A Discrete Hamilton–Jacobi Theory

for Contact Hamiltonian Dynamics

Oğul Esen111E-mail<EMAIL_ADDRESS>
Department of Mathematics,

Gebze Technical University, 41400 Gebze, Kocaeli, Turkey.

Cristina Sardón222E-mail<EMAIL_ADDRESS>
Department of Applied Mathematics

Universidad Politécnica de Madrid

C/ José Gutiérrez Abascal, 2, 28006, Madrid. Spain.

Marcin Zajac333E-mail<EMAIL_ADDRESS>
Department of Mathematical Methods in Physics,

Faculty of Physics. University of Warsaw,

ul. Pasteura 5, 02-093 Warsaw, Poland.

###### Abstract

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete)
Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we
first provide a novel geometric Hamilton–Jacobi theory for continuous contact
Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian
formulation, we obtain the discrete equations for Hamiltonian dynamics by the
discrete Legendre transformation. Based on the discrete contact Hamilton
equation, we construct a discrete Hamilton–Jacobi equation for contact
Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is
related to the continuous Hamilton–Jacobi theory presented in this work. Then,
we propose geometric foundations of the discrete Hamilton–Jacobi equations on
contact manifolds in terms of discrete contact flows. At the end of the paper
we provide a numerical example to test the theory.

Keywords: Hamilton-Jacobi theory; Discrete dynamics; Contact manifolds;
Discrete Hamilton–Jacobi.
MSC2020: 65P10; 37J55; 70H20.

##  1 Introduction

This work lies in the intersection among the geometric Hamilton-Jacobi
(abbreviated as HJ) theory, discrete dynamics and contact geometry. A HJ
theory for discrete Hamiltonian dynamics on contact manifolds was missing in
the literature, this is why our aim in this work is to fill this gap by
introducing a discrete HJ equation for discrete contact Hamiltonian dynamics.
Additionally, we shall examine the geometric foundations of the discrete HJ
equation. Accordingly, we shall propose a geometric discrete HJ theorem in the
contact framework.

The Hamilton-Jacobi equation was first given for classical and continuous
Hamiltonian dynamics on symplectic manifolds [1, 26]. More recently, a
geometrization of the HJ equation was established in [12]. Since this work,
various applications and generalizations of the geometric HJ theory have been
exhibited. We refer to two recent surveys [20, 45] (and references therein)
for a contemporary view of the current geometric HJ theory.

On the other hand, in recent years there has been a growing trend in providing
proper discrete analogs of continuous differential equations and designing
numerical methods adapted to the type of discretization pursued, as well as
the type of differential equations that are dealt with. Numerical methods have
shown their utility in solving equations that cannot be managed analytically
[2, 35]. In the branch of geometric mechanics, we deal with a plethora of
geometric structures that provide different underlying geometric properties to
dynamical systems. This is why when one discretizes a dynamical system, one
has to make sure that the discretization is compatible with the geometric
structure and that we are applying specific methods that preserve the
geometric structure. These specific methods are known as geometric integrators
[4, 31]. For example, in classical mechanics we propose numerical methods that
preserve the symplectic structure when we work on a phase space [11, 51],
others methods are energy-preserving numerical methods [43], momentum-
preserving methods [36], etc.

Contact geometry is a popular theme in the recent literature [7, 13, 16, 17,
18], being widely used to describe mechanical dissipative systems, dissipative
field theories and generalizations of the Hamilton principle [32, 33]. Some of
the main uses of contact geometry and its main characteristics can be
consulted in [5]. The dissipative character of the formalism provides an
important geometric foundation for irreversible dynamics, especially
thermodynamics. Here is an incomplete list of works related to contact
mechanics and its role in thermodynamics [6, 25, 27, 28, 41, 44].

To propose a discrete Hamilton–Jacobi theory on a contact manifold as we
shall, one needs to review first the discrete formulation of mechanics on the
Lagrangian side [40]. This leads to the discretization of Lagrangian and
Hamiltonian systems, as well as the variational principles for dynamical
systems and principles of critical action on both the tangent and cotangent
bundle [30, 39]. Such discretizations led to discretized versions of Noether’s
theorem, Legendre transformations, infinitesimal symmetries, etc. The
discretization of the Hamiltonian formulation gave rise to optimal control
problems by developing a discrete maximum principle that yields discrete
necessary conditions for optimality. Furthermore, discrete Hamiltonian
theories have been particularly useful in distributed network optimization and
derivation of variational integrators [37]. The geometry of the space is also
a key point to performing better discretizations. For this matter, it is
important to rely on symmetries and invariants of the geometric space [10]. In
this work we preserve the contact structure under discretization, minimizing
the error in the approximation. Some very recent works addressing discrete
Lagrangian and Hamiltonian dynamics on contact settings are [8, 9, 47, 48,
50]. In these works, one can see the discrete generalized Lagrangian
(Herglotz) dynamics on the extended tangent bundle as well as the discrete
Hamiltonian dynamics on a contact manifold. Since these works are fundamental
for the present study, we shall give a quick review of the theories in the
main body of the paper.

We refer to [42] for the discrete HJ equation on a symplectic manifold. In
[42], the HJ equation is derived by employing discrete symplectic flows. In
the present work, in similar fashion, we carry this discussion to contact
geometry. The role of the discrete symplectic flows will be played by discrete
contact flows analogously.

More recently, in [19], the geometrization of the discrete HJ equation in [42]
has been established in the symplectic category. In this regard, it is
possible to consider the present work as a continuation or an extension of the
works [19, 42] to contact geometry and discrete contact Hamiltonian dynamics.
In the present paper, we both derive the discrete HJ equation on contact
manifolds and then proceed with its geometrization.

The main body of this work contains three sections. In the upcoming Section 2,
we shall first review the fundamental principles of contact manifolds as well
as Lagrangian and Hamiltonian dynamics on contact manifolds, in order to fix
the notation. In Subsection 2.4, we shall provide a HJ theorem for contact
Hamiltonian dynamics. We shall start Section 3 by presenting discrete
Lagrangian and Hamiltonian contact dynamics. In subsection 3.3, we shall
present our main result which is a Hamilton–Jacobi equation for discrete
contact Hamiltonian dynamics. We shall prove this result in terms of discrete
contact flows. In Subsection 3.4, a geometrization of the HJ is provided. In
Section 4, we conclude this work by proposing a numerical example applied to
the well-known parachute equation in contact dynamics. To avoid mathematical
conflict and without loss of generalization, we assume all objects to be
smooth and globally defined unless stated otherwise. Manifolds are connected
and differentiable.

##  2 Fundamentals of Continuous Contact Dynamics

In this section, we first recall briefly the main definitions and results of
the theory of Lagrangian and Hamiltonian dynamics on contact manifolds
following [7, 16, 17]. Later we introduce a geometric Hamilton-Jacobi theory
for Hamiltonian dynamics on contact manifolds. This novel theorem will be the
continuous version of the discrete HJ theorem that will be presented in the
upcoming section.

###  2.1 Contact Manifolds

We call a contact manifold a pair $(M,\eta)$, where $M$ is an odd-dimensional
manifold, say $(2n+1)$-dimensional, with a contact form $\eta$, i.e., a one-
form on $M$ such that $\eta\wedge\mathrm{d}\eta^{n}\neq 0$ is a volume form.
This type of manifold has a distinguished vector field, the Reeb vector field
$\mathcal{R}$, which is the unique vector field that satisfies the two
following identities.

$i_{\mathcal{R}}\mathrm{d}\eta=0,\qquad\eta(\mathcal{R})=1.$ (2.1)

There exists a Darboux coordinate system
$(\mathbf{q},\mathbf{p},s)=(q^{i},p_{i},s)$ on $M$ (with $i$ ranging from 1 to
the dimension of $M$) such that the contact one-form reads

$\eta=\mathrm{d}s-\mathbf{p}\cdot\mathrm{d}\mathbf{q}.$ (2.2)

In these coordinates, we have $\mathcal{R}=\nabla_{s}=\partial/\partial s$.
This local observation provides an example of a contact manifold as the
extended cotangent bundle

$\big{(}T^{*}Q\times\mathbb{R},\eta=\mathrm{d}s-\theta_{Q}\big{)}$ (2.3)

where $\theta_{Q}$ is the pullback of the tautological one-form of $T^{*}Q$.

Musical Morphisms. For a contact manifold $(M,\eta)$, there is a musical
isomorphism

$\flat:TM\longrightarrow T^{*}M,\qquad
v\mapsto\iota_{v}\mathrm{d}\eta+\eta(v)\eta.$ (2.4)

This mapping takes the Reeb field $\mathcal{R}$ to the contact one-form
$\eta$. We denote the inverse of this mapping by $\sharp$. Referring to this,
we define a bivector field $\Lambda$ on $M$ as

$\Lambda(\alpha,\beta)=-\mathrm{d}\eta(\sharp\alpha,\sharp\beta).$ (2.5)

Then referring to $\Lambda$ we introduce the following musical mapping

$\sharp_{\Lambda}:T^{*}M\longrightarrow
TM,\qquad\alpha\mapsto\Lambda(\alpha,\bullet)=\sharp\alpha-\alpha(\mathcal{R})\mathcal{R}.$
(2.6)

Evidently, the mapping $\sharp_{\Lambda}$ fails to be an isomorphism. Notice
that the kernel is spanned by the contact one-form $\eta$.

