Discrete Nonholonomic Lagrangian Systems on Lie Groupoids
DISCRETE NONHOLONOMIC LAGRANGIAN SYSTEMS ON LIE
GROUPOIDS
DAVID IGLESIAS, JUAN C. MARRERO, DAVID MARTÍN DE DIEGO,
AND EDUARDO MARTÍNEZ
Abstract. This paper studies the construction of geometric integrators for
nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange
equations in a setting which permits to deduce geometric integrators for con-
tinuous nonholonomic systems (reduced or not). The formalism is given in
terms of Lie groupoids, specifying a discrete Lagrangian and a constraint sub-
manifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie
algebroid associated to the Lie groupoid. We also discuss the existence of non-
holonomic evolution operators in terms of the discrete nonholonomic Legendre
transformations and in terms of adequate decompositions of the prolongation
of the Lie groupoid. The characterization of the reversibility of the evolution
operator and the discrete nonholonomic momentum equation are also consid-
ered. Finally, we illustrate with several classical examples the wide range of
application of the theory (the discrete nonholonomic constrained particle, the
Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on
a rotating table and the two wheeled planar mobile robot).
Contents
1. Introduction 2
2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 5
2.1. Lie algebroids 5
2.2. Lie groupoids 6
2.3. Discrete Unconstrained Lagrangian Systems 9
3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie
groupoids 11
3.1. Discrete Generalized Hölder’s principle 11
3.2. Discrete Nonholonomic Legendre transformations 14
3.3. Nonholonomic evolution operators and regular discrete nonholonomic
Lagrangian systems 20
3.4. Reversible discrete nonholonomic Lagrangian systems 22
3.5. Lie groupoid morphisms and reduction 23
3.6. Discrete nonholonomic Hamiltonian evolution operator 24
3.7. The discrete nonholonomic momentum map 24
4. Examples 26
4.1. Discrete holonomic Lagrangian systems on a Lie groupoid 26
4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid 27
This work has been partially supported by MICYT (Spain) Grants MTM 2006-03322, MTM
2004-7832, MTM 2006-10531 and S-0505/ESP/0158 of the CAM. D. Iglesias thanks MEC for a
“Juan de la Cierva” research contract.
2 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
4.3. Discrete nonholonomic Lagrangian systems on a Lie group 29
4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid 32
4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid 35
4.6. Discrete Chaplygin systems 39
5. Conclusions and Future Work 43
References 43
1. Introduction
In the paper of Moser and Veselov [40] dedicated to the complete integrability
of certain dynamical systems, the authors proposed a discretization of the tangent
bundle TQ of a configuration space Q replacing it by the product Q×Q, approx-
imating a tangent vector on Q by a pair of ‘close’ points (q0, q1). In this sense,
the continuous Lagrangian function L : TQ −→ R is replaced by a discretization
Ld : Q×Q −→ R. Then, applying a suitable variational principle, it is possible to
derive the discrete equations of motion. In the regular case, one obtains an evolu-
tion operator, a map which assigns to each pair (qk−1, qk) a pair (qk, qk+1), sharing
many properties with the continuous system, in particular, symplecticity, momen-
tum conservation and a good energy behavior. We refer to [32] for an excellent
review in discrete Mechanics (on Q×Q) and its numerical implementation.
On the other hand, in [40, 44], the authors also considered discrete Lagrangians
defined on a Lie group G where the evolution operator is given by a diffeomorphism
of G.
All the above examples led to A. Weinstein [45] to study discrete mechanics on
Lie groupoids. A Lie groupoid is a geometric structure that includes as particular
examples the case of cartesian products Q × Q as well as Lie groups and other
examples as Atiyah or action Lie groupoids [26]. In a recent paper [27], we studied
discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids, deriving from
a variational principle the discrete Euler-Lagrange equations. We also introduced
a symplectic 2-section (which is preserved by the Lagrange evolution operator)
and defined the Hamiltonian evolution operator, in terms of the discrete Legendre
transformations, which is a symplectic map with respect to the canonical symplectic
2-section on the prolongation of the dual of the Lie algebroid of the given groupoid.
These techniques include as particular cases the classical discrete Euler-Lagrange
equations, the discrete Euler-Poincaré equations (see [5, 6, 29, 30]) and the discrete
Lagrange-Poincaré equations. In fact, the results in [27] may be applied in the
construction of geometric integrators for continuous Lagrangian systems which are
invariant under the action of a symmetry Lie group (see also [18] for the particular
case when the symmetry Lie group is abelian).
From the perspective of geometric integration, there are a great interest in intro-
ducing new geometric techniques for developing numerical integrators since stan-
dard methods often introduce some spurious effects like dissipation in conservative
systems [16, 42]. The case of dynamical systems subjected to constraints is also
of considerable interest. In particular, the case of holonomic constraints is well
established in the literature of geometric integration, for instance, in simulation of
molecular dynamics where the constraints may be molecular bond lengths or angles
and also in multibody dynamics (see [16, 20] and references therein).
DISCRETE NONHOLONOMIC MECHANICS 3
By contrast, the construction of geometric integrators for the case of nonholo-
nomic constraints is less well understood. This type of constraints appears, for
instance, in mechanical models of convex rigid bodies rolling without sliding on a
surface [41]. The study of systems with nonholonomic constraints goes back to the
XIX century. The equations of motion were obtained applying either D’Alembert’s
principle of virtual work or Gauss principle of least constraint. Recently, many
authors have shown a new interest in that theory and also in its relation to the
new developments in control theory and robotics using geometric techniques (see,
for instance, [2, 3, 4, 8, 19, 22, 24]).
Geometrically, nonholonomic constraints are globally described by a submanifold
M of the velocity phase space TQ. If M is a vector subbundle of TQ, we are dealing
with the case of linear constraints and, in the case M is an affine subbundle, we are in
the case of affine constraints. Lagrange-D’Alembert’s or Chetaev’s principles allow
us to determine the set of possible values of the constraint forces only from the set
of admissible kinematic states, that is, from the constraint manifold M determined
by the vanishing of the nonholonomic constraints φa. Therefore, assuming that the
dynamical properties of the system are mathematically described by a Lagrangian
function L : TQ −→ R and by a constraint submanifold M, the equations of motion,
following Chetaev’s principle, are[
δqi = 0 ,
where δqi denotes the virtual displacements verifying
δqi = 0. By using the
Lagrange multiplier rule, we obtain that
= λ̄a
, (1.1)
with the condition q̇(t) ∈ M, λ̄a being the Lagrange multipliers to be determined.
