# Discrete Dynamics on Locally Conformal Framework

###### Abstract

In this paper, we address the globalization problem of discrete Lagrangian and
Hamiltonian dynamics in locally conformal framework.

Oğul<EMAIL_ADDRESS>Ayten<EMAIL_ADDRESS>corresponding author*, Hasan<EMAIL_ADDRESS>

*Department of Mathematics, Gebze Technical University,

41400 Gebze-Kocaeli, Turkey

** Department of Mathematics, Yeditepe University,

34755 Ataşehir-İstanbul, Turkey

###### Contents

  1. 1 Introduction
  2. 2 Locally Conformally Continuous Dynamics
    1. 2.1 Locally Conformally Symplectic Geometry
    2. 2.2 Locally Conformally Lagrangian Dynamics
  3. 3 Locally Conformally Discrete Dynamics
    1. 3.1 Discrete Lagrangian and Discrete Hamiltonian Dynamics
    2. 3.2 Locally Conformally Discrete Lagrangian Dynamics
    3. 3.3 Locally Conformally Discrete Hamiltonian Dynamics
  4. 4 Conclusion and Future Work

##  1 Introduction

Discrete dynamics is gaining attention from both theoretical and practical
perspectives due to its deep geometrical and algebraic foundations, as well as
its effectiveness in solving equations numerically [5, 23]. Notably, discrete
Hamiltonian dynamics [38, 43, 44] and discrete Euler-Lagrange equations [22,
34] are two significant approaches within the field of discrete geometry. The
Hamilton-Jacobi theorem is also addressed in discrete models [35, 36]. In its
most abstract form, Lie groupoids play a crucial role in discrete dynamics
[32]. In this framework, the coupling problem of two mutually interacting
system is examined in [20]. Additionally, the study of implicit discrete
dynamics and the Tulczyjew triplet for singular dynamics has been explored in
detail in [26].

To clarify the purpose of this paper, let’s briefly review some geometric
constructions. Jacobi manifolds444Kirillov called this geometry as local Lie
algebra [24]. represent generalizations of Poisson manifolds obtained by
relaxing the Leibniz rule [29, 30, 31, 40]. These manifolds have collected
attention for their suitability in describing irreversible dynamic systems
such as thermodynamics and dissipative models [9, 41]. Jacobi manifolds are
completely integrable, where the odd-dimensional leaves of the Jacobi manifold
are regarded as contact manifolds [1, 28], while the even-dimensional leaves
are considered as locally conformally symplectic (LCS) manifolds [3, 39]. The
discrete dynamics within contact geometry recently announced in the
literature, see, for example, [6, 7, 19, 42].

In the present work, we investigate discrete dynamics on locally conformally
symplectic geometry. As we shall be more concrete in the main body of the
paper, the study of discrete dynamics within the locally conformal symplectic
framework addresses the globalization problem encountered in variational and
symplectic integrators. Specifically, this paper explores the geometric
aspects of local dynamical behaviour that fails to be extended globally.

The paper is structured as follows: To arrive at a proper discretization, we
begin by applying the inverse Legendre transformation to the continuous
Hamiltonian dynamics on the LCS manifolds. This leads us to arrive at a novel
framework called locally conformally Euler-Lagrange (LCEL) dynamics. The LCEL
system represents a generalization of the classical Euler-Lagrange equations.
Subsequently, we replace the action integral with a discrete sum and apply
discrete variation. As a result, we introduce discrete locally conformally
Euler-Lagrange (dLCEL) dynamics, which generalizes the discrete Euler-Lagrange
equations (dEL). Finally, we employ the Legendre transformation to obtain the
discrete locally conformally symplectic (dLCS) Hamilton’s equation within
discrete geometry.

Accordingly, the paper is organized into two main sections. In the following
section, following [18], we present a quick summary of LCS geometry on the
cotangent bundle. We further extend this locally conformal characterization to
the tangent bundle and derive locally conformally Euler-Lagrange LCEL
equations. In section 3, we present the discretizations of LCS Hamiltonian
dynamics and LCEL equations. These novel dynamics will be obtained on a purely
geometrical basis, making them suitable for further numerical as well as
theoretical analysis.

##  2 Locally Conformally Continuous Dynamics

###  2.1 Locally Conformally Symplectic Geometry

A manifold $M$ is symplectic if it possesses a closed, non-degenerate two-form
$\Omega$, called the symplectic two-form [28, 33]. On a symplectic manifold,
Hamilton’s equation for a given Hamiltonian function $H$ is defined to be

$\iota_{X_{H}}\Omega=dH,$ (2.1)

where $X_{H}$ is the Hamiltonian vector field.

Consider the cotangent bundle $T^{*}Q$ of a manifold $Q$, where the canonical
(Liouville) one-form $\Theta_{Q}$ exists. The canonical symplectic two-form on
$T^{*}Q$ is given by the negative exterior derivative

$\Omega_{Q}=-d\Theta_{Q}$ (2.2)

of the canonical one-form $\Theta_{Q}$. This construction provides a
foundational understanding of symplectic structures since locally every
symplectic manifold is looking like a cotangent bundle.

Locally Conformally Symplectic Manifolds. A manifold $M$ is called a locally
conformally symplectic (LCS) manifold if it admits a non-degenerate two-form
$\Omega$ so that

$d\Omega=\varphi\wedge\Omega$ (2.3)

for a closed (called as Lee) one-form $\varphi$, see [4, 25, 27, 39]. We
denote an LCS manifold by $(M,\Omega,\varphi)$. In terms of the Lichnerowicz-
deRham differential555For a closed one-form $B$, the Lichnerowicz-deRham
differential is defined to be $d_{B}\Omega=d\Omega-B\wedge\Omega$, see [21].
we can write the equality (2.3) as

$d_{\varphi}\Omega=0.$ (2.4)

If the dimension of an LCS manifold is $2n$ then we determine the LCS volume-
form as

$d\mu=\frac{(-1)^{n(n-1)/2}}{n!}\Omega\wedge\dots\wedge\Omega.$ (2.5)

The local picture for an LCS manifold is as follows. On every local chart of
the manifold $M$ there exists local symplectic two-form. We record a
symplectic chart as $(U_{\alpha},\Omega_{\alpha})$. On the intersection
$U_{\alpha}\cap U_{\beta}$ of two local charts, the local symplectic two-forms
$\Omega_{\alpha}$ and $\Omega_{\beta}$ are related with

$e^{\sigma_{\alpha}}\Omega_{\alpha}=e^{\sigma_{\beta}}\Omega_{\beta},$ (2.6)

where the family of real-valued functions $\\{\sigma_{\alpha}\\}$ are the
conformal factors. It is noteworthy to see that the conformal factors satisfy
the cocycle condition, allowing us to assemble local symplectic two-forms
$\Omega_{\alpha}$ into a line bundle-valued two-form on $M$. We define a real-
valued global two-form $\Omega$ using the following local characterization:

$\Omega|_{\alpha}=e^{\sigma_{\alpha}}\Omega_{\alpha},$ (2.7)

where $\Omega|_{\alpha}$ is the restriction of the global two-form to
$U_{\alpha}$. The exterior derivative of this two-form is computed to be

$d\Omega|_{\alpha}=d\sigma_{\alpha}\wedge\Omega|_{\alpha},$ (2.8)

which is the same as the LCS two-form condition in (2.3) with the Lee one-form
$\varphi|_{\alpha}=d\sigma_{\alpha}$ defined as the exterior derivative of the
conformal factor.

Cotangent Bundle and Coordinate Realization. Generic examples of LCS manifolds
are cotangent bundles [10]. Consider a closed one-form $\vartheta$ on a
manifold $Q$. We pull back this one-form to the cotangent bundle using the
cotangent bundle projection and obtain a closed and semi-basic one-form

$\varphi=(\pi_{Q})^{*}\vartheta.$ (2.9)

The negative LdR differential of the canonical one-form $\Theta_{Q}$ is a LCS
two-form

$\Omega_{\varphi}=-d_{\varphi}\Theta_{Q}=-d\Theta_{Q}+\varphi\wedge\Theta_{Q}=\Omega_{Q}+\varphi\wedge\Theta_{Q}.$
(2.10)

Therefore, the three-tuple $(T^{*}Q,\Omega_{\varphi},\varphi)$ is an LCS
manifold.

