). The Jacobi bracket is locally expressed as

$\\{f,g\\}=\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial
q^{i}}-\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}+p_{i}\left(\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial
z}-\frac{\partial g}{\partial p_{i}}\frac{\partial f}{\partial
z}\right)+g\frac{\partial f}{\partial z}-f\frac{\partial g}{\partial z}.$

The morphism induced by the Jacobi tensor $\Lambda$ satisfies

$\operatorname{Ker}\sharp_{\Lambda}=\langle\eta\rangle,\,\,\,\,\operatorname{Im}\sharp_{\Lambda}=\mathcal{H}.$

######  Remark 7.1.

Although notation between cosymplectic and contact manifolds is similar, they
are different in nature. In cosymplectic geometry, we had a closed $1$-form
$\theta$, and a closed $2$-form $\Omega$ satisfying the non-degeneracy
condition. In contact geometry we have a $1$-form $\eta$ and a (closed)
$2$-form $d\eta$ which also satisfies the non-degeneracy condition. In each
tangent space, these structures will be isomorphic. Indeed, we can always
assume that $p_{i}=0$ at certain $q\in M$, which would give $\eta=dz$ in said
point. However, they are far from locally isomorphic. In cosymplectic
manifolds, the horizontal distribution is involutive, but in contact manifolds
is not (this is the key to obtain dissipative dynamics). In Section 8 we will
enphasize on the differences between these geoemtries by studying the
variational principle in contact mechanics.

###  7.1. Hamiltonian and evolution vector fields as Lagrangian and Legendrian
submanifolds

######  Definition 7.2 (Hamiltonian vector field).

Let $H\in\mathcal{C}^{\infty}(M)$. Define the Hamiltonian vector field of $H$
as

$X_{H}:=\sharp_{\Lambda}(dH)-H\mathcal{R}=\sharp_{\eta}(dH)-(\mathcal{R}(H)+H)\mathcal{R}.$

Locally, it has the local expression

$X_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\left(\frac{\partial H}{\partial q^{i}}+p_{i}\frac{\partial H}{\partial
z}\right)\frac{\partial}{\partial p_{i}}+\left(p_{i}\frac{\partial H}{\partial
p_{i}}-H\right)\frac{\partial}{\partial z}.$

The dynamics corresponding to a Hamiltonian vector field $X_{H}$ are
determined by

$\displaystyle\frac{dq^{i}}{dt}$ $\displaystyle=\frac{\partial H}{\partial
p_{i}},$ $\displaystyle\frac{dp_{i}}{dt}$ $\displaystyle=-\frac{\partial
H}{\partial q^{i}}-p_{i}\frac{\partial H}{\partial z},$
$\displaystyle\frac{dz}{dt}$ $\displaystyle=p_{i}\frac{\partial H}{\partial
p_{i}}-H.$

For instance, if we take as Hamiltonian

$H:=\frac{p^{2}}{2m}+\frac{\kappa^{2}mq^{2}}{2}+\gamma z,$

the equations determined by $X_{H}$ are

$\displaystyle\frac{dq}{dt}$ $\displaystyle=\frac{p}{m},$
$\displaystyle\frac{dp}{dt}$ $\displaystyle=-\kappa^{2}mq-\gamma p,$
$\displaystyle\frac{dz}{dt}$
$\displaystyle=\frac{p^{2}}{2m}-\frac{\kappa^{2}mq^{2}}{2}-\gamma z,$

which are precisely the equations for the damped harmonic oscillator.

We can define a symplectic structure in $TM$ taking
$\Omega_{0}:=\flat_{\eta}^{*}\Omega_{M}$, where $\Omega_{M}$ is the canonical
symplectic structure in $T^{*}M$. In local coordinates
$(q^{i},p_{i},z,\dot{q}^{i},\dot{p}_{i},\dot{z})$, it has the expression

$\displaystyle\Omega_{0}=\,$ $\displaystyle p_{i}p_{j}dq^{i}\wedge
d\dot{q}^{j}+\left((1+\delta_{i}^{j})p_{i}\dot{q}^{j}-\delta_{i}^{j}\dot{z}\right)dq^{i}\wedge
dp_{j}-dq^{i}\wedge d\dot{p}_{i}-p_{i}dq^{i}\wedge d\dot{z}$
$\displaystyle+dp_{i}\wedge d\dot{q}^{i}+dz\wedge d\dot{z}-p_{i}dz\wedge
d\dot{q}^{i}-\dot{q}^{i}dz\wedge dp_{i}$ $\displaystyle=\,$ $\displaystyle
dq^{i}\wedge\left(p_{i}p_{i}dq^{j}+(1+\delta_{i}^{j})p_{i}\dot{q}^{j}dp_{j}-\dot{z}dp_{i}-d\dot{p}_{i}-p_{i}d\dot{z}\right)$
$\displaystyle+dp_{i}\wedge
d\dot{q}^{i}+dz\wedge\left(d\dot{z}-p_{i}d\dot{q}^{i}-\dot{q}^{i}dp_{i}\right)$

######  Definition 7.3 (Gradient vector field).

Given a Hamiltonian $H$ on $M$, define the gradient vector field of $H$ as

$\operatorname{grad}H:=\sharp_{\eta}(dH).$

Locally, it is given by

$\operatorname{grad}H=\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial q^{i}}-\left(\frac{\partial H}{\partial
q^{i}}+p_{i}\frac{\partial H}{\partial z}\right)\frac{\partial}{\partial
p_{i}}+\left(p_{i}\frac{\partial H}{\partial p_{i}}+\frac{\partial H}{\partial
z}\right)\frac{\partial}{\partial z}.$

We have the following relation between both vector fields

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

Just like in the previous sections, a vector field $X:M\rightarrow TM$ is
locally a gradient vector field if and only if it defines a Lagrangian
submanifold in $(TM,\Omega_{0})$. The proof is straight-forward, checking that

$X^{*}\Omega_{0}=-dX^{\flat_{\eta}}.$

We can also interpret the Hamiltonian vector field $X_{H}$ as a Lagrangian
submanifold of $TM$, but we need to modify slightly the symplectic form. It is
easy to verify that

$X_{H}^{*}\Omega_{0}=\operatorname{d}(\mathcal{R}(H)\eta+H\eta),$

therefore, taking

$\Omega_{H}:=\Omega_{0}-\operatorname{d}(\mathcal{R}(H)\eta+H\eta)^{v},$

we have

$X_{H}^{*}\Omega_{H}=0.$

It is clear that $\Omega_{H}$ is a symplectic form. We have proved:

######  Proposition 7.1.

The Hamiltonian vector field $X_{H}:M\rightarrow TM$ defines a Lagrangian
submanifold of the symplectic manifold $(TM,\Omega_{H})$.

Now we study evolution vector fields, an important vector field in the
application of contact geometry to thermodynamics.

######  Definition 7.4.

Given a Hamiltonian $H$, we define the evolution vector field as

$\mathcal{E}_{H}:=X_{H}+H\mathcal{R}.$

Locally, the evolution vector field is written

$\mathcal{E}_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\left(\frac{\partial H}{\partial q^{i}}+p_{i}\frac{\partial H}{\partial
z}\right)\frac{\partial}{\partial p_{i}}+p_{i}\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial z}.$

Let us see how we can modify the symplectic form $\Omega_{0}$ in such a way
that $\mathcal{E}_{H}$ defines a Lagrangian submanifold. We have

$\mathcal{E}_{H}^{*}\Omega_{0}=X_{H}^{*}\Omega_{0}+(H\mathcal{R})^{*}\Omega_{0}=\operatorname{d}(\mathcal{R}(H)\eta+H\eta)-\operatorname{d}(H\eta)=\operatorname{d}(\mathcal{R}(H)\eta),$

and thus, $\mathcal{E}_{H}$ defines a Lagrangian submanifold of
$(TM,\widetilde{\Omega}_{H}),$ where

$\widetilde{\Omega}_{H}=\Omega_{0}-\operatorname{d}(\mathcal{R}(H)\eta).$

We can also interpret Hamiltonian and evolution vector fields as Legendrian
submanifolds of a certain contact structure defined on $TM\times\mathbb{R}.$

######  Definition 7.5.

Let $(M,\eta)$ be a contact manifold. Define the contact form on
$TM\times\mathbb{R}$ as

$\hat{\eta}:=\eta^{c}+t\eta^{v},$

where $\eta^{c}$, $\eta^{v}$ are the complete and vertical lifts
[de2011methods]. It is easily checked that $\hat{\eta}$ defines a contact
structure [de2019contact].

In local coordinates it has the expression:

$\hat{\eta}=d\dot{z}-\dot{p}_{i}dq^{i}-p_{i}d\dot{q}^{i}+tdz-tp_{i}dq^{i}$

We have the following [de2019contact]:

######  Proposition 7.2.

Let $X_{H}:M\rightarrow TM$ be a Hamiltonian vector field. Then, the
submanifold defined by the immersion

$i:M\hookrightarrow
TM\times\mathbb{R};\,\,\,\,p\mapsto(X_{H}(p),\mathcal{R}(H))$

is a Legendrian submanifold of $(TM\times\mathbb{R},\hat{\eta})$.