Legendrian Submanifolds. Let $(M,\eta)$ be a contact manifold. Recall the
associated bivector field $\Lambda$ defined in (2.5). Consider a linear
subbundle $\Xi$ of the tangent bundle $TM$ (that is, a distribution on $M$).
We define the contact complement of $\Xi$ as

$\Xi^{\perp}:=\sharp_{\Lambda}(\Xi^{o}),$ (2.7)

where the sharp map on the right hand side is the one in (2.6) and $\Xi^{o}$
is the annihilator of $\Xi$. We say that $N$ is Legendrian if $TN^{\perp}=TN$.

Consider the contact manifold $T^{*}Q\times\mathbb{R}$ in (2.3) and let $W$ be
a real valued function on the base manifold $Q$. Its first prolongation is

$\rm{J}^{1}W:Q\longrightarrow
T^{*}Q\times\mathbb{R},\qquad\mathbf{q}\mapsto\big{(}\mathbf{q},W_{\mathbf{q}},W(\mathbf{q})\big{)}.$
(2.8)

The image space of the first prolongation $\rm{J}^{1}W$ is a Legendrian
submanifold of $T^{*}Q\times\mathbb{R}$. The converse of this assertion is
also true, that is, if the image space of a section of
$T^{*}Q\times\mathbb{R}\mapsto Q$ is a Legendrian submanifold then it is the
first prolongation of a function $W$.

Contact Diffeomorphisms. Let $(M,\eta)$ and $(\widehat{M},\widehat{\eta})$ be
two contact manifolds. A diffeomorphism $\varphi$ from $M$ to $\widehat{M}$ is
said to be a contact diffeomorphism (or contactomorphism) if it preserves the
contact structures, that is, $T\varphi(\ker\eta)=\ker\widehat{\eta}$. In terms
of contact forms, a contact diffeomorphism $\varphi$ is the one satisfying

$\varphi^{*}\widehat{\eta}=\mu\eta.$ (2.9)

where $\mu$ is a conformal factor. If we additionally impose that the
conformal factor $\mu$ in definition (2.9) has to be equal to one, we arrive
at the conservation of the contact form. We call such a mapping a strict
contact diffeomorphism (or quantomorphism).

Consider two contact manifolds $(M,\eta)$ and $(\widehat{M},\widehat{\eta})$.
A contact product is the product manifold $\widehat{M}\times
M\times\mathbb{R}$ with a contact one-form $\tau\widehat{\eta}\ominus\eta$
where $\tau$ is a global coordinate on $\mathbb{R}$ [3]. It is possible to
validate that the graph of a contact diffeomorphism $\varphi$ is a Legendrian
submanifold of the contact product $\widehat{M}\times M\times\mathbb{R}$.

Generating Functions For Legendrian Submanifolds. In particular, consider two
same dimensional extended cotangent bundles $T^{*}Q\times\mathbb{R}$ and
$T^{*}\widehat{Q}\times\widehat{\mathbb{R}}$ equipped with Darboux’
coordinates $(\mathbf{q},\mathbf{p},s)$ and
$(\widehat{\mathbf{q}},\widehat{\mathbf{p}},\widehat{s})$, respectively. Then,
we determine the contact product
$\big{(}T^{*}\widehat{Q}\times\widehat{\mathbb{R}}\big{)}\times\big{(}T^{*}Q\times\mathbb{R}\big{)}\times\mathbb{R}$
of these two contact manifolds and consider it as a fiber bundle over the
product manifold $\widehat{Q}\times Q\times\mathbb{R}$ given by

$\big{(}T^{*}\widehat{Q}\times\widehat{\mathbb{R}}\big{)}\times\big{(}T^{*}Q\times\mathbb{R}\big{)}\times\mathbb{R}\longrightarrow\widehat{Q}\times
Q\times\mathbb{R},\qquad(\widehat{\mathbf{q}},\widehat{\mathbf{p}},\widehat{s};\mathbf{q},\mathbf{p},s;\tau)\mapsto(\widehat{\mathbf{q}},\mathbf{q},\widehat{s}),$
(2.10)

where $\tau$ is the coordinate on the extended $\mathbb{R}$. In the contact
product manifold, the product contact one-form is defined to be

$\tau\widehat{\eta}-\eta=\tau(\mathrm{d}\widehat{s}-\widehat{\mathbf{p}}\cdot\mathrm{d}\widehat{\mathbf{q}})-(\mathrm{d}s-\mathbf{p}\cdot\mathrm{d}\mathbf{q}).$
(2.11)

Determining the isomorphism

$\big{(}T^{*}\widehat{Q}\times\widehat{\mathbb{R}}\big{)}\times\big{(}T^{*}Q\times\mathbb{R}\big{)}\times\mathbb{R}\simeq
T^{*}(\widehat{Q}\times Q\times\mathbb{R})\times\mathbb{R},$ (2.12)

we provide Darboux coordinates for the contact product manifold as

$(\widehat{\mathbf{q}},\mathbf{q},\widehat{s};\widehat{\boldsymbol{\pi}},\boldsymbol{\pi},\widehat{\pi};z)=(\widehat{\mathbf{q}},\mathbf{q},\widehat{s};\tau\widehat{\mathbf{p}},-\mathbf{p},-\tau;-s).$
(2.13)

This enables us to recast the contact one-form (2.11) in the canonical form
given in (2.2). So that the one-form (2.11) turns out to be

$\tau\widehat{\eta}-\eta=\mathrm{d}z-\widehat{\boldsymbol{\pi}}\cdot\mathrm{d}\widehat{\mathbf{q}}-\boldsymbol{\pi}\cdot\mathrm{d}\mathbf{q}-\widehat{\pi}\mathrm{d}\widehat{s}.$
(2.14)

On the base manifold $\widehat{Q}\times Q\times\mathbb{R}$, we define a smooth
function in form

$W(\widehat{\mathbf{q}},\mathbf{q},\widehat{s})=\widehat{s}+S(\widehat{\mathbf{q}},\mathbf{q},\widehat{s}).$
(2.15)

for a function $S$. The theory manifests that the image space of the first jet
$\rm{J}^{1}W$ is a Legendrian submanifold of the contact product manifold
(2.10). The contact diffeomorphism determined by the Legendrian submanifold
$im(\rm{J}^{1}W)$ is computed to be

$T^{*}Q\times\mathbb{R}\longrightarrow
T^{*}\widehat{Q}\times\widehat{\mathbb{R}},\qquad(\mathbf{q},\mathbf{p},s)\mapsto(\widehat{\mathbf{q}},\widehat{\mathbf{p}},\widehat{s})$
(2.16)

where one has the following identifications

$\mathbf{p}=D_{2}S(\widehat{\mathbf{q}},\mathbf{q},\widehat{s}),\qquad
s=\widehat{s}+S(\widehat{\mathbf{q}},\mathbf{q},\widehat{s}),\qquad\widehat{\mathbf{p}}=-\frac{D_{1}S(\widehat{\mathbf{q}},\mathbf{q},\widehat{s})}{1+D_{3}S(\widehat{\mathbf{q}},\mathbf{q},\widehat{s})},$
(2.17)

where $D_{i}S$ represents the partial derivative of $S$ with respect to its
$i$-th entry.

###  2.2 Contact Lagrangian Dynamics

Now we review the Lagrangian picture of contact systems. In [15] we give a
more comprehensive description which also covers the case of singular
Lagrangians. Let $Q$ be a $n$-dimensional configuration manifold and consider
the extended tangent bundle $TQ\times\mathbb{R}$. If $\mathbf{q}=(q^{i})$ is a
local coordinate system, then the induced coordinates on the $(2n+1)$\-
dimensional manifold $TQ\times\mathbb{R}$ are
$(\mathbf{q},\dot{\mathbf{q}},s)$.

Herglotz’s Action and Herglotz’s Equation. Consider two points
$\mathbf{q}_{0}$ and $\mathbf{q}_{T}$ on $Q$, a Lagrangian $L$ on the extended
tangent bundle $TQ\times\mathbb{R}$ and the following initial value problem

$\frac{\mathrm{d}s}{\mathrm{d}t}=L\Big{(}\mathbf{q},\frac{\mathrm{d}\mathbf{q}}{\mathrm{d}t},s\Big{)},\qquad
s(0)=s_{0},$ (2.18)

where $\mathbf{q}=\mathbf{q}(t)$ is the curve containing $\mathbf{q}_{0}$ and
$\mathbf{q}_{T}$ as initial and final points, i.e.,
$\mathbf{q}(0)=\mathbf{q}_{0}$ and $\mathbf{q}(T)=\mathbf{q}_{T}$, with $0\leq
t\leq T$. Evidently, the Cauchy problem (2.18) will be different for different
curves $\mathbf{q}(t)$. So that a solution $s=s(t)$ of the problem depends on
the curve $\mathbf{q}(t)$ substituted in the Lagrangian function.