Recently, J. Cortés et al [9] (see also [11, 38, 39]) proposed a unified framework for
nonholonomic systems in the Lie algebroid setting that we will use along this paper
generalizing some previous work for free Lagrangian mechanics on Lie algebroids
(see, for instance, [23, 33, 34, 35]).
The construction of geometric integrators for Equations (1.1) is very recent. In
fact, in [37] appears as an open problem:
...The problem for the more general class of non-holonomic con-
straints is still open, as is the question of the correct analogue
of symplectic integration for non-holonomically constrained La-
grangian systems...
Numerical integrators derived from discrete variational principles have proved their
adaptability to many situations: collisions, classical field theory, external forces...[28,
32] and it also seems very adequate for nonholonomic systems, since nonholonomic
equations of motion come from Hölder’s variational principle which is not a stan-
dard variational principle [1], but admits an adequate discretization. This is the
procedure introduced by J. Cortés and S. Mart́ınez [8, 10] and followed by other
authors [12, 14, 15, 36] extending, moreover, the results to nonholonomic systems
defined on Lie groups (see also [25] for a different approach using generating func-
tions).
In this paper, we tackle the problem from the unifying point of view of Lie
groupoids (see [9] for the continuous case). This technique permits to recover all
the previous methods in the literature [10, 14, 36] and consider new cases of great
4 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
importance in nonholonomic dynamics. For instance, using action Lie groupoids,
we may discretize LR-nonholonomic systems such as the Veselova system or us-
ing Atiyah Lie groupoids we find discrete versions for the reduced equations of
nonholonomic systems with symmetry.
The paper is structured as follows. In section 2 we review some basic results on
Lie algebroids and Lie groupoids. In particular, we describe the prolongation of a
Lie groupoid [43], which has a double structure of Lie groupoid and Lie algebroid.
Then, we briefly expose the geometric structure of discrete unconstrained mechanics
on Lie groupoids: Poincaré-Cartan sections, Legendre transformations... The main
results of the paper appear in section 3, where the geometric structure of discrete
nonholonomic systems on Lie groupoids is considered. In particular, given a discrete
Lagrangian Ld : Γ → R on a Lie groupoid Γ, a constraint distribution Dc in the
Lie algebroid EΓ of Γ and a discrete constraint submanifold Mc in Γ, we obtain
the nonholonomic discrete Euler-Lagrange equations from a discrete Generalized
Hölder’s principle (see section 3.1). In addition, we characterize the regularity of the
nonholonomic system in terms of the nonholonomic Legendre transformations and
decompositions of the prolongation of the Lie groupoid. In the case when the system
is regular, we can define the nonholonomic evolution operator. An interesting
situation, studied in in Section 3.4, is that of reversible discrete nonholonomic
Lagrangian systems, where the Lagrangian and the discrete constraint submanifold
are invariants with respect to the inversion of the Lie groupoid. The particular
example of reversible systems in the pair groupoid Q×Q was first studied in [36].
We also define the discrete nonholonomic momentum map. In order to give an
idea of the breadth and flexibility of the proposed formalism, several examples are
discussed, including their regularity and their reversibility:
- Discrete holonomic Lagrangian systems on a Lie groupoid, which are a
generalization of the Shake algorithm for holonomic systems [16, 20, 32];
- Discrete nonholonomic systems on the pair groupoid, recovering the equa-
tions first considered in [10]. An explicit example of this situation is the
discrete nonholonomic constrained particle.
- Discrete nonholonomic systems on Lie groups, where the equations that
are obtained are the so-called discrete Euler-Poincaré-Suslov equations (see
[14]). We remark that, although our equations coincide with those in [14],
the technique developed in this paper is different to the one in that paper.
Two explicit examples which we describe here are the Suslov system and
the Chaplygin sleigh.
- Discrete nonholonomic Lagrangian systems on an action Lie groupoid.
This example is quite interesting since it allows us to discretize a well-
known nonholonomic LR-system: the Veselova system (see [44]; see also
[13]). For this example, we obtain a discrete system that is not reversible
and we show that the system is regular in a neighborhood around the
manifold of units.
- Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid.
With this example, we are able to discretize reduced systems, in particular,
we concentrate on the example of the discretization of the equations of
motion of a rolling ball without sliding on a rotating table with constant
angular velocity.
- Discrete Chaplygin systems, which are regular systems (Ld,Mc,Dc) on
the Lie groupoid Γ ⇒ M , for which (α, β) ◦ iMc : Mc → M × M is
a diffeomorphism and ρ ◦ iDc : Dc → TM is an isomorphism of vector
bundles, (α, β) being the source and target of the Lie groupoid Γ and ρ
DISCRETE NONHOLONOMIC MECHANICS 5
being the anchor map of the Lie algebroid EΓ. This example includes a
discretization of the two wheeled planar mobile robot.
We conclude our paper with future lines of work.
2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids
2.1. Lie algebroids. A Lie algebroid E over a manifold M is a real vector bundle
τ : E →M together with a Lie bracket [[·, ·]] on the space Sec(τ) of the global cross
sections of τ : E → M and a bundle map ρ : E → TM , called the anchor
map, such that if we also denote by ρ : Sec(τ) → X(M) the homomorphism of
C∞(M)-modules induced by the anchor map then
[[X, fY ]] = f [[X,Y ]] + ρ(X)(f)Y, (2.1)
for X,Y ∈ Sec(τ) and f ∈ C∞(M) (see [26]).
If (E, [[·, ·]], ρ) is a Lie algebroid over M then the anchor map ρ : Sec(τ) →
X(M) is a homomorphism between the Lie algebras (Sec(τ), [[·, ·]]) and (X(M), [·, ·]).