###### Proof.

Using the properties of complete and vertical lifts we have

$(X_{H}\times\mathcal{R}(H))^{*}\hat{\eta}=\mathcal{L}_{X_{H}}\eta+\mathcal{R}(H)\eta.$

Using Lemma 7.2 it will be sufficient to see that
$\mathcal{L}_{X_{H}}\eta=-\mathcal{R}(H).$ This is a straight-forward
verification using

$\mathcal{L}_{X_{H}}\eta=di_{X_{H}}\eta+i_{X_{H}}d\eta=-dH+dH-\mathcal{R}(H)\eta.$

∎

###  7.2. An overview of the use of contact geometry in thermodynamics

A thermodynamical system is characterized by the following variables

$(H,T,S,P_{i},V^{i}),$

corresponding to energy, temperature, entropy, and the generalized pressures
and volumes, respectively. We will denote $q^{i}=V^{i}$, following the
notation used so far. The energy of the system is a function

$\displaystyle H:$
$\displaystyle\;T^{\ast}Q\times\mathbb{R}\rightarrow\mathbb{R};\,\,(q^{i},p_{i},S)\mapsto
H(q^{i},p_{i},S).$

The first law of thermodynamics can be written as

$dH=\delta\mathcal{Q}-\delta\mathcal{W},$

where $\delta\mathcal{Q}$ and $\delta\mathcal{W}$ are one forms representing
the heat and work, respectively. We will assume that these forms have local
expressions

$\delta\mathcal{Q}=TdS,\,\,\delta\mathcal{W}=P_{i}dq^{i},$

for some functions $P_{i},T$ that physically represent the conjugate variables
to $q^{i}$ (pressure, if $q^{i}$ represented volume), and $S$ (temperature, if
$S$ was the entropy), respectively. Then, the first law of thermodynamics
reads

$dH=TdS-P_{i}dq^{i}.$

If we were studying an isolated system, energy would be conserved, that is

$0=TdS-P_{i}dq^{i}$

or, dividing by $T$,

$0=dS-\frac{P_{i}}{T}dq^{i}.$

Identifying $p_{i}=P_{i}/T$, after a change of variables if necessary, we
obtain

$0=dS-p_{i}dq^{i}.$

Defining

$\eta:=dS-p_{i}dq^{i},$

the canonical contact form in $T^{\ast}Q\times\mathbb{R},$ we conclude that
isolated processes take place in Legendrian submanifolds of
$T^{\ast}Q\times\mathbb{R},$ motivating their study.

It turns out that the integral curves of the evolution vector field
$\mathcal{E}_{H}$ can be interpreted as an isolated system

######  Proposition 7.3.

The integral curves of $\mathcal{E}_{H}$ satisfy

$\frac{dH}{dt}=0.$

Furthermore, locally

$\frac{dS}{dt}=p_{i}dq^{i}=\frac{P_{i}}{T}dq^{i}.$

###### Proof.

Indeed, by definition we know that

$i_{\operatorname{grad}H}d\eta+\eta(H)\eta=dH.$

Therefore,

$\displaystyle\frac{dH}{dt}$ $\displaystyle=dH\cdot\mathcal{E}_{H}=dH\cdot
X_{H}+H\mathcal{R}(H)=dH\cdot\operatorname{grad}H-(\mathcal{R}(H)+H)\mathcal{R}(H)+H\mathcal{R}(H)$
$\displaystyle=(i_{\operatorname{grad}H}d\eta+\eta(H)\eta)\cdot\operatorname{grad}H-(\mathcal{R}(H))^{2}=0.$

Now, the last part is a consequence of the equality

$dH=dS-p_{i}dq^{i}.$

∎

For more details on the subject, we refer to [ArnoldThermo,
GeometrySomeThermo, simoes2020contact, simoes2021ThermodynamicSystems,
MRUGALAcontactstrcuture, MRUGALAcontinuouscontact], (see also two modern
approaches [bravetti1, bravetti2]).

###  7.3. Coisotropic reduction

Coisotropic reduction in contact manifolds has been developed in
[de2019contact] (see also [Tortorella2017, le2019deformations]).

The following definition will result useful. Given a subspace
$\Delta_{q}\subseteq T_{q}M$, we define the $d\eta$-orthogonal complement as

$\Delta_{q}^{\perp_{d\eta}}:=\\{v\in T_{q}M,\,|\,\,d\eta(v,w)=0\,\,\forall
w\in\Delta_{q}\\}.$

######  Proposition 7.4.

Let $\Delta_{q}\subseteq T_{q}M$ be a subspace. Then

$\Delta_{q}^{\perp_{d\eta}}\cap\mathcal{H}_{q}\subseteq\Delta_{q}^{\perp_{\Lambda}}.$

Furthermore, if $\mathcal{R}(q)\in\Delta_{q}$ or
$\Delta_{q}\subseteq\mathcal{H}_{q}$, the equality holds.

###### Proof.

Let $v\in\Delta_{q}^{\perp_{d\eta}}\cap\mathcal{H}_{q}$ and take
$\alpha:=i_{v}d\eta$. It is clear that $\alpha\in\Delta_{q}^{0}$. We will
prove that $\sharp_{\Lambda}(-\alpha)=v$. Indeed, for $\beta\in T^{*}_{q}M$,
since $b_{\eta}(v)=\alpha$ (as a direct calculation shows), we have

$\displaystyle\langle\beta,\sharp_{\Lambda}(\alpha)\rangle$
$\displaystyle=\Lambda(\alpha,\beta)=\Omega(\sharp_{\eta}(\alpha),\sharp_{\eta}(\beta))=\Omega(v,\sharp_{\eta}(\beta))$
$\displaystyle=-\Omega(\sharp_{\eta}(\beta),v)-\eta(\sharp_{\Lambda}(\beta))\eta(v)=-\langle\flat_{\eta}(\sharp_{\eta}(\beta)),v\rangle=-\langle\beta,v\rangle,$

that is, $v=\sharp_{\Lambda}(-\alpha)$.

Now, if $\mathcal{R}(q)\in\Delta_{q}$, we compare dimensions. Since
$\operatorname{Ker}\sharp_{\Lambda}=\langle\eta\rangle$ and
$\eta\not\in\Delta_{q}^{0}$, we have

$\operatorname{dim}\Delta_{q}^{\perp_{\Lambda}}=\operatorname{dim}\Delta_{q}^{0}=\operatorname{dim}M-\operatorname{dim}\Delta_{q}.$

Furthermore, $\Delta^{\perp_{d\eta}}\cap\mathcal{H}_{q}$ has the same
dimension, since

$\displaystyle\Delta_{q}^{\perp_{d\eta}}\cap\mathcal{H}_{q}$
$\displaystyle=(\Delta_{q}\cap\mathcal{H}_{q}\oplus\mathcal{V}_{q})^{\perp_{d\eta}}\cap\mathcal{H}_{q}=(\Delta_{q}\cap\mathcal{H}_{q})^{\perp_{d\eta}}\cap\mathcal{V}_{q}^{\perp_{d\eta}}\cap\mathcal{H}_{q}$
$\displaystyle=(\Delta_{q}\cap\mathcal{H}_{q})^{\perp_{d\eta}}\cap\mathcal{H}_{q}.$

This latter is just the symplectic complement in
$(\mathcal{H}_{q},d\eta|_{\mathcal{H}})$ and hence,

$\operatorname{dim}(\Delta_{q}^{\perp_{d\eta}}\cap\mathcal{H}_{q})=\operatorname{dim}\mathcal{H}_{q}-\operatorname{dim}(\Delta_{q}\cap\mathcal{H}_{q})=\operatorname{dim}M-1-(\operatorname{dim}\Delta_{q}-1).$

Now, if $\Delta_{q}\subseteq\mathcal{H}_{q}$, $\eta\in\Delta_{q}^{0}$ and,
thus,

$\operatorname{dim}\Delta_{q}^{\perp_{d\eta}}=\operatorname{dim}M-\operatorname{dim}\Delta_{q}-1.$

Since $\Delta_{q}^{\perp_{d\eta}}\cap\mathcal{H}_{q}$ is just the symplectic
complement of $\Delta_{q}$ we have

$\operatorname{dim}(\Delta_{q}^{\perp_{d\eta}}\cap\mathcal{H}_{q})=\operatorname{dim}\mathcal{H}_{q}-\operatorname{dim}\Delta_{q}=\operatorname{dim}M-1-\operatorname{dim}\Delta_{q}.$

∎

This proposition allows us to characterize Legendrian submanifolds:

######  Lemma 7.1.

If $L\hookrightarrow M$ is a Legendrian submanifold, then $L$ is horizontal
and $\operatorname{dim}L=n$ (where $\operatorname{dim}M=2n+1$). Furthermore,
if $L$ is horizontal and isotropic (or coisotropic) with
$\operatorname{dim}L=n$, $L$ is Legendrian.

###### Proof.

Since $\sharp_{\Lambda}$ takes values in $\mathcal{H}$, it is clear that every
Legendrian submanifold is horizontal. Since $L$ is horizontal,

$\operatorname{dim}T_{q}L^{\perp_{\Lambda}}=\operatorname{dim}M-\operatorname{dim}L-1.$

From the previous equation and using that $T_{q}L^{\perp_{\Lambda}}=T_{q}L$,
we deduce that
$\operatorname{dim}L=\operatorname{dim}M-\operatorname{dim}L-1$. This implies
$\operatorname{dim}L=n$. The last property is easily seen via a direct
comparison of dimensions. ∎

We will also need characterization of isotropic submanifolds in contact
geometry:

######  Lemma 7.2.

A submanifold $N\hookrightarrow M$ is isotropic if and only if $i^{*}\eta=0$.

###### Proof.

Necessity is clear, since $\sharp_{\Lambda}$ takes values in $\mathcal{H}$.
Now suppose that $N$ is horizontal. We have $i^{*}\eta=0$ and thus,
$i^{*}d\eta=0$. This implies that $T_{q}N\subseteq
T_{q}N^{\perp_{d\eta}}\cap\mathcal{H}_{q}\subseteq T_{q}N^{\perp_{\Lambda}},$
using Proposition 7.4. ∎

######  Proposition 7.5.

Let $i:N\hookrightarrow M$ be a coisotropic submanifold such that
$\mathcal{R}(q)\in T_{q}N$ for every $q\in N$ or
$T_{q}N\subseteq\mathcal{H}_{q}$ for every $q\in N$. Define
$\eta_{0}:=i^{*}\eta$. Then

$T_{q}N^{\perp_{\Lambda}}=\operatorname{Ker}{d\eta_{0}}\cap\operatorname{Ker}\eta_{0}.$

###### Proof.

Let $q\in N$. Proposition 7.4 implies that

$T_{q}N^{\perp_{\Lambda}}=T_{q}N^{\perp_{\eta}}\cap\mathcal{H}_{q}.$

But, since $T_{q}N$ is coisotropic, then it is just
$\operatorname{Ker}d\eta_{0}\cap\operatorname{Ker}\eta_{0}.$ ∎

We then have the following result:

######  Proposition 7.6.

Let $i:N\hookrightarrow M$ be a coisotropic subamnifold such that
$\mathcal{R}(q)\in T_{q}N$ for every $q\in N$ or
$T_{q}N\subseteq\mathcal{H}_{q}$ for every $q\in N$. Then, the distribution
$TN^{\perp_{\Lambda}}$ defined by $q\mapsto T_{q}N^{\perp_{\Lambda}}$ is
involutive.