Let $\mathcal{C}^{\infty}(Q)$ be the space of all smooth curves on $Q$ joining
$\mathbf{q}_{0}$ and $\mathbf{q}_{T}$. This space depends on the end points
but we omit this fact in the notation. $\mathcal{C}^{\infty}(Q)$ is an
infinite dimensional manifold. According to the Cauchy problem in (2.18), for
every initial value $s(0)=s_{0}$, we determine the Herglotz action as a map
from the product space $\mathcal{C}^{\infty}(Q)\times\mathbb{R}$ to the real
numbers as follows

$\mathcal{C}^{\infty}(Q)\times\mathbb{R}\rightarrow\mathbb{R},\qquad(\mathbf{q},s_{0})\mapsto
s(T)-s(0)=\int_{0}^{T}\frac{\mathrm{d}s}{\mathrm{d}t}dt=\int_{0}^{T}L\left(\mathbf{q},\frac{\mathrm{d}\mathbf{q}}{\mathrm{d}t},s\right)dt.$
(2.19)

The extreme values of the action are the curves $\mathbf{q}=\mathbf{q}(t)$
solving the Herglotz’s (generalized Euler-Lagrange) equations [32, 33, 29]

$\frac{\mathrm{d}}{\mathrm{d}t}L_{\dot{\mathbf{q}}}-L_{\mathbf{q}}=L_{s}L_{\dot{\mathbf{q}}}.$
(2.20)

###  2.3 Contact Hamiltonian Dynamics

Consider a contact manifold $(M,\eta)$. For a Hamiltonian function $H$, the
contact Hamiltonian vector field is defined in terms of the contact one-form
$\eta$ as

$\iota_{X_{H}}\eta=-H,\qquad\iota_{X_{H}}\mathrm{d}\eta=\mathrm{d}H-\mathcal{R}(H)\eta,$
(2.21)

where $\mathcal{R}$ is the Reeb vector field. A direct computation determines
that a Hamiltonian flow does not preserve the contact one-form

$\mathcal{L}_{X_{H}}\eta=\mathrm{d}\iota_{X_{H}}\eta+\iota_{X_{H}}\mathrm{d}\eta=-\mathcal{R}(H)\eta.$
(2.22)

Notice that, according to (2.22), the flow of a contact Hamiltonian system
preserves the contact structure which is defined to be the kernel of the
contact one-form. $X_{H}$ does not preserve the Hamiltonian function, nor the
volume form $\mathrm{d}\eta^{n}\wedge\eta$. Instead, we obtain

${\mathcal{L}}_{X_{H}}\,H=-\mathcal{R}(H)H,\qquad{\mathcal{L}}_{X_{H}}\,(\mathrm{d}\eta^{n}\wedge\eta)=-(n+1)\mathcal{R}(H)\mathrm{d}\eta^{n}\wedge\eta.$
(2.23)

As manifested previously, all contact manifolds locally resemble the extended
cotangent bundle $T^{*}Q\times\mathbb{R}$. In this local view
$(\mathbf{q},\mathbf{p},s)$, the Hamiltonian vector field turns out to be

$X_{H}=H_{\mathbf{p}}\cdot\nabla_{\mathbf{q}}-\big{(}H_{\mathbf{q}}+\mathbf{p}H_{s}\big{)}\cdot\nabla_{\mathbf{p}}+\big{(}\mathbf{p}\cdot
H_{\mathbf{p}}-H\big{)}\nabla_{s}$ (2.24)

Thus, an integral curve $(\mathbf{q}(t),\mathbf{p}(t),s(t))$ of $X_{H}$
satisfies the contact Hamilton equations

$\frac{\mathrm{d}\mathbf{q}}{\mathrm{d}t}=H_{\mathbf{p}},\qquad\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}=-\big{(}H_{\mathbf{q}}+\mathbf{p}H_{s}\big{)},\qquad\frac{\mathrm{d}s}{\mathrm{d}t}=\mathbf{p}\cdot
H_{\mathbf{p}}-H.$ (2.25)

The Legendre Transformation. The fiber derivative $\mathbb{F}L$ of a
Lagrangian function $L$ determines a mapping from the extended tangent bundle
to the extended cotangent bundle as

$\mathbb{F}L:TQ\times\mathbb{R}\longrightarrow
T^{*}Q\times\mathbb{R},\qquad(\mathbf{q},\dot{\mathbf{q}},s)\mapsto(\mathbf{q},L_{\dot{\mathbf{q}}},s).$
(2.26)

See that, for a regular Lagrangian, $\mathbb{F}L$ is a contactomorphism from
the contact manifold $TM\times\mathbb{R}$ equipped with the contact one-form
$\eta_{L}=(\mathbb{F}L)^{*}\eta$ to the contact manifold
$T^{*}Q\times\mathbb{R}$ equipped with $\eta$ in (2.3). In this situation, we
can define the corresponding Hamiltonian function as

$H:T^{*}Q\times\mathbb{R}\to\mathbb{R},\qquad H=\dot{\mathbf{q}}\cdot
L_{\dot{\mathbf{q}}}-L.$ (2.27)

A direct computation proves that the contact Hamilton equations are the
generalized Euler-Lagrange equations, they coincide for regular Lagrangian
functions. For the Legendre transformation of nonregular cases, we cite [23,
15, 13].

###  2.4 A Continuous Geometric Hamilton-Jacobi Theory on Contact Manifolds

We start with the fibration $Q\mapsto\mathbb{R}$ whose first jet bundle is the
extended cotangent bundle $T^{*}Q\times\mathbb{R}$. In this case, the
fibration is given by the target map

$\rho:T^{*}Q\times\mathbb{R}\longrightarrow
Q,\qquad(\mathbf{z},s)\mapsto\pi_{Q}(\mathbf{z}),$ (2.28)

where $\pi_{Q}$ is the cotangent bundle projection. The first prolongation of
a smooth function $F$ on $Q$ is a section of the projection $\rho$. We write
the first prolongation as

$\gamma=\rm{J}^{1}F:Q\longrightarrow
T^{*}Q\times\mathbb{R},\qquad\mathbf{q}\mapsto(\mathbf{q},F_{\mathbf{q}},F(\mathbf{q})).$
(2.29)

Notice that $T^{*}Q\times\mathbb{R}$ is a contact manifold and the image space
of the first prolongation $\gamma$ is a Legendrian submanifold of this space.

Consider a section $\gamma$ in form (2.29) and a contact Hamiltonian vector
field $X_{H}$ given as in (2.24). We define a vector field

$X_{H}^{\gamma}:=T\rho\circ X_{H}\circ\gamma,$ (2.30)

on the base manifold $Q$ according to the commutativity of the following
diagram

$\textstyle{T^{*}Q\times\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{X_{H}}$$\textstyle{T(T^{*}Q\times\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T{\rho}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{X^{\gamma}_{H}}$$\textstyle{TQ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\gamma}$
(2.31)

where $T{\rho}$ is the tangent mapping of the fibration $\rho$ in (2.28).
Next, we state a geometric Hamilton-Jacobi theorem for contact Hamiltonian
dynamics.

######  Theorem 1.

For the first prolongation $\gamma$ of a function $F$ on $Q$, the following
two conditions are equivalent:

  1. 1.

The vector fields $X_{H}$ and $X^{\gamma}_{H}$ are $\gamma$-related, that is

$T\gamma\circ X_{H}^{\gamma}=X_{H}\circ\gamma$ (2.32)

where $T\gamma:TQ\mapsto T(T^{*}Q\times\mathbb{R})$ is the tangent mapping of
the section $\gamma$.

  2. 2.

The equation

$d(H\circ\gamma)=0$ (2.33)

is fulfilled.