Moreover, one may define the differential d of E as follows:
dµ(X0, . . . , Xk) =
(−1)iρ(Xi)(µ(X0, . . . , X̂i, . . . , Xk))
(−1)i+jµ([[Xi, Xj ]], X0, . . . , X̂i, . . . , X̂j , . . . , Xk),
(2.2)
for µ ∈ Sec(∧kτ∗) and X0, . . . , Xk ∈ Sec(τ). d is a cohomology operator, that is,
d2 = 0. In particular, if f : M −→ R is a real smooth function then df(X) = ρ(X)f,
for X ∈ Sec(τ).
Trivial examples of Lie algebroids are a real Lie algebra of finite dimension (in
this case, the base space is a single point) and the tangent bundle of a manifold M.
On the other hand, let (E, [[·, ·]], ρ) be a Lie algebroid of rank n over a manifold
M of dimension m and π : P →M be a fibration. We consider the subset of E×TP
TEP = { (a, v) ∈ E × TP | (Tπ)(v) = ρ(a) } ,
where Tπ : TP → TM is the tangent map to π. Denote by τπ : TEP → P the
map given by τπ(a, v) = τP (v), τP : TP → P being the canonical projection. If
dimP = p, one may prove that TEP is a vector bundle over P of rank n + p −m
with vector bundle projection τπ : TEP → P .
A section X̃ of τπ : TEP → P is said to be projectable if there exists a section X
of τ : E →M and a vector field U ∈ X(P ) which is π-projectable to the vector field
ρ(X) and such that X̃(p) = (X(π(p)), U(p)), for all p ∈ P . For such a projectable
section X̃, we will use the following notation X̃ ≡ (X,U). It is easy to prove that
one may choose a local basis of projectable sections of the space Sec(τπ).
The vector bundle τπ : TEP → P admits a Lie algebroid structure ([[·, ·]]π, ρπ).
Indeed, if (X,U) and (Y, V ) are projectable sections then
[[(X,U), (Y, V )]]π = ([[X,Y ]], [U, V ]), ρπ(X,U) = U.
(TEP, [[·, ·]]π, ρπ) is the E-tangent bundle to P or the prolongation of E
over the fibration π : P →M (for more details, see [23]).
Now, let (E, [[·, ·]], ρ) (resp., (E′, [[·, ·]]′, ρ′)) be a Lie algebroid over a manifold M
(resp., M ′) and suppose that Ψ : E → E′ is a vector bundle morphism over the map
Ψ0 : M →M ′. Then, the pair (Ψ,Ψ0) is said to be a Lie algebroid morphism if
d((Ψ,Ψ0)
∗φ′) = (Ψ,Ψ0)
∗(d′φ′), for all φ′ ∈ Sec(∧k(τ ′)∗) and for all k, (2.3)
6 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
where d (resp., d′) is the differential of the Lie algebroid E (resp., E′) (see [23]). In
the particular case when M = M ′ and Ψ0 = Id then (2.3) holds if and only if
[[Ψ ◦X,Ψ ◦ Y ]]′ = Ψ[[X,Y ]], ρ′(ΨX) = ρ(X), for X,Y ∈ Sec(τ).
2.2. Lie groupoids. A Lie groupoid over a differentiable manifold M is a differ-
entiable manifold Γ together with the following structural maps:
• A pair of submersions α : Γ → M , the source, and β : Γ → M, the
target. The maps α and β define the set of composable pairs
Γ2 = { (g, h) ∈ G×G | β(g) = α(h) } .
• A multiplication m : Γ2 → Γ, to be denoted simply by m(g, h) = gh,
such that
– α(gh) = α(g) and β(gh) = β(h).
– g(hk) = (gh)k.
• An identity section � : M → Γ such that
– �(α(g))g = g and g�(β(g)) = g.
• An inversion map i : Γ → Γ, to be simply denoted by i(g) = g−1, such
– g−1g = �(β(g)) and gg−1 = �(α(g)).
A Lie groupoid Γ over a set M will be simply denoted by the symbol Γ ⇒ M .
On the other hand, if g ∈ Γ then the left-translation by g and the right-
translation by g are the diffeomorphisms
lg : α−1(β(g)) −→ α−1(α(g)) ; h −→ lg(h) = gh,
rg : β−1(α(g)) −→ β−1(β(g)) ; h −→ rg(h) = hg.
Note that l−1g = lg−1 and r
g = rg−1 .
A vector field X̃ on Γ is said to be left-invariant (resp., right-invariant) if
it is tangent to the fibers of α (resp., β) and X̃(gh) = (Thlg)(X̃h) (resp., X̃(gh) =
(Tgrh)(X̃(g))), for (g, h) ∈ Γ2.
Now, we will recall the definition of the Lie algebroid associated with Γ.
We consider the vector bundle τ : EΓ → M , whose fiber at a point x ∈ M is
(EΓ)x = V�(x)α = Ker(T�(x)α). It is easy to prove that there exists a bijection
between the space Sec(τ) and the set of left-invariant (resp., right-invariant) vector
fields on Γ. If X is a section of τ : EΓ →M , the corresponding left-invariant (resp.,
right-invariant) vector field on Γ will be denoted
X (resp.,
X ), where
X (g) = (T�(β(g))lg)(X(β(g))), (2.4)
X (g) = −(T�(α(g))rg)((T�(α(g))i)(X(α(g)))), (2.5)
for g ∈ Γ. Using the above facts, we may introduce a Lie algebroid structure
([[·, ·]], ρ) on EΓ, which is defined by
←−−−−
[[X,Y ]] = [
Y ], ρ(X)(x) = (T�(x)β)(X(x)), (2.6)
for X,Y ∈ Sec(τ) and x ∈M . Note that
−−−−→
[[X,Y ]] = −[
Y ], [
Y ] = 0, (2.7)
(for more details, see [7, 26]).
Given two Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′, a morphism of Lie groupoids
is a smooth map Φ : Γ→ Γ′ such that
(g, h) ∈ Γ2 =⇒ (Φ(g),Φ(h)) ∈ (Γ′)2
DISCRETE NONHOLONOMIC MECHANICS 7
Φ(gh) = Φ(g)Φ(h).