###### Proof.

Denote $\eta_{0}:=i^{*}\eta$ and let $X,Y$ be vector fields along $N$ taking
values in $TN^{\perp_{\Lambda}}$. Proposition 7.5 implies that

$i_{X}d\eta_{0}=i_{Y}d\eta_{0}=0;\,\,\,\,i_{X}\eta_{0}=i_{Y}\eta_{0}=0.$

It suffices to check that

$i_{[X,Y]}d\eta_{0}=0;\,\,\,\,i_{[X,Y]}\eta_{0}=0.$

Indeed, taking $Z$ an arbitrary vector field in $N$, we have

$\displaystyle 0$
$\displaystyle=\operatorname{d}^{2}\eta_{0}(X,Y,Z)=X(d\eta_{0}(Y,Z))-Y(d\eta_{0}(X,Z))+Z(d\eta_{0}(X,Y))$
$\displaystyle-d\eta_{0}([X,Y],Z)+d\eta_{0}([X,Z],Y)-d\eta_{0}([Y,Z],X)=-d\eta_{0}([X,Y],Z),$

where we have used that $X,Y\in\operatorname{Ker}d\eta_{0}.$ In a similar way
we obtain

$\displaystyle
0=d\eta_{0}(X,Y)=X(\eta_{0}(Y))-Y(\eta_{0}(X))-\eta_{0}([X,Y])=-\eta_{0}([X,Y]),$

that is,
$[X,Y]\in\operatorname{Ker}d\eta_{0}\cap\operatorname{Ker}\eta_{0}=TN^{\perp_{\Lambda}}.$
∎

###  7.4. Vertical coisotropic reduction

We will restrict the study to vertical submanifolds, that is, submanifolds
satisfying $\mathcal{R}(q)\in T_{q}N$, for every $q\in N$. Notice that if $N$
is a coisotropic vertical submanifold, the distribution
$(TN)^{\perp_{\Lambda}}$ is regular of rank

$\operatorname{rank}\,(TN)^{\perp_{\Lambda}}=\operatorname{dim}M-\operatorname{dim}N.$

######  Theorem 7.1 (Vertical coisotropic reduction in the contact setting).

Let $(M,\eta)$ be a contact manifold and $i:N\hookrightarrow M$ be a
coisotropic submanifold such that $\mathcal{R}(q)\in T_{q}N$ for every $q\in
N$. If the space of all leaves $N/\mathcal{F}$ admits a manifold structure
such that the projection $\pi:N\rightarrow N/\mathcal{F}$ is a submersion,
then there exists a unique $1$-form $\eta_{N}$ in $N/\mathcal{F}$ such that
$\pi^{*}\eta_{N}=i_{*}\eta$ and $(N/\mathcal{F},\eta_{N})$ is a contact
manifold.

###### Proof.

Denote $\eta_{0}:=i^{*}\eta$. Uniqueness is clear from the imposed relation
since it forces us to define

$\eta_{N}([u]):=\eta_{0}(u).$

It only remains to check well-definedness and that it defines a contact
manifold. That this definition does not depend on the chosen representative
vector is clear since a vector tangent to the distribution is necessarily in
the kernel of $\eta$. Furthermore, if $X$ is a vector field tangent to the
distribution $TN^{\perp_{\Lambda}}$, Proposition 7.5 implies

$\mathcal{L}_{X}\eta_{0}=0=di_{X}\eta_{0}+i_{X}d\eta_{0}=0,$

since $X\in\operatorname{Ker}d\eta_{0}\cap\operatorname{Ker}\eta_{0}$.

To check that it is a contact manifold, we calculate the dimension of
$N/\mathcal{F}$. We have that
$\operatorname{rank}\,(T_{q}N)^{\perp_{\Lambda}}=\operatorname{dim}M-\operatorname{dim}N$.
We conclude, taking $k:=\operatorname{dim}N$, that

$\operatorname{dim}N/\mathcal{F}=2\operatorname{dim}N-\operatorname{dim}M=2(k-n-1)+1$

and therefore, $(N/\mathcal{F},\eta_{N})$ is a contact manifold if and only if

$\eta_{N}\wedge(d\eta_{N})^{k-n-1}\neq 0.$

Since $\pi$ is a submersion, this is equivalent to
$\eta_{0}\wedge(d\eta_{0})^{k-n-1}\neq 0$. This is straightforward using
Proposition 7.5. ∎

####  7.4.1. Projection of Legendrian submanifolds

Now we check that the image of a Legendrian submanifold $L\hookrightarrow M$
under the projection $\pi:N\rightarrow N/\mathcal{F}$ is again a Legendrian
submanifold.

######  Proposition 7.7.

Let $L\hookrightarrow M$ be a Legendrian submanifold such that $L$ and $N$
have clean intersection. If $L_{N}:=\pi(L\cap N)$ is a submanifold of
$N/\mathcal{F}$, then $L_{N}$ is Legendrian.

###### Proof.

It suffices to check that $L_{N}$ is horizontal, isotropic and
$\operatorname{dim}L_{N}=\operatorname{dim}N-n-1$ using Lemma 7.2. Since $L$
is horizontal, $L_{N}$ is horizontal and thus, $L_{N}$ is isotropic.

Comparing dimensions, we have

$\operatorname{dim}T_{[q]}L_{N}=\operatorname{dim}T_{q}L\cap
T_{q}N-\operatorname{dim}T_{q}L\cap T_{q}N^{\perp_{\Lambda}}.$ (10)

Now, since $(T_{q}L\cap
T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}=T_{q}L+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}}$
and $(T_{q}L\cap T_{q}N)^{\perp_{\Lambda}}$ is horizontal, we have

$\operatorname{dim}(T_{q}L+(T_{q}N^{\perp_{\Lambda}})^{\perp_{\Lambda}})=\operatorname{dim}M-\operatorname{dim}(T_{q}L\cap
T_{q}N^{\perp_{\Lambda}})-1.$ (11)

Using $\operatorname{dim}(T_{q}L\cap
T_{q}N^{\perp_{\Lambda}})=\operatorname{dim}(T_{q}L+T_{q}N)-1$ and
substituting (11)) in (10), we obtain

$\displaystyle\operatorname{dim}L_{N}$ $\displaystyle=\operatorname{dim}L\cap
N-\left(\operatorname{dim}M-\operatorname{dim}(T_{q}L+T_{q}N)\right)$
$\displaystyle=\operatorname{dim}L\cap
N-\operatorname{dim}M+\operatorname{dim}N+\operatorname{dim}L-\operatorname{dim}L\cap
N$ $\displaystyle=\operatorname{dim}N-2n-1+n=\operatorname{dim}N-n-1.$

∎

###  7.5. Horizontal coisotropic reduction

We will restrict the study to horizontal coisotropic submanifolds
$N\rightarrow M$, that is, manifolds satisfying
$T_{q}N\subseteq\mathcal{H}_{q}$ for every $q\in N$.

######  Remark 7.2.

Notice that in this case reduction is trivial, since the only coisotropic
horizontal submanifolds of a contact manifold are those that are Legendrian.
This would imply

$\operatorname{dim}N/\mathcal{F}=0,$

making the resulting manifold trivial.

Given an arbitrary coisotropic submanifold $N\hookrightarrow M$, we cannot
guarantee the well-definedness of the $2$-form in the quotient $N/\mathcal{F}$
(actually, in the contact setting, we cannot even guarantee the integrability
of $TN^{\perp_{\Lambda}}$) so this time (referring to horizontal reduction in
cosymplectic geometry) we cannot obtain a foliation of $N$ in symplectic
leaves, since $TN\cap\mathcal{H}|N$ is not integrable in the general setting.

######  Remark 7.3.

The triviality of this case makes the projection of Lagrangian submanifolds
trivial.

##  8\. An interlude: Different geometries provide different dynamics

In this section we will show how different geometric structures on the same
phase space can provide different dynamics for the same Lagrangian or
Hamiltonian function. This fact explains the different equations of motion
between the cosymplectic case (time-dependent Lagrangians) and the contact
case (action-dependent Lagrangians). At the end of the section we will also
see that Hamilton’s principle can be generalised to the so-called Herglotz
principle, which gives a new way to obtain these different dynamics.

###  8.1. The Lagrangian picture

In order to distinguish the different dynamics discussed in this paper and
their physical nature, it is useful to recall their Lagrangian formulation.

Assume that $L:TQ\longrightarrow\mathbb{R}$ is a Lagrangian function, where
$Q$ is a $n$-dimensional configuration manifold. Then,
$L=L(q^{i},\dot{q}^{i})$, where $(q^{i})$ are coordinates in $Q$ and
$(q^{i},\dot{q}^{i})$ are the induced bundle coordinates in $TQ$ (positions
and velocities). We will assume that $L$ is regular, that is, the Hessian
matrix

$\left(\frac{\partial^{2}L}{\partial\dot{q}^{i}\partial\dot{q}^{j}}\right)$

is regular. Using the canonical endomorphism $S$ on $TQ$ locally defined by

$S=dq^{i}\otimes\frac{\partial}{\partial\dot{q}^{i}}$

one can construct a 1-form $\lambda_{L}$ defined by

$\lambda_{L}=S^{*}(dL)$

and the 2-form

$\omega_{L}=-d\lambda_{L}$

Then, $\omega_{L}$ is symplectic if and only if $L$ is regular.

In that case, we have the corresponding vector bundle isomorphism

$\displaystyle\flat_{\omega_{L}}:T(TQ)\longrightarrow T^{*}(TQ)$
$\displaystyle\flat_{\omega_{L}}(v)=i_{v}\,\omega_{L}$

and the Hamiltonian vector field

$\xi_{L}=X_{E_{L}},$

defined by

$\flat_{\omega_{L}}(\xi_{L})=dE_{L},$

where $E_{L}=\Delta(L)-L$ is the energy, and
$\Delta=\dot{q}^{i}\,\frac{\partial}{\partial\dot{q}^{i}}$ is the Liouville
vector field on $TQ$.

The vector field $\xi_{L}$ (the Euler-Lagrange vector field) is a SODE (second
order differential equation), that is, its integral curves are just the
tangent lifts of its projections to $Q$. These projections are called the
solutions of $\xi_{L}$, and satisfy the usual Euler-Lagrange equations

$\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{q}^{i}}\right)-\frac{\partial L}{\partial q^{i}}=0.$ (12)

Next, we recall here the geometric formalism for time-dependent Lagrangian
systems. In this case, we also have a regular Lagrangian
$L:TQ\times\mathbb{R}\longrightarrow\mathbb{R}$, and we consider the
cosymplectic structure given by the pair $(\Omega_{L},dz)$ (in this case $z$
represents time), where

$\Omega_{L}=-d\lambda_{L}$

It is esay to check that $L$ is regular if and only if

$dz\wedge\Omega_{L}^{n}\not=0.$

In that case, we have a cosymplectic structure and, defining $(W^{ij})$ to be
the inverse matrix of

$(W_{ij})=\left(\frac{\partial^{2}L}{\partial\dot{q}^{i}\partial\dot{q}^{j}}\right),$

the Reeb vector field of the cosymplectic manifold $(TQ,\Omega_{L},dz)$ is
locally given by

$\mathcal{R}=\frac{\partial}{\partial
z}-W^{ij}\frac{\partial^{2}L}{\partial\dot{q}^{j}\partial
z}\,\frac{\partial}{\partial\dot{q}^{i}}.$

Recall that we had the vector bundle isomorphism

$\displaystyle{\flat}_{dz,\Omega_{L}}:T(TQ\times\mathbb{R})\longrightarrow
T^{*}(TQ\times\mathbb{R})$
$\displaystyle{\flat}_{dz,\Omega_{L}}(v)=i_{v}\,\Omega_{L}+dz(v)\,dz$

which gives the vector fields defined in Section 5. In particular, we have the
evolution vector field

$\mathcal{E}_{E_{L}}=X_{E_{L}}+\mathcal{R},$

where $E_{L}$ is the energy of the system,

$E_{L}=\Delta(L)-L=\dot{q}^{i}\frac{\partial L}{\partial q^{i}}-L.$

Now, if $(q^{i}(t),\dot{q}^{i}(t),z(t))$ is an integral curve of
${\mathcal{E}}_{L}$, then its projection to $Q$ satisfies the usual Euler-
Lagrange equations

$\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{q}^{i}}\right)-\frac{\partial L}{\partial q^{i}}=0,$ (13)

since $z=t+constant$.

Finally, let $L:TQ\times\mathbb{R}\longrightarrow\mathbb{R}$ be a Lagrangian
function, $L=L(q^{i},\dot{q}^{i},z)$, where $z$ is a global coordinate on
$\mathbb{R}$, representing action.

We will assume that $L$ is regular, that is, the Hessian matrix

$\left(\frac{\partial^{2}L}{\partial\dot{q}^{i}\partial\dot{q}^{j}}\right)$

is regular. So, we construct a 1-form $\lambda_{L}$ defined by

$\lambda_{L}=S^{*}(dL)$

where now $S$ and $S^{*}$ are the natural extension of $S$.

Now, the 1-form

$\eta_{L}=dz-\frac{\partial L}{\partial\dot{q}^{i}}\,dq^{i}$

is a contact form on $TQ\times\mathbb{R}$ if and only if $L$ is regular, and
then

$\eta_{L}\wedge(d\eta_{L})^{n}\not=0.$

The corresponding Reeb vector field is

${\mathcal{R}}=\frac{\partial}{\partial
z}-W^{ij}\frac{\partial^{2}L}{\partial\dot{q}^{j}\partial
z}\,\frac{\partial}{\partial\dot{q}^{i}}.$

The energy of the system is defined as in the precedent cases by

$E_{L}=\Delta(L)-L$

where $\Delta=\dot{q}^{i}\,\frac{\partial}{\partial\dot{q}^{i}}$ is the
Liouville vector field on $TQ$ extended in the usual way to
$TQ\times\mathbb{R}$. The vector bundle isomorphism of contact manifolds
defined in Section 7 is given by

${\flat_{\eta_{L}}}:T(TQ\times\mathbb{R})\longrightarrow
T^{*}(TQ\times\mathbb{R});\,\,\,{\flat_{\eta_{L}}}(v)=i_{v}(d\eta_{L})+(i_{v}\eta_{L})\cdot\eta_{L}.$

We shall denote its inverse by
${\sharp_{\eta_{L}}}=({\flat_{\eta_{L}})}^{-1}$.

Denote by ${\xi}_{L}$ the unique vector field defined by the equation

${\flat_{\eta_{L}}}({\xi}_{L})=dE_{L}-(\mathcal{R}(E_{L})+E_{L})\,\eta_{L}$
(14)

Note that this is precisely the Hamiltonian vector field of $E_{L}$ defined in
Section 7. A direct computation from eq. (14) shows that, if
$(q^{i}(t),\dot{q}^{i}(t),z(t))$ is an integral curve of $\bar{\xi}_{L}$, we
obtain

${\ddot{q}}^{i}\,\frac{\partial}{\partial\dot{q}^{i}}\left(\frac{\partial
L}{\partial\dot{q}^{j}}\right)+\dot{q}^{i}\,\frac{\partial}{\partial
q^{i}}\left(\frac{\partial
L}{\partial\dot{q}^{j}}\right)+\dot{z}\frac{\partial}{\partial
z}\left(\frac{\partial L}{\partial\dot{q}^{i}}\right)-\frac{\partial
L}{\partial q^{i}}=\frac{\partial L}{\partial\dot{q}^{i}}\frac{\partial
L}{\partial z}$

with an additional equation $\dot{z}=L$. These equations correspond to the
generalized Euler-Lagrange equations considered by G. Herglotz in 1930.

$\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{q}^{i}}\right)-\frac{\partial L}{\partial
q^{i}}=\frac{\partial L}{\partial\dot{q}^{i}}\frac{\partial L}{\partial z}$
(15)

Notice that Herglotz equations depend on the action, so this type of
Lagrangians are called in physics dependent on the action.

###  8.2. The Herglotz principle

In order to give additional differences between the usual Hamilton principle
and Herglotz principle, it is interesting to recall briefly the last one and
how it is a natural generalization of the former one.

Let $L:TQ\times\mathbb{R}\to\mathbb{R}$ be a Lagrangian function.

Fix $q_{1},q_{2}\in Q$ and an interval $[a,b]\subset\mathbb{R}$. We denote by
$\Omega(q_{1},q_{2},[a,b])\subseteq({\cal C}^{\infty}([a,b]\to Q))$ the space
of smooth curves $\xi$ such that $\xi(a)=q_{1}$ and $\xi(b)=q_{2}$. This space
has the structure of an infinite dimensional smooth manifold whose tangent
space at $\xi$ is given by the set of vector fields over $\xi$ that vanish at
the endpoints, that is,

$\displaystyle T_{\xi}\Omega(q_{1},q_{2},[a,b])=\\{$ $\displaystyle
v_{\xi}\in{\cal C}^{\infty}([a,b]\to TQ)\mid$ (16) $\displaystyle\tau_{Q}\circ
v_{\xi}=\xi,\,v_{\xi}(a)=0,\,v_{\xi}(b)=0\\}.$

We will consider the following maps. Fix an initial action $c\in\mathbb{R}$.
Let

$\mathcal{Z}:\Omega(q_{1},q_{2},[a,b])\to{\cal
C}^{\infty}([a,b]\to\mathbb{R})$ (17)

be the operator that assigns to each curve $\xi$ the curve $\mathcal{Z}(\xi)$
that solves the following ODE:

$\frac{d\mathcal{Z}(\xi)(t)}{dt}=L(\xi(t),\dot{\xi}(t),\mathcal{Z}(\xi)(t)),\quad\mathcal{Z}(\xi)(a)=c.$
(18)

Now we define the _action functional_ as the map which assigns to each curve
the solution to the previous ODE evaluated at the endpoint:

$\displaystyle\mathcal{A}:\Omega(q_{1},q_{2},[a,b])$
$\displaystyle\to\mathbb{R},$ (19) $\displaystyle\xi$
$\displaystyle\mapsto\mathcal{Z}(\xi)(b),$

that is, $\mathcal{A}=ev_{b}\circ\mathcal{Z}$, where
$ev_{b}:\zeta\mapsto\zeta(b)$ is the evaluation map at $b$. We have

######  Theorem 8.1.

Let $L:TQ\times\mathbb{R}\to\mathbb{R}$ be a Lagrangian function and let
$\xi\in\Omega(q_{1},q_{2},[a,b])$ be a curve in $Q$. Then,
$(\xi,\dot{\xi},\mathcal{Z}(\xi))$ satisfies the Herglotz’s equations if and
only if $\xi$ is a critical point of $\mathcal{A}$.

######  Remark 8.1.

This theorem generalizes Hamilton’s Variational Principle. In the case that
the Lagrangian is independent of the $\mathbb{R}$ coordinate (i.e.,
$L(q^{i},\dot{q}^{i},z)=\hat{L}(q^{i},\dot{q}^{i})$) the contact Lagrange
equations reduce to the usual Euler-Lagrange equations. In this situation, we
can integrate the ODE of (19) and we get

$\mathcal{A}(\xi)=\int_{a}^{b}\hat{L}(\xi(t),\dot{\xi}(t))dt+\frac{c}{b-a},$
(20)

that is, the usual Euler-Lagrange action up to a constant.

###  8.3. The Legendre transformation and the Hamiltonian picture

Given a Lagrangian function $L:TQ\times\mathbb{R}\longrightarrow\mathbb{R}$ we
can define the Legendre transformation

$\mathbb{F}L:TQ\times\mathbb{R}\longrightarrow T^{*}Q\times\mathbb{R}$

given by

$\mathbb{F}L(q^{i},\dot{q}^{i},z)=(q^{i},\hat{p}_{i},z)$

where

$\hat{p}_{i}=\frac{\partial L}{\partial\dot{q}^{i}}$

A direct computation shows that

$\mathbb{F}L^{*}\lambda_{Q}=\lambda_{L},$

where $\lambda_{Q}$ is the canonical Liouville form on $T^{*}Q$, here extended
to the product manifold $T^{*}Q\times\mathbb{R}$. If the Lagrangian does not
depend on time, then the Legendre transformation reduces to a bundle morphism

$\mathbb{F}L:TQ\longrightarrow T^{*}Q$

and then the pull-back of the canonical symplectic form $\omega_{Q}$ on
$T^{*}Q$ is just $\omega_{L}$. In addition, the energy $E_{L}$ corresponds via
the Legendre transformation to the Hamiltonian energy $H$ such that

$(\mathbb{F}L)^{*}(H)=E_{L}.$

Since the corresponding geometric structures (symplectic, cosymplectic and
contact) on both sides are preserved by the Legendre transformation, one
concludes that corresponding dynamics are $\mathbb{F}L$-related (see [canaria]
for the details).

##  9\. Coisotropic reduction in cocontact geometry

Cocontact manifolds have been introduced in [de2022time] just to provide a
setting for dissipative systems which also depend on time. In geometric terms,
we are combining cosymplectic and contact structures.

######  Definition 9.1 (Cocontact manifold).

A cocontact manifold is a triple $(M,\theta,\eta)$, where $M$ is a
$(2n+2)$-dimensional manifold, $\theta$ is a closed $1$-form, $\eta$ is a
$1$-form and, $\theta\wedge\eta\wedge(d\eta)^{n}\neq 0$ is a volume form.

The bundle isomorphism in this case is defined as

$\flat_{\theta,\eta}:TM\rightarrow
T^{*}M;\,\,v\mapsto\theta(v)\theta+i_{v}d\eta+\eta(v)\eta,$

and its inverse is denoted by $\sharp_{\theta,\eta}=\flat_{\theta,\eta}^{-1}.$

In cocontact geometry there exists as well a set of canonical coordinates
$(q^{i},p_{i},z,t)$, which will be called Darboux coordinates, such that

$\eta=dz-p_{i}dq^{i};\,\,\theta=dt.$

We can define as well the Reeb vector fields as

$\mathcal{R}_{z}:=\sharp_{\theta,\eta}(\eta);\,\,\mathcal{R}_{t}=\sharp_{\theta,\eta}(\theta),$

which can be expressed locally as

$\mathcal{R}_{z}=\frac{\partial}{\partial
z};\,\,\mathcal{R}_{t}=\frac{\partial}{\partial t}.$

We also have vertical and horizontal distributions:

  * $i)$

The $z$-horizontal distribution, $\mathcal{H}_{z}:=\operatorname{Ker}\eta$;

  * $ii)$

The $t$-horizontal distribution, $\mathcal{H}_{t}:=\operatorname{Ker}\theta$;

  * $iii)$

The $tz$-horizontal distribution
$\mathcal{H}_{tz}:=\mathcal{H}_{t}\cap\mathcal{H}_{z}$;

  * $iv)$

The $t$-vertical distribution,
$\mathcal{V}_{t}:=\langle\mathcal{R}_{t}\rangle$;

  * $v)$

The $z$-vertical distribution,
$\mathcal{V}_{z}:=\langle\mathcal{R}_{z}\rangle.$

###  9.1. Hamiltonian vector fields as Lagrangian and Legendrian submanifolds

Just like in previous sections, define the gradient vector field of certain
Hamiltonian $H\in\mathcal{C}^{\infty}(M)$ as

$\operatorname{grad}H=\sharp_{\theta,\eta}(dH).$

Locally, the gradient vector field is expressed:

$\operatorname{grad}H=\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial q^{i}}-\left(\frac{\partial H}{\partial
q^{i}}+p_{i}\frac{\partial H}{\partial z}\right)\frac{\partial}{\partial
p_{i}}+\left(p_{i}\frac{\partial H}{\partial p_{i}}+\frac{\partial H}{\partial
z}\right)\frac{\partial}{\partial z}+\frac{\partial H}{\partial
t}\frac{\partial}{\partial t}.$

We can define a symplectic structure in $TM$ taking

$\Omega_{0}:=\flat_{\theta,\eta}^{*}\Omega_{M},$

where $\Omega_{M}$ is the canonical symplectic form on the cotangent bundle.
In the induced coordinates $(q^{i},p_{i},z,\dot{q}^{i},\dot{p}_{i},\dot{z}),$
$\Omega_{0}$ takes the form

$\displaystyle\Omega_{0}=\,$ $\displaystyle
dq^{i}\wedge\left(p_{i}p_{i}dq^{j}+(1+\delta_{i}^{j})p_{i}\dot{q}^{j}dp_{j}-\dot{z}dp_{i}-d\dot{p}_{i}-p_{i}d\dot{z}\right)$
$\displaystyle+dp_{i}\wedge
d\dot{q}^{i}+dz\wedge\left(d\dot{z}-p_{i}d\dot{q}^{i}-\dot{q}^{i}dp_{i}\right)+dt\wedge
d\dot{t}.$

It is easy to verify that a vector field $X:M\rightarrow TM$ is a locally
gradient vector field if and only if it defines a Lagrangian submanifold in
$(TM,\Omega_{0}).$

######  Definition 9.2 (Hamiltonian vector field).

Given a Hamiltonian $H$ on $M$, define its Hamiltonian vector field as

$X_{H}:=\sharp_{\theta,\eta}(dH)-(\mathcal{R}_{z}(H)+H)\mathcal{R}_{z}+(1-\mathcal{R}_{t}(H))\mathcal{R}_{t}.$

The Hamiltonian vector field has the local expression

$X_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\left(\frac{\partial H}{\partial q^{i}}+p_{i}\frac{\partial H}{\partial
z}\right)\frac{\partial}{\partial p_{i}}+\left(p_{i}\frac{\partial H}{\partial
p_{i}}-H\right)\frac{\partial}{\partial z}+\frac{\partial}{\partial t}.$

Integral curves of this vector field satisfy

$\displaystyle\frac{dq^{i}}{d\lambda}$ $\displaystyle=\frac{\partial
H}{\partial p_{i}},$ $\displaystyle\frac{dp_{i}}{d\lambda}$
$\displaystyle=-\frac{\partial H}{\partial q^{i}}-p_{i}\frac{\partial
H}{\partial z},$ $\displaystyle\frac{dz}{d\lambda}$
$\displaystyle=p_{i}\frac{\partial H}{\partial p_{i}}-H,$
$\displaystyle\frac{dt}{d\lambda}$ $\displaystyle=1,$

which are equivalent to the time-dependent version of the contact equations:

$\displaystyle\frac{dq^{i}}{dt}$ $\displaystyle=\frac{\partial H}{\partial
p_{i}},$ $\displaystyle\frac{dp_{i}}{dt}$ $\displaystyle=-\frac{\partial
H}{\partial q^{i}}-p_{i}\frac{\partial H}{\partial z},$
$\displaystyle\frac{dz}{dt}$ $\displaystyle=p_{i}\frac{\partial H}{\partial
p_{i}}-H.$

In general, $X_{H}$ does not define a Lagrangian submanifold of
$(TM,\Omega_{0})$; but, just like in the cosymplectic and contact scenario, we
can achieve this by modifying the symplectic form. Indeed, since

$X_{H}^{*}\Omega_{0}=-dX_{H}^{\flat_{\theta,\eta}}=d\left((\mathcal{R}_{z}(H)+H)\eta\right)-d\left((1-\mathcal{R}_{t}(H))\theta\right),$

defining

$\Omega_{H}:=\Omega_{0}-\operatorname{d}((\mathcal{R}_{z}(H)+H)\eta)+\operatorname{d}((1-\mathcal{R}_{t}(H))\theta),$

we have that $X_{H}$ defines a Lagrangian submanifold of $(TM,\Omega_{H}).$

Now we interpret the Hamiltonian vector field $X_{H}$ as a Legendrian
submanifold of $TM\times\mathbb{R}\times\mathbb{R}$ with the cocontact
structure given by the forms

$\widetilde{\eta}:=\eta^{c}+s\eta^{v}+\theta^{c}+e\theta^{v};\,\,\,\widetilde{\theta}=\theta^{c},$

where $(s,e)$ are the parameters in $\mathbb{R}\times\mathbb{R}$. In local
coordinates $(q^{i},p_{i},z,t,\dot{q}^{i},\dot{p}_{i},\dot{z},\dot{t},s,e)$,
these forms have the expression:

$\displaystyle\widetilde{\eta}$
$\displaystyle=d\dot{z}-\dot{p}_{i}dq^{i}-p_{i}d\dot{q}^{i}+sdz-
sp_{i}dq^{i}+d\dot{t}+edt,$ $\displaystyle\widetilde{\theta}$
$\displaystyle=d\dot{t}.$

It is easy to see that these forms define a cocontact structure. Now, given a
vector field $X:M\rightarrow TM$ and two functions $f,g$ on $M$, define

$X\times f\times g:M\rightarrow
TM\times\mathbb{R}\times\mathbb{R};\,\,\,p\mapsto(X(p),f(p),g(p)).$

Applying the properties of complete and vertical lifts, namely
$X^{*}\alpha^{c}=\mathcal{L}_{X}(\alpha)$, we have

$(X\times f\times
g)^{*}\widetilde{\eta}=\mathcal{L}_{X}\eta+f\eta+\mathcal{L}_{X}\theta+g\theta$

and

$(X\times f\times g)^{*}\widetilde{\theta}=d\theta(X).$

######  Proposition 9.1.

Let $H$ be a Hamiltonian on $M$. Then $X_{H}\times\mathcal{R}_{z}(H)\times 0$
defines a Legendrian submanifold of
$(TM\times\mathbb{R}\times\mathbb{R},\widetilde{\theta},\widetilde{\eta}).$

###### Proof.

Using the observation above and Lemma 9.1, it is sufficient to observe that

$\mathcal{L}_{X_{H}}\eta=-\mathcal{R}_{z}(H)\eta,\,\,\mathcal{L}_{X_{H}}\theta=0.$

∎

###  9.2. Coisotropic reduction

A cocontact manifold is also a Jacobi manifold defining

$\Lambda(\alpha,\beta):=-d\eta(\sharp(\alpha),\sharp(\beta)),\,\,E=-\mathcal{R}_{z},$

and thus, we have the $\Lambda$ -orthogonal and the corresponding definitions
of isotropic, coisotropic or Legendrian submanifolds and distributions.

Notice that $\mathcal{H}_{t}$ is an integrable distribution and that each leaf
of its foliation inherits a contact structure. Indeed, $\mathcal{H}_{t}$ is
the characteristic distribution defined by the Jacobi structure,
$\mathcal{S}$.

Now we give a symplectic interpretation of the $\Lambda$-orthogonal. Notice
that the restriction of $d\eta$ to $\mathcal{H}_{tz}$ defines a symplectic
structure on the distribution. Denote by $\perp_{d\eta|}$ its symplectic
orthogonal. The $\Lambda$-orthogonal is just the symplectic orthogonal of the
intersection with $\mathcal{H}_{tz}$.

######  Proposition 9.2.

Given a distribution $\Delta$ on a cocontact manifold $(M,\eta,\theta)$,

$\Delta^{\perp_{\Lambda}}=(\Delta\cap\mathcal{H}_{tz})^{\perp_{d\eta|}}.$

###### Proof.

We check one inclusion and compare dimensions:
Let $\alpha\in\Delta^{0}_{q}$ and
$u\in\Delta^{0}_{q}\cap(\mathcal{H}_{tz})_{q}.$ We will see that
$d\eta_{q}(u,\sharp_{\Lambda}(\alpha))=0.$ Indeed,

$\displaystyle d\eta_{q}(u,\sharp_{\Lambda}(\alpha))$
$\displaystyle=d\eta_{q}(u,\sharp_{\Lambda}(\alpha))+\theta_{q}(u)\theta_{q}(\sharp_{\Lambda}(\alpha))+\eta_{q}(u)\eta_{q}(\sharp_{\Lambda}(\alpha))$
$\displaystyle=\langle\flat(u),\sharp_{\Lambda}(\alpha)\rangle=\Lambda_{q}(\alpha,\flat(u))=-d\eta_{q}(\sharp(\alpha),\sharp(\flat(u)))$
$\displaystyle=-d\eta_{q}(\sharp(\alpha),u)=-d\eta_{q}(\sharp(\alpha),u)-\theta_{q}(\sharp(\alpha))\theta_{q}(u)-\eta_{q}(\sharp(\alpha))\eta_{q}(u)$
$\displaystyle=-\langle\flat(\sharp(\alpha)),u\rangle=-\alpha(u)=0,$

that is,
$\Delta^{\perp_{\Lambda}}\subseteq(\Delta\cap(\mathcal{H}_{tz})_{q})^{\perp_{d\eta|}}.$

Now we compare both dimensions. Let
$k:=\operatorname{dim}\Delta,r_{q}:=\operatorname{dim}(\Delta_{q}^{0}\cap\langle\theta_{q},\eta_{q}\rangle).$
Since

$\Delta_{q}^{0}\cap\langle\theta_{q},\eta_{q}\rangle=(\Delta_{q}\oplus(\mathcal{H}_{tz})_{q})^{0},$

we have

$\displaystyle r_{q}$
$\displaystyle=\operatorname{dim}(\Delta_{q}^{0}\cap\langle\theta_{q},\eta_{q}\rangle)=\operatorname{dim}(\Delta_{q}\oplus(\mathcal{H}_{tz})_{q})^{0}=2n+2-\operatorname{dim}(\Delta_{q}\oplus(\mathcal{H}_{tz})_{q})$
$\displaystyle=2n+2-(\operatorname{dim}\Delta_{q}+\operatorname{dim}(\mathcal{H}_{tz})_{q}-\operatorname{dim}(\Delta_{q}\cap(\mathcal{H}_{tz})_{q}))=2n+2-k-2n+\operatorname{dim}(\Delta_{q}\cap(\mathcal{H}_{tz})_{q})$
$\displaystyle=2+\operatorname{dim}(\Delta_{q}\cap(\mathcal{H}_{tz})_{q})-k,$

which implies that

$\operatorname{dim}(\Delta_{q}\cap(\mathcal{H}_{tz})_{q})=k+r_{q}-2.$

It only remains to observe that

$\operatorname{dim}\Delta^{\perp_{\Lambda}}=2n+2-k-r_{q}=\operatorname{dim}\Delta^{0}-\operatorname{dim}(\Delta^{0}\cap\operatorname{Ker}\sharp_{\Lambda})$

and that

$\operatorname{dim}(\Delta\cap(\mathcal{H}_{tz})_{q})^{\perp_{d\eta|}}=2n+2-k-r_{q}=2n-\operatorname{dim}(\Delta\cap(\mathcal{H}_{tz})_{q}).$

∎

Now we can give a characterization of Legendrian submanifolds:

######  Lemma 9.1.

$i:L\rightarrow M$ is a Legendrian submanifold ($(TL)^{\perp_{\Lambda}}=TL$)
if and only if $\operatorname{dim}L=n$ and

$i^{*}\theta=0,i^{*}\eta=0.$

###### Proof.

Necessity is clear, since $L$ is necessarily horizontal. Sufficiency follows
from $i^{*}d\eta=0$ which, together with $\operatorname{dim}L=n$, implies that
$T_{q}L$ is a Lagrangian submanifold of $(\mathcal{H}_{tz})_{q}$, for every
$q\in L$. Using Proposition 9.2, we have the equivalence. ∎

This allows us to express the $\Lambda$-orthogonal complement of a coisotropic
distribution in a more convenient way:

######  Corollary 9.1.

Let $\Delta$ be a coisotropic distribution on $M$
($\Delta^{\perp_{\Lambda}}\subseteq\Delta$), then

$\Delta^{\perp_{\Lambda}}=\operatorname{Ker}d\eta_{0}|_{\Delta\cap\mathcal{H}_{tz}}\cap\operatorname{Ker}\eta_{0}\cap\operatorname{Ker}\theta_{0},$

where $\eta_{0}$ and $\theta_{0}$ are the restrictions of $\eta$ and $\theta$
to $\Delta$, respectively.

######  Corollary 9.2.

Let $i:N\hookrightarrow M$ be a coisotropic submanifold of $(M,\theta,\eta).$
Then the distribution $TN^{\perp_{\Lambda}}$ is involutive.

###### Proof.

Denote $\eta_{0}:=i^{*}\eta$, $\theta_{0}:=i^{*}\theta$ and let $X,Y$ be
vector fields on $N$ tangent to the distribution $TN^{\perp_{\Lambda}}$ and
$Z$ be an arbitrary $tz$-horizontal vector field on $N$. Using Corollary 9.2,
we only need to check that
$[X,Y]\in\operatorname{Ker}d\eta_{0}|_{\Delta\cap\mathcal{H}_{tz}}\cap\operatorname{Ker}\eta_{0}\cap\operatorname{Ker}\theta_{0}.$
Indeed, expanding the expressions $0=\operatorname{d}^{2}\eta_{0}(X,Y,Z)$,
$0=d\eta_{0}(X,Y)$, $0=d\theta_{0}(X,Y)$; we obtain

$\displaystyle 0$ $\displaystyle=-d\eta_{0}([X,Y],Z),$ $\displaystyle 0$
$\displaystyle=-\eta_{0}([X,Y]),$ $\displaystyle 0$
$\displaystyle=-\theta_{0}([X,Y]).$

∎

Now, given a coisotropic submanifold $N\hookrightarrow M$, since
$TN^{\perp_{\Lambda}}$ is involutive, it provides a maximal foliation,
$\mathcal{F}$. We assume that $N/\mathcal{F}$ inherits a manifold structure
such that the canonical projection $\pi:N\rightarrow N/\mathcal{F}$ is a
submersion.

Just like in the previous cases, for the well-definedness and non-degeneracy
of the forms in the quotient, we need to restrict the coisotropic submanifolds
we are studying. Consequently, we will say that a submanifold
$N\hookrightarrow M$ is:

  * $i)$

$t$-vertical (resp. $z$-vertical) if $\mathcal{V}_{t}\subseteq TN$ (resp.
$\mathcal{V}_{z}\subseteq TN$).

  * $ii))$

$tz$-vertical, if it is both $t$-vertical and $z$-vertical.

  * $iii)$

$t$-horizontal (resp. $z$-horizontal) if $\mathcal{H}_{t}\subseteq TN$ (resp.
$\mathcal{H}_{z}\subseteq TN$).

  * $iv)$

$tz$-horizontal if it is both $t$-horizontal and $z$-horizontal, that is, if
$TN\subseteq\mathcal{H}_{tz}$.

###  9.3. $tz-$vertical reduction

Let $i:N\hookrightarrow M$ be a $tz$-vertical submanifold. It is easy to check
that under these conditions

$TN^{\perp_{\Lambda}}=\operatorname{Ker}d\eta_{0}\cap\operatorname{Ker}\eta_{0}\cap\operatorname{Ker}\theta_{0},$

and that $(TN)^{\perp_{\Lambda}}$ is a regular distribution of rank

$\operatorname{rank}\,(TN)^{\perp_{\Lambda}}=\operatorname{dim}M-\operatorname{dim}N.$

######  Theorem 9.1 ($tz$-vertical coisotropic reduction).

Let $i:N\hookrightarrow M$ be a $tz$-vertical submanifold of a cocontact
manifold $(M,\theta,\eta).$ Denote by $\mathcal{F}$ the maximal foliation
induced by the integrable distribution $TN^{\perp_{\Lambda}}$ on $N$. If
$N/\mathcal{F}$ admits a manifold structure such that the canonical projection
$\pi:N\rightarrow N/\mathcal{F}$ defines a submersion, then there exists
unique forms $\theta_{N},\eta_{N}$ on $N/\mathcal{F}$ such that
$(N/\mathcal{F},\theta_{N},\eta_{N})$ defines a cocontact structure and

$i^{*}\theta=\pi^{*}\theta_{N},\,\,i^{*}\eta_{N}=\pi^{*}\theta_{N}.$

Before proving the theorem, let us calculate the dimension of the quotient.
Let $k+2:=\operatorname{dim}N$. We have

$\operatorname{rank}\,TN^{\perp_{\Lambda}}=2n+2-(k+2)=2n-k$

and, therefore,

$\operatorname{dim}N/\mathcal{F}=2(k-n)+2.$

###### Proof.

Uniqueness is clear, since $\pi$ is a submersion. We only need to check the
well-definedness taking

$\theta_{N}([u]):=\theta_{0}(u),\,\,\eta_{N}([u]):=\eta_{0}(u).$

Independence of the vector is clear, using Proposition 9.2. For independence
on the point, let $X$ be a vector field on $N$ tangent to the distribution. It
is easy to check that

$\mathcal{L}_{X}\eta_{0}=0,\mathcal{L}_{X}\theta_{0}=0;$

and thus, well-definedness follows. For non-degeneracy, it is enough to proof
that

$\theta_{0}\wedge\eta_{0}\wedge(d\eta_{0})^{k-n}\neq 0.$

This follows easily from
$TN^{\perp_{\Lambda}}=\operatorname{Ker}d\eta_{0}\cap\operatorname{Ker}\eta_{0}\cap\operatorname{Ker}\theta_{0}.$
∎

####  9.3.1. Projection of Legendrian submanifolds

######  Proposition 9.3.

Let $L\hookrightarrow M$ be a Legendrian submanifold and $i:L\hookrightarrow
M$ be a $tz$-vertical coisotropic submanifold. If $L$ and $N$ have clean
intersection and $L_{N}=\pi(L\cap N)$ is a submanifold in $N/\mathcal{F}$,
$L_{N}$ is Legendrian.

###### Proof.

Using Lemma 9.1, $L_{N}$ is clearly isotropic. Now we need to check that
$\operatorname{dim}L_{N}=k-n$, given that
$\operatorname{dim}N/\mathcal{F}=2(k-n)+2.$ We have

$\operatorname{dim}\pi(L\cap N)=\operatorname{dim}(L\cap
N)-\operatorname{rank}\,(TL\cap(TN)^{\perp_{\Lambda}}).$ (21)

Furthermore, since
$(TL\cap(TN)^{\perp_{\Lambda}})=TL+(TN^{\perp_{\Lambda}})^{\perp_{\Lambda}}$
and $TL\cap(TN)^{\perp_{\Lambda}}$ is $tz$-horizontal,

$\operatorname{rank}\,(TL+(TN^{\perp_{\Lambda}})^{\perp_{\Lambda}})=2n-\operatorname{rank}\,(TL\cap
TN^{\perp_{\Lambda}}).$ (22)

Now, using the Grassman formula:

$\displaystyle\operatorname{rank}\,(TL+(TN^{\perp_{\Lambda}})^{\perp_{\Lambda}})$
$\displaystyle=\operatorname{dim}L+\operatorname{rank}\,(TN^{\perp_{\Lambda}})^{\perp_{\Lambda}}-\operatorname{rank}\,(TL\cap(TN^{\perp_{\Lambda}})^{\perp_{\Lambda}})$
(23)
$\displaystyle=\operatorname{dim}L+(\operatorname{dim}N-2)-\operatorname{dim}(L\cap
N)$ (24) $\displaystyle=n+k-\operatorname{dim}(L\cap N).$ (25)

From (22) and (25) we obtain

$\operatorname{rank}\,(TL\cap
TN^{\perp_{\Lambda}})=n-k+\operatorname{dim}(L\cap N).$ (26)

Substituting in (21) yields $\operatorname{dim}\pi(L\cap N)=k-n.$ ∎

###  9.4. $t-$vertical, $z-$horizontal reduction

Suppose $i:N\hookrightarrow M$ is a $t$-vertical and $z$-horizontal
coisotropic submanifold. This time we have

$(TN)^{\perp_{\Lambda}}=\operatorname{Ker}\theta_{0},$

since $\eta_{0}=0$ implies $d\eta_{0}=0$. We conclude that
$(TN)^{\perp_{\Lambda}}=TN\cap\mathcal{H}_{tz}$, which implies that

$\operatorname{dim}N/\mathcal{F}=1.$

This means that reduction is trivial, leaving the trivial cosymplectic
submanifold of dimension $1$:

######  Theorem 9.2.

Let $i:N\hookrightarrow M$ be a $t$-vertical, $z$-horizontal coisotropic
submanifold of a coconatct manifold $(M,\theta,\eta)$. Denote by $\mathcal{F}$
the maximal foliation defined by the distribution $(TN)^{\perp_{\Lambda}}$. If
$N/\mathcal{F}$ has a manifold structure such that the canonical projection
$\pi:N\rightarrow N/\mathcal{F}$ defines a submersion, then $N/\mathcal{F}$ is
one-dimensional and there exists and unique volume form $\theta_{N}$ on
$N/\mathcal{F}$ such that

$i^{*}\theta=\pi^{*}\theta_{N}.$

######  Remark 9.1.

Given the triviality of reduction in the $t$-vertical and $z$-horizontal case,
projection of Legendrian submanifolds in $M$ will always result in
$0$-dimensional Lagrangian submanifolds in $N/\mathcal{F}$.

###  9.5. $z-$vertical, $t-$horizontal reduction

Let $i:N\rightarrow M$ be a $z$-vertical and $t$-horizontal coisotropic
submanifold. It is easy to check that this time we have the equality

$TN=\operatorname{Ker}d\eta_{0}\cap\operatorname{Ker}\eta_{0}.$

Since $\mathcal{H}_{t}$ is integrable, we have that coisotropic reduction of
$N$ is actually happening in one of the leaves of the foliation that inhertits
a contact structure from the cocontact structure. We conclude, from Theorem
7.1:

######  Theorem 9.3.

Let $i:N\rightarrow M$ be a $z$-vertical and $t$-horizontal coisotropic
submanifold of a cocontact manifold $(M,\theta,\eta)$. Denote by $\mathcal{F}$
the maximal foliation on $N$ defined by the distribution
$TN^{\perp_{\Lambda}}$. If $N/\mathcal{F}$ has a manifold structure such that
the canonical projection $\pi:N\rightarrow N/\mathcal{F}$ defines a
submersion, then there exists an unique form $\eta_{N}$ such that
$(N/\mathcal{F},\eta_{N})$ is a contact manifold and

$i^{*}\eta=\pi^{*}\eta_{N}.$

####  9.5.1. Projection of Legendrian submanifolds

######  Proposition 9.4.

Let $L\hookrightarrow M$ be a Legendrian submanifold. If $L$ and $N$ have
clean intersection and $L_{N}=\pi(L\cap N)$ is a submanifold in
$N/\mathcal{F}$, $L_{N}$ is Legendrian in $(N/\mathcal{F},\theta_{N})$.

###### Proof.

It is clearly horizontal and, therefore, using Lemma 7.2, it is isotropic.
Now, supposing $k=\operatorname{dim}N$, we only need to check that

$\operatorname{dim}L_{N}=k-n,$

since $\operatorname{dim}N/\mathcal{F}=2(k-n)+1.$ This is straight-forward,
following the same steps given in Proposition 9.3. ∎

###  9.6. $tz-$horizontal reduction

Let $i:N\hookrightarrow M$ be a $tz$-horizontal coisotropic submanifold. Since
$\eta_{0}=0$, $d\eta_{0}=0$ and $\theta_{0}=0$, we have

$(TN)^{\perp_{\Lambda}}=TN,$

wich implies that

$\operatorname{dim}N/\mathcal{F}=0,$

leaving a trivial symplectic manifold, having as many points as path
components of $N$. This means that if $N/\mathcal{F}$ admits a manifold
structure, it will be a symplectic manifold.

######  Remark 9.2.

Proceeding as in the previous cases we immediately obtain that the projection
of Legendrian submanifolds is trivial.

##  10\. Coisotropic reduction in stable Hamiltonian structures

There were several attempts to combine cosymplectic and contact structures.
The first one is due to Albert [albert1989theoreme], using a combination of a
1-form and a 2-form; however, the setting is not useful for us since the lack
of integrability. The second attempt in this direction is due to Acakpo
[acakpo2022stable], which is studied in this section.

######  Definition 10.1 (Stable Hamiltonian structure).

A stable Hamiltonian structure (SHS) is a triple $(M,\omega,\lambda)$ where
$M$ is a $2n+1$ dimensional manifold, $\omega$ is a closed $2$-form and
$\lambda$ is a $1$-form such that

$\lambda\wedge\omega^{n}\neq
0,\,\,\,\operatorname{Ker}\omega\subseteq\operatorname{Ker}d\lambda.$

There exists, just like in the previous cases, a natural isomorphism

$\flat_{\lambda,\omega}:TM\rightarrow T^{*}M;\,\,\,v_{q}\mapsto
i_{vq}\omega+\lambda(v_{q})\cdot\lambda.$

and its inverse $\sharp_{\lambda,\omega}:=\flat_{\lambda,\omega}^{-1}.$ Let us
perform some calculations in coordinates. Since $d\omega$ is closed and of
constant range $2n$, around any point, there exists coordinates
$(q^{i},p_{i},z)$ such that

$\omega=dq^{i}\wedge dp_{i}$

(see [godbillon1969geometrie]). In this coordinate chart $\lambda$ will have
an expression of the form

$\lambda=a_{i}dq^{i}+b^{i}dp_{i}+cdz.$

Since

$0\neq\lambda\wedge\omega^{n}=cdz\wedge\omega^{n},$

we conclude that $c\neq 0$. Let $\varphi_{t}(q^{i},p_{i},z)$ be the (local)
flow of the vector field $\displaystyle{\frac{1}{c}\frac{\partial}{\partial
z}}$. Fix some value $z_{0}$, and define the map

$\psi(q^{i},p_{i},t):=(q^{i},p_{i},z(\varphi_{t}(q^{i},p_{i},z_{0}))).$

It is clear that this defines a local diffeomorphism. Take the new set of
coordinates to be

$(q^{i},p_{i},t):=\psi^{-1}\circ(q^{i},p_{i},z).$

We have

$dz=\frac{\partial z}{\partial q^{i}}dq^{i}+\frac{\partial z}{\partial
p_{i}}dp_{i}+\frac{\partial z}{\partial t}dt=\frac{\partial z}{\partial
q^{i}}dq^{i}+\frac{\partial z}{\partial p_{i}}dp_{i}+\frac{1}{c}dt.$

Therefore, in the new coordinate chart,

$\lambda=\left(a_{i}+\frac{\partial z}{\partial
q^{i}}\right)dq^{i}+\left(b_{i}+\frac{\partial z}{\partial
p_{i}}\right)dp_{i}+dt.$

We conclude:

######  Proposition 10.1.

Around every point of $M$ there exists a coordinate chart $(q^{i},p_{i},z)$
such that

$\omega=dq^{i}\wedge dp_{i},\lambda=a_{i}dq^{i}+b^{i}dp_{i}+dz.$

We call these coordinates Darboux coordinates.

In Darboux coordinates, the condition $\operatorname{Ker}d\omega\subseteq
d\lambda$ translates to

$\frac{\partial a_{i}}{\partial z}=\frac{\partial b^{i}}{\partial z}=0.$

Also, the musical isomorphisms take the expression:

$\displaystyle\flat_{\lambda,\omega}\left(\frac{\partial}{\partial
q^{i}}\right)$
$\displaystyle=dp_{i}+a_{i}a_{j}dq^{j}+a_{i}b^{j}dp_{j}+a_{i}dz,$
$\displaystyle\flat_{\lambda,\omega}\left(\frac{\partial}{\partial
p_{i}}\right)$
$\displaystyle=-dq^{i}+b^{i}a_{j}dq^{j}+b^{i}b^{j}dp_{j}+b^{i}dz,$
$\displaystyle\flat_{\lambda,\omega}\left(\frac{\partial}{\partial z}\right)$
$\displaystyle=a_{i}dq^{i}+b^{i}dp_{i}+dz,$
$\displaystyle\sharp_{\lambda,\omega}\left(dq^{i}\right)$
$\displaystyle=-\frac{\partial}{\partial
p_{i}}+{b^{i}}\frac{\partial}{\partial z},$
$\displaystyle\sharp_{\lambda,\omega}\left(dp_{i}\right)$
$\displaystyle=\frac{\partial}{\partial q^{i}}-{a_{i}}\frac{\partial}{\partial
z},$ $\displaystyle\sharp_{\lambda,\omega}\left(dz\right)$
$\displaystyle=\frac{\partial}{\partial z}+{a_{i}}\frac{\partial}{\partial
p_{i}}-{b^{i}}\frac{\partial}{\partial q^{i}}.$

Imitating the definitions in the contact case, we can define a bivector field
on $M$ as

$\Lambda_{q}(\alpha_{q},\beta_{q}):=\omega_{q}(\sharp_{\lambda,\omega}(\alpha_{q}),\sharp_{\lambda,\omega}(\beta_{q})),$

and the morphism

$\sharp_{\Lambda}:T^{*}M\rightarrow TM;\,\,\,\alpha_{q}\mapsto
i_{\alpha_{q}}\Lambda$

with the induced $\Lambda$-orthogonal complement for distributions

$\Delta_{q}^{\perp_{\Lambda}}:=\sharp_{\Lambda}(\Delta_{q}^{0}).$

In coordinates $(q^{i},p_{i},z)$ the bivector field $\Lambda$ takes the local
form:

$\Lambda=\frac{\partial}{\partial q^{i}}\wedge\frac{\partial}{\partial
p_{i}}+\left({a_{i}}\frac{\partial}{\partial
p_{i}}-{b^{i}}\frac{\partial}{\partial
q^{i}}\right)\wedge\frac{\partial}{\partial z}.$

We also have the distributions

  * i)

$\mathcal{H}_{q}:=\operatorname{Ker}\lambda_{q},$

  * ii)

$\mathcal{V}_{q}:=\operatorname{Ker}\omega_{q},$

and the Reeb vector field
$\mathcal{R}_{q}:=\sharp_{\lambda,\omega}(\lambda_{q}).$ Locally,

$\displaystyle\mathcal{H}$ $\displaystyle=\langle\frac{\partial}{\partial
q^{i}}-{a_{i}}\frac{\partial}{\partial z},\frac{\partial}{\partial
p_{i}}-{b^{i}}\frac{\partial}{\partial z}\rangle,$ (27)
$\displaystyle\mathcal{V}$ $\displaystyle=\langle\frac{\partial}{\partial
z}\rangle,$ (28) $\displaystyle\mathcal{R}$
$\displaystyle=\frac{\partial}{\partial z}.$ (29)

A natural question to ask is wether the bivector field $\Lambda$ arises from a
Jacobi bracket. We have

$[\Lambda,\Lambda]=2\frac{\partial}{\partial p_{i}}\wedge\left(\frac{\partial
a_{j}}{\partial q^{i}}\frac{\partial}{\partial p_{j}}-\frac{\partial
b^{j}}{\partial q^{i}}\frac{\partial}{\partial
q^{j}}\right)\wedge\frac{\partial}{\partial z}-2\frac{\partial}{\partial
q^{i}}\wedge\left(\frac{\partial a_{j}}{\partial
p_{i}}\frac{\partial}{\partial p_{j}}-\frac{\partial b^{j}}{\partial
p_{i}}\frac{\partial}{\partial q^{j}}\right)\wedge\frac{\partial}{\partial
z}.$

Taking an arbitrary vector field

$E=X^{i}\frac{\partial}{\partial q^{i}}+Y_{i}\frac{\partial}{\partial
p_{i}}+Z\frac{\partial}{\partial z},$

$(\Lambda,E)$ defines a Jacobi structure if and only if

$[\Lambda,\Lambda]=2E\wedge\Lambda,\,\,\,[E,\Lambda]=0.$

It is easily checked that the first equality holds when

$\displaystyle\frac{\partial a_{j}}{\partial q^{i}}-\frac{\partial
a_{i}}{\partial q^{j}}=\frac{\partial b^{j}}{\partial p_{i}}-\frac{\partial
b^{i}}{\partial p_{j}}=0,$ (30) $\displaystyle\frac{\partial b^{i}}{\partial
q^{j}}-\frac{\partial a_{j}}{\partial p_{i}}=0,\,\,i\neq j,$ (31)
$\displaystyle\frac{\partial b^{i}}{\partial q_{i}}-\frac{\partial
a_{i}}{\partial p_{i}}=f,$ (32) $\displaystyle X^{i}=Y_{i}=0,$ (33)
$\displaystyle Z=f,$ (34)

for certain local unique funtion $f$. It is easy to check that this relations
translate intrinsically to

$d\lambda=f\omega,\,\,E=f\mathcal{R}.$

Now, let us compute $[E,\Lambda]$ for
$E=f\displaystyle{\frac{\partial}{\partial z}}.$

$[E,\Lambda]=\left(\frac{\partial f}{\partial q^{i}}-a_{i}\frac{\partial
f}{\partial z}\right)\frac{\partial}{\partial
p_{i}}\wedge\frac{\partial}{\partial z}+\left(-\frac{\partial f}{\partial
p_{i}}+b^{i}\frac{\partial f}{\partial z}\right)\frac{\partial}{\partial
q^{i}}\wedge\frac{\partial}{\partial z}.$

Therefore, $[E,\Lambda]=0$ if and only if

$\displaystyle\frac{\partial f}{\partial q^{i}}-a_{i}\frac{\partial
f}{\partial z}=-\frac{\partial f}{\partial p_{i}}+b^{i}\frac{\partial
f}{\partial z}=0.$ (35)

This is easily seen to be equivalent to

$\sharp_{\lambda,\omega}(df)\in\mathcal{V}.$

We have concluded the following:

######  Proposition 10.2.

The bivector field $\Lambda$ arises from a Jacobi structure if and only if
there exists some $f\in\mathcal{C}^{\infty}(M)$ such that

$d\lambda=f\omega,\,\,\ \sharp_{\lambda,\omega}(df)\in\mathcal{V}.$

And, in that case, the Jacobi structure is defined by the pair
$(\Lambda,f\mathcal{R})$.

######  Remark 10.1.

Notice that we recover the cosymplectic scenario when $f=0$ and the contact
scenario when $f=1$ (because the definition of $\Lambda$ in contact geometry
is the opposite of the definition we gave in SHS).

Let us return to the study of coisotropic reduction. It is easy to see that
$\omega$ induces a symplectic form in $\mathcal{H}$, $\omega|_{\mathcal{H}}$.
This induces the symplectic orthogonal for $\Delta_{q}\subset\mathcal{H}_{q}$:

$\Delta_{q}^{\perp_{\omega|_{\mathcal{H}}}}.=\\{v\in\mathcal{H}_{q}\,|\,\omega(v,w)=0\,\forall
w\in\Delta_{q}\\}.$

A distribution $\Delta$ in $M$ will be called:

  * i)

Isotropic if $\Delta\subseteq\Delta^{\perp_{\Lambda}};$

  * ii)

Coisotropic if $\Delta^{\perp_{\Lambda}}\subseteq\Delta;$

  * iii)

Lagrangian if
$\Delta^{\perp_{\Lambda}}=\Delta^{\perp_{\Lambda}}\cap\mathcal{H}.$

We have the following equality:

######  Proposition 10.3.

Let $\Delta$ be a distribution on $M$. Then

$\Delta^{\perp_{\Lambda}}=(\Delta\cap\mathcal{H})^{\perp_{\omega|_{\mathcal{H}}}}.$

###### Proof.

The proof follows the same lines as that of Proposition 5.3 ∎

Just like in previous sections, we say that a Lagrangian submanifold
$L\hookrightarrow M$ is horizontal if $T_{q}L\subseteq\mathcal{H}_{q}\,\forall
q\in L$ and say that it is non-horizontal if
$T_{q}L\not\subseteq\mathcal{H}_{q}\,\forall q\in L.$ We have the following
characterization:

######  Lemma 10.1.

Let $L\hookrightarrow M$ be an isotropic (or coisotropic) submanifold. We have

  * i)

If $L$ is horizontal and $\operatorname{dim}L=n$, then $L$ is Lagrangian.

  * ii)

If $L$ is non-horizontal and $\operatorname{dim}L=n+1$, then $L$ is
Lagrangian.

###### Proof.

The proof is similar to the proof of Lemma 5.2, since we only need to check
the condition at each tangent space. ∎

Now, given a coisotropic submanifold $N\hookrightarrow M$ (that is,
$(T_{q}N)^{\perp_{\Lambda}}\subseteq T_{q}N$), the distribution
$(TN)^{\perp_{\Lambda}}$ is not necessarily integrable and we shall assume it
in what follows:

###  10.1. Gradient and Hamiltonian vector fields as Lagrangian submanifolds

We can define a symplectic structure on $TM$ taking

$\Omega_{0}:=\flat_{\lambda,\omega}^{*}\Omega_{M},$

where $\Omega_{M}$ is the canonical symplectic form on $T^{*}M$. In Darboux
coordinates it has the expression:

$\displaystyle\Omega_{0}=$ $\displaystyle dq^{j}\wedge
d\left(\dot{q}^{i}a_{i}a_{j}+\dot{p}_{i}b^{i}a_{j}+\dot{z}a_{j}-\dot{p}_{j}\right)+$
$\displaystyle dp_{j}\wedge
d\left(\dot{q}^{i}a_{i}b^{j}+\dot{p_{i}}b^{i}b^{j}+\dot{z}b^{j}+\dot{q}^{j}\right)+$
$\displaystyle dz\wedge