###### Proof.

We prove this theorem in local coordinates. The restriction of the Hamiltonian
vector field $X_{H}$ to the image of $\gamma$ is computed to be

$\begin{split}X_{H}\circ\gamma(\mathbf{q})&=\Big{(}\mathbf{q},F_{\mathbf{q}},F(\mathbf{q});H_{\mathbf{p}}\Big{|}_{im(\gamma)},-H_{\mathbf{q}}\Big{|}_{im(\gamma)}-H_{s}F_{\mathbf{q}}\Big{|}_{im(\gamma)},F_{\mathbf{q}}\cdot
H_{\mathbf{q}}\Big{|}_{im(\gamma)}-H\Big{|}_{im(\gamma)}\Big{)}.\end{split}$
(2.34)

Using $T\rho$, we map this vector field down to the tangent bundle of $Q$.
This is the projected vector field (2.30) that is

$X^{\gamma}_{H}(\mathbf{q})=H_{\mathbf{p}}\Big{|}_{im(\gamma)}\cdot\nabla_{\mathbf{q}}.$
(2.35)

On the other hand, the tangent mapping of the section $\gamma$ is computed to
be

$T\gamma(\mathbf{q};\mathbf{\dot{q}})=\big{(}\mathbf{q},F_{\mathbf{q}}(\mathbf{q}),F(\mathbf{q});\mathbf{\dot{q}},F_{\mathbf{qq}}(\mathbf{q})\mathbf{\dot{q}},F_{\mathbf{q}}(\mathbf{q})\cdot\mathbf{\dot{q}}\big{)},$
(2.36)

where the notation $F_{\mathbf{qq}}(\mathbf{q})\mathbf{\dot{q}}$ stands for
the multiplication of the Hessian matrix $F_{\mathbf{qq}}$ with the column
vector $\mathbf{\dot{q}}$. Accordingly, the left hand side of the equation
(2.32) is computed to be

$\begin{split}T\gamma\circ
X^{\gamma}_{H}(\mathbf{q})&=\big{(}\mathbf{q},F_{\mathbf{q}}(\mathbf{q}),F(\mathbf{q});H_{\mathbf{p}}\Big{|}_{im(\gamma)},F_{\mathbf{qq}}H_{\mathbf{p}}\Big{|}_{im(\gamma)},F_{\mathbf{q}}\cdot
H_{\mathbf{q}}\Big{|}_{im(\gamma)}\big{)}.\end{split}$ (2.37)

To satisfy (2.32), the expressions (2.34) and (2.37) must be same. The first
three entries of these local realizations are the same. The first of the fiber
variables (that coincides with the fourth entries) are also the same. See that
the fifth and the sixth entries of (2.34) and (2.37) are not equal. The fifth
entries are equal if and only if

$F_{\mathbf{qq}}H_{\mathbf{p}}\Big{|}_{im(\gamma)}+H_{\mathbf{q}}\Big{|}_{im(\gamma)}+H_{s}F_{\mathbf{q}}\Big{|}_{im(\gamma)}=0.$
(2.38)

See that, the identity (2.38) can be compactly written as $d(H\circ\gamma)=0$.
This gives that $H$ turns out to be a constant if it is restricted to the
image space of the first prolongation $\gamma$. On the other hand, the sixth
entries are the same if and only if $H$ is not only constant but further it
vanishes on the image space of the first prolongation $\gamma$. This is the
second condition (2.33). The inverse of the assertion is proved by reversing
the arguments. This completes the proof. ∎

There exist alternative versions of the geometric HJ theory for contact
manifolds where the base space is considered to be the extended configuration
space $Q\times\mathbb{R}$. To check these versions, we refer to [24, 14, 19].

##  3 Discrete Contact Dynamics

In this section, we first recall discrete Lagrangian and Hamiltonian dynamics
on the contact framework. The definitions and the approach we adopt are the
ones displayed in [9, 8, 48, 47]. We also wish to cite [50]. Then we introduce
the HJ theory for discrete Hamiltonian dynamics. This is the contact version
of the approach performed in [42] for the case of symplectic manifolds. In
accordance with this, we state a geometric Hamilton-Jacobi theory. This is a
contact generalization of the one presented in [19].

###  3.1 Discrete Contact Lagrangian Dynamics

In the framework of discrete dynamics the role of differentiable curves in a
configuration manifold $Q$ is played by the finite sequences of points in $Q$.
Therefore, we are interested in the space of sequences consisting of $N+1$
points, denoted by $\mathcal{C}^{N}(Q)$. It is interesting to note that
$\mathcal{C}^{N}(Q)$ is a manifold isomorphic to $N+1$ number of copies of
$Q$, denoted by $Q^{N+1}$. We denote such a sequence by a set
$[\mathbf{q}]=\\{\mathbf{q}_{0},\dots,\mathbf{q}_{N}\\}$ which is a point in
$Q^{N+1}$. Here, $\mathbf{q}_{0}$ is the initial point of the sequence,
whereas $\mathbf{q}_{N}$ is the end point. We use the subindex
$\mathbf{q}_{\kappa}$ to denote the location of the point in the sequence from
$\kappa=0$ to $\kappa=N$. We call $[\mathbf{q}]$ a discrete curve.

Herglotz’s Action and Herglotz’s Equation. A discrete Lagrangian in this
framework is a real valued function $L_{d}$ defined on the product space
$Q\times Q\times\mathbb{R}$. Comparing with the continuous case, we see that
in this geometry the role of the tangent bundle $TQ$ is played by the pair
$Q\times Q$. More technically, the discrete Lagrangian $L_{d}$ is an
approximation of the exact contact discrete Lagrangian $L_{d}^{ex}$ which is
defined to be

$L_{d}^{ex}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})=\int_{t_{\kappa}}^{t_{\kappa+1}}{L\left(\mathbf{q},\frac{\mathrm{d}\mathbf{q}}{\mathrm{d}t},s\right)dt}$
(3.1)

where $\mathbf{q}$ is a solution of the continuous generalized Euler-Lagrange
equation (2.20) with boundary conditions
$\mathbf{q}(t_{\kappa})=\mathbf{q}_{\kappa}$ and
$\mathbf{q}(t_{\kappa+1})=\mathbf{q}_{\kappa+1}$. To each discrete curve
$[\mathbf{q}]$ and for a fixed initial value point $s_{0}$, we introduce the
discrete Cauchy problem as

$s_{\kappa}-s_{\kappa-1}=L_{d}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa},s_{\kappa-1}),\qquad
s_{0}=s(0),$ (3.2)

where $\kappa$ runs from $1$ to $N$. We determine the discrete Herglotz action
as a map from the product space $\mathcal{C}^{N}(Q)\times\mathbb{R}$ to the
real numbers that is

$\mathcal{C}^{N}(Q)\times\mathbb{R}\longrightarrow\mathbb{R},\qquad([\mathbf{q}],s_{0})\mapsto
s_{N}-s_{0}=\sum_{\kappa=0}^{N-1}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa}).$
(3.3)

The extreme values of the discrete action (3.3) are curves solving the
discrete Herglotz’s (generalized discrete Euler-Lagrange) equations

$\begin{split}&D_{1}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})+\big{(}1+D_{3}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})\big{)}D_{2}L_{d}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa},s_{\kappa-1})=0\\\
&s_{\kappa}-s_{\kappa-1}=L_{d}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa},s_{\kappa-1}),\qquad
s_{0}=s(0),\end{split}$ (3.4)

provided that $1+D_{3}L_{d}\neq 0$. Here, $D_{i}L_{d}$ refers to the partial
derivative of the discrete Lagrangian with respect to its $i$-th entry.

The Discrete Legendre Transformation. Assume a discrete Lagrangian function
$L_{d}$ satisfying the condition $1+D_{3}L_{d}\neq 0$. We define the following
Legendre transformations from the extended discrete space $Q\times
Q\times\mathbb{R}$ to the extended cotangent bundle $T^{*}Q\times\mathbb{R}$
as

$\begin{split}\mathbb{F}L_{d}^{+}&:Q\times Q\times\mathbb{R}\longrightarrow
T^{*}Q\times\mathbb{R},\\\ &\hskip
56.9055pt(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})\mapsto\big{(}\mathbf{q}_{\kappa+1},D_{2}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa}),s_{\kappa}+L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})\big{)}\\\
\mathbb{F}L_{d}^{-}&:Q\times Q\times\mathbb{R}\longrightarrow
T^{*}Q\times\mathbb{R},\\\ &\hskip
56.9055pt(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})\mapsto\left(\mathbf{q}_{\kappa},-\frac{D_{1}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})}{1+D_{3}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})},s_{\kappa}\right).\end{split}$
(3.5)

So that we have two alternative definitions of the momenta given as

$\mathbf{p}^{+}_{\kappa-1,\kappa,\kappa}=D_{2}L_{d}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa},s_{\kappa}),\qquad\mathbf{p}^{-}_{\kappa,\kappa+1,\kappa}=-\frac{D_{1}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})}{1+D_{3}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})}.$
(3.6)

In the light of the discrete Herglotz equations (3.4), one can directly
establish the momentum matching equation

$\mathbb{F}L_{d}^{+}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})=\mathbb{F}L_{d}^{-}(\mathbf{q}_{\kappa+1},\mathbf{q}_{\kappa+2},s_{\kappa+1}).$
(3.7)

So that, in this case, one can define the momenta

$\mathbf{p}_{\kappa}:=\mathbf{p}^{+}_{\kappa-1,\kappa,\kappa}=\mathbf{p}^{-}_{\kappa,\kappa+1,\kappa+1}.$
(3.8)

A discrete Lagrangian $L_{d}$ is called regular if the Hessian matrix obtained
by taking the partial derivatives of $L_{d}$ is nondegenerate, that is

$\det[D_{2}D_{2}L_{d}(\mathbf{q}_{\kappa-1},\mathbf{q}_{\kappa},s_{\kappa})]\neq
0.$ (3.9)

Equivalently, a discrete Lagrangian is a regular Lagrangian if the Legendre
transformations in (3.5) become invertible.

Diffeomorphisms. Consider a regular discrete Lagrangian function $L_{d}$,
referring to the discrete Legendre mappings in (3.5), one can define a local
diffeomorphism on the extended discrete space as

$\begin{split}\Phi&:Q\times Q\times\mathbb{R}\longrightarrow Q\times
Q\times\mathbb{R},\\\ &\hskip
56.9055pt(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})\mapsto(\mathbf{q}_{\kappa+1},\mathbf{q}_{\kappa+2},s_{\kappa+1}):=(\mathbb{F}L_{d}^{-})^{-1}\circ\mathbb{F}L_{d}^{+}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa}).\end{split}$
(3.10)

Similarly, one can define a diffeomorphism on the extended cotangent bundle as
follows

$\tilde{\Phi}:=\mathbb{F}L_{d}^{+}\circ(\mathbb{F}L_{d}^{-})^{-1}.$ (3.11)

A direct comparison shows that the diffeomorphism $\Phi$ given in (3.10) and
the diffeomorphism $\tilde{\Phi}$ given in (3.11) are related by the Legendre
transformations

$\tilde{\Phi}=\mathbb{F}L_{d}^{+}\circ\Phi\circ(\mathbb{F}L_{d}^{+})^{-1},\qquad\tilde{\Phi}=\mathbb{F}L_{d}^{-}\circ\Phi\circ(\mathbb{F}L_{d}^{-})^{-1}.$
(3.12)

In order to be more specific about the diffeomorphism we plot the following
commutative diagram.

|  |  |
---|---|---|---
$\textstyle{(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{F}L_{d}^{-}}$$\scriptstyle{\mathbb{F}L_{d}^{+}}$$\scriptstyle{\Phi}$$\textstyle{(\mathbf{q}_{\kappa+1},\mathbf{q}_{\kappa+2},s_{\kappa+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{F}L_{d}^{-}}$$\scriptstyle{\mathbb{F}L_{d}^{+}}$$\textstyle{(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa},s_{\kappa})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\Phi}}$$\textstyle{(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa+1},s_{\kappa+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\Phi}}$$\textstyle{(\mathbf{q}_{\kappa+2},\mathbf{p}_{\kappa+2},s_{\kappa+2})}$
(3.13)

See that in this diagram in the bottom row we have the flow on the extended
discrete cotangent bundle $T^{*}Q\times\mathbb{R}$, whereas the top row
corresponds with the flow on the extended discrete space $Q\times
Q\times\mathbb{R}$. Here, the Legendre transformations establish the
equivalency between the flows.

###  3.2 Discrete Contact Hamiltonian Dynamics

To derive Hamiltonian dynamics, we use that a discrete contact Lagrangian is
essentially a generating function of type one [1] and that we can apply the
defined Legendre transformations to the discrete Lagrangian to find a discrete
Hamiltonian [1, 26]. We start with the definition of momentum
$\mathbf{p}_{\kappa+1}$ and the local inversion operation

$\mathbf{p}_{\kappa+1}=D_{2}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa}),\qquad\mathbf{q}_{\kappa+1}=\Psi(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa}),$
(3.14)

provided that the regularity condition holds, i.e., the operator
$D_{2}D_{2}L_{d}$ does not vanish. Referring to these local identifications,
we introduce the right discrete Hamiltonian function on the extended cotangent
bundle as

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

By taking the partial derivatives of the Hamiltonian function $H^{+}_{d}$ with
respect to its arguments (applying the chain rule referring to (3.14)), we
arrive at the following expressions

$\begin{split}D_{1}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})&=-D_{1}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa})\\\
D_{2}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})&=\mathbf{q}_{\kappa+1}\\\
D_{3}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})&=-D_{3}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa}).\end{split}$
(3.16)

Notice that, these equations hold under the identifications (3.14). The second
equation in the list determines $\mathbf{q}_{\kappa+1}$. We substitute the
first and third identities in (3.16) into the discrete Euler-Lagrange equation
(3.4), and using (3.14), we get

$\mathbf{p}_{\kappa}=\frac{D_{1}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})}{1-D_{3}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})}.$
(3.17)

Considering the discrete Cauchy problem (3.2) as part of the set, we arrive at
the following system of equations, which we call right discrete contact
Hamilton equations,

$\begin{split}\mathbf{q}_{\kappa+1}&=D_{2}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa}),\\\
\mathbf{p}_{\kappa}&=\frac{D_{1}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})}{(1-D_{3}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa}))},\\\
s_{\kappa+1}&=s_{\kappa}+\mathbf{p}_{\kappa+1}\cdot
D_{2}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})-H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa}).\end{split}$
(3.18)

Left Hamilton Equations. The left discrete Hamiltonian function on the
extended cotangent bundle is

$H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})=-\mathbf{p}_{\kappa}\cdot\mathbf{q}_{\kappa}-L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1}).$
(3.19)

By taking the partial derivatives of the Hamiltonian function $H^{-}_{d}$, we
arrive at the following expressions

$\begin{split}D_{1}H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})&=-\mathbf{q}_{\kappa}\\\
D_{2}H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})&=\mathbf{p}_{\kappa+1}\\\
D_{3}H^{+}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})&=-D_{3}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1}).\end{split}$
(3.20)

The second equation in the list determines $\mathbf{p}_{\kappa+1}$.

$\mathbf{p}_{\kappa+1}=D_{2}H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})=-D_{2}L_{d}(\mathbf{q}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1}).$
(3.21)

Considering the discrete Cauchy problem (3.2) as part of the set, we arrive at
the following system of equations, which we call the left discrete contact
Hamilton equations,

$\begin{split}&\mathbf{q}_{\kappa}=-D_{1}H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})\\\
&\mathbf{p}_{\kappa+1}=D_{2}H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})\\\
&s_{\kappa}=s_{\kappa+1}-\mathbf{p}_{\kappa}\cdot
D_{1}H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})+H^{-}_{d}(\mathbf{p}_{\kappa},\mathbf{q}_{\kappa+1},s_{\kappa+1})\end{split}$
(3.22)

From now on, we shall focus on the right discrete dynamics, since everything
can be reenacted in terms of left discrete approach straightforwardly.

###  3.3 Discrete Contact Hamilton–Jacobi theory

The discrete contact Hamilton–Jacobi equation can be derived in terms of a
generating function of a coordinate transformation that trivializes the
dynamics, as it is done in the classical continuous theory [26]. For this we
need a generating function to introduce a contact diffeomorphism from the
contact manifold $T^{*}Q\times\mathbb{R}$ to the contact manifold
$T^{*}\widehat{Q}\times\widehat{\mathbb{R}}$ that is

$T^{*}Q\times\mathbb{R}\longrightarrow
T^{*}\widehat{Q}\times\widehat{\mathbb{R}},\qquad(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa},s_{\kappa})\rightarrow(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa},\widehat{s}_{\kappa}).$
(3.23)

Recall from (2.17) that, in Section 2.1, we have obtained a generating
function defined on the product space $\widehat{Q}\times
Q\times\widehat{\mathbb{R}}$ realizing the transformation. The main result for
the generating function is described in the next theorem, following the lines
of [42] in the symplectic case.

######  Theorem 2.

Consider the right discrete contact Hamilton equations (3.18) and a discrete
phase space $(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},s_{\kappa})$.
Consider also a transformation (3.23) that satisfies:

  1. 1.

The old and new coordinates are related by a generating function
$S^{\kappa}:\widehat{Q}\times
Q\times\widehat{\mathbb{R}}\rightarrow\mathbb{R}$ of the type

$\mathbf{p}_{\kappa}=D_{2}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa}),\quad
s_{\kappa}=\widehat{s}_{\kappa}+S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa}),\quad\widehat{\mathbf{p}}_{\kappa}=-\frac{D_{1}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})}{1+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})}.$
(3.24)

  2. 2.

The dynamics in the new coordinates
$(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa},\widehat{s}_{\kappa})$
is rendered trivial, i.e.,

$(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa},\widehat{s}_{\kappa})=(\widehat{\mathbf{q}}_{\kappa+1},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa+1}).$
(3.25)

Then, the set of functions $\\{S^{\kappa}\\}$ satisfies the right discrete
contact Hamilton–Jacobi equation:

$\begin{split}S^{\kappa+1}(\widehat{\mathbf{q}}_{0},\mathbf{q}_{\kappa+1},\widehat{s}_{0})&-S^{\kappa}(\widehat{\mathbf{q}}_{0},\mathbf{q}_{\kappa},\widehat{s}_{0})-D_{2}S^{\kappa+1}(\widehat{\mathbf{q}}_{0},\mathbf{q}_{\kappa+1},\widehat{s}_{0})\cdot\mathbf{q}_{\kappa+1}\\\
&+H^{+}_{d}(\mathbf{q}_{\kappa},D_{2}S^{\kappa+1}(\widehat{\mathbf{q}}_{0},\mathbf{q}_{\kappa+1},\widehat{s}_{0}),\widehat{s}_{0}+S^{\kappa}(\widehat{\mathbf{q}}_{0},\mathbf{q}_{\kappa},\widehat{s}_{0}))=0.\end{split}$
(3.26)

###### Proof.

We establish the proof in four steps.

Step 1. Consider the right discrete contact Hamilton equations (3.18) in the
new variables
$(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa},\widehat{s}_{\kappa})$,
that is

$\begin{split}\widehat{\mathbf{q}}_{\kappa+1}&=D_{2}\widehat{H}^{+}_{d}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa}),\\\
\widehat{\mathbf{p}}_{\kappa}&=D_{1}\widehat{H}^{+}_{d}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})/\big{(}1-D_{3}\widehat{H}^{+}_{d}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\big{)},\\\
\widehat{s}_{\kappa+1}&=\widehat{s}_{\kappa}+\widehat{\mathbf{p}}_{\kappa+1}\cdot
D_{2}\widehat{H}^{+}_{d}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})-\widehat{H}^{+}_{d}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa}).\end{split}$
(3.27)

These equations can be recast in the form of a total derivative of the right
discrete contact Hamiltonian in the following way

$\begin{split}\mathrm{d}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})&=D_{1}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}+D_{2}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\cdot\mathrm{d}\widehat{\mathbf{p}}_{\kappa+1}\\\
&\hskip
170.71652pt+D_{3}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\mathrm{d}\widehat{s_{\kappa}}\\\
&=\widehat{\mathbf{p}}_{\kappa}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}+\widehat{\mathbf{q}}_{\kappa+1}\cdot\mathrm{d}\widehat{\mathbf{p}}_{\kappa+1}+D_{3}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}\end{split}$
(3.28)

where we have employed (3.27) in the total derivative. More explicitly we have
replaced $D_{1}{\widehat{H}^{+}_{d}}$ from the second equation in (3.27).
Notice that we have used the following notation for the discrete contact form

$\widehat{\eta}_{\kappa}=\mathrm{d}\widehat{s}_{\kappa}-\widehat{\mathbf{p}}_{\kappa}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}.$
(3.29)

Step 2. Start with the generating function
$S=S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})$
satisfying (3.24). The total derivative of the generating function as

$\begin{split}\mathrm{d}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})&=D_{1}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}+D_{2}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\cdot
d\mathbf{q}_{\kappa}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\mathrm{d}\widehat{s}_{\kappa}\\\
&=-\widehat{\mathbf{p}}_{\kappa}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}+\mathbf{p}_{\kappa}\cdot
d\mathbf{q}_{\kappa}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}\end{split}$
(3.30)

in which we can introduce $D_{1}S^{k}$ and $D_{2}S^{k}$ from (3.24).
Recursively, we can write the expression for $\mathrm{d}S^{\kappa+1}$, as:

$\mathrm{d}S^{\kappa+1}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})=-\widehat{\mathbf{p}}_{\kappa+1}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa+1}+\mathbf{p}_{\kappa+1}\cdot
d\mathbf{q}_{\kappa+1}+D_{3}S^{\kappa+1}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})\widehat{\eta}_{\kappa+1}$
(3.31)

where
$\widehat{\eta}_{\kappa+1}=\mathrm{d}\widehat{s}_{\kappa+1}-\widehat{\mathbf{p}}_{\kappa+1}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa+1}$.

Coming back to the expression of the total derivative of the Hamiltonian
(3.28), realize that the second term
$\widehat{\mathbf{q}}_{\kappa+1}\cdot\mathrm{d}\widehat{\mathbf{p}}_{\kappa+1}$
can be written as
$d(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1})-\widehat{\mathbf{p}}_{\kappa+1}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa+1}$.
Then we substitute the expressions (3.30) and (3.31) into the total derivative
of the Hamiltonian function. In the light of these, we can continue the
calculation (3.28) as follows

$\begin{split}\mathrm{d}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},&\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})=\widehat{\mathbf{p}}_{\kappa}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}+\widehat{\mathbf{q}}_{\kappa+1}\cdot\mathrm{d}\widehat{\mathbf{p}}_{\kappa+1}+D_{3}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}\\\
&=\widehat{\mathbf{p}}_{\kappa}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}-\widehat{\mathbf{p}}_{\kappa+1}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa+1}+\mathrm{d}(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1})+D_{3}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}\\\
&=\mathbf{p}_{\kappa}\cdot\mathrm{d}\mathbf{q}_{\kappa}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}-\mathrm{d}S^{\kappa}\\\
&\qquad-\big{(}\mathbf{p}_{\kappa+1}\cdot\mathrm{d}\mathbf{q}_{\kappa+1}+D_{3}S^{\kappa+1}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})\widehat{\eta}_{\kappa+1}-\mathrm{d}S^{\kappa+1}\big{)}\\\
&\qquad+\mathrm{d}(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1})+D_{3}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}.\end{split}$
(3.32)

Step 3. We take the exterior derivative of the second identity in (3.24). See
that

$\begin{split}ds_{\kappa}&=\mathrm{d}\widehat{s}_{\kappa}+\mathrm{d}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\\\
&=\mathrm{d}\widehat{s}_{\kappa}+D_{1}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}+D_{2}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\cdot\mathrm{d}\mathbf{q}_{\kappa}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\cdot\mathrm{d}\widehat{s}_{\kappa}\\\
&=\mathrm{d}\widehat{s}_{\kappa}-(1+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa}))\widehat{\mathbf{p}}_{\kappa}\cdot\mathrm{d}\widehat{\mathbf{q}}_{\kappa}+\mathbf{p}_{\kappa}\cdot\mathrm{d}\mathbf{q}_{\kappa}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\cdot\mathrm{d}\widehat{s}_{\kappa}\\\
&=\big{(}1+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\big{)}\widehat{\eta}_{\kappa}+\mathbf{p}_{\kappa}\cdot\mathrm{d}\mathbf{q}_{\kappa}.\end{split}$
(3.33)

So that we arrive at two expressions. One is the relationship between the
contact forms in terms of the generating function

$\eta_{\kappa}=\big{(}1+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\big{)}\widehat{\eta}_{\kappa}.$
(3.34)

Additionally, we have the followings identities

$\begin{split}&\mathbf{p}_{\kappa}\cdot\mathrm{d}\mathbf{q}_{\kappa}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}=ds_{\kappa}-\widehat{\eta}_{\kappa},\\\
&\mathbf{p}_{\kappa+1}\cdot\mathrm{d}\mathbf{q}_{\kappa+1}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})\widehat{\eta}_{\kappa+1}=\mathrm{d}s_{\kappa+1}-\widehat{\eta}_{\kappa+1}.\end{split}$
(3.35)

We continue the calculation (3.32) by substituting these equations.
Accordingly, we have that

$\begin{split}\mathrm{d}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})&=\mathbf{p}_{\kappa}\cdot\mathrm{d}\mathbf{q}_{\kappa}+D_{3}S^{\kappa}(\widehat{\mathbf{q}}_{\kappa},\mathbf{q}_{\kappa},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}-\mathrm{d}S^{\kappa}\\\
&\qquad-\big{(}\mathbf{p}_{\kappa+1}\cdot\mathrm{d}\mathbf{q}_{\kappa+1}+D_{3}S^{\kappa+1}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})\widehat{\eta}_{\kappa+1}-\mathrm{d}S^{\kappa+1}\big{)}\\\
&\qquad+d(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1})+D_{3}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}.\\\
&=\mathrm{d}s_{\kappa}-\widehat{\eta}_{\kappa}-\mathrm{d}S^{\kappa}-(\mathrm{d}s_{\kappa+1}-\widehat{\eta}_{\kappa+1}-\mathrm{d}S^{\kappa+1})+\mathrm{d}(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1})\\\
&\qquad+D_{3}{\widehat{H}^{+}_{d}}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})\widehat{\eta}_{\kappa}.\end{split}$
(3.36)

Since on image space, i.e, the space with coordinates
$(\widehat{\mathbf{q}},\widehat{\mathbf{p}},\widehat{s})$, the dynamics is
rendered trivial (3.25), we take the Hamiltonian function as

$\widehat{H}^{+}_{d}(\widehat{\mathbf{q}}_{\kappa},\widehat{\mathbf{p}}_{\kappa+1},\widehat{s}_{\kappa})=\widehat{\mathbf{q}}_{\kappa}\cdot\widehat{\mathbf{p}}_{\kappa+1}.$
(3.37)

This gives that the Hamiltonian function is independent of
$\widehat{s}_{\kappa}$ so that $D_{3}{\widehat{H}^{+}_{d}}$ vanishes
identically. We substitute this into (3.36) and arrive at

$\begin{split}\mathrm{d}{\widehat{H}^{+}_{d}}&=\mathrm{d}s_{\kappa}-\widehat{\eta}_{\kappa}-\mathrm{d}S^{\kappa}-(\mathrm{d}s_{\kappa+1}-\widehat{\eta}_{\kappa+1}-\mathrm{d}S^{\kappa+1})+\mathrm{d}(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1})+D_{3}{\widehat{H}^{+}_{d}}\widehat{\eta}_{\kappa}.\\\
\mathrm{d}(\widehat{\mathbf{q}}_{\kappa}\cdot\widehat{\mathbf{p}}_{\kappa+1})&=\mathrm{d}s_{\kappa}-\widehat{\eta}_{\kappa}-\mathrm{d}S^{\kappa}-(ds_{\kappa+1}-\widehat{\eta}_{\kappa+1}-\mathrm{d}S^{\kappa+1})+\mathrm{d}(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1}).\end{split}$
(3.38)

Step 4. Since the dynamics is trivial on the image space, the discrete contact
forms coincide

$\widehat{\eta}_{\kappa}=\widehat{\eta}_{\kappa+1}$ (3.39)

and so do the coupling functions

$\widehat{\mathbf{q}}_{\kappa}\cdot\widehat{\mathbf{p}}_{\kappa+1}=\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1}=\widehat{\mathbf{q}}_{0}\cdot\widehat{\mathbf{p}}_{0}.$
(3.40)

Now, we take the exterior derivative of the third equation in (3.18) and
replace the second equation in (3.18). This gives

$\begin{split}\mathrm{d}s_{\kappa+1}-\mathrm{d}s_{\kappa}&=\mathrm{d}(\mathbf{p}_{\kappa+1}\cdot\mathbf{q}_{\kappa+1})-\mathrm{d}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})\\\
&=\mathrm{d}(D_{2}S^{\kappa+1}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})\cdot\mathbf{q}_{\kappa+1})-\mathrm{d}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})\end{split}$
(3.41)

We collect all the equations in (3.39), (3.40) and (3.41) and substitute them
into (3.38). These read the following

$\begin{split}\mathrm{d}(\widehat{\mathbf{q}}_{\kappa}\cdot\widehat{\mathbf{p}}_{\kappa+1})&=\mathrm{d}s_{\kappa}-\widehat{\eta}_{\kappa}-\mathrm{d}S^{\kappa}-(\mathrm{d}s_{\kappa+1}-\widehat{\eta}_{\kappa+1}-\mathrm{d}S^{\kappa+1})+\mathrm{d}(\widehat{\mathbf{q}}_{\kappa+1}\cdot\widehat{\mathbf{p}}_{\kappa+1})\\\
\mathrm{d}(\widehat{\mathbf{q}}_{0}\cdot\widehat{\mathbf{p}}_{0})&=\mathrm{d}s_{\kappa}-\widehat{\eta}_{\kappa}-\mathrm{d}S^{\kappa}-(\mathrm{d}s_{\kappa+1}-\widehat{\eta}_{\kappa+1}-\mathrm{d}S^{\kappa+1})+\mathrm{d}(\widehat{\mathbf{q}}_{0}\cdot\widehat{\mathbf{p}}_{0})\\\
0&=\mathrm{d}s_{\kappa}-\mathrm{d}s_{\kappa+1}+\mathrm{d}S^{\kappa+1}-\mathrm{d}S^{\kappa}\\\
0&=-\mathrm{d}(D_{2}S^{\kappa+1}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})\cdot\mathbf{q}_{\kappa+1})+\mathrm{d}H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})+\mathrm{d}S^{\kappa+1}-\mathrm{d}S^{\kappa}\\\
0&=\mathrm{d}\big{(}S^{\kappa+1}-S^{\kappa}-D_{2}S^{\kappa+1}(\widehat{\mathbf{q}}_{\kappa+1},\mathbf{q}_{\kappa+1},\widehat{s}_{\kappa+1})\cdot\mathbf{q}_{\kappa+1}+H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})\big{)}.\end{split}$
(3.42)

So that we have obtained a Hamilton-Jacobi equation (3.26) for discrete
contact Hamiltonian dynamics. ∎

Now, fix the initial point
$(\widehat{\mathbf{q}}_{0},\mathbf{q}_{0},\widehat{s}_{0})$. For the
generating function (3.24) presented in the previous subsection, we introduce
a new notation as follows

$S^{\kappa}_{d}(\mathbf{q}_{\kappa}):=\widehat{s}_{0}+S^{\kappa}(\widehat{\mathbf{q}}_{0},\mathbf{q}_{\kappa},\widehat{s}_{0}).$
(3.43)

This denotation determines a function from $Q$ to the real numbers. In this
notation, the discrete contact Hamilton-Jacobi equation (3.26) turns out to be

$\begin{split}S^{\kappa+1}_{d}(\mathbf{q}_{\kappa+1})-S^{\kappa}_{d}(\mathbf{q}_{\kappa})-\mathrm{d}S^{\kappa+1}_{d}(\mathbf{q}_{\kappa+1})\cdot\mathbf{q}_{\kappa+1}+H^{+}_{d}(\mathbf{q}_{\kappa},\mathrm{d}S^{\kappa+1}_{d}(\mathbf{q}_{\kappa+1}),S^{\kappa}_{d}(\mathbf{q}_{\kappa}))=0.\end{split}$
(3.44)

One can obtain a similar result in terms of the left discrete Hamiltonian.
Using the new notation (3.43), we present the left discrete contact
Hamilton–Jacobi equation

$\begin{split}S^{\kappa+1}_{d}(\mathbf{q}_{\kappa+1})-S^{\kappa}_{d}(\mathbf{q}_{\kappa})+\mathrm{d}S^{\kappa}_{d}(\mathbf{q}_{\kappa})\cdot\mathbf{q}_{\kappa}+H^{-}_{d}(\mathbf{q}_{\kappa+1},\mathrm{d}S^{\kappa}_{d}(\mathbf{q}_{\kappa}),S^{\kappa+1}_{d}(\mathbf{q}_{\kappa+1}))=0.\end{split}$
(3.45)

###  3.4 A Geometric Discrete Hamilton-Jacobi Theory on Contact Manifolds

Having obtained a discrete contact Hamilton–Jacobi equation (3.44) in the
previous section, it is easy to see that that the differential of the discrete
generating function (3.43) is precisely the discrete version of the continuous
section $\gamma$ in Theorem . Here, we aim at interpreting the discrete
contact Hamilton–Jacobi equation (3.44) in terms of discrete flows.

Projection of Discrete Flow. Consider a Hamiltonian function
$H^{+}_{d}=H^{+}_{d}(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa+1},s_{\kappa})$.
Referring to the commutativity of diagram (3.13) and in the light of the
discrete contact Hamiltonian dynamics (3.18), we compute the Hamiltonian flow
on the extended cotangent bundle as

$\tilde{\Phi}:\Big{(}\mathbf{q}_{\kappa},\frac{D_{1}H^{+}_{d}}{1-D_{3}H^{+}_{d}},s_{\kappa}\Big{)}\longrightarrow\Big{(}D_{2}H^{+}_{d},\mathbf{p}_{\kappa+1},s_{\kappa}+\mathbf{p}_{\kappa+1}\cdot
D_{2}H^{+}_{d}-H^{+}_{d}\Big{)}$ (3.46)

Recall the projection $\rho:T^{*}Q\times\mathbb{R}\mapsto Q\times\mathbb{R}$
given in (2.28). For a function $S_{d}=S_{d}(\mathbf{q}_{\kappa})$ on the base
space, we define local sections of the projection $\rho$ as

$\begin{split}\gamma^{\kappa}&:Q\longrightarrow
T^{*}Q\times\mathbb{R},\qquad\mathbf{q}_{\kappa}\mapsto(\mathbf{q}_{\kappa},\mathrm{d}S_{d}^{\kappa}(\mathbf{q}_{\kappa}),S_{d}^{\kappa}(\mathbf{q}_{\kappa})),\\\
\gamma^{\kappa+1}&:Q\longrightarrow
T^{*}Q\times\mathbb{R},\qquad\mathbf{q}_{\kappa+1}\mapsto(\mathbf{q}_{\kappa+1},\mathrm{d}S_{d}^{\kappa}(\mathbf{q}_{\kappa+1}),S_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1})).\end{split}$
(3.47)

According to the commutative diagram

$\textstyle{(\mathbf{q}_{\kappa},\mathbf{p}_{\kappa},s_{\kappa})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{*}}$$\scriptstyle{\tilde{\Phi}}$$\textstyle{(\mathbf{q}_{\kappa+1},\mathbf{p}_{\kappa+1},s_{\kappa+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{*}}$$\textstyle{(\mathbf{q}_{\kappa},s_{\kappa})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{\kappa}}$$\scriptstyle{\tilde{\Phi}^{\gamma^{\kappa}}}$$\textstyle{(\mathbf{q}_{\kappa+1},s_{\kappa+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{\kappa+1}}$
(3.48)

we pull down the flow $\tilde{\Phi}$ to the base manifold $Q$ and obtain the
following projected discrete flow

$\tilde{\Phi}^{\gamma}:Q\longrightarrow
Q,\qquad\mathbf{q}_{\kappa}\mapsto\mathbf{q}_{\kappa+1}=D_{2}H^{+}_{d}\big{(}\mathbf{q}_{\kappa},\mathrm{d}s_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1}),S_{d}^{\kappa}(\mathbf{q}_{\kappa})\big{)}.$
(3.49)

This procedure reduces the number of dependent variables to the coordinates
$\mathbf{q}$ on the base manifold $Q$. By means of the section $\gamma$, one
can lift a solution of the projected flow to the extended cotangent bundle
level. The lifted solution becomes a solution to the Hamiltonian flow if and
only if the commutation relation

$\tilde{\Phi}\circ\gamma^{\kappa}=\gamma^{\kappa+1}\circ\tilde{\Phi}^{\gamma}.$
(3.50)

holds.

######  Theorem 3.

For a section $\gamma$ admitting the local forms in (3.47), the following
conditions are equivalent:

  1. 1.

The flows $\tilde{\Phi}$ and $\tilde{\Phi}^{\gamma}$ commute, i.e.,
$\tilde{\Phi}\circ\gamma^{\kappa}=\gamma^{\kappa+1}\circ\tilde{\Phi}^{\gamma}.$

  2. 2.

$S$ solves the HJ equation (3.44).

###### Proof.

In terms of the local coordinates, the commutation condition (that is the
first of the condition in the statement of the theorem) gives rise to the
following equations

$\begin{split}S_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1})&=S_{d}^{\kappa}(\mathbf{q}_{\kappa})+\mathrm{d}s_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1})\cdot\mathbf{q}_{\kappa+1}-H^{+}_{d}\big{(}\mathbf{q}_{\kappa},\mathrm{d}s_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1}),S_{d}^{\kappa}(\mathbf{q}_{\kappa})\big{)},\\\
\mathrm{d}S_{d}^{\kappa}(\mathbf{q}_{\kappa})&={D_{1}H^{+}_{d}\big{(}\mathbf{q}_{\kappa},\mathrm{d}S_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1}),S_{d}^{\kappa}(\mathbf{q}_{\kappa})\big{)}}\big{/}\big{(}1-D_{3}H^{+}_{d}\big{(}\mathbf{q}_{\kappa},\mathrm{d}S_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1}),S_{d}^{\kappa}(\mathbf{q}_{\kappa})\big{)}\big{)}.\end{split}$
(3.51)

A direct comparison gives that the second condition in (3.51) is the
infinitesimal version of the first one. Indeed, if we take the derivative of
the first equation in (3.51) with respect to $\mathbf{q}_{\kappa}$, we have
that

$0=\mathrm{d}S_{d}^{\kappa}(\mathbf{q}_{\kappa})-D_{1}H^{+}_{d}\big{(}\mathbf{q}_{\kappa},\mathrm{d}S_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1}),S_{d}^{\kappa}(\mathbf{q}_{\kappa})\big{)}-D_{3}H^{+}_{d}\big{(}\mathbf{q}_{\kappa},\mathrm{d}S_{d}^{\kappa+1}(\mathbf{q}_{\kappa+1}),S_{d}^{\kappa}(\mathbf{q}_{\kappa})\big{)}\mathrm{d}S_{d}^{\kappa}(\mathbf{q}_{\kappa})$
(3.52)

By collecting all the terms involving $\mathrm{d}S_{d}^{\kappa}$ in the left
hand side, it is immediate to see that it is precisely the second condition in
(3.51). We remark that the first equation is the discrete Hamilton-Jacobi
equation (3.44). The inverse of the assertion is straightforwardly proved by
reversing the arguments. ∎

##  4 An Application: The Parachute Equation

In [25] the so-called parachute equation was introduced. This equation
describes the vertical motion of a particle falling in a fluid under the
action of constant gravity. The friction is modeled by the drag equation and
it is proportional to the square of the velocity. Here we present the
Hamiltonian associated with the discrete parachute equation on the extended
phase space $T^{*}Q\times\mathbb{R}$ where $Q$ is a one-dimensional manifold
with dynamic variable $q$ that is discretized as:
$q\rightarrow(q_{1},\dots,q_{N})$, so is the associated momentum $p$ as
$p\rightarrow(p_{1},\dots,p_{N})$. So, the discrete right contact Hamiltonian
for the parachute reads:

$H^{+}_{d}(q_{\kappa},p_{\kappa+1},s_{\kappa})=\frac{1}{2m}{(p_{\kappa+1}+2\lambda
s_{\kappa})}^{2}+\frac{mg}{2\lambda}(e^{2\lambda q_{\kappa}}-1),$ (4.1)

where $\lambda,g\in\mathbb{R}$. Calculating the discrete right contact
Hamiltonian equations we have the following discrete dynamics:

$\displaystyle q_{\kappa+1}$
$\displaystyle=\frac{1}{m}\left(p_{\kappa+1}+2\lambda s_{\kappa}\right),$
$\displaystyle p_{\kappa}$ $\displaystyle=\frac{mge^{2\lambda
q_{\kappa}}}{1-\frac{2\lambda}{m}\left(p_{\kappa+1}+2\lambda
s_{\kappa}\right)},$ $\displaystyle s_{\kappa+1}$
$\displaystyle=s_{\kappa}+\frac{p_{\kappa+1}^{2}}{2m}-\frac{2\lambda^{2}s_{\kappa}}{m}-\frac{mg}{2\lambda}\left(e^{2\lambda
q_{\kappa}}-1\right)$

This dynamic is represented in the following diagram obtained by plotting the
points $(q(\kappa),p(\kappa),s(\kappa))$ which have been integrated
recursively for different values of $\kappa$. For $\lambda=-0.01$, $m=1$,
$g=10$ and initial condition $(q(0),p(0),s(0))=(100,1,1)$ we obtain the
dynamics of the parachute as follows.

$\begin{matrix}\includegraphics[scale={0.4}]{qps3.jpg}&\includegraphics[scale={0.4}]{DSk.jpg}\\\
\text{Figure 1: Dynamics of the parachute}&\text{Figure 2: Evolution of
$DS(k)$}\end{matrix}$ (4.2)

Here it is easy to depict from the figure 4.2 that the position coordinate
(red line on the left) descends from the initial position $q(0)=100$ until the
parachute reaches the ground. The momentum (blue line on the left) grows until
it crosses the red line. This crossing point represents the instant in which
the parachute opens and the momentum decreases until the parachute touches the
ground, i.e., the parachute decelerates so the parachuter can touch the ground
safely. Nonetheless, as it happens, the velocity is not zero when the
parachuter touches the ground.

Now, in order to prove the accuracy of the discrete contact Hamilton–Jacobi
equation, we plot the differential of the generating function, i.e.,
$DS^{\kappa}(q_{\kappa})$, which according to the first equation in (3.24), it
should match the dynamic of the blue line in the diagram depicted above. Let
us corroborate this fact by plotting $\gamma=DS^{\kappa}(q_{\kappa})$ for
different discrete values of $\kappa$. For $\lambda=-0.01$, $m=1$, $g=10$ and
initial condition $(q(0),S(0),s(0))=(100,1,1)$ we obtain:

It is easy to check that the line representing $DS(k)$ (blue line on the right
figure (4.2)) is very similar to the blue line depicted in blue line on the
left. Indeed, both lines should be equal (see that for $k=2$ both reach the
maximum momentum $p$) since in the vicinity of the discrete Hamilton–Jacobi
equation, $p(k)=DS(k)$.

##  5 Conclusions

In this paper, we have proposed a discrete Hamilton–Jacobi equation (stated in
Theorem 2) for contact discrete Hamiltonian dynamics. We have presented its
geometric foundations in terms of contact discrete flows in Theorem 3. We have
exhibited the relationship between the continuous geometric HJ Theorem 3.4 for
contact Hamiltonian dynamics and Theorem 3 for discrete contact Hamiltonian
dynamics.

We wish to continue in the following directions:

  * •

Contact Hamiltonian dynamics does not preserve the Hamiltonian function. There
exists an alternative characterization of Hamiltonian dynamics on contact
manifolds that preserves the energy, and it is known as evolution dynamics
[46, 47]. We wish to examine the discretization of evolution dynamics and its
HJ formulation.

  * •

For the extended cotangent bundle $T^{*}Q\times\mathbb{R}$, the continuous HJ
theory for contact Hamiltonian dynamics was presented in [14, 19, 24]. The
authors consider the base manifold as the extended configuration space
$Q\times\mathbb{R}$. In this work, we consider the base manifold as $Q$. We
wish to write a discrete HJ equation on contact manifolds with base manifold
$Q\times\mathbb{R}$.

  * •

If a Lagrangian is degenerate, then one cannot arrive at explicit Euler-
Lagrange equations. In this case, the Legendre transformation is not
immediate. Tulczyjew’s triple is a geometric formulation that allows us to
achieve this in singular cases as well [49]. In a discrete framework,
Tulczyjew’s triple was constructed in [38]. This determines a proper geometry
for implicit discrete Lagrangian and Hamiltonian dynamics [34]. For the
continuous case, a geometric HJ theory has been recently given in [21, 22] in
the symplectic framework and in [24] in the contact framework. We wish to
concentrate on generalizing the discrete HJ theories both for symplectic and
contact geometry including the implicit case. On the other hand, the Tulczyjew
triple for contact geometry has been recently constructed in [23]. In the
future, we aim at constructing a discrete contact Tulczyjew’s triple.

## Acknowledgements

This work has been partially supported by MINECO MTM 2013-42-870-P and the
ICMAT Severo Ochoa project SEV-2011-0087. Marcin Zajac and Cristina Sardón
acknowledge the funding from Universidad Politécnica de Madrid and the
department of Applied Mathematics at ETSII for hosting Prof. Marcin Zajac to
work alongside Cristina Sardón.

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