From now on we intereted in the LCS manifold
$(T^{*}Q,\Omega_{\varphi},\varphi)$. Let us be a bit more precise about this
construction. Assume an open covering for a manifold $Q$ with dimension $n$:

$Q=\bigsqcup_{\alpha}V_{\alpha}$ (2.11)

This induces manifold structures on the tangent and the cotangent bundle as

$TQ=\bigsqcup_{\alpha}TV_{\alpha}=\bigsqcup_{\alpha}V_{\alpha}\times\mathbb{R}^{n},\qquad
T^{*}Q=\bigsqcup_{\alpha}T^{*}V_{\alpha}=\bigsqcup_{\alpha}V_{\alpha}\times(\mathbb{R}^{n})^{*},$
(2.12)

respectively. The Darboux coordinates $(q^{i},p_{i})$ on $T^{*}V_{\alpha}$
reads the global canonical one-form $\Theta_{Q}$ and the global symplectic
two-form $\Omega_{Q}$ as

$\Theta_{Q}|_{\alpha}=p_{i}dq^{i},\qquad\Omega_{Q}|_{\alpha}=dq^{i}\wedge
dp_{i},$ (2.13)

respectively. On the other hand, there exist Darboux coordinates
$(q^{i},r_{i})$ that yield the local canonical one-form $\Theta_{\alpha}$
(which is not global) and the local canonical symplectic two-form
$\Omega_{\alpha}$ (which is also not global) as

$\Theta_{\alpha}=r_{i}dq^{i},\qquad\Omega_{\alpha}=dq^{i}\wedge dr_{i},$
(2.14)

respectively. The relationship between local momentum coordinates is

$r_{i}=e^{-\sigma_{\alpha}}p_{i}.$ (2.15)

As presented in (2.10), the Lee one-form is a semi-basic one-form on the
cotangent bundle framework. So, without a loss of generality, we can assume
that the conformal factors depend only on the coordinates $(q^{i})$ of the
base manifold $Q$. That is $\sigma_{\alpha}=\sigma_{\alpha}(q)$.

Accordingly, we can write the local symplectic two-form $\Omega_{\alpha}$ in
(2.13) in terms of the coordinates $(q^{i},p_{i})$ as

$\Omega_{\alpha}=-d\Theta_{\alpha}=dq^{i}\wedge
dr_{i}=e^{-\sigma_{\alpha}}\Big{(}dq^{i}\wedge
dp_{i}+\frac{1}{2}A_{ij}dq^{i}\wedge dq^{j}\big{)},$ (2.16)

where we have used the following notations:

$A_{ij}:=\varphi_{i}p_{j}-\varphi_{j}p_{i},\qquad\varphi_{i}:=\frac{\partial\sigma_{\alpha}}{\partial
q^{i}}.$ (2.17)

Thus LCS two-form $\Omega_{\varphi}$ defined in (2.10) is written in terms of
the coordinates $(q^{i},p_{i})$ as

$\Omega_{\varphi}|_{\alpha}=dq^{i}\wedge dp_{i}+\frac{1}{2}A_{ij}dq^{i}\wedge
dq^{j}.$ (2.18)

See that the first term on the right-hand side is the canonical symplectic
two-form $\Omega_{Q}$ whereas the second one is precisely
$\varphi\wedge\Theta_{Q}$. Therefore, the local realization (2.18) fits well
with the global definition of $\Omega$ in (2.10). The following diagram
summarizes the situation in one picture:

$\textstyle{{\begin{array}[]{c}\Omega_{Q}|_{\alpha}\\\ \eqref{bar-
bar}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
twist~{}\eqref{Omega-
phi}}$$\textstyle{{\begin{array}[]{c}\Theta_{Q}|_{\alpha}\\\ \eqref{bar-
bar}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-d_{\varphi}}$$\scriptstyle{-d}$$\textstyle{{\begin{array}[]{c}\Omega_{\varphi}|_{\alpha}\\\
\eqref{twist}\end{array}}}$$\textstyle{{\begin{array}[]{c}\Theta_{\alpha}\\\
\eqref{r-p}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-d}$$\scriptstyle{e^{\sigma_{\alpha}}}$$\textstyle{{\begin{array}[]{c}\Omega_{\alpha}\\\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\eqref{r-p}\end{array}}}$$\scriptstyle{e^{\sigma_{\alpha}}}$

We cite [37] for Darboux-Weinstein theorem for LCS manifolds.

LCS Hamiltonian Dynamics. On a local symplectic space
$(T^{*}V_{\alpha},\Omega_{\alpha})$ with conformal factors
$(\sigma_{\alpha})$, for a local Hamiltonian function $H_{\alpha}$, we write
local Hamilton’s equation as

$\iota_{X_{\alpha}}\Omega_{\alpha}=dH_{\alpha},$ (2.19)

where $X_{\alpha}$ is the local Hamiltonian vector field. Assuming the local
equality

$e^{\sigma_{\alpha}}H_{\alpha}=e^{\sigma_{\beta}}H_{\beta}$ (2.20)

on the intersection $T^{*}V_{\alpha}\cap T^{*}V_{\beta}$, we define a global
function $H$ on $T^{*}Q$ so that its restriction reads

$H|_{\alpha}=e^{\sigma_{\alpha}}H_{\alpha}.$ (2.21)

In accordance with, we glue the local Hamilton’s equation (2.19) to the LCS
Hamilton’s equation

$\iota_{\xi_{H}}\Omega=d_{\varphi}H.$ (2.22)

Here, the LCS Hamiltonian vector field $\xi_{H}$ is obtained by gluing
together the local Hamiltonian vector fields $X_{\alpha}$ from (2.19), such
that $\xi_{H}|_{\alpha}=X_{\alpha}$. In terms of the Darboux coordinates
$(q^{i},p_{i})$ of the canonical symplectic structure on the cotangent bundle
$T^{*}Q$, the LCS Hamilton’s equation (2.22) is expressed as:

$\frac{dq^{i}}{dt}=\frac{\partial H|_{\alpha}}{\partial
p_{i}},\qquad\frac{dp_{i}}{dt}=-\frac{\partial H|_{\alpha}}{\partial
q^{i}}-A_{ij}\frac{\partial H|_{\alpha}}{\partial
p_{j}}+H|_{\alpha}\varphi_{i},$ (2.23)

where the covariant quantity $A_{ij}$ is defined in (2.17). We record also the
divergence of the LCS Hamiltonian vector field:

$\mathrm{div}(\xi_{H})=\frac{1}{2}n\langle\varphi.\xi_{H}\rangle.$ (2.24)

Notice that if there is no nontrivial conformal factor then LCS Hamilton’s
equation in (2.23) reduces to Hamilton’s equation

$\frac{dq^{i}}{dt}=\frac{\partial H|_{\alpha}}{\partial
p_{i}},\qquad\frac{dp_{i}}{dt}=-\frac{\partial H|_{\alpha}}{\partial q^{i}}.$
(2.25)

###  2.2 Locally Conformally Lagrangian Dynamics

We follow the notations and definitions presented in the previous section.
Consider a Lagrangian function $L_{\alpha}=L_{\alpha}(q^{i},\dot{q}^{i})$
define on the local tangent bundle $TV_{\alpha}$ and define the Legendre
transformation

$\mathbb{F}L_{\alpha}:TV_{\alpha}\longrightarrow
T^{*}V_{\alpha},\qquad(q^{i},\dot{q}^{i})\mapsto(q^{i},r_{i})=\big{(}q^{i},\frac{\partial
L_{\alpha}}{\partial\dot{q}^{i}}\big{)}.$ (2.26)

We pull back the local canonical one-form and local symplectic two-form with
this Legendre transformation and obtain local Poincaré-Cartan one-form and
local Poincaré-Cartan two-form given by

$\begin{split}\theta_{\alpha}&=\mathbb{F}L^{*}_{\alpha}\Theta_{\alpha}=\frac{\partial
L_{\alpha}}{\partial\dot{q}^{i}}dq^{i}\\\
\omega_{\alpha}&=\mathbb{F}L^{*}_{\alpha}\Omega_{\alpha}=\frac{\partial^{2}L_{\alpha}}{\partial\dot{q}^{i}\partial
q^{j}}dq^{i}\wedge
dq^{j}+\frac{\partial^{2}L_{\alpha}}{\partial\dot{q}^{i}\partial\dot{q}^{j}}dq^{i}\wedge
d\dot{q}^{j},\end{split}$ (2.27)

respectively. The local two-form $\omega_{\alpha}$ is symplectic if the rank
of the Hessian matrix
$[\partial^{2}L_{\alpha}/\partial\dot{q}^{i}\partial\dot{q}^{j}]$ is the
maximum that is if the Lagrangian is regular.

The local energy function $E_{L_{\alpha}}$ for a given local Lagrangian
function $L_{\alpha}$ is defined by means of the Liouville vector field
$\Delta$ as

$E_{L_{\alpha}}=\Delta(L_{\alpha})-L_{\alpha},\qquad\Delta=\dot{q}^{i}\frac{\partial}{\partial\dot{q}i}.$
(2.28)

We introduce the following covariance relation

$\iota_{Y_{\alpha}}\omega_{\alpha}=dE_{L_{\alpha}}$ (2.29)

on the tangent bundle $TV_{\alpha}$. This equation reads the local Euler-
Lagrange equations

$\frac{d}{dt}\frac{\partial L_{\alpha}}{\partial\dot{q}^{i}}=\frac{\partial
L_{\alpha}}{\partial q^{i}}.$ (2.30)

Locally conformally Poincaré-Cartan Forms. Consider the tangent bundle $TQ$
covered by local charts $TV_{\alpha}$, each equipped with a local Lagrangian
function. It is assumed that these local Lagrangian functions satisfy a
conformal relation on intersections, denoted as $TV_{\alpha}\cap TV_{\beta}$,
given by

$e^{\sigma_{\alpha}}L_{\alpha}=e^{\sigma_{\beta}}L_{\beta},$ (2.31)

where ${\sigma_{\alpha}}$ represents a family of smooth functions. This
construction implies that the local Lagrangian functions ${L_{\alpha}}$ cannot
be seamlessly combined into a single real-valued Lagrangian function on the
tangent bundle $TQ$. Instead, these local functions define a line-bundle-
valued function on $TQ$. However, by multiplying the local functions by
conformal factors, as indicated in (2.31), a global real-valued function $L$
on $TQ$ emerges. The restriction of $L$ to an open chart $TV_{\alpha}$ is
given by

$L|_{\alpha}=e^{\sigma_{\alpha}}L_{\alpha}.$ (2.32)

This local-to-global realization is also true for local Poincaré-Cartan forms.
Notice that local Poincaré-Cartan forms defined as (2.27) are related with
conformal factors on the intersection $TV_{\alpha}\cap TV_{\beta}$. That is

$\begin{split}e^{\sigma_{\alpha}}\theta_{\alpha}&=e^{\sigma_{\beta}}\theta_{\beta},\\\
e^{\sigma_{\alpha}}\omega_{\alpha}&=e^{\sigma_{\beta}}\omega_{\beta},\end{split}$
(2.33)

respectively. These local forms can only be glued up to a line bundle-valued
one-form. Referring to (2.33), we define global one-form $\theta_{L}$ and
$\omega_{\varphi}$, in a respected manner, as follows

$\begin{split}\theta_{L}|_{\alpha}&=e^{\sigma_{\alpha}}\theta_{\alpha},\\\
\omega_{\varphi}|_{\alpha}&=e^{\sigma_{\alpha}}\omega_{\alpha}.\end{split}$
(2.34)

Note that the negative exterior derivative of $\theta_{L}|_{\alpha}$ gives the
following two-form

$\omega_{L}|_{\alpha}=-d\theta_{L}|_{\alpha}=-d\sigma_{\alpha}\wedge
e^{\sigma_{\alpha}}\theta_{\alpha}+e^{\sigma_{\alpha}}\omega_{\alpha}.$ (2.35)

The global realizations of each term give that

$\omega_{\varphi}=\omega_{L}+\varphi\wedge\theta_{L},\qquad\omega_{\varphi}=-d_{\varphi}\theta_{L}.$
(2.36)

We call $\omega_{\varphi}$ as locally conformally Euler-Poincaré two-form. We
present the following diagram to summarize the differential forms obtained in
this subsection:

$\textstyle{{\begin{array}[]{c}\omega_{L}|_{\alpha}\\\ \eqref{glo-PC-
rest}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
twist~{}\eqref{conf-
EP}}$$\textstyle{{\begin{array}[]{c}\theta_{L}|_{\alpha}\\\ \eqref{glo-glue-
forms}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-d_{\varphi}}$$\scriptstyle{-d}$$\textstyle{{\begin{array}[]{c}\omega_{\varphi}|_{\alpha}\\\
\eqref{glo-glue-
forms}\end{array}}}$$\textstyle{{\begin{array}[]{c}\theta_{\alpha}\\\
\eqref{loc-
PC}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-d}$$\scriptstyle{e^{\sigma_{\alpha}}}$$\textstyle{{\begin{array}[]{c}\omega_{\alpha}\\\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\eqref{loc-
PC}\end{array}}}$$\scriptstyle{e^{\sigma_{\alpha}}}$ (2.37)

Assume that the energy function is also locally conformal that is

$E_{L}|_{\alpha}=e^{\sigma_{\alpha}}E_{L_{\alpha}}.$ (2.38)

By glueing the local Euler-Lagrange equation (2.29), we arrive at the locally
conformally Euler-Lagrange (LCEL) equation

$\iota_{\xi_{L}}\omega_{\varphi}=d_{\varphi}E_{L}$ (2.39)

where the vector field $\xi_{L}$ is the global picture of the vector field
$Y_{\alpha}$ that is $\xi_{L}|_{\alpha}=Y_{\alpha}$. In the induced
coordinates $(q^{i},\dot{q}^{i})$ on the tangent bundle $TQ$, the locally
conformally Euler-Lagrange equations (2.39) is written as [18]

$\frac{d}{dt}(\frac{\partial L|_{\alpha}}{\partial\dot{q}^{i}})-\frac{\partial
L|_{\alpha}}{\partial q^{i}}=\varphi_{j}\dot{q}^{j}(\frac{\partial
L|_{\alpha}}{\partial\dot{q}^{i}})-\varphi_{i}L|_{\alpha}.$ (2.40)

See that when the conformal factor is zero, we obtain the Euler-Lagrange
equation:

$\frac{d}{dt}(\frac{\partial L|_{\alpha}}{\partial\dot{q}^{i}})-\frac{\partial
L|_{\alpha}}{\partial q^{i}}=0$ (2.41)

##  3 Locally Conformally Discrete Dynamics

###  3.1 Discrete Lagrangian and Discrete Hamiltonian Dynamics

In continuous dynamics, Lagrangian function is defined on the tangent bundle
$TQ$. In this case, solving the Euler-Lagrange equations (2.41) involves
determining a smooth curve $q=q(t)$ in $Q$. This curve, along with its
velocity $(q(t),\dot{q}(t))$, is substituted into the Lagrangian function to
achieve an extremum of the action integral. We cite [34, 35, 36] for more
details on the discrete formulations presented in this subsection.