A morphism of Lie groupoids Φ : Γ → Γ′ induces a smooth map Φ0 : M → M ′ in
such a way that
α′ ◦ Φ = Φ0 ◦ α, β′ ◦ Φ = Φ0 ◦ β, Φ ◦ � = �′ ◦ Φ0,
α, β and � (resp., α′, β′ and �′) being the source, the target and the identity section
of Γ (resp., Γ′).
Suppose that (Φ,Φ0) is a morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒
M ′ and that τ : EΓ → M (resp., τ ′ : EΓ′ → M ′) is the Lie algebroid of Γ (resp.,
Γ′). Then, if x ∈ M we may consider the linear map Ex(Φ) : (EΓ)x → (EΓ′)Φ0(x)
defined by
Ex(Φ)(v�(x)) = (T�(x)Φ)(v�(x)), for v�(x) ∈ (EΓ)x. (2.8)
In fact, we have that the pair (E(Φ),Φ0) is a morphism between the Lie algebroids
τ : EΓ →M and τ ′ : EΓ′ →M ′ (see [26]).
Trivial examples of Lie groupoids are Lie groups and the pair or banal groupoid
M ×M , M being an arbitrary smooth manifold. The Lie algebroid of a Lie group
Γ is just the Lie algebra g of Γ. On the other hand, the Lie algebroid of the pair
(or banal) groupoid M ×M is the tangent bundle TM to M .
Apart from the Lie algebroid EΓ associated with a Lie groupoid Γ ⇒ M , other
interesting Lie algebroids associated with Γ are the following ones:
• The EΓ- tangent bundle to E∗Γ:
Let TEΓE∗Γ be the EΓ-tangent bundle to E
Γ, that is,
E∗Γ =
(vx, XΥx) ∈ (EΓ)x × TΥxE
∣∣ (TΥxτ∗)(XΥx) = (T�(x)β)(vx)}
for Υx ∈ (E∗Γ)x, with x ∈M. As we know, T
EΓE∗Γ is a Lie algebroid over E
We may introduce the canonical section Θ of the vector bundle (TEΓE∗Γ)
∗ → E∗Γ
as follows:
Θ(Υx)(ax, XΥx) = Υx(ax),
for Υx ∈ (E∗Γ)x and (ax, XΥx) ∈ T
E∗Γ. Θ is called the Liouville section as-
sociated with EΓ. Moreover, we define the canonical symplectic section Ω
associated with EΓ by Ω = −dΘ, where d is the differential on the Lie algebroid
TEΓE∗Γ → E
Γ. It is easy to prove that Ω is nondegenerate and closed, that is, it is
a symplectic section of TEΓE∗Γ (see [23]).
Now, if Z is a section of τ : EΓ → M then there is a unique vector field Z∗c on
E∗Γ, the complete lift of X to E
Γ, satisfying the two following conditions:
(i) Z∗c is τ∗-projectable on ρ(Z) and
(ii) Z∗c(X̂) = ̂[[Z,X]]
for X ∈ Sec(τ) (see [23]). Here, if X is a section of τ : EΓ → M then X̂ is the
linear function X̂ ∈ C∞(E∗) defined by
X̂(a∗) = a∗(X(τ∗(a∗))), for all a∗ ∈ E∗.
Using the vector field Z∗c, one may introduce the complete lift Z∗c of Z as the
section of τ τ
: TEΓE∗Γ → E
Γ defined by
Z∗c(a∗) = (Z(τ∗(a∗)), Z∗c(a∗)), for a∗ ∈ E∗. (2.9)
8 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Z∗c is just the Hamiltonian section of Ẑ with respect to the canonical symplectic
section Ω associated with EΓ. In other words,
iZ∗cΩ = dẐ, (2.10)
where d is the differential of the Lie algebroid τ τ
: TEΓE∗Γ → E
Γ (for more details,
see [23]).
• The Lie algebroid τ̃Γ : TΓΓ→ Γ :
Let TΓΓ be the Whitney sum V β ⊕Γ V α of the vector bundles V β → Γ and
V α → Γ, where V β (respectively, V α) is the vertical bundle of β (respectively,
α). Then, the vector bundle τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ admits a Lie algebroid
structure ([[·, ·]]T
ΓΓ, ρT
ΓΓ). The anchor map ρT
ΓΓ is given by
ΓΓ)(Xg, Yg) = Xg + Yg
and the Lie bracket bracket [[·, ·]]T
ΓΓ on the space Sec(τ̃Γ) is characterized for the
following relation
Y ), (
Y ′)]]T
ΓΓ = (−
−−−−−→
[[X,X ′]],
←−−−−
[[Y, Y ′]]),
for X,Y,X ′, Y ′ ∈ Sec(τ) (for more details, see [27]).
On other hand, if X is a section of τ : EΓ → M , one may define the sections
X(1,0), X(0,1) (the β and α-lifts) and X(1,1) (the complete lift) of X to τ̃Γ : TΓΓ→ Γ
as follows:
X(1,0)(g) = (
X (g), 0g), X
(0,1)(g) = (0g,
X (g)), and X(1,1)(g) = (−
X (g),
X (g)).
We have that
[[X(1,0), Y (1,0)]]T
ΓΓ = −[[X,Y ]](1,0) [[X(0,1), Y (1,0)]]T
ΓΓ = 0,
[[X(0,1), Y (0,1)]]T
ΓΓ = [[X,Y ]](0,1),
and, as a consequence,
[[X(1,1), Y (1,0)]]T
ΓΓ = [[X,Y ]](1,0), [[X(1,1), Y (0,1)]]T
ΓΓ = [[X,Y ]](0,1),
[[X(1,1), Y (1,1)]]T
ΓΓ = [[X,Y ]](1,1).
Now, if g, h ∈ Γ one may introduce the linear monomorphisms (1,0)h : (EΓ)
(TΓhΓ)
∗ ≡ V ∗h β ⊕ V
h α and
(0,1)
g : (EΓ)∗β(g) → (T
∗ ≡ V ∗g β ⊕ V ∗g α given by
(1,0)
h (Xh, Yh) = γ(Th(i ◦ rh−1)(Xh)), (2.11)
γ(0,1)g (Xg, Yg) = γ((Tglg−1)(Yg)), (2.12)
for (Xg, Yg) ∈ TΓg Γ and (Xh, Yh) ∈ TΓhΓ.
Thus, if µ is a section of τ∗ : E∗Γ → M , one may define the corresponding lifts
µ(1,0) and µ(0,1) as the sections of τ̃Γ
∗ : (TΓΓ)∗ → Γ given by
µ(1,0)(h) = µ(1,0)h , for h ∈ Γ,
µ(0,1)(g) = µ(0,1)g , for g ∈ Γ.
Note that if g ∈ Γ and {XA} (respectively, {YB}) is a local basis of Sec(τ) on an
open subset U (respectively, V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V )
then {X(1,0)A , Y
(0,1)
B } is a local basis of Sec(τ̃Γ) on the open subset α
−1(U)∩β−1(V ).
In addition, if {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively,
{YB}) then {(XA)(1,0), (Y B)(0,1)} is the dual basis of {X
(1,0)
A , Y
(0,1)
DISCRETE NONHOLONOMIC MECHANICS 9
2.3. Discrete Unconstrained Lagrangian Systems. (See [27] for details) A
discrete unconstrained Lagrangian system on a Lie groupoid consists of a
Lie groupoid Γ ⇒ M (the discrete space) and a discrete Lagrangian Ld : Γ→
2.3.1. Discrete unconstrained Euler-Lagrange equations. An admissible sequence
of order N on the Lie groupoid Γ is an element (g1, . . . , gN ) of ΓN ≡ Γ× · · · ×Γ
such that (gk, gk+1) ∈ Γ2, for k = 1, . . . , N − 1.
An admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete
unconstrained Euler-Lagrange equations for Ld if
do[Ld ◦ lgk + Ld ◦ rgk+1 ◦ i](�(xk))|(EΓ)xk = 0
where β(gk) = α(gk+1) = xk and do is the standard differential on Γ, that is, the
differential of the Lie algebroid τΓ : TΓ→ Γ (see [27]).
The discrete unconstrained Euler-Lagrange operator DDELLd : Γ2 → E∗Γ
is given by
(DDELLd)(g, h) = d
o[Ld ◦ lg + Ld ◦ rh ◦ i](�(x))|(EΓ)x = 0,
for (g, h) ∈ Γ2, with β(g) = α(h) = x ∈M (see [27]).
Thus, an admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete
unconstrained Euler-Lagrange equations if and only if
(DDELLd)(gk, gk+1) = 0, for k = 1, . . . , N − 1.
2.3.2. Discrete Poincaré-Cartan sections. Consider the Lie algebroid τ̃Γ : TΓΓ ≡
V β ⊕Γ V α → Γ, and define the Poincaré-Cartan 1-sections Θ−Ld ,Θ
Sec((τ̃Γ)∗) as follows
Θ−Ld(g)(Xg, Yg) = −Xg(Ld), Θ
(g)(Xg, Yg) = Yg(Ld), (2.13)
for each g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα.
Since dLd = Θ
−Θ−Ld and so, using d
2 = 0, it follows that dΘ+Ld = dΘ
. This
means that there exists a unique 2-section ΩLd = −dΘ
= −dΘ−Ld , which will be
called the Poincaré-Cartan 2-section. This 2-section will be important to study
the symplectic character of the discrete unconstrained Euler-Lagrange equations.
If g is an element of Γ such that α(g) = x and β(g) = y and {XA} (respectively,
{YB}) is a local basis of Sec(τ) on the open subset U (respectively, V ) of M , with
x ∈ U (respectively, y ∈ V ), then on α−1(U) ∩ β−1(V ) we have that
Θ−Ld = −
XA(L)(XA)(1,0), Θ
YB(L)(Y B)(0,1),
ΩLd = −
YB(Ld))(XA)(1,0) ∧ (Y B)(0,1)
(2.14)
where {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively, {YB})
(for more details, see [27]).
2.3.3. Discrete unconstrained Lagrangian evolution operator. Let Υ : Γ → Γ be a
smooth map such that:
- graph(Υ) ⊆ Γ2, that is, (g,Υ(g)) ∈ Γ2, for all g ∈ Γ (Υ is a second order
operator) and
- (g,Υ(g)) is a solution of the discrete unconstrained Euler-Lagrange equa-
tions, for all g ∈ Γ, that is, (DDELLd)(g,Υ(g)) = 0, for all g ∈ Γ.
10 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
In such a case ←−
X (g)(Ld)−
X (Υ(g))(Ld) = 0, (2.15)
for every section X of τ : EΓ → M and every g ∈ Γ. The map Υ : Γ→ Γ is called
a discrete flow or a discrete unconstrained Lagrangian evolution operator
for Ld.
Now, let Υ : Γ → Γ be a second order operator. Then, the prolongation TΥ :
TΓΓ ≡ V β ⊕Γ V α → TΓΓ ≡ V β ⊕Γ V α of Υ is the Lie algebroid morphism over
Υ : Γ→ Γ defined as follows (see [27]):
TgΥ(Xg, Yg) = ((Tg(rgΥ(g) ◦ i))(Yg), (TgΥ)(Xg)
+(TgΥ)(Yg)− Tg(rgΥ(g) ◦ i)(Yg)),
(2.16)
for all (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα. Moreover, from (2.4), (2.5) and (2.16), we
obtain that
X (g),
Y (g)) = (−
Y (Υ(g)), (TgΥ)(
X (g) +
Y (g)) +
Y (Υ(g))), (2.17)
for all X,Y ∈ Sec(τ).
Using (2.16), one may prove that (see [27]):
(i) The map Υ is a discrete unconstrained Lagrangian evolution operator for
Ld if and only if (TΥ,Υ)∗Θ
= Θ+Ld .
(ii) The map Υ is a discrete unconstrained Lagrangian evolution operator for
Ld if and only if (TΥ,Υ)∗Θ
−Θ−Ld = dLd.
(iii) If Υ is discrete unconstrained Lagrangian evolution operator then
(TΥ,Υ)∗ΩLd = ΩLd .
2.3.4. Discrete unconstrained Legendre transformations. Given a Lagrangian Ld :
Γ → R we define the discrete unconstrained Legendre transformations
F−Ld : Γ→ E∗Γ and F
+Ld : Γ→ E∗Γ by (see [27])
(F−Ld)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ (EΓ)α(h),
(F+Ld)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ (EΓ)β(g).
Now, we introduce the prolongations TΓF−Ld : TΓΓ ≡ V β ⊕Γ V α → TEΓE∗Γ and
TΓF+Ld : TΓΓ ≡ V β ⊕Γ V α→ TEΓE∗Γ by
−Ld(Xh, Yh) = (Th(i ◦ rh−1)(Xh), (ThF−Ld)(Xh) + (ThF−Ld)(Yh)),(2.18)
TΓg F
+Ld(Xg, Yg) = ((Tglg−1)(Yg), (TgF+Ld)(Xg) + (TgF+Ld)(Yg)), (2.19)
for all h, g ∈ Γ and (Xh, Yh) ∈ TΓhΓ ≡ Vhβ ⊕ Vhα and (Xg, Yg) ∈ T
g Γ ≡ Vgβ ⊕
Vgα (see [27]). We observe that the discrete Poincaré-Cartan 1-sections and 2-
section are related to the canonical Liouville section of (TEΓE∗Γ)
∗ → E∗Γ and the
canonical symplectic section of ∧2(TEΓE∗Γ)
∗ → E∗Γ by pull-back under the discrete
unconstrained Legendre transformations, that is (see [27]),
(TΓF−Ld,F−Ld)∗Θ = Θ−Ld , (T
ΓF+Ld,F+Ld)∗Θ = Θ+Ld , (2.20)
(TΓF−Ld,F−Ld)∗Ω = ΩLd , (T
ΓF+Ld,F+Ld)∗Ω = ΩLd . (2.21)
2.3.5. Discrete regular Lagrangians. A discrete Lagrangian Ld : Γ→ R is said to be
regular if the Poincaré-Cartan 2-section ΩLd is nondegenerate on the Lie algebroid
τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ (see [27]). In [27], we obtained some necessary and
sufficient conditions for a discrete Lagrangian on a Lie groupoid Γ to be regular
that we summarize as follows:
Ld is regular ⇐⇒ The Legendre transformation F+Ld is a local diffeomorphism
⇐⇒ The Legendre transformation F−Ld is a local diffeomorphism
DISCRETE NONHOLONOMIC MECHANICS 11
Locally, we deduce that Ld is regular if and only if for every g ∈ Γ and every
local basis {XA} (respectively, {YB}) of Sec(τ) on an open subset U (respectively,
V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V ) we have that the matrix
Y B(Ld))) is regular on α−1(U) ∩ β−1(V ).
Now, let Ld : Γ→ R be a discrete Lagrangian and g be a point of Γ. We define
the R-bilinear map GLdg : (EΓ)α(g) ⊕ (EΓ)β(g) → R given by
GLdg (a, b) = ΩLd(g)((−T�(α(g))(rg ◦ i)(a), 0), (0, (T�(β(g))lg)(b))). (2.22)
Then, using (2.14), we have that
Proposition 2.1. The discrete Lagrangian Ld : Γ → R is regular if and only if
GLdg is nondegenerate, for all g ∈ Γ, that is,
GLdg (a, b) = 0, for all b ∈ (EΓ)β(g) ⇒ a = 0
(respectively, GLdg (a, b) = 0, for all a ∈ (EΓ)α(g) ⇒ b = 0).
On the other hand, if Ld : Γ → R is a discrete Lagrangian on a Lie groupoid Γ
then we have that
τ∗ ◦ F−Ld = α, τ∗ ◦ F+Ld = β,
where τ∗ : E∗Γ → M is the vector bundle projection. Using these facts, (2.18) and
(2.19), we deduce the following result.
Proposition 2.2. Let Ld : Γ → R be a discrete Lagrangian function. Then, the
following conditions are equivalent:
(i) Ld is regular.
(ii) The linear map TΓhF
−Ld : Vhβ ⊕ Vhα → TEΓF−Ld(h)E
Γ is a linear isomor-
phism, for all h ∈ Γ.
(iii) The linear map TΓg F
+Ld : Vgβ ⊕ Vgα → TEΓF+Ld(g)EΓ
∗ is a linear isomor-
phism, for all g ∈ Γ.
Finally, let Ld : Γ→ R be a regular discrete Lagrangian function and (g0, h0) ∈
Γ×Γ be a solution of the discrete Euler-Lagrange equations for Ld. Then, one may
prove (see [27]) that there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0
and h0 ∈ V0, and there exists a (local) discrete unconstrained Lagrangian evolution
operator ΥLd : U0 → V0 such that:
(i) ΥLd(g0) = h0,
(ii) ΥLd is a diffeomorphism and
(iii) ΥLd is unique, that is, if U
0 is an open subset of Γ, with g0 ∈ U ′0, and
Υ′Ld : U
0 → Γ is a (local) discrete Lagrangian evolution operator then
ΥLd|U0∩U ′0 = Υ
Ld|U0∩U ′0
3. Discrete Nonholonomic (or constrained) Lagrangian systems on
Lie groupoids
3.1. Discrete Generalized Hölder’s principle. Let Γ be a Lie groupoid with
structural maps
α, β : Γ→M, � : M → Γ, i : Γ→ Γ, m : Γ2 → Γ.
Denote by τ : EΓ → M the Lie algebroid associated to Γ. Suppose that the rank
of EΓ is n and that the dimension of M is m.
A generalized discrete nonholonomic (or constrained) Lagrangian system on Γ
is determined by:
12 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
- a regular discrete Lagrangian Ld : Γ −→ R,
- a constraint distribution, Dc, which is a vector subbundle of the bundle
EΓ → M of admissible directions. We will denote by τDc : Dc → M the
vector bundle projection and by iDc : Dc → EΓ the canonical inclusion.
- a discrete constraint embedded submanifold Mc of Γ, such that
dimMc = dimDc = m + r, with r ≤ n. We will denote by iMc : Mc → Γ
the canonical inclusion.
Remark 3.1. Let Ld : Γ→ R be a regular discrete Lagrangian on a Lie groupoid
Γ and Mc be a submanifold of Γ such that �(M) ⊆ Mc. Then, dimMc = m + r,
with 0 ≤ r ≤ m. Moreover, for every x ∈M , we may introduce the subspace Dc(x)
of EΓ(x) given by
Dc(x) = T�(x)Mc ∩ EΓ(x).
Since the linear map T�(x)α : T�(x)Mc → TxM is an epimorphism, we deduce that
dimDc(x) = r. In fact, Dc =
x∈M Dc(x) is a vector subbundle of EΓ (over M) of
rank r. Thus, we may consider the discrete nonholonomic system (Ld,Mc,Dc) on
the Lie groupoid Γ. �
For g ∈ Γ fixed, we consider the following set of admissible sequences of order
CNg =
(g1, . . . , gN ) ∈ ΓN
∣∣ (gk, gk+1) ∈ Γ2, for k = 1, .., N − 1 and g1 . . . gN = g } .
Given a tangent vector at (g1, . . . , gN ) to the manifold CNg , we may write it as the
tangent vector at t = 0 of a curve in CNg , t ∈ (−ε, ε) ⊆ R −→ c(t) which passes
through (g1, . . . , gN ) at t = 0. This type of curves is of the form
c(t) = (g1h1(t), h
1 (t)g2h2(t), . . . , h
N−2(t)gN−1hN−1(t), h
N−1(t)gN )
where hk(t) ∈ α−1(β(gk)), for all t, and hk(0) = �(β(gk)) for k = 1, . . . , N − 1.
Therefore, we may identify the tangent space to CNg at (g1, . . . , gN ) with
T(g1,g2,..,gN )C
g ≡ { (v1, v2, . . . , vN−1) | vk ∈ (EΓ)xk and xk = β(gk), 1 ≤ k ≤ N − 1 } .
Observe that each vk is the tangent vector to the curve hk at t = 0.
The curve c is called a variation of (g1, . . . , gN ) and (v1, v2, . . . , vN−1) is called
an infinitesimal variation of (g1, . . . , gN ).
Now, we define the discrete action sum associated to the discrete Lagrangian
Ld : Γ −→ R as
SLd : CNg −→ R
(g1, . . . , gN ) 7−→
Ld(gk).
We define the variation δSLd : T(g1,...,gN )C
g → R as
δSLd(v1, . . . , vN−1) =
SLd(c(t))
Ld(g1h1(t)) + Ld(h
1 (t)g2h2(t))
+ . . .+ Ld(h
N−2(t)gN−1hN−1(t)) + Ld(h
N−1(t)gN )
do(Ld ◦ lgk)(�(xk))(vk) + d
o(Ld ◦ rgk+1 ◦ i)(�(xk))(vk)
where do is the standard differential on Γ, i.e., do is the differential of the Lie
algebroid τΓ : TΓ → Γ. It is obvious from the last expression that the definition
DISCRETE NONHOLONOMIC MECHANICS 13
of variation δSLd does not depend on the choice of variations c of the sequence g
whose infinitesimal variation is (v1, . . . , vN−1).
Next, we will introduce the subset (Vc)g of T(g1,...,gN )C
g defined by
(Vc)g =
(v1, . . . , vN−1) ∈ T(g1,...,gN )C
∣∣ ∀k ∈ {1, . . . , N − 1}, vk ∈ Dc } .
Then, we will say that a sequence in CNg satisfying the constraints determined
by Mc is a Hölder-critical point of the discrete action sum SLd if the restriction
of δSLd to (Vc)g vanishes, i.e.
(Vc)g
Definition 3.2 (Discrete Hölder’s principle). Given g ∈ Γ, a sequence (g1, . . . , gN )
∈ CNg such that gk ∈ Mc, 1 ≤ k ≤ N , is a solution of the discrete nonholo-
nomic Lagrangian system determined by (Ld,Mc,Dc) if and only if (g1, . . . , gN ) is
a Hölder-critical point of SLd.
If (g1, . . . , gN ) ∈ CNg ∩ (Mc × · · · ×Mc) then (g1, . . . , gN ) is a solution of the
nonholonomic discrete Lagrangian system if and only if
(do(Ld ◦ lgk) + d
o(Ld ◦ rgk+1 ◦ i))(�(xk))|(Dc)xk = 0,
where β(gk) = α(gk+1) = xk. For N = 2, we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc)
(with β(g) = α(h) = x) is a solution if
do(Ld ◦ lg + Ld ◦ rh ◦ i)(�(x))|(Dc)x = 0.
These equations will be called the discrete nonholonomic Euler-Lagrange
equations for the system (Ld,Mc,Dc).
Let (g1, . . . , gN ) be an element of CNg . Suppose that β(gk) = α(gk+1) = xk,
1 ≤ k ≤ N − 1, and that {XAk} = {Xak, Xαk} is a local adapted basis of Sec(τ)
on an open subset Uk of M , with xk ∈ Uk. Here, {Xak}1≤a≤r is a local basis of
Sec(τDc) and, thus, {Xαk}r+1≤α≤n is a local basis of the space of sections of the
vector subbundle τD0c : D
c →M , where D0c is the annihilator of Dc and {Xak, Xαk}
is the dual basis of {Xak, Xαk}. Then, the sequence (g1, . . . , gN ) is a solution of
the discrete nonholonomic equations if (g1, . . . , gN ) ∈Mc×· · ·×Mc and it satisfies
the following closed system of difference equations
gk)(Ld)−
gk+1)(Ld)
〈dLd, (Xak)(0,1)〉(gk)− 〈dLd, (Xak)(1,0)〉(gk+1)
for 1 ≤ a ≤ r, d being the differential of the Lie algebroid πτ : TΓΓ ≡ V β⊕ΓV α −→
Γ. For N = 2 we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc) (with β(g) = α(h) = x) is a
solution if
Xa(g)(Ld)−
Xa(h)(Ld) = 0
where {Xa} is a local basis of Sec(τDc) on an open subset U of M such that x ∈ U .
Next, we describe an alternative version of these difference equations. First
observe that using the Lagrange multipliers the discrete nonholonomic equations
are rewritten as
do [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))(v) = λαXα(x)(v),
14 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
for v ∈ (EΓ)x, with (g, h) ∈ Γ2 ∩ (Mc ×Mc) and β(g) = α(h) = x. Here, {Xα} is
a local basis of sections of the annihilator D0c .
Thus, the discrete nonholonomic equations are:
Y (g)(Ld)−
Y (h)(Ld) = λα(X
α)(Y )|β(g), (g, h) ∈ Γ2 ∩ (Mc ×Mc),
for all Y ∈ Sec(τ) or, alternatively,
〈dLd − λα(Xα)(0,1), Y (0,1)〉(g)− 〈dLd, Y (1,0)〉(h) = 0, (g, h) ∈ Γ2 ∩ (Mc ×Mc),
for all Y ∈ Sec(τ).
On the other hand, we may define the discrete nonholonomic Euler-Lagrange
operator DDEL(Ld,Mc,Dc) : Γ2 ∩ (Mc ×Mc)→ D∗c as follows
DDEL(Ld,Mc,Dc)(g, h) = d
o [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))|(Dc)x ,
for (g, h) ∈ Γ2 ∩ (Mc ×Mc), with β(g) = α(h) = x ∈M .
Then, we may characterize the solutions of the discrete nonholonomic equations
as the sequences (g1, . . . , gN ), with (gk, gk+1) ∈ Γ2 ∩ (Mc × Mc), for each k ∈
{1, . . . , N − 1}, and
DDEL(Ld,Mc,Dc)(gk, gk+1) = 0.
Remark 3.3. (i) The set Γ2 ∩ (Mc ×Mc) is not, in general, a submanifold
of Mc ×Mc.
(ii) Suppose that αMc : Mc → M and βMc : Mc → M are the restrictions
to Mc of α : Γ → M and β : Γ → M , respectively. If αMc and βMc are
submersions then Γ2∩(Mc×Mc) is a submanifold of Mc×Mc of dimension
m+ 2r.
3.2. Discrete Nonholonomic Legendre transformations. Let (Ld,Mc,Dc) be
a discrete nonholonomic Lagrangian system. We define the discrete nonholo-
nomic Legendre transformations
F−(Ld,Mc,Dc) : Mc → D∗c and F
+(Ld,Mc,Dc) : Mc → D∗c
as follows:
F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ Dc(α(h)),(3.1)
F+(Ld,Mc,Dc)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ Dc(β(g)).(3.2)
If F−Ld : Γ→ E∗Γ and F
+Ld : Γ→ E∗Γ are the standard Legendre transformations
associated with the Lagrangian function Ld and i∗Dc : E
Γ → D
c is the dual map of
the canonical inclusion iDc : Dc → EΓ then
F−(Ld,Mc,Dc) = i∗Dc ◦ F
−Ld ◦ iMc , F
+(Ld,Mc,Dc) = i
◦ F+Ld ◦ iMc . (3.3)
Remark 3.4. (i) Note that
τ∗Dc ◦ F
−(Ld,Mc,Dc) = αMc , τ
◦ F+(Ld,Mc,Dc) = βMc . (3.4)
(ii) If DDEL(Ld,Mc,Dc) is the discrete nonholonomic Euler-Lagrange opera-
tor then
DDEL(Ld,Mc,Dc)(g, h) = F+(Ld,Mc,Dc)(g)− F−(Ld,Mc,Dc)(h),
for (g, h) ∈ Γ2 ∩ (Mc ×Mc).
DISCRETE NONHOLONOMIC MECHANICS 15
On the other hand, since by assumption Ld : Γ → R is a regular discrete La-
grangian function, we have that the discrete Poincaré-Cartan 2-section ΩLd is sym-
plectic on the Lie algebroid τ̃Γ : TΓΓ → Γ. Moreover, the regularity of L is equiv-
alent to the fact that the Legendre transformations F−Ld and F+Ld to be local
diffeomorphisms (see Subsection 2.3.5).
Next, we will obtain necessary and sufficient conditions for the discrete non-
holonomic Legendre transformations associated with the system (Ld,Mc,Dc) to be
local diffeomorphisms.
Let F be the vector subbundle (over Γ) of τ̃Γ : TΓΓ→ Γ whose fiber at the point
h ∈ Γ is
(1,0)
∣∣∣ γ ∈ Dc(α(h))0 }0 ⊆ TΓhΓ.
In other words,
F 0h =
(1,0)
∣∣∣ γ ∈ Dc(α(h))0 } .
Note that the rank of F is n+ r.
We also consider the vector subbundle F̄ (over Γ) of τ̃Γ : TΓΓ→ Γ of rank n+ r
whose fiber at the point g ∈ Γ is
F̄g =
γ(0,1)g
∣∣∣ γ ∈ Dc(β(g))0 }0 ⊆ TΓg Γ.
Lemma 3.5. F (respectively, F̄ ) is a coisotropic vector subbundle of the symplectic
vector bundle (TΓΓ,ΩLd), that is,
F⊥h ⊆ Fh, for every h ∈ Γ
(respectively, F̄⊥g ⊆ F̄g, for every g ∈ Γ), where F⊥h (respectively, F̄
g ) is the
symplectic orthogonal of Fh (respectively, F̄g) in the symplectic vector space (TΓhΓ,
ΩLd(h)) (respectively, (T
g Γ,ΩLd(g))).
Proof. If h ∈ Γ we have that
F⊥h = [
ΩLd (h)
(F 0h ),
[ΩLd (h)
: TΓhΓ→ (T
∗ being the canonical isomorphism induced by the symplectic
form ΩLd(h). