On the geometry of discrete contact mechanics
Alexandre Anahory Simoes David Martín de Diego Manuel de León Manuel Lainz Valcázar
math-ph cs.NA math.MP math.NA Primary 34C25, Secondary 37E40, 70K40

	
	



On the geometry of discrete contact mechanics
    
		Alexandre Anahory Simoes
		


		Instituto de Ciencias Matematicas 

		Calle Nicolas Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain

		


		David Martín de Diego
		


		Instituto de Ciencias Matematicas (CSIC-UAM-UC3M-UCM) 

		Calle Nicolas Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain

		


		Manuel Lainz Valcázar
		


		Instituto de Ciencias Matematicas 

		Calle Nicolas Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain

		


		Manuel de León
		


		Instituto de Ciencias Matematicas and Real Academia Española de Ciencias

		Calle Nicolas Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain

	
    December 30, 2023
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





In this paper, we continue the construction of variational integrators adapted to contact geometry started in <cit.>, in particular, we introduce a discrete Herglotz Principle and the corresponding discrete Herglotz Equations for a discrete Lagrangian in the contact setting. This allows us to develop convenient numerical integrators for contact Lagrangian systems that are conformally contact by construction. The existence of an exact Lagrangian function is also discussed. 



[AMS Mathematics Subject Classification (2010). Primary 34C25; Secondary  37E40,
	70K40.

	Keywords. Contact geometry, exact discrete Lagrangian, variational integrator, dissipation
	]





§ INTRODUCTION


Contact Hamiltonian and Lagrangian systems have deserved a lot of attention in recent years <cit.> or <cit.>.
One of the most relevant features of contact dynamics is the absence of conservative properties
contrarily to the conservative character of the energy in symplectic dynamics; indeed, 
we have a dissipative behavior. This fact suggests that contact geometry may be the appropriate framework to model many physical and mathematical problems with dissipation we find in thermodynamics, statistical physics, quantum mechanics, gravity or control theory, among many others. Consequently, it becomes an important necessity to develop numerical methods adapted to the contact setting for applications in the above mentioned subjects.
The idea is to develop geometric integrators, that is,  numerical methods for differential equations which preserve geometric properties like contact structure, symmetries, configuration space...  This preservation of structural properties is often desirable to achieve correct qualitative behavior and long time stability <cit.>.


As far as we know, the first attempt to develop geometric integrators for the contact case is in the paper <cit.> (see also <cit.>), 
where the authors present geometric numerical integrators for contact flows that stem from a discretization of Herglotz variational principle.

Our goal in the current paper is to go further in the discrete description of contact dynamics,
so we will mention some of the new and relevant results that the reader can find in the next pages.
Instead of deriving the discrete Herglotz equations by an heuristic argument, they are directly
obtained from a clear discrete variational principle. In addition, to develop the discrete algorithm
we use the natural discretization Q × Q ×ℝ, which preserves all the contact geometry flavor.

Another relevant point is the discussion of the existence of an exact discrete Lagrangian function <cit.>, which will lead us to define the contact exponential map and prove its existence. This construction is essential to develop a complete theory of variational error analysis for contact Lagrangian systems. 

Finally, we consider a discrete version of the infinitesimal symmetries discussed in <cit.>, jointly
with the corresponding dissipated quantities.

The paper is structured as follows. Section 2 is devoted to a quick review of contact Hamiltonian and Lagrangian systems in the continuous setting.
In particular, we recall the Herglotz variational principle, since it will be the motivation to develop the corresponding discrete version. 
Section 3 is devoted to construct the discrete version of contact Lagrangian dynamics for a discrete Lagrangian L_d : Q × Q ×ℝ→ℝ, where Q is the configuration manifold. We consider the discrete Herglotz principle to obtain the so-called discrete Herglotz equations.
The Legendre transformations F^-L_d and  F^+L_d are defined, and consequently the
discrete flow (at the Lagrangian and Hamiltonian levels); the main result is that the discrete flow is a conformal
contactomorphism. In Section 4 we define the contact exponential map for the Herglotz vector field and prove that it is a local diffeomorphism.
This result permits to study the existence of an exact Lagrangian function.
Finally, we consider several examples to illustrate our theoretical developments.



§ CONTINUOUS CONTACT MECHANICS




 §.§ Contact manifolds and Hamiltonian systems

In this section we will recall the main definitions and results on the theory of contact manifolds and Hamiltonian system. See <cit.> for a more detailed overview.

A contact manifold (M,η) is an (2n+1)-dimensional manifold with a contact form η <cit.>. That is, η is a 1-form on M such that η∧η^n is a volume form. This type of  manifolds have a distinguished vector field: the so-called Reeb vector field , which is the unique vector field that satisfies:

    η = 0,   η()=1.


On a contact manifold (M,η), we  define the following isomorphism of vector bundles:

    ♭:      TM    ⟶    T^* M,
          v    ⟼   vη + η(v)η.

Notice that ♭() = η.

There is a Darboux theorem for contact manifolds. In a neighborhood of each point in M one can find local coordinates (q^i, p_i, z) such that

    η =  z - p_i   q^i.

In these coordinates, we have

    = ∂/∂ z.


An example of a contact manifold is T^*Q×. Here, the contact form is given by 

    η_Q =  z - θ_Q =  z - p_i  q^i,

where θ_Q is pullback the tautological 1-form of T^*Q, (q^i,p_i) are natural coordinates on T^*Q and z is the -coordinate.

We say that a (local) diffeomorphism between two contact manifolds F:(M,η) → (N,τ) is a (local) contactomorphism if F^* τ = η. We say that F is a (local) conformal contactomorphism if F^* τ = η or, equivalently, F^* τ = ση, where σ: M →∖0 is the conformal factor.

We say that a vector field X on M is an infinitesimal (conformal) contactomorphism if its flow F_t consists of (conformal) contactomorphisms.

From the general identify, where F_t is a flow and X is its infinitesimal generator

    t F_t^* η  =
    	 F_t^* Xη,

we deduce that X is infinitesimal contactomorphism if and only if

    Xη = 0.

Furthermore, X is a conformal contactomorphism if and only if 

    Xη = a η,

for some a:M→. The function a is related to the conformal factors σ_t of the conformal contactomorphisms F_t by

    σ_t (x)= ∫_0^t exp(a(F_τ(x))) τ.


Given a smooth function f:M→, its Hamiltonian vector field X_f is given by

    ♭(X_f) =   f - (f + (f)) η.


A vector field X is the Hamiltonian vector field of some function f if and only if it is an infinitesimal conformal contactomorphism. In that case X=X_f for f = - η(X). Moreover, Xη  = - (f) η. Hence X is an infinitesimal contactomorphism if and only if X = X_f for some function f such that (f) = 0.

We call the triple (M, η, H) a contact Hamiltonian system, where (M,η) is a contact manifold and H:M → is the Hamiltonian function.

In contrast to their symplectic counterpart, contact Hamiltonian vector fields do not preserve the Hamiltonian. In fact

    X_H(H) = -(H) H.





 §.§ Contact Lagrangian systems

Now we review the Lagrangian picture of contact systems. In <cit.> we give a more comprehensive description which also covers the case of singular Lagrangians.

Let Q be an n-dimensional configuration manifold and consider the extended phase space TQ × and a Lagrangian function  
 L:TQ×→. In this paper, we will assume that the Lagrangian is regular, that is, the Hessian matrix with respect to the velocities (W_ij) is regular where
 
    W_ij = ∂^2 L/∂q̇^i ∂q̇^j ,

 and (q^i, q̇^i,z) are bundle coordinates for TQ ×. Equivalently, L is regular if and only if the one-form

    η_L =  z - θ_L

is a contact form. Here,

    θ_L    = S^* ( L) = Lq̇^iq^i,

 where S is the canonical vertical endomorphism on TQ extended to TQ ×, that is, in local TQ × bundle coordinates,

    S =  q^i ⊗∂/∂q̇^i.


The energy of the system is defined by

    E_L = Δ(L) - L = q̇^i Lq̇^i - L,

where Δ is the Liouville vector field on TQ extended to TQ× in the natural way.

The Reeb vector field of η_L, which we will denoted by _L is given by

    _L = ∂/∂ z - 
        W^ij∂^2 L/∂q̇^i ∂ zq̇^j,

where (W^ij) is the inverse of the Hessian matrix with respect to the velocities W^ij (Equation (<ref>)).

The Hamiltonian vector field of the energy E_L will be denoted ξ_L = X_E_L, hence

    ♭_L(ξ_L) =  E_L - (_L(E_L)+E_L)η_L,

where ♭_L(v) = vη_L + η_L (v) η_L is the isomorphism defined in Equation (<ref>) for this particular contact structure.

ξ_L is a second order differential equation (SODE) (that is, S(ξ_L) = Δ) and its solutions are just the ones of the Herglotz equations (also called generalized Euler-Lagrange equations) for L (see <cit.>):

    t(Lq̇^i) - Lq^i = Lq̇^iLz.


There exists a Legendre transformation for contact Lagrangian systems. Given the vector bundle TQ×→ Q ×, one can consider the fiber derivative L of L:TQ ×→, which has the following coordinate expression in natural coordinates:

    L:TQ ×   → T^*Q ×
    
    		(q^i,q̇^i,z)    ↦ (q^i, Lq̇^i,z).

If we consider the contact structure η_Q (<ref>) on T^*Q ×, and η_L on TQ× then L is a local contactomorphism. 

In the case that L is a global contactomorphism, then we say that L is hyperregular. In this situation, we can define a Hamiltonian H:T^*Q×→ such that E_L = H ∘ L and the Lagrangian and Hamiltonian dynamics are L-related, that is, L_* ξ_L = X_H.



  §.§.§ Herglotz variational principle

Equations (<ref>) can be derived from a modified variational principle <cit.>. In contrast to the symplectic case, the action is not a definite integral. The contact action is the value at the endpoint of solution to a non-autonomous ODE.

In <cit.> we defined the action on the space of curves with fixed endpoints. However, for our purposes here it is more convenient to define the action on the space of all curves and all initial conditions and then restrict it to the appropriate submanifold.

Let Ω be the (infinite dimensional) manifold of curves on Q, c:[0,1]→ Q. We denote by  Ω(q_0, q_1) ⊆Ω, where q_0,q_1 ∈ Q, the submanifold whose elements are the smooth curves c ∈Ω such that c(0)=q_0, c(1)=q_1. The tangent space of Ω at a curve c is given by vector fields over c. In the case of T_c Ω(q_0,q_1), the vector fields over c vanish a the endpoints. That is,


    T_c Ω   =  δ v ∈^∞([0,1] → TQ) |τ_Q ∘δ v = c, 
    
            T_c Ω(q_0,q_1)    =  δ c ∈  T_c Ω|δ c(0)=0,  δ c(1)=0 
                .


We define the operator 

    𝒵:Ω×→^∞ ([0,1] →),

 which assigns to each curve and initial condition (c,z_0) the curve 𝒵_z_0(c) that solves the following ODE:

    𝒵_z_0(c)t   = L(c, ċ, 𝒵_z_0(c)),
    𝒵_z_0(c)(0)    = z_0.


Now we define the contact action functional as the map which assigns to each curve c and initial condition z_0, the solution to the previous ODE evaluated at the endpoint:

    𝒜: Ω×   →,
    
            (c,z_0)    ↦𝒵_z_0(c)(1).


When restricted to Ω(q_0,q_1)×z_0, the critical points of 𝒜 are the solutions to Herglotz equation. More precisely,

    Let L: TQ ×→ be a Lagrangian function and let c∈Ω(q_0, q_1) and z_0 ∈. Then, (c,ċ, 𝒵_z_0(c)) satisfies the Herglotz equations (<ref>) if and only if c is a critical point of 𝒜_z_0|_Ω(q_0, q_1).




    We will compute T𝒵. Let c ∈Ω(q_0, q_1) be a curve and consider some tangent vector δ c ∈ T_c Ω. We will first compute the partial derivative with respect to c by fixing z_0∈, and then we will fix the curve and compute the partial derivative with respect to the initial condition z_0. In order to simplify the notation, let χ=(c,ċ, 𝒵_z_0(c)) and let ψ = T_c 𝒵_z_0(δ v).
    
    Consider a curve c_λ∈Ω (that is, a smoothly parametrized family of curves) such that
    
    δ c=c_λλ|_λ=0
 
    Since 𝒵_z_0(c_λ)(0)=z_0 for all λ, then ψ(0)=0.
    
    We compute the derivative of ψ by interchanging the order of the derivatives using the ODE defining 𝒵:
    
    ψ̇(t)    = 
            λt𝒵_z_0(c_λ(t))|_λ=0
       =
            λL(c_λ(t), ċ_λ(t),𝒵(c_λ)(t))|_λ=0
       =
            Lq^i(χ(t)) δ c ^i(t) +
            Lq̇^i(χ(t)) δċ^i(t) + 
            Lz(χ(t)) ψ(t).


Hence, the function ψ is the solution to the ODE above. Explicitly

    ψ(t) = 1/σ(t)∫_0^t σ(τ) (Lq^i(χ(τ)) δ c ^i(τ) + Lq̇^i(χ(τ)) δċ^i(τ)
            ) τ,

where

    σ(t) = exp(-∫_0^t Lz(χ(τ)) τ) > 0.


Integrating by parts and using that ψ(0)=0, we get the following expression

    ψ(t)     = δc^i(t) Lq̇^i(χ(t)) 
        + 1/σ(t)∫_0^t  δ c^i (t) (
    		σ(t) Lq^i(χ(t)) - 
    		t(σ(t)Lq̇^i(χ(t))) 
    		)  t 
        = δc^i(t) Lq̇^i(χ(t)) 
       + δc^i(t) Lq̇^i(χ(t)) 
        + 1/σ(t)∫_0^t  δ c ^i(τ) σ(τ) (
    		Lq^i(χ(τ)) - 
    		τLq̇^i(χ(τ))+  Lq̇^i(χ(τ)) Lz(χ(τ)) ) τ.


Now we compute the partial derivative with respect to the initial condition z_0. We interchange the order of the derivatives

    t𝒵_z_0(c)z_0 = 
        Lz (c,ċ, 𝒵(c)) 𝒵_z_0(c)z_0



If we solve for 𝒵_z_0(c)z_0 the ODE above using that 𝒵_z_0(c)z_0(0)=1, we notice that

    𝒵_z_0(c)z_0(t) = 
        exp(∫_0^t Lz(χ(τ)) τ) = 1/σ(t) ,

where σ is defined in (<ref>).



  §.§.§ Symmetries and dissipated quantities on contact Lagrangian systems

As explained in <cit.>, given a symmetry on a contact system, one does not obtain a conserved quantity, but a quantity f that dissipates at the same rate as the Lagrangian.

Given a contact Hamiltonian system (M,η, H), we say that a quantity f:M→ is dissipated if 

    X_H f = -(H) f,

or, equivalently,

    ϕ_t^*(f) = σ_t,

where ϕ is the flow of X_H and σ_t, its conformal factor.

Notice that the quotient of two dissipated quantities (if it is well defined) is a conserved quantity. 

We end this section by stating a Noether theorem in this setting, which provides a link between symmetries of the Lagrangian and conserved quantities.

Let L:TQ ×→ be a regular Lagrangian. Let G be a Lie group  acting on Q

    Φ: G× Q → Q.


We defined the lifted action as

    Φ̃: G× TQ×→ TQ×,

given by Φ̃(g, v_q, z) = (T_qΦ(v_q), z) where v_q∈ T_qQ. We denote by ξ_TQ× to the vector field on TQ× which is the infinitesimal generator by the lifted action of an element ξ of the Lie algebra  𝔤 of G.


We define the momentum map J_L:

    J_L: TQ ×   →𝔤^*,
    ⟨ J_L(v_q,z), ξ⟩    = - η_L(ξ_TQ ×).

and we define Ĵ(ξ): TQ×→ by  Ĵ(ξ)(v_q, z) =⟨ J_L(v_q,z), ξ⟩.

Then we have the following <cit.>

	Let the lifted action Φ̃ preserve the Lagrangian L, then Φ̃ acts by contactomorphisms on (TQ×,η_L,E_L) and Ĵ(ξ)	is a dissipated quantity for every ξ∈𝔤.

 


§ DISCRETE CONTACT MECHANICS



In this section, we will extend the approach to discrete mechanics as in<cit.> to the case of contct dynamics (see also <cit.>).

Let L_d:Q× Q ×→ be a discrete Lagrangian function. In our point of view Q× Q × will be the discrete space corresponding to the manifold TQ×, where continuous contact Lagrangian mechanics takes place. We fix a time-step h>0, on which L_d depends, though we will omit this explicit dependence.

For each N ∈, let us define the discrete path space as the space containing sequences on Q with length N+1, i.e.,

    _d^N(Q)={ (q_0,q_1, ..., q_N ) | q_k∈ Q, k=0,…, N }.

The set _d^N(Q) is a manifold and it is canonically identified with the product space Q^N+1.

To each q_d∈_d^N(Q)  and each z_0∈ we will associate another sequence (z_k)∈^N+1 defined by

    z_k+1-z_k=L_d(q_k,q_k+1,z_k),    k=0,..., N-1.

In the sequel, for each 1⩽ k ⩽ N, we will denote by 𝒵_k the function 𝒵_k:Q × Q ×⟶

    𝒵_k(q_k-1,q_k,z_k-1)=z_k-1+L_d(q_k-1,q_k,z_k-1).


We define the contact discrete action to be the functional that for each point q_d∈_d^N(Q) and each real number z_0 returns as output the real number z_N obtained recursively from (<ref>), i.e.,

    𝒜_d:  _d^N(Q)×   ⟶
    
    		(q_d,z_0)    ↦ z_N.


A variation of a sequence q_d∈_d^N(Q) is a curve q_d:(-ϵ,ϵ)→_d^N(Q) satisfying q_d(0)=q_d. Given such a variation, we will define its infinitesimal variation by

    δ q_d := . /ϵ|_ϵ=0q_d(ϵ)=(δ q_0, ..., δ q_N),

where δ q_k:=. /ϵ|_ϵ=0q_k (ϵ).


Let L_d be a smooth discrete Lagrangian. Then, if we fix z_0∈, we obtain the functional

    𝒜_d,z_0:  _d^N(Q)    ⟶
    
    		q_d   ↦𝒜_d(q_d,z_0).

The differential of the functional 𝒜_d,z_0 is the following
	
    𝒜_d,z_0(q_d)=   σ_N···σ_2∂𝒵_1/∂ q_0(q_0,q_1,z_0) q_0
        +∑_k=1^N-1∏_j=k+2^Nσ_j·( ∂𝒵_k+1/∂ q_k+∂𝒵_k+1/∂ z_k∂𝒵_k/∂ q_k)  q_k
        +∂𝒵_N/∂ q_N(q_N-1,q_N,z_N-1) q_N,

where we are using the identification of _d^N(Q) with Q^N+1 and for each 1 ⩽ j⩽ N

    σ_j=∂𝒵_j/∂ z_j-1(q_j-1,q_j,z_j-1).






    Using the identification of _d^N(Q) with Q^N+1, note that the discrete action may be rewritten as
    
    𝒜_d,z_0(q_d)=𝒵_N(q_N-1,q_N,𝒵_N-1(q_N-2,q_N-1,𝒵_N-2(...𝒵_1(q_0,q_1,z_0)...)).

    Using that
    
    𝒜_d,z_0(q_d)=∂𝒜_d,z_0/∂ q_0 q_0+∑_k=1^N-1∂𝒜_d,z_0/∂ q_k q_k+∂𝒜_d,z_0/∂ q_N q_N.

    and applying the chain rule, we deduce that
    
    ∂𝒜_d,z_0/∂ q_0=∂𝒵_N/∂ z_N-1···∂𝒵_2/∂ z_1∂𝒵_1/∂ q_0,

    since the function 𝒵_1 is the only one that depends on q_0 among all the N functions 𝒵_k.
    It is also clear that
    
    ∂𝒜_d,z_0/∂ q_N=∂𝒵_N/∂ q_N,

    since none of the functions 𝒵_k depend on q_N except the function 𝒵_N.
    Finally if 1⩽ k ⩽ N-1 we have that
    
    ∂𝒜_d,z_0/∂ q_k=∂𝒵_N/∂ z_N-1···∂𝒵_k+2/∂ z_k+1(∂𝒵_k+1/∂ q_k+∂𝒵_k+1/∂ z_k∂𝒵_k/∂ q_k),

    where we applied the chain rule and the fact that the functions 𝒵_k+1 and 𝒵_k are the only ones that depend on q_k. Hence, we finished the proof.



Let us see the special case N=2, where we can directly compute the differential of the action:

Let L_d be a smooth discrete Lagrangian. In the case where N=2, the differential of the discrete action function satisfies:
	
    𝒜_d,z_0 =  ( D_1L_d (q_1,q_2,z_1)+(1+D_zL_d(q_1,q_2,z_1) D_2L_d(q_0,q_1,z_0) )  q_1
          + D_2L_d(q_1,q_2,z_1)  q_2 + (1+D_zL_d(q_1,q_2,z_1))D_1L_d(q_0,q_1,z_0)  q_0.


























Given z_0∈, a discrete path q_d=(q_0,..., q_N) in _d^N(Q) is said to satisfy the Discrete Herglotz Principle if q_d is a critical value of the discrete action functional 𝒜_d,z_0 among all paths in _d^N(Q) with fixed end points q_0,q_N.


We will now obtain as a sufficient and necessary condition for a path to satisfy the discrete Herglotz principle, a set of equations called Discrete Herglotz equations <cit.>.

Let L_d be a discrete Lagrangian function such that 1+D_zL_d is non-vanishing everywhere. Given z_0∈, a discrete path q_d∈_d^N(Q) satisfies the discrete Herglotz principle if and only if it satisfies

    D_1L_d (q_k,q_k+1,z_k)+(1+D_zL_d(q_k,q_k+1,z_k)) D_2L_d(q_k-1,q_k,z_k-1)=0,
       z_k-z_k-1=L_d(q_k-1,q_k,z_k-1),

for k=1,...,N-1.



    Let q_d(ϵ) be a variation of q_d∈_d^N(Q) with fixed end-points q_0 and q_N. Then q_d is a critical value of the discrete action functional if and only if
    
    ./ϵ|_ϵ=0(𝒜_d,z_0(q_d(ϵ)))=𝒜_d,z_0(δ q_d)=0.

    
    By (<ref>) the last expression is equivalent to
    
    ∑_k=1^N-1∏_j=k+2^Nσ_j·( ∂𝒵_k+1/∂ q_k+∂𝒵_k+1/∂ z_k∂𝒵_k/∂ q_k) δ q_k = 0.

    Since the infinitesimal variations δ q_k, 1≤ k≤ N-1, are arbitrary we deduce
    
    ∏_j=k+2^Nσ_j·( ∂𝒵_k+1/∂ q_k+∂𝒵_k+1/∂ z_k∂𝒵_k/∂ q_k) = 0.

    Note that,
    
    σ_j=∂𝒵_j/∂ z_j-1(q_j-1,q_j,z_j-1)=1+D_zL_d(q_j-1,q_j,z_j-1)

    is non-vanishing by hypothesis and
    
    ∂𝒵_k+1/∂ q_k+∂𝒵_k+1/∂ z_k∂𝒵_k/∂ q_k= D_1L_d(q_k,q_k+1,z_k)+σ_k+1D_2L_d(q_k-1,q_k,z_k-1),

    from where the result follows.



    The discrete principle introduced in <cit.> is just the condition
    
    ∂𝒵_k+1/∂ q_k+∂𝒵_k+1/∂ z_k∂𝒵_k/∂ q_k=0,

    afer rewriting it in our notation. For discrete Lagrangian functions where 1+D_zL_d is non-vanishing, the condition above is equivalent to the Herglotz discrete principle.






 §.§ Discrete Lagrangian flows and discrete Legendre transforms


Given a discrete contact Lagrangian L_d, if 1+D_zL_d(q_0,q_1,z_0) does not vanish, we can define two maps called discrete Legendre transforms: ^±L_d:Q × Q ×→ T^*Q × 

    ^+L_d(q_0,q_1,z_0)=(q_1,D_2L_d(q_0,q_1,z_0),z_0+L_d(q_0, q_1, z_0)) 
       ^-L_d(q_0,q_1,z_0)=(q_0,-D_1L_d(q_0,q_1,z_0)/1+D_zL_d(q_0,q_1,z_0),z_0).



^+L_d is a local diffeomorphism if and only if ^- L_d is a local diffeomorphism.


	It is a direct consequence of the implicit function theorem.  


The Legendre transforms allow us to rewrite discrete Herglotz equations (<ref>) as a momentum matching equations as in <cit.>. Indeed, provided 1+D_zL_d(q_0,q_1,z_0) is not zero, we may write

    ^+L_d(q_0,q_1,z_0)=^-L_d(q_1,q_2,z_1).


We have the following theorem

Suppose 1+D_zL_d(q_0,q_1,z_0) does not vanish. Given a discrete Lagrangian L_d:Q× Q ×→, there is a well-defined discrete Lagrangian flow Φ_d:Q× Q×→ Q× Q × if and only if ^-L_d
is a local diffeomorphism where 

    Φ_d=(^-L_d)^-1∘^+L_d



	Given (q_0, q_1, z_0)∈ Q× Q×, from the definition
	Φ_d=(^-L_d)^-1∘^+L_d it is easy to show that 
 the point (q_2, z_1)∈  Q× required to satisfy
	
    Φ_d (q_0, q_1, z_0) =(q_1, q_2, z_1)

is defined by Equations  (<ref>), and so q_2 and z_1  is uniquely defined as a
	function of (q_0, q_1, z_0) if and only if ^-L_d is locally an isomorphism.
	

Inspired by the last theorem, we say that a discrete contact Lagrangian is regular if 1+D_zL_d(q_0,q_1,z_0) does not vanish and its negative discrete Legendre transform ^-L_d is a local diffeomorphism.

The discrete Legendre transforms also allow us to define an associated discrete Hamiltonian flow on T^*Q×. 
Indeed, considering a regular discrete Lagrangian function L_d, let Φ_d:T^*Q×→ T^*Q× be defined by

    Φ_d=^+L_d∘Φ_d∘ (^+L_d)^-1.

It is not difficult to show that the discrete Hamiltonian flow admits the alternative expressions

    Φ_d=^-L_d∘Φ_d∘ (^-L_d)^-1    or    Φ_d=^+L_d∘ (^-L_d)^-1.



    [column sep=1]
                                                                    Q × Q ×ℝ[rr, "Φ_d"] [ld, "𝔽^-L_d"] [rd, "𝔽^+L_d"]        Q × Q ×ℝ[ld, "𝔽^-L_d"] [rd, "𝔽^+L_d"]    
    
    T^*Q×ℝ[rr, "Φ_d"']        T^*Q×ℝ[rr, "Φ_d"']        T^*Q×ℝ


We may define the one-forms


    η^+=(^+L_d)^*η,   η^-=(^-L_d)^*η,

where η is the canonical contact form on T^*Q×. These are contact forms on Q× Q ×. If we chose natural coordinates (q^i,p_i,z) on T^*Q× where η= z-p_i q^i, the discrete 1-forms may be locally written as the pullback

    η^+   = z_0+ L_d(q_0,q_1,z_0)-D_2L_d(q_0,q_1,z_0) q_1, 
    η^-   = z_0+D_1L_d(q_0,q_1,z_0)/1+D_zL_d(q_0,q_1,z_0) q_0,

by the corresponding discrete Legendre transform. The one-form η^+ is further simplified to


    η^+=(1+D_zL_d(q_0,q_1,z_0)) z_0+D_1L_d(q_0,q_1,z_0) q_0.


Given a discrete Lagrangian L_d, let σ_d:Q× Q×→ be the smooth function given by

    σ_d(q_0,q_1,z_0)=1+D_zL_d(q_0,q_1,z_0)

then we have that:


The discrete contact forms η^± satisfy


  (i) η^+=σ_d·η^-;

  (ii) (Φ_d)^*η^-=η^+.




	For the first item, observe that (<ref>) is equivalent to
	
    η^+=(1+D_zL_d(q_0,q_1,z_0))η^-.

	For the second one, note that
	
    (Φ_d)^*η^-=	(Φ_d)^*∘ (^-L_d)^*η=(^-L_d∘Φ_d)^*η=(^+L_d)^*η

	by applying Theorem <ref>.


As a consequence of the last Lemma we have the following theorem:


Let L_d be a regular discrete Lagrangian function. The discrete flow Φ_d associated to L_d is a conformal contactomorphism with respect to both contact structures η^±. In particular, it satisfies 
	
    (Φ_d)^*η^+=(σ_d∘Φ_d)·η^+,    (Φ_d)^*η^-=σ_d·η^-

Likewise, the discrete Hamiltonian flow Φ_d is also a conformal contactomorphism satisfying

    (Φ_d)^*η=(σ_d∘ (^-L_d)^-1) ·η.




	These are trivial consequences of Lemma <ref>. Combining the two statements of the Lemma we get
	
    (Φ_d)^*η^-=σ_d·η^-.

	Then, also
	
    (Φ_d)^*η^+=(Φ_d)^*(σ_d·η^-)=(σ_d∘Φ_d) · (Φ_d)^*η^-=(σ_d∘Φ_d) ·η^+.

	Observing that the discrete Hamiltonian flow satisfies Φ_d=^+L_d∘Φ_d∘ (^+L_d)^-1 by definition, then
	
    (Φ_d)^*η =     ((^+L_d)^-1)^*∘ (Φ_d)^*η^+=((^+L_d)^-1)^*((σ_d∘Φ_d) ·η^+) 
    
    			=     (σ_d∘Φ_d∘ (^+L_d)^-1) · ((^+L_d)^-1)^*η^+,

	where the last equality comes from the properties of the pullback. Since we have that
	
    Φ_d∘ (^+L_d)^-1=(^-L_d)^-1  and   ((^+L_d)^-1)^*η^+=η,

	the desired result follows.
	
	Moreover, since the discrete Lagrangian function L_d is regular, the function σ_d does not vanish. Hence, the discrete flows Φ_d and Φ_d are conformal contact.




 §.§ Discrete symmetries and dissipated quantities


Let G be a Lie group acting on Q through the map Φ:G× Q→ Q. We define the lifted action on Q× Q × to be the diagonal action on Q× Q and the identity on , so that

    Φ:G× Q× Q×→ Q× Q×,   Φ_g(q_0,q_1,z_0)=(Φ_g(q_0),Φ_g(q_1),z_0).

Let us denote by ξ_Q∈𝔛(Q) the infinitesimal generator associated to a Lie algebra element ξ∈𝔤  and by ξ∈𝔛(Q× Q×) the corresponding infinitesimal generator on Q× Q×.

Notice that, since pr_3(Φ_g(q_0,q_1,z_0))=z_0 is constant for all g∈ G, where pr_3:Q× Q×→ is the projection onto the third factor, then we have that

    T_(q_0,q_1,z_0)pr_3(ξ(q_0,q_1,z_0))=0.

In fact, the infinitesimal generator may be identified with

    ξ(q_0,q_1,z_0)=(ξ_Q(q_0),ξ_Q(q_1),0_z_0)∈ T_q_0Q× T_q_1Q× T_z_0,

where 0:→ T is the zero section of T.


If L_d:Q× Q×→ is an invariant discrete Lagrangian function, i.e., L_d∘Φ_g=L_d for all g∈ G, then it satisfies the equation
    
    D_1L_d(q_0,q_1,z_0)ξ_Q(q_0)+D_2L_d(q_0,q_1,z_0)ξ_Q(q_1)=0.




    Since the discrete Lagrangian function is invariant for the lifted action, it satisfies
    
    ⟨ L_d(q_0,q_1,z_0), ξ(q_0,q_1,z_0)⟩=0,  ∀  (q_0,q_1,z_0)∈ Q× Q×.

    Then using equation (<ref>), one immediately gets the desired expression.


Now consider the discrete momentum map J_d given by

    J_d: Q× Q ×   →𝔤^*,
    ⟨ J_d(q_0,q_1,z_0), ξ⟩    = ⟨η^-, ξ(q_0,q_1,z_0)⟩ .



 Let L_d be an invariant discrete Lagrangian function for the lifted action Φ. Then Φ acts by contactomorphisms on Q× Q× and the function Ĵ_d(ξ):Q× Q×→ given by
 
    Ĵ_d(ξ)(q_0,q_1,z_0)=⟨ J_d(q_0,q_1,z_0), ξ⟩

 is dissipated along the discrete flow of Herglotz equations in the sense that
 
    Ĵ_d(ξ)(Φ_d (q_0,q_1,z_0))= σ_d(q_0,q_1,z_0) Ĵ_d(ξ)(q_0,q_1,z_0),

 where σ_d(q_0,q_1,z_0)=1+D_zL_d(q_0,q_1,z_0).



    The fact that Φ acts by contactomorphisms is immediately checked by computing the pullback of either the 1-forms η^±:
    
    (Φ_g)^*η^±=
            
            η^±.

as a direct consequence of the G-invariance of L_d (see Subsection 1.3.3 in <cit.>).

In order to simplify the notation, let P_0=(q_0,q_1,z_0) and P_1=Φ_d (q_0,q_1,z_0). By definition we have that
    
    Ĵ_d(ξ)(P_1)=⟨η^-(P_1), ξ(P_1)⟩.

Now applying the definition of η^- and equation (<ref>) we get
    
    Ĵ_d(ξ)(P_1)=1/σ_d(P_1)⟨ D_1L_d(P_1), ξ_Q(q_1)⟩.

Using the discrete Herglotz equations, the right-hand side reduces to
    
    Ĵ_d(ξ)(P_1)=-⟨ D_2L_d(P_0),ξ_Q(q_1)⟩.

From the infinitesimal symmetry formula in equation (<ref>), we deduce
   
    Ĵ_d(ξ)(P_1)=⟨ D_1L_d(P_0), ξ_Q(q_0)⟩ .

Now inserting σ_d(P_0) so that
    
    Ĵ_d(ξ)(P_1)=σ_d(P_0) ⟨D_1L_d(P_0)/σ_d(P_0),ξ_Q(q_0)⟩,

    we deduce
    
    Ĵ_d(ξ)(P_1)=σ_d(P_0)⟨η^-(P_0)(ξ(P_0)⟩

    and so we have proved that
    
    Ĵ_d(ξ)(P_1)=σ_d(P_0) Ĵ_d(ξ)(P_0).







§ EXACT DISCRETE LAGRANGIAN FOR CONTACT SYSTEMS




 §.§ The contact exponential map


Given a contact regular Lagrangian L:TQ×→, consider the corresponding Lagrangian vector field ξ_L and denote its flow by ϕ_t^ξ_L.

Define the open subset U_h of TQ× given by

    U_h={(q_0,q̇_0,z_0)∈ TQ × |  ϕ_t^ξ_L is defined for t∈ [0,h]}

and let the contact exponential map be defined by

    exp_h^ξ_L:      U_h⊆ TQ ×→ Q × Q ×
        (q_0,q̇_0,z_0)↦ (q_0,q_1,z_0),

where q_1=p_Q∘ϕ_h^ξ_L(q_0,q̇_0,z_0) and p_Q:TQ×→ Q is the projection onto Q given by p_Q(v_q,z)=q for v_q∈ T_q Q.

We will prove that the contact exponential map is a local diffeomorphism, using the fact that the non-holonomic exponential map, i.e., the exponential map of a non-holonomic system is a local embedding (see <cit.>). In fact, to every regular contact system, one can associate a non-holonomic Lagrangian system on T(Q×) with nonlinear constraints.

Consider the singular Lagrangian function

    L:T(Q×) →,   L=L∘π,

where π:T(Q×) → TQ× is a projection onto TQ×. Also, we take the non-linear constraints

    M_L={ (q,z,q̇,ż)∈ T(Q×)  |  ż=L(q,q̇,z) }.

Observe that  M_L is the zero level set of the real-valued function Φ:T(Q×) → given by Φ(q,z,q̇,ż)=ż-L(q,q̇,z).

The pair (L,M_L) forms a Lagrangian non-holonomic system with non-linear constraints determined by the submanifold M_L and dynamics given by Chetaev's principle (see <cit.> and references therein). According to this principle the equations of motion are

    / t( ∂L/∂q̇^i) - ∂L/∂ q^i=λ∂Φ/∂q̇^i
       / t( ∂L/∂ż) - ∂L/∂ z=λ∂Φ/∂ż
       Φ(q^i,z,q̇^i,ż)=0,

with Lagrange multiplier λ. As L does not depend on ż it is straightforward to check that the Lagrange multiplier is just

    λ=-∂ L/∂ z

and that equations (<ref>) are equivalent to the Herglotz equations for L.

Moreover, since L is regular, we can define a SODE vector field Γ_(L,M_L) ∈𝔛(M_L) as the unique vector field on M_L whose integral curves satisfy equations (<ref>). Hence, we deduce

    Tπ(Γ_(L,M_L))=ξ_L∘π.


Let us denote the flow of the vector field Γ_(L,M_L) by ϕ_t^Γ_(L,M_L):M_L→ M_L.

Consider now the submanifold of _L given by

    M_L,h={(q_0,q̇_0,z_0,ż_0)∈ T(Q ×)  |  ϕ_t^Γ_(L,M_L) is defined for t∈ [0,h]}.

We define the non-holonomic exponential map to be

    exp_h^Γ_(L,M_L):  M_L,h⊆ M_L   ⟶ (Q ×) × (Q ×) 
    
    						(q_0,z_0,q̇_0,ż_0)    ↦ (q_0,z_0,q_1,z_1),

where (q_1,z_1)=τ_Q×∘ϕ_h^Γ_(L,M_L)(q_0,z_0,q̇_0,ż_0), with τ_Q×:T(Q×) → Q× the tangent bundle projection.

In <cit.>  the authors prove that there is an open subset N_h⊆ M_L,h such that the non-holonomic exponential map exp_h^Γ_(L,M_L)|_N_h is a smooth embedding and, hence, a diffeomorphism into its image, which we will denote by M_d.


 There exists an open set V_h⊆ U_h such that the contact exponential map exp_h^ξ_L|_V_h is a diffeomorphism.



Let us consider the non-holonomic system (L,M_L) defined previously.

According to equation (<ref>), the vector fields ξ_L and Γ_(L,M_L) are π-related therefore, its flows satisfy

    π∘ϕ_t^Γ_(L,M_L)=ϕ_t^ξ_L∘π.

We remark that π|_M_L is a diffeomorphism, since M_L is diffeomorphic to the graph of the Lagrangian function L. As such, we can also write

    ϕ_t^Γ_(L,M_L)=(π|_M_L)^-1∘ϕ_t^ξ_L∘π|_M_L.

Thus, we can write the non-holonomic exponential map in terms of the contact dynamics in the following way

    exp_h^Γ_(L,M_L)(q_0,z_0,q̇_0,ż_0)=(q_0,z_0,q_1,z_1),

with (q_1,z_1)=τ_Q×∘ (π|_M_L)^-1∘ϕ_h^ξ_L∘π|_M_L(q_0,z_0,q̇_0,ż_0) where ż_0=L(q_0, q̇_0, z_0).

Also note that τ_Q×∘ (π|_M_L)^-1=p_Q×, where

    p_Q×:TQ×→ Q×,    p_Q×(v_q,z)=(q,z).

In Diagram (<ref>) we show the different projections we can define on the manifolds involved in this section.


    T(Q ×ℝ) [ld, "π"'] [rd, "τ_Q×ℝ"]    
    
    TQ×ℝ[rr, "p_Q×ℝ"] [rd, "p_Q"']        Q×ℝ[ld, "pr_1"] 
        Q


With these projections we can also write the contact exponential map as

    exp_h^ξ_L(q_0,q̇_0,z_0)=(q_0,q_1,z_0),

with q_1=pr_1∘ p_Q×∘ϕ_h^ξ_L (q_0,q̇_0,z_0). Hence, we can write it as

    exp_h^ξ_L = pr_1∘exp_h^Γ_(L,M_L)∘ (π|_M_L)^-1,

    with

    pr_1:(Q×) × (Q ×)    ⟶ Q × Q ×
    
            (q_0,z_0,q_1,z_1)    ↦ (q_0, pr_1(q_1,z_1),z_0).


Therefore, if pr_1|_M_d is a local diffeomorphism then, by equation (<ref>), the contact exponential map exp_h^ξ_L|_V_h is a diffeomorphism if we choose 
    V_h=π|_M_L(N_h),

where N_h is the open subset where exp_h^Γ_(L,M_L)|_N_h is an embedding.

We are going to prove in the next Lemma that pr_1|_M_d is a local diffeomorphism.




Using the same notation as in the previous theorem, pr_1|_M_d is a local diffeomorphism.



    All we must prove is that pr_1|_M_d is a local submersion (immersion) since, by dimensional reasons, this forces pr_1|_M_d to be also a local immersion (submersion).
    
    Let x∈ M_d. Any vector in the kernel of T_xpr_1|_M_d must be the tangent vector of a curve of the form
    
    Z(s)=(q_0,z_0,q_1,w · s)∈ M_d,    w∈.

    Let γ_s(t)=ϕ_t^Γ_(L,M_L)∘ (exp_h^Γ_(L,M_L))^-1(Z(s)). For each fixed value of s, this is an integral curve of Γ_(L,M_L) satisfying
    
    τ_Q×∘γ_s(0)=(q_0,z_0),   τ_Q×∘γ_s(h)=(q_1,w · s).

    
    Moreover, note that the projection of γ_s(t) to TQ×, i.e., the curve π∘γ_s(t) is an integral curve of ξ_L with endpoints q_0 and q_1 for each fixed value of s and so π∘γ_0(t) must satisfy Herglotz' principle. Note that the action over the curves π∘γ_s(t) is given by
    
    𝒜(p_Q∘π∘γ_s(t))=p_∘π∘γ_s(h)=w · s,

    where p_:TQ×→ is the projection onto the second factor.
    
    Therefore, p_Q∘π∘γ_0(t) is a critical value of the action if and only if w=0. Therefore, T_xpr_1|_M_d is trivial and pr_1|_M_d must be a local diffeomorphism in a neighbourhood of each point.


Since the contact exponential map is a local diffeomorphism we can define a local inverse called the exact retraction and denote it by R_h^e-:Q × Q ×→ TQ ×. We will also use its translation by the flow

    R_h^e+:Q × Q ×→ TQ ×,    R_h^e+:=ϕ_h^ξ_L∘ R_h^e-.




 §.§ The exact discrete Lagrangian function


Let L_h^e: Q × Q ×→ and defined by

    L_h^e (q_0,q_1,z_0)=∫_0^h L∘ϕ_t^ξ_L∘ R_h^e-(q_0,q_1,z_0)  t

is called the exact discrete Lagrangian function.

We will need the following classical result in the proof of the next theorem: the solution of the first order linear equation ẏ=a(t)+ b/ t(t)y is

    y(t)=e^b(t)( ∫_0^t a(s)e^-b(s)  s + y(0) ).



The Legendre transforms of a regular Lagrangian L:TQ×→ are related to the discrete Legendre transforms of the corresponding exact discrete Lagrangian L_h^e:Q× Q×→ in the following way
	
    ^+L_h^e= L∘ R_h^e+,   ^-L_h^e= L∘ R_h^e-.




	
	We will prove in local computations that the derivatives of the exact discrete Lagrangian function satisfy
	
    D_1 L_h^e (q_0,q_1,z_0) = -∂ L/∂q̇(q_0,q̇_0,z_0)e^b(h); 
        D_2 L_h^e (q_0,q_1,z_0) = ∂ L/∂q̇(q_1,q̇_1,z_1); 
        D_z  L_h^e (q_0,q_1,z_0) = e^b(h)-1.

	where
	
    (q_0,q̇_0,z_0)=R_h^e-(q_0,q_1,z_0),    (q_1,q̇_1,z_1)= ϕ_h^ξ_L∘ R_h^e-(q_0,q_1,z_0), 
       and   b(t)=∫_0^t∂ L/∂ z(ϕ_s^ξ_L∘ R_h^e-(q_0,q_1,z_0))   s

	
To simplify the notation in the proof we will use the notation γ_0,1(t)=(q_0,1(t),q̇_0,1(t),z_0,1(t)):=ϕ_t^ξ_L∘ R_h^e-(q_0,q_1,z_0). Under this convention we will have

    L_h^e (q_0,q_1,z_0)=∫_0^h L(γ_0,1(t))  t.

Note first that any variation of the exact discrete Lagrangian will take the form

    δ L_h^e (q_0,q_1,z_0) = . / s|_s=0L_h^e(q_0(s),q_1(s),z_0(s)) 
        =∫_0^h∂ L/∂ q(γ_0,1(t))δ q_0,1 +∂ L/∂q̇(γ_0,1(t))δq̇_0,1+∂ L/∂ z(γ_0,1(t))δ z_0,1  t.

Since γ_0,1(t) is a solution of Euler-Lagrange equations, it satisfies

    ż_0,1=L(q_0,1(t),q̇_0,1(t),z_0,1(t)).

Therefore, any variation of z_0,1 satisfies the variational equation

    δż_0,1=∂ L/∂ q(γ_0,1(t))δ q_0,1 +∂ L/∂q̇(γ_0,1(t))δq̇_0,1+∂ L/∂ z(γ_0,1(t))δ z_0,1.

Hence, any variation of the exact discrete Lagrangian reduces to

    δ L_h^e (q_0,q_1,z_0) = δ z_0,1(h)-δ z_0,1(0).

Moreover, we can solve the function δ z_0,1 explicitly, by solving the differential equation (<ref>)

    δ z_0,1(h)=e^b(h)(∫_0^h a(s)e^-b(s)  s + δ z_0,1(0) ),

with

    b(t)     = ∫_0^t∂ L/∂ z(γ_0,1(s))   s, 
    
    a(t)     = ∂ L/∂ q(γ_0,1(t))δ q_0,1 +∂ L/∂q̇(γ_0,1(t))δq̇_0,1.

Let us compute the integration in the expression of δ z_0,1:

    ∫_0^h a(s)e^-b(s)  s     = ∫_0^h( ∂ L/∂ qδ q_0,1 +∂ L/∂q̇δq̇_0,1)e^-b(t)  t 
        = ∫_0^h( ∂ L/∂ q - d/dt∂ L/∂q̇+∂ L/∂q̇∂ L/∂ z)δ q_0,1 e^-b(t)  t 
        + ∂ L/∂q̇(γ_0,1(h))e^-b(h)δ q_0,1(h)-∂ L/∂q̇(γ_0,1(0))δ q_0,1(0),

where we are using integration by parts. Note that the term between brackets is zero, since we are over solutions of Euler-Lagrange equations. Therefore,

    δ z_0,1(h)=∂ L/∂q̇(γ_0,1(h))δ q_0,1(h)-∂ L/∂q̇(γ_0,1(0))e^b(h)δ q_0,1(0) + e^b(h)δ z_0,1(0).


Note that the differentials of the discrete Lagrangian D_1L_h^e, D_2L_h^e and D_z L_h^e are instances of particular variations. Therefore, we have that

    D_1 L_h^e(q_0,q_1,z_0)     = (∂ L/∂q̇(γ_0,1(h))∂ q_0,1(h)/∂ q_0^i -∂ L/∂q̇(γ_0,1(0))e^b(h)∂ q_0,1(0)/∂ q_0^i. 
       .+ (e^b(h)-1)∂ z_0,1(0)/∂ q_0^i)  q_0^i
        = -∂ L/∂q̇^i(γ_0,1(0))e^b(h) q_0^i,

since q_0,1(h)≡ q_1 and so its derivative with respect to q_0 vanishes, q_0,1(0)≡ q_0 and so its derivative with respect to q_0 is the identity and, finally, z_0,1(0)≡ z_0 does not depend upon q_0. Likewise, the next derivative follows from applying similar arguments. Indeed, we have that

    D_2 L_h^e(q_0,q_1,z_0)     = (∂ L/∂q̇(γ_0,1(h))∂ q_0,1(h)/∂ q_1^i -∂ L/∂q̇(γ_0,1(0))e^b(h)∂ q_0,1(0)/∂ q_1^i. 
       .+ (e^b(h)-1)∂ z_0,1(0)/∂ q_1^i)  q_1^i
        = ∂ L/∂q̇^i(γ_0,1(h)) q_1^i.

Analogously, we also deduce

    D_z L_h^e(q_0,q_1,z_0)     = (∂ L/∂q̇(γ_0,1(h))∂ q_0,1(h)/∂ z_0 -∂ L/∂q̇(γ_0,1(0))e^b(h)∂ q_0,1(0)/∂ z_0. 
       .+ (e^b(h)-1)∂ z_0,1(0)/∂ z_0)  z_0
        = (e^b(h)-1) z_0.

Now, the result follows by the definition of the discrete Legendre transforms.


The commutativity of the following diagram summarizes the statement of the previous theorem

    Q× Q×ℝ[r, "R^e±_h"] [rd, "𝔽 ^± L^e_h"']     TQ×ℝ[d, "𝔽 L"] 
        T^*Q×ℝ


Now, we are going to relate the continuous contact Lagrangian flow with its discrete counterpart, when we take as discrete Lagrangian the corresponding exact discrete Lagrangian.


Take a regular Lagrangian L:TQ→ and fix a time step h>0. Then we have that:

 
  * L_h^e is a regular discrete Lagrangian function;
 
  * If H is the Hamiltonian function corresponding to L introduced at the end of Section <ref> and ϕ_t^X_H is its contact Hamiltonian flow, we have that
 
    ^+ L_h^e=ϕ_h^X_H∘^- L_h^e.

 
  * If (q,z):[0,Nh]→ Q× is a solution of the Herglotz equations, then it is related to the solution of the discrete Herglotz equations { (q_0,z_0),(q_1,z_1),...,(q_N,z_N) } for the corresponding exact discrete Lagrangian with (q(0),q(h),z(0)) as initial conditions in the following way:
 	
    q_k=q(kh),    z_k=z(kh)   for k=0,..., N.





	Item 1. is a consequence of of the theorem before, since ^-L_h^e is a composition of two local diffeomorphisms it is itself a local diffeomorphism. Item 2. comes from unyielding the definitions:
	
    ^+ L_h^e= L ∘ R^e+_h= L ∘ϕ_h^Γ_L∘ R^e-_h=ϕ_h^X_H∘ L∘ R^e-_h=ϕ_h^X_H∘^- L_h^e.

	For item 3. observe that, by discrete Herglotz equations, for every k=1,...,N-1 we have that
	
    ^+ L_h^e(q(k-1),q(k),z(k-1))=^- L_h^e(q(k),q(k+1),z(k))

	so that { (q_0,z_0),(q_1,z_1),...,(q_N,z_N) } is indeed the solution of these equations.






§ NUMERICAL EXAMPLES


Given a mechanical contact Lagrangian with a euclidean metric and a potential function V:Q→ of the type

    L(q,q̇,z)=1/2q̇^2-V(q)+γ z,    (q,q̇,z)∈ TQ×,   γ < 0.

one usually approximates the exact discrete Lagrangian associated to L by means of a quadrature rule. Note that the restriction of γ to negative values is necessary to model dissipative dynamics, though we could define the integrator for any value of γ∈. If we use the middle point rule to approximate the positions, i.e., q≈q_1+q_0/2, one may define the discrete Lagrangian L_d:Q× Q×→ in the following way

    L_d(q_0,q_1,z_0)=1/2h(q_1-q_0)^2-hV(q_1+q_0/2)+hγ z_0.

We remark that the value of h should be chosen small enough so that the function σ_d does not vanish anywhere. In this case, the discrete Herglotz equations are of the type

    q_1-q_0/h-h/2∂ V/∂ q(q_1+q_0/2)=1/(1+hγ)(q_2-q_1/h+h/2∂ V/∂ q(q_2+q_1/2))  
        z_1=L_d(q_0,q_1,z_0)=1/2h(q_1-q_0)^2-hV(q_1+q_0/2)+(hγ+1) z_0



	The free single particle contact Lagrangian is
	
    L(q,q̇,z)=1/2q̇^2+γ z,    (q,q̇,z)∈ TQ×.

	A simple discretization of this Lagrangian would be
	
    L_d(q_0,q_1,z_0)=1/2h(q_1-q_0)^2+hγ z_0.

	Then, choosing h small enough so that the function σ_d is non-vanishing, the discrete Herglotz equations for L_d are locally given by
	
    q_1-q_0/h=q_2-q_1/h(1+hγ)  ⇒   q_2=(hγ+2)q_1-(hγ+1)q_0
        z_1=1/2h(q_1-q_0)^2+(hγ+1) z_0


	The discrete flow obtained by solving these equations is plotted in Fig. <ref>.

	In this case, one can also compute the exact discrete Lagrangian and solve the exact dynamics.
	
    L_h^e(q_0,q_1,z_0) = γ(q_1- q_0)^2 e^γ h/2 e^γ h - 2 - z_0(e^γ h - 1).


	



	The damped harmonic oscillator is described by the Lagrangian
	
    L(q,q̇,z)=1/2q̇^2-1/2q^2+γ z,    (q,q̇,z)∈ TQ×.

	Using a middle point discretization, i.e., q≈q_1+q_0/2, one may define the discrete Lagrangian
	
    L_d(q_0,q_1,z_0)=1/2h(q_1-q_0)^2-h/8(q_1+q_0)^2+hγ z_0.

	In this case, after choosing h small enough, the discrete Herglotz equations hold
	
    q_1-q_0/h-h/4(q_1+q_0)=1/(1+hγ)(q_2-q_1/h+h/4(q_2+q_1))  
        z_1=1/2h(q_1-q_0)^2-h/8(q_1+q_0)^2+(hγ+1) z_0,

	which can be solved explicitly for q_2
	
    q_2=-(h^3γ+4hγ+h^2+4)q_0+(h^3γ-4hγ+2h^2-8)q_1/h^2+4.


	The discrete flow obtained by solving these equations is plotted in Fig. <ref>.

	In this case, the exact discrete Lagrangian and the exact discrete dynamics can be computed with the aid of a Computer Algebra system, but the analytic expressions are complicated, so we only include their graph in Fig. <ref>.
	






§ CONCLUSIONS AND FUTURE WORK

In this paper, we went deeper in the geometry of discrete contact mechanics following, as a starting point, the results by <cit.>. We have done a detailed study of the discrete Herglotz principle and its geometric properties, including the discrete Legendre transforms and the associated discrete Lagrangian and Hamiltonian flows. Moreover, we have analyzed the existence of dissipated quantities related with symmetries of the system and the construction of the exact discrete Lagrangian function given the correspondence between the discrete and continuous system.  

In future work, we will study the variational error analysis allowing us to estimate the error order of the proposed methods just from the error of approximation of the exact discrete Lagrangian function, that is, how well the discrete Lagrangian function matches the exact discrete Lagrangian function <cit.>. Moreover, we will introduce higher-order methods for contact Lagrangian systems extending the theory of Morse functions to Legendrian submanifolds (see <cit.>). For instance, this theory will give a complete geometric explanation of other possible discretizations of the phase space, as for instance,  the one used by
Vermeeren et al which is Q × Q ×ℝ^2 instead of Q× Q×.


 

§ ACKNOWLEDGMENTS
Manuel Lainz wishes to thank MICINN and ICMAT for a FPI-Severo Ochoa predoctoral contract PRE2018-083203

The authors are  supported  by Ministerio de Ciencia e Innovación ( Spain) under grants  MTM2016-76702-P and “Severo Ochoa Programme for Centres of Excellence” in R&D (SEV-2015-0554). A.Simoes is supported by the FCT (Portugal) research fellowship SFRH/BD/129882/2017. M. Lainz wishes to thank MICINN and ICMAT for a FPI-Severo Ochoa predoctoral contract PRE2018-083203.


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