To discretize the Lagrangian dynamics we replace the role of the smooth curve
with a finite sequence of points. For example, let $a,b\in\mathbb{R}$ and
$a<b$ then define the step $h=(b-a)/N$, where $N$ is the number of divisions,
that is the number of elements in the finite sequence. We determine such a
sequence by

$[\mathbf{q}]=(\mathbf{q}_{a}=\mathbf{q}_{0},\dots,\mathbf{q}_{b}=\mathbf{q}_{N}).$
(3.1)

Here, the bold notation stands for $\mathbf{q}=(q^{i})$. In the discrete
framework, the Lagrangian function is substituted by a discrete Lagrangian
function $L^{d}:Q\times Q\rightarrow\mathbb{R}$ which is an approximation to
the exact discrete Lagrangian

$L^{d-ex}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})=\int_{t_{\kappa}}^{t_{\kappa+1}}{L(\mathbf{q}(t),{\bf\dot{q}}(t))dt},$
(3.2)

where $\mathbf{q}:[t_{\kappa},t_{\kappa+1}]\rightarrow Q$ is the solution of
the continuous Euler-Lagrange equation with boundary conditions
$\mathbf{q}(t_{\kappa})=\mathbf{q}_{\kappa}$ and
$\mathbf{q}(t_{\kappa+1})=\mathbf{q}_{\kappa+1}$. In the following proposition
we state discrete Euler-Lagrange equations, see, for example, [34].

###### Proposition 3.1

The discrete Euler-Lagrange (dEL) equation, arising from a discrete Lagrangian
function $L^{d}$, is

$D_{2}L^{d}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa})+D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})=0,$
(3.3)

where $D_{1}$ denotes the partial derivative with respect to the first
argument whereas $D_{2}$ is the partial derivative with respect to the second
argument.

###### Proof 3.2

Let us write action sum for discrete Lagrangian $L^{d}$ as the following
finite sum:

$S^{d}([\mathbf{q}])=\sum_{\kappa=0}^{N-1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).$
(3.4)

We take the variation of this action and endow the discrete variational
principle $\delta S^{d}(\mathbf{q})=0$. This gives us the equality:

$\begin{split}\delta
S^{d}(\mathbf{q})=\sum_{\kappa=0}^{N-1}(D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa}+D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa+1})=0.\end{split}$
(3.5)

Considering that at the boundary the variations are identically zero that is
$\delta\mathbf{q}_{0}=\delta\mathbf{q}_{N}=0$, we have the following
expression:

$\sum_{\kappa=1}^{N-1}D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa}+\sum_{\kappa=0}^{N-2}D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa+1}=0.$
(3.6)

In the second term, we change $\kappa$ with $\kappa-1$ in order to take the
whole expression into the parenthesis of $\delta\mathbf{q}_{\kappa}$. This
reads

$\sum_{\kappa=1}^{N-1}(D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})+D_{2}L^{d}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa}))\delta\mathbf{q}_{\kappa}=0.$
(3.7)

For each point in the sequence, we get the discrete Euler-Lagrange equation
(3.3).

The Discrete Legendre Transformation. As in the case of continuous dynamics,
we employ the Legendre transformation to pass from the discrete Lagrangian
dynamics to the discrete Hamiltonian dynamics. In the discrete case, we have
the right and left Legendre transformations

$\mathbb{F}^{\pm}L^{d}:Q\times Q\longrightarrow T^{*}Q$ (3.8)

given respectively by

$\begin{split}\mathbb{F}^{+}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})&=(\mathbf{q}_{\kappa+1},D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})),\\\
\mathbb{F}^{-}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})&=(\mathbf{q}_{\kappa},-D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})).\end{split}$
(3.9)

We determine the momenta as the second entry of the image space of the
discrete Legendre transformations. Accordingly, we have the right and the left
momenta as

$\begin{split}&\mathbf{p}^{+}_{\kappa,\kappa+1}=D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}),\\\
&\mathbf{p}^{-}_{\kappa,\kappa+1}=-D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}),\end{split}$
(3.10)

respectively. See that from the discrete Euler-Lagrange equation (3.3) we can
define momenta depending on a single point as

$\mathbf{p}_{\kappa}:=\mathbf{p}^{+}_{\kappa-1,\kappa}=\mathbf{p}^{-}_{\kappa,\kappa+1}.$
(3.11)

Accordingly, we can write

$\begin{split}&\mathbf{p}_{\kappa}=-D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}),\\\
&\mathbf{p}_{\kappa+1}=D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).\end{split}$
(3.12)

In the continuous case, the Legendre transformation is non-degenerate if the
Lagrangian function is regular, meaning that the rank of the Hessian of the
Lagrangian function is at its maximum. To analyze this phenomenon in the
discrete case, we introduce discrete (right and left) Poincaré-Cartan one-
forms:

$\begin{split}&\theta^{+}=D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa+1},\\\
&\theta^{-}=-D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa}.\end{split}$
(3.13)

The exterior derivatives of these one-forms coincide and determine the
discrete Poincaré-Cartan two-form as

$\omega^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})=-d\theta^{\pm}=-D_{1}D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\
d\mathbf{q}_{\kappa}\wedge d\mathbf{q}_{\kappa+1}.$ (3.14)

The discrete Legendre transformation (3.8) is a local diffeomorphism if the
discrete Poincaré-Cartan two-form is non-degenerate (hence symplectic). In
this case, the discrete Lagrangian function is called regular. We determine
the discrete Hamiltonian dynamics under this assumption.

For a regular discrete Lagrangian function $L^{d}$, referring to the Darboux
coordinates, we define the right discrete Hamiltonian function as the
difference between the coupling function and the discrete Lagrangian function
as

$H^{d+}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1})=\mathbf{p}_{\kappa+1}\cdot\mathbf{q}_{\kappa+1}-L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).$
(3.15)

On the other hand, we define the left discrete Hamiltonian function as the
difference between the minus of the coupling function and the discrete
Lagrangian function as

$H^{d-}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa})=-\mathbf{p}_{\kappa}\cdot\mathbf{q}_{\kappa}-L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).$
(3.16)

The following proposition presents the discrete Hamiltonian dynamics for both
the left and right Hamiltonian functions.

###### Proposition 3.3

The right discrete Hamilton’s equations generated by the right Hamiltonian
function (3.15) are

$\begin{cases}&\mathbf{q}_{\kappa+1}=D_{2}H^{d+}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1}),\\\
&\mathbf{p}_{\kappa}=D_{1}H^{d+}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1}),\end{cases}$
(3.17)

whereas the left discrete Hamilton’s equations generated by the left
Hamiltonian function (3.16) are

$\begin{cases}&\mathbf{q}_{\kappa}=-D_{2}H^{d-}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa}),\\\
&\mathbf{p}_{\kappa+1}=-D_{1}H^{d-}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa}),\end{cases}$
(3.18)

where $D_{a}$ denotes the partial derivative with respect to the $a$-th entry.

###### Proof 3.4

We consider the right Hamiltonian function (3.15), then take the partial
derivatives of it with respect to its first entry $\mathbf{q}_{\kappa}$ and
its second entry $\mathbf{p}_{\kappa+1}$. This gives us

$\begin{split}&D_{1}H^{d+}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1})=-D_{1}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\\\
&D_{2}H^{d+}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1})=\mathbf{q}_{\kappa+1}.\end{split}$
(3.19)

Substituting the momentum definition from (3.12), we obtain the right discrete
Hamilton’s equations (LABEL:rd-Hameq).

On the other hand, in the left case, we compute the partial derivatives of the
left Hamiltonian function (3.16) with respect to its first entry
$\mathbf{q}_{\kappa+1}$ and its second entry $\mathbf{p}{\kappa}$, yielding:

$\begin{split}&D_{1}H^{d-}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa})=-D_{2}L^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\\\
&D_{2}H^{d-}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa})=-\mathbf{q}_{\kappa}\end{split}$
(3.20)

Substituting (3.12), we derive the left discrete Hamilton’s equations (3.18).

###  3.2 Locally Conformally Discrete Lagrangian Dynamics

In this subsection, we generalize the discrete dynamics discussions presented
in the previous subsection to the locally conformal framework. In this novel
construction, we start with elaborating the discretization of the locally
conformally Euler-Lagrange (LCEL) dynamics given in Subsection 2.2.

For a manifold $Q$, we assume an open covering as in (2.11), consisting of
local charts $\\{V_{\alpha}\\}$. For each local chart $V_{\alpha}$, we define
the discrete local Lagrangian function $L^{d}_{\alpha}$ on the product vector
space $V_{\alpha}\times V_{\alpha}$ as follows

$L^{d}_{\alpha}:V_{\alpha}\times V_{\alpha}\longrightarrow\mathbb{R}.$ (3.21)

Referring to Proposition 3.1 and the dynamical equation (3.3), we express the
local discrete dynamics generated by this local discrete Lagrangian as

$D_{2}L^{d}_{\alpha}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa})+D_{1}L^{d}_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})=0.$
(3.22)

Recalling from the construction in Subsection 2.2, we assume locally conformal
character determined by the family of scalar functions $\\{\sigma_{\alpha}\\}$
so that the local discrete Lagrangian functions are satisfying

$e^{\sigma_{\alpha}}L^{d}_{\alpha}=e^{\sigma_{\beta}}L^{d}_{\beta}$ (3.23)

on the intersection $V_{\alpha}\cap V_{\beta}$. Following (2.32) in Subsection
2.2, we define a global discrete Lagrangian function $L^{d}$ on the whole
manifold $Q$ whose local realization is

$L^{d}|_{\alpha}=e^{\sigma_{\alpha}}L^{d}_{\alpha}.$ (3.24)

The following proposition establishes discrete dynamics in terms of the global
discrete Lagrangian function $L^{d}$, presenting the discretization of the
locally conformal Euler-Lagrange equations (2.40).

###### Proposition 3.5

The discrete dynamics generated by the global discrete Lagrangian function
$L^{d}$ defined in (3.24) is given by:

$\begin{split}&e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})-\sigma_{\alpha}(\mathbf{q}_{\kappa-1})}D_{2}L^{d}|{\alpha}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa})\\\
&\qquad\qquad=D\sigma_{\alpha}(\mathbf{q}_{\kappa})\cdot
L^{d}|{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})-D_{1}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}),\end{split}$
(3.25)

where $D_{a}$ denotes the partial derivative with respect to the $a$-th entry
whereas $D\sigma_{\alpha}$ is the partial derivative of the function
$\sigma_{\alpha}$.

###### Proof 3.6

We begin with the discrete action

$S^{d}([\mathbf{q}])=\sum_{\kappa=0}^{N-1}L_{\alpha}^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})$
(3.26)

where $L^{d}_{\alpha}$ represents the local discrete Lagrangian function. To
write the action sum for the global discrete Lagrangian function, we
substitute (3.24) into the discrete action (3.26) yields

$S^{d}([\mathbf{q}])=\sum_{\kappa=0}^{N-1}e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).$
(3.27)

See that the action sum takes different values for different sequences. The
variation of the action is computed to be

$\begin{split}\delta
S^{d}([\mathbf{q}])&=\sum_{\kappa=0}^{N-1}(-e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D\sigma_{\alpha}(\mathbf{q}_{\kappa})L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa}\\\
&\qquad\qquad+e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D_{1}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa}\\\
&\qquad\qquad+e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa+1}).\end{split}$
(3.28)

In order to extremize the action sum, we set $\delta S^{d}([\mathbf{q}])=0$
while fixing the boundary conditions: $\delta\mathbf{q}_{0}=0$ and
$\delta\mathbf{q}_{N}=0$. Accordingly, we obtain

$\begin{split}&\sum_{\kappa=1}^{N-1}e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}(-D\sigma_{\alpha}(\mathbf{q}_{\kappa})L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})+D_{1}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}))\delta\mathbf{q}_{\kappa}\\\
&\qquad\qquad+\sum_{\kappa=0}^{N-2}e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\delta\mathbf{q}_{\kappa+1}=0.\end{split}$
(3.29)

By relabeling $\kappa$ with $\kappa-1$ in the last term we write

$\begin{split}&\sum_{\kappa=1}^{N-1}\Big{(}e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}(-D\sigma_{\alpha}(\mathbf{q}_{\kappa})L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})+D_{1}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}))\\\
&\qquad\qquad+e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa-1})}D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa})\Big{)}\delta\mathbf{q}_{\kappa}=0.\end{split}$
(3.30)

Thus for arbitrary $\delta\mathbf{q}_{\kappa}$, the equality holds if (3.25)
is satisfied.

We call (3.25) as discrete locally conformally Euler-Lagrange (dLCEL)
equations. It is noteworthy that if there is no locally conformal character
within the setting, meaning $L|_{\alpha}=L_{\alpha}$, then Proposition 3.5
aligns with Proposition 3.1, and the discrete locally conformally Euler-
Lagrange equation (3.25) reduces to the discrete Euler-Lagrange equation
(3.3). The following diagram summarizes this discussion.

$\textstyle{{\begin{array}[]{c}LCEL\\\
\eqref{LCEL}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{discretization}$$\scriptstyle{\varphi=0}$$\textstyle{{\begin{array}[]{c}~{}EL\\\
\eqref{EL}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{discretization}$$\textstyle{{\begin{array}[]{c}dLCEL\\\
\eqref{dLCEL}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi=0}$$\textstyle{{\begin{array}[]{c}dEL\\\
\eqref{dEL}\end{array}}}$ (3.31)

The Discrete Legendre Transformation. For a local chart $V_{\alpha}$ we
introduce the local right and left Legendre Transformations as

$\mathbb{F}^{\pm}L^{d}_{\alpha}:V_{\alpha}\times V_{\alpha}\longrightarrow
T^{*}V_{\alpha}$ (3.32)

given explicitly by

$\begin{split}\mathbb{F}^{+}L^{d}_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})&=(\mathbf{q}_{\kappa+1},r^{+}_{\kappa,\kappa+1}=D_{2}L^{d}_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})),\\\
\mathbb{F}^{-}L^{d}_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})&=(\mathbf{q}_{\kappa},r^{-}_{\kappa,\kappa+1}=-D_{1}L^{d}_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})),\end{split}$
(3.33)

respectively. Here, $r^{+}_{\kappa,\kappa+1}$ represents the right local
momenta, while $r^{-}_{\kappa,\kappa+1}$ denotes the left local momenta. To
express the discrete Legendre transformations in terms of the global discrete
Lagrangian function $L^{d}$, we utilize the identification (3.24) and
substitute it into the definitions of local momenta. Consequently, we obtain

$\begin{split}\mathbf{r}^{+}_{\kappa,\kappa+1}&=e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}),\\\
\mathbf{r}^{-}_{\kappa,\kappa+1}&=e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}\big{(}D\sigma_{\alpha}(\mathbf{q}_{\kappa})\cdot
L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})-D_{1}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})\big{)}.\end{split}$
(3.34)

Under dLCEL equation (3.25), and after a parameter shift, the right and the
left momenta turn out to be the same and this results in the following
definition

$\mathbf{r}_{\kappa}:=\mathbf{r}^{+}_{\kappa-1,\kappa}=\mathbf{r}^{-}_{\kappa,\kappa+1}.$
(3.35)

Recall that in the continuous case, there exist two sets of momenta in a
relation given by $r_{i}=e^{-\sigma_{\alpha}}p_{i}$. This is depicted in
(2.14). Motivated by this, and following the order in (3.34), we define two
new sets of local momenta

$\begin{split}\mathbf{p}^{+}_{\kappa,\kappa+1}&=e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})}r^{+}_{\kappa,\kappa+1}=D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).\\\
\mathbf{p}^{-}_{\kappa,\kappa+1}&=e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})}r^{-}_{\kappa,\kappa+1}=D\sigma_{\alpha}(\mathbf{q}_{\kappa})\cdot
L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})-D_{1}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).\end{split}$
(3.36)

A naive shift of parameters is not enough to define a momenta $\mathbf{p}$
depending on a single point. More precisely one has that

$\mathbf{p}_{\kappa}:=e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})-\sigma_{\alpha}(\mathbf{q}_{\kappa-1})}\mathbf{p}^{+}_{\kappa-1,\kappa}=\mathbf{p}^{-}_{\kappa,\kappa+1}.$
(3.37)

It is immediate to see that $\mathbf{r}$-momenta in (3.35) and
$\mathbf{p}$-momenta in (3.37) are related as

$\mathbf{r}_{\kappa}=e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}\mathbf{p}_{\kappa}$
(3.38)

as we expected.

Poincaré-Cartan Forms. Referring to (3.13), we pull back the local canonical
one-form $\Theta_{\alpha}$ in (2.14), defined on the cotangent bundle
$T^{*}V_{\alpha}$, to the product space $V_{\alpha}\times V_{\alpha}$ via the
right and left Legendre transformations (3.33). Then, in a respective manner,
we define the right and left local Poincaré-Cartan one-forms for the local
Lagrangian function $L_{\alpha}$ as follows

$\begin{split}\theta_{\alpha}^{+}&=\mathbb{F}^{+}L^{d}_{\alpha}(\Theta_{\alpha})=D_{2}L_{\alpha}^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa+1},\\\
\theta^{-}_{\alpha}&=\mathbb{F}^{-}L^{d}_{\alpha}(\Theta_{\alpha})=-D_{1}L^{d}_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa}.\end{split}$
(3.39)

Exterior derivatives of these one-forms coincide and determine local Poincaré-
Cartan two-form

$\omega^{+}_{\alpha}=-d\theta^{\pm}_{\alpha}$ (3.40)

This is a symplectic two-form if the local Lagrangian function is regular.

Let us use the discrete Legendre transformations to elucidate the locally
conformal character of Proposition 3.5 and the dLCEL equation presented in
(3.25). We do this for the right Poincaré-Cartan one-form the left case is
more or less following the same lines of calculation. Referring to (3.24), we
write the right Poincaré-Cartan one-form in terms of the global discrete
Lagrangian function as

$\begin{split}\theta^{+}_{\alpha}&=D_{2}(e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}))d\mathbf{q}_{\kappa+1}\\\
&=e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa+1}.\end{split}$
(3.41)

This one-form fails to be global as in the continuous case. To have a global
one we introduce a one-form $\theta^{d+}$ whose restriction to $V_{\alpha}$ is

$\theta^{d+}|_{\alpha}=e^{\sigma_{\alpha}}\theta^{+}_{\alpha}=D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa+1}.$
(3.42)

Taking the negative exterior derivative of this one-form we obtain a global
two-form

$\begin{split}\omega^{d+}|_{\alpha}&=-d\theta^{d+}|_{\alpha}=-e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})}d\theta^{+}_{\alpha}-e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})}d\sigma_{\alpha}(\mathbf{q}_{\kappa})\wedge\theta^{+}_{\alpha}\\\
&=e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})}\big{(}\omega^{+}_{\alpha}-d\sigma_{\alpha}(\mathbf{q}_{\kappa})\wedge\theta^{+}_{\alpha}\big{)}.\end{split}$
(3.43)

We introduce discrete locally conformally Poincaré-Cartan two-form as

$\omega_{\varphi}^{d+}|_{\alpha}=e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})}\omega^{+}_{\alpha},$
(3.44)

where $\omega_{\varphi}^{d+}|_{\alpha}$ denotes the restriction of
$\omega_{\varphi}^{d+}$ to local chart $TV_{\alpha}$. Referring to (3.43), it
is a direct observation to see that

$\omega_{\varphi}^{d+}|_{\alpha}=\omega^{d+}|_{\alpha}+d\sigma_{\alpha}(\mathbf{q}_{\kappa})\wedge\theta^{+}|_{\alpha}.$
(3.45)

is satisfying the LCS two-form construction in (2.10).

###### Proposition 3.7

The discrete locally conformally Poincaré-Cartan two-form
$\omega_{\varphi}^{d+}$ defined in (3.45) is a LCS two-form with the Lee one-
form is $d\sigma_{\alpha}$.

###### Proof 3.8

Let us now perform the following calculation to assure once more that the
discrete locally conformally Poincaré-Cartan two-form $\omega_{\varphi}^{d+}$
is indeed an LCS two-form:

$\begin{split}\omega_{\varphi}^{d+}|_{\alpha}&=-d(D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa+1})\\\
&\qquad\qquad+d\sigma_{\alpha}(\mathbf{q}_{\kappa})\wedge
D_{2}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})d\mathbf{q}_{\kappa+1}\\\
&=-\frac{\partial^{2}L^{d}|_{\alpha}}{\partial\mathbf{q}^{i}_{\kappa}\partial\mathbf{q}^{j}_{\kappa+1}}d\mathbf{q}^{i}_{\kappa}\wedge
d\mathbf{q}^{j}_{\kappa+1}-\frac{\partial^{2}L^{d}|_{\alpha}}{\partial\mathbf{q}^{i}_{\kappa+1}\partial\mathbf{q}^{j}_{\kappa+1}}d\mathbf{q}^{i}_{\kappa+1}\wedge
d\mathbf{q}^{j}_{\kappa+1}\\\ &\hskip
56.9055pt+\frac{\partial\sigma_{\alpha}}{\partial\mathbf{q}^{i}_{\kappa}}\frac{\partial
L^{d}|_{\alpha}}{\partial\mathbf{q}^{j}_{\kappa+1}}d\mathbf{q}^{i}_{\kappa}\wedge
d\mathbf{q}^{j}_{\kappa+1}.\end{split}$ (3.46)

Then we take the exterior derivative of this explicit expression as

$\begin{split}d\omega_{\varphi}^{d+}|_{\alpha}&=\frac{\partial\sigma_{\alpha}}{\partial\mathbf{q}^{i}_{\kappa}}\frac{\partial^{2}L^{d}|_{\alpha}}{\partial\mathbf{q}^{l}_{\kappa}\partial\mathbf{q}^{j}_{\kappa+1}}d\mathbf{q}^{l}_{\kappa}\wedge
d\mathbf{q}^{i}_{\kappa}\wedge d\mathbf{q}^{j}_{\kappa+1}\\\
&=\frac{\partial\sigma_{\alpha}}{\partial\mathbf{q}^{i}_{\kappa}}d\mathbf{q}^{i}_{\kappa}\wedge(-\frac{\partial^{2}L^{d}|_{\alpha}}{\partial\mathbf{q}^{l}_{\kappa}\partial\mathbf{q}^{j}_{\kappa+1}}d\mathbf{q}^{l}_{\kappa}\wedge
d\mathbf{q}^{j}_{\kappa+1})\\\
&=d\sigma_{\alpha}(\mathbf{q}_{\kappa})\wedge\omega^{+}|_{\alpha}.\end{split}$
(3.47)

reading that that the discrete locally conformally Poincaré-Cartan two-form
satisfies the LCS two-form condition in (2.8).

###  3.3 Locally Conformally Discrete Hamiltonian Dynamics

We have defined the discrete locally conformal Euler-Lagrange equations (3.25)
in Proposition 3.5. We have also determined the right and left Legendre
transformations $\mathbb{F}^{\pm}L^{d}_{\alpha}$ in (3.33). In this
subsection, our goal is to derive discrete versions of the right and the left
discrete Hamiltonian dynamics exhibited in Proposition 3.3.

Let us start with the local picture first by concentrating on the plus
momenta. Accordingly, we pass to the Hamiltonian picture applying the right
discrete Legendre transformation to the discrete locally conformally Euler-
Lagrange equations (3.25). At first, we introduce the Hamiltonian function for
$\mathbf{r}$-momenta as

${H_{\alpha}^{d}}^{+}(\mathbf{q}_{\kappa},\mathbf{r}_{\kappa+1})=\mathbf{r}_{\kappa+1}\cdot\mathbf{q}_{\kappa+1}-L^{d}_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).$
(3.48)

A direct computation reads the local right discrete Hamiltonian dynamics in
terms of the $\mathbf{r}$-momenta as

$\begin{cases}&\mathbf{q}_{\kappa+1}=D_{2}{H_{\alpha}^{d}}^{+}(\mathbf{q}_{\kappa},{\mathbf{r}}_{\kappa+1}),\\\
&{\mathbf{r}}_{\kappa}=D_{1}{H_{\alpha}^{d}}^{+}(\mathbf{q}_{\kappa},{\mathbf{r}}_{\kappa+1}).\end{cases}$

We assume the locally conformal character for the local discrete Hamiltonian
functions. Motivated by the continuous case depicted in (2.20), we take

$e^{\sigma_{\alpha}}H^{d+}_{\alpha}=e^{\sigma_{\beta}}H^{d+}_{\beta}$ (3.49)

on the intersection of the tangent bundles $TV_{\alpha}\cap TV_{\beta}$. The
equality determines an obstruction to glue up local functions
$\\{H^{d}_{\alpha}\\}$ to a real-valued function on the cotangent bundle
$T^{*}Q$. However, the same identity determines a global discrete Hamiltonian
function $H^{d}$ whose local characterization is given by in terms of the
$\mathbf{p}$-momenta as

$H^{d+}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1}):=e^{\sigma_{\alpha}(\mathbf{q}_{\kappa})}{H_{\alpha}^{d}}^{+}(\mathbf{q}_{\kappa},\mathbf{r}_{\kappa+1}).$
(3.50)

Here the $\mathbf{r}$-momenta and the $\mathbf{p}$-momenta are related as
shown in (3.38).

###### Proposition 3.9

The right discrete locally conformal Hamiltonian dynamics generated by the
right discrete Hamiltonian function $H^{d+}$ given in (3.50) is

$\begin{cases}&\mathbf{q}_{\kappa+1}=e^{\sigma_{\alpha}(\mathbf{q}_{\kappa+1})-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D_{2}H^{d+}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1}),\\\
&\mathbf{p}_{\kappa}=D_{1}H^{d+}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1})-D\sigma_{\alpha}(\mathbf{q}_{\kappa})\cdot
H^{d+}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1}).\end{cases}$
(3.51)

###### Proof 3.10

Referring to the global discrete Lagrangian function $L^{d}$ in (3.24), the
equation (3.48) can be written as

$\begin{split}&e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}H^{d+}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1})\\\
&\qquad\qquad=e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa+1})}\mathbf{p}_{\kappa+1}\cdot\mathbf{q}_{\kappa+1}-e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).\end{split}$
(3.52)

Taking the partial derivative of this equation with respect to its first entry
$\mathbf{q}_{\kappa}$, we arrive at

$\begin{split}&-e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D\sigma_{\alpha}(\mathbf{q}_{\kappa})H^{d+}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1})+e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}D_{1}H^{d+}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1}))\\\
&\qquad\qquad=e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}(D\sigma_{\alpha}(\mathbf{q}_{\kappa})L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})-D_{1}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}))\end{split}$
(3.53)

then use the definition $\mathbf{p}_{\kappa}$ in (3.37) we get the second line
in (3.51). In a similar manner, to obtain the first line in (3.51), we
differentiate (3.52) with respect to the second entry $\mathbf{p}_{\kappa+1}$,
and a direct calculation yields the result.

Notice that if one assumes that the conformal functions
$\sigma_{\alpha}(\mathbf{q}_{\kappa})$ are constant functions, then the right
discrete locally conformally symplectic Hamiltonian dynamics in (3.51) reduce
to the discrete Hamilton’s equation (LABEL:rd-Hameq).

Equivalently, with the left Legendre transformation, we can obtain the local
left discrete Hamiltonian

${H_{\alpha}^{d}}^{-}(\mathbf{q}_{\kappa+1},\mathbf{r}_{\kappa})=-{\mathbf{r}}_{\kappa}\mathbf{q}_{\kappa}-L_{\alpha}^{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1})$
(3.54)

and the local left discrete Hamilton’s equations

$\begin{cases}&\mathbf{q}_{\kappa}=-D_{2}{H_{\alpha}^{d}}^{-}(\mathbf{q}_{\kappa+1},{\mathbf{p}}_{\kappa}),\\\
&{\mathbf{r}}_{\kappa+1}=-D_{1}{H_{\alpha}^{d}}^{-}(\mathbf{q}_{\kappa+1},{\mathbf{p}}_{\kappa}).\end{cases}$

We express the relation (2.20) for the left local discrete Hamiltonian
function to define the left global discrete Hamiltonian function $H^{d-}$ with
the following local expression:

$H^{d-}|_{\alpha}(\mathbf{q}{\kappa+1},\mathbf{p}{\kappa}):=e^{\sigma_{\alpha}(\mathbf{q}{\kappa+1})}{H\alpha^{d}}^{-}(\mathbf{q}{\kappa+1},\mathbf{r}{\kappa}).$
(3.55)

This leads us to write (3.54) in terms of the global discrete Hamiltonian
function as

$e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa+1})}H^{d-}|_{\alpha}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa})=-e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}\mathbf{p}_{\kappa}\cdot\mathbf{q}_{\kappa}-e^{-\sigma_{\alpha}(\mathbf{q}_{\kappa})}L^{d}|_{\alpha}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1}).$
(3.56)

By performing similar calculations done for the case of the right Hamiltonian,
we obtain the left discrete LCS Hamilton’s equation stated in the following
proposition.

###### Proposition 3.11

The left discrete LCS Hamilton’s equation generated by the left discrete
Hamiltonian function $H^{d-}$ in (3.55) is

$\begin{cases}&\mathbf{q}_{\kappa}=-e^{\sigma{\alpha}(\mathbf{q}_{\kappa})-\sigma{\alpha}(\mathbf{q}_{\kappa+1})}D_{2}H^{d-}|_{\alpha}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa}),\\\
&\mathbf{p}_{\kappa+1}=e^{\sigma{\alpha}(\mathbf{q}_{\kappa})-\sigma{\alpha}(\mathbf{q}_{\kappa+1})}(D\sigma{\alpha}(\mathbf{q}_{\kappa+1})\cdot
H^{d-}|{\alpha}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa})\\\
&\qquad\qquad\qquad-
D_{1}H^{d-}|_{\alpha}(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa})).\end{cases}$
(3.57)

The following diagram gives the hierarchy of discrete and continuous
Hamiltonian dynamics discussed in this work:

$\textstyle{{\begin{array}[]{c}LCS~{}HE\\\
\eqref{LCSHE}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{discretization}$$\scriptstyle{\varphi=0}$$\textstyle{{\begin{array}[]{c}~{}HE\\\
\eqref{HE}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{discretization}$$\textstyle{{\begin{array}[]{c}dLCS~{}HE\\\
right:\eqref{r-dLCH}\\\
left:\eqref{l-dLCH}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi=0}$$\textstyle{{\begin{array}[]{c}dHE\\\
right:\eqref{rd-Hameq}\\\ left:\eqref{ld-Hameq}\end{array}}}$ (3.58)

where $d$ refers to term discrete whereas $HE$ stands for an abbreviation of
Hamilton’s equation.

##  4 Conclusion and Future Work

In this work, we have examined the globalization problem of discrete
Lagrangian and Hamiltonian dynamics in the locally conformal framework. We
first presented the locally conformal Euler-Lagrange dynamics (2.40) and
discretized it in (3.25). Diagram 3.31 demonstrates this at a glance.
Additionally, we provided the discretization of LCS Hamiltonian dynamics
(3.51) and (3.57), illustrated in Diagram 3.58.

As a future work, we plan to generalize the present approach to include time-
dependent Lagrangian dynamics. Although discretization techniques for such
Lagrangian and Hamiltonian systems are available in the literature, see, for
example, [12, 13], the globalization problem remains unexplored. Cosymplectic
manifolds are well-suited for studying time-dependent dynamics [8]. To address
the globalization problem in this context, locally conformally cosymplectic
manifolds, as described in [2, 11], could provide a suitable framework.
Another future work to is write geometric Hamilton-Jacobi theory for the
discrete locally conformal dynamics. For the continuous locally conformal
dynamics the Hamilton-Jacobi problem is studied in a series of papers [14, 15,
16, 17] ranging from the particle motion to the fields theories.

## References

  * [1] V. I. Arnold.  Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1989.  Translated from the Russian by K. Vogtmann and A. Weinstein.
  * [2] B. Ateşli, O. Esen, M. de León, and C. Sardón.  On locally conformally cosymplectic Hamiltonian dynamics and Hamilton-Jacobi theory.  J. Phys. A, 56(1):Paper No. 015204, 40, 2023.
  * [3] A. Banyaga.  Some properties of locally conformal symplectic structures.  Commentarii Mathematici Helvetici, 77:383–398, 01 2002.
  * [4] G. Bazzoni.  Locally conformally symplectic and Kähler geometry.  EMS Surv. Math. Sci., 5(1-2):129–154, 2018.
  * [5] S. Blanes and F. Casas.  A concise introduction to geometric numerical integration.  Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016.
  * [6] A. Bravetti, M. Seri, M. Vermeeren, and F. Zadra.  Numerical integration in celestial mechanics: a case for contact geometry.  Celestial Mech. Dynam. Astronom., 132(1):Paper No. 7, 29, 2020.
  * [7] A. Bravetti, M. Seri, and F. Zadra.  New directions for contact integrators.  In International Conference on Geometric Science of Information, pages 209–216. Springer, 2021.
  * [8] B. Cappelletti-Montano, A. De Nicola, and I. Yudin.  A survey on cosymplectic geometry.  Rev. Math. Phys., 25(10):1343002, 55, 2013.
  * [9] J. F. Carinena and P. Guha.  Geometry of non-standard Hamiltonian structures of liénard equations and contact structure.  International Journal of Geometric Methods in Modern Physics, page 2440005, 2023.
  * [10] B. Chantraine and E. Murphy.  Conformal symplectic geometry of cotangent bundles.  Journal of Symplectic Geometry, 17(3):639 – 661, 2019.
  * [11] D. Chinea, M. de León, and J. C. Marrero.  Locally conformal cosymplectic manifolds and time-dependent Hamiltonian systems.  Comment. Math. Univ. Carolin., 32(2):383–387, 1991.
  * [12] L. Colombo, M. G. Fernández, and D. Martín de Diego.  Variational integrators for non-autonomous Lagrangian systems.  J. Comput. Appl. Math., 424:Paper No. 114966, 15, 2023.
  * [13] M. de León and D. M. de Diego.  Variational integrators and time-dependent Lagrangian systems.  Rep. Math. Phys., 49(2-3):183–192, 2002.
  * [14] O. Esen, M. de León, M. Lainz, C. Sardón, and M. Zajac.  Reviewing the geometric Hamilton-Jacobi theory concerning Jacobi and Leibniz identities.  J. Phys. A, 55(40):Paper No. 403001, 62, 2022.
  * [15] O. Esen, M. de León, C. Sardón, and M. Zajac.  Cauchy data space and multisymplectic formulation of conformal classical field theories.  Ann. Physics, 434:Paper No. 168616, 26, 2021.
  * [16] O. Esen, M. de León, C. Sardón, and M. Zajac.  The globalization problem of the Hamilton-Dedonder-Weyl equations on a local $k$-symplectic framework.  Mediterr. J. Math., 18(1):Paper No. 26, 25, 2021.
  * [17] O. Esen, M. de León, C. Sardón, and M. Zajac.  Hamilton-Jacobi formalism on locally conformally symplectic manifolds.  J. Math. Phys., 62(3):Paper No. 033506, 15, 2021.
  * [18] O. Esen, A. Gezici, and H. Gümral.  Variatinal aspects and kinetic theory of locally conformal dynamics.  (Under Construction).
  * [19] O. Esen, C. Sardón, and M. Zajac.  A discrete Hamilton–Jacobi theory for contact Hamiltonian dynamics.  arXiv preprint arXiv:2209.05922, 2022.
  * [20] O. Esen and S. Sütlü.  Discrete dynamical systems over double cross-product Lie groupoids.  International Journal of Geometric Methods in Modern Physics, 18(04):2150057, 2021.
  * [21] F. Guedira and A. Lichnerowicz.  Géométrie des algèbres de Lie locales de Kirillov.  J. Math. Pures Appl. (9), 63(4):407–484, 1984.
  * [22] V. M. Guibout and A. M. Bloch.  Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems.  arXiv preprint math/0409296, 2004.
  * [23] E. Hairer, C. Lubich, and G. Wanner.  Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics.  Springer, Heidelberg, 2010.  Structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition.
  * [24] A. A. Kirillov.  Local Lie algebras.  Uspehi Mat. Nauk, 31(4(190)):57–76, 1976.
  * [25] H.-C. Lee.  A kind of even-dimensional differential geometry and its application to exterior calculus.  Amer. J. Math., 65:433–438, 1943.
  * [26] M. Leok and T. Ohsawa.  Discrete Dirac structures and implicit discrete Lagrangian and Hamiltonian systems.  In XVIII International Fall Workshop on Geometry and Physics, volume 1260 of AIP Conf. Proc., pages 91–102. Amer. Inst. Phys., Melville, NY, 2010.
  * [27] P. Libermann.  Sur les structures presque complexes et autres structures infinitésimales régulières.  Bull. Soc. Math. France, 83:195–224, 1955.
  * [28] P. Libermann and C. M. Marle.  Symplectic geometry and analytical mechanics, volume 35.  Springer Science & Business Media, 2012.
  * [29] A. Lichnerowicz.  Sur la reductivite de certaines algebres d’atomorphismes.  C.R. Acad. Sci. Paris, 253:1302–1304, 1961.
  * [30] A. Lichnerowicz.  Les variétés de Jacobi et leurs algèbres de Lie associées.  J. Math. Pures Appl. (9), 57(4):453–488, 1978.
  * [31] C.-M. Marle.  On Jacobi manifolds and Jacobi bundles, volume 20 of Math. Sci. Res. Inst. Publ., pages 227–246.  Springer, New York, 1991.
  * [32] J. C. Marrero, D. Martín de Diego, and E. Martínez.  Corrigendum: “Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids” [Nonlinearity 19 (2006), no. 6, 1313–1348; mr2230001].  Nonlinearity, 19(12):3003–3004, 2006.
  * [33] J. E. Marsden and T. S. Ratiu.  Introduction to mechanics and symmetry, volume 17 of Texts in Applied Mathematics.  Springer-Verlag, New York, second edition, 1999.  A basic exposition of classical mechanical systems.
  * [34] J. E. Marsden and M. West.  Discrete mechanics and variational integrators.  Acta Numer., 10:357–514, 2001.
  * [35] T. Ohsawa, A. M. Bloch, and M. Leok.  Discrete Hamilton-Jacobi theory and discrete optimal control.  In 49th IEEE Conference on Decision and Control (CDC), pages 5438–5443. IEEE, 2010.
  * [36] T. Ohsawa, A. M. Bloch, and M. Leok.  Discrete Hamilton-Jacobi theory.  SIAM J. Control Optim., 49(4):1829–1856, 2011.
  * [37] A. Otiman and M. Stanciu.  Darboux-weinstein theorem for locally conformally symplectic manifolds.  J. Geom. Phys., 111:1–5, 2017.
  * [38] J. M. Sanz-Serna.  Symplectic integrators for Hamiltonian problems: an overview.  Acta numerica, 1:243–286, 1992.
  * [39] I. Vaisman.  Locally conformal symplectic manifolds.  Internat. J. Math. Math. Sci., 8(3):521–536, 1985.
  * [40] I. Vaisman.  Jacobi manifolds.  Selected topics in Geometry and Mathematical Physics, 1:81–100, 2002.
  * [41] A. van der Schaft and B. Maschke.  Geometry of thermodynamic processes.  Entropy, 20(12):Paper No. 925, 23, 2018.
  * [42] M. Vermeeren, A. Bravetti, and M. Seri.  Contact variational integrators.  J. Phys. A, 52(44):445206, 28, 2019.
  * [43] H. Yoshida.  Construction of higher order symplectic integrators.  Phys. Lett. A, 150(5-7):262–268, 1990.
  * [44] G. Zhong and J. E. Marsden.  Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators.  Phys. Lett. A, 133(3):134–139, 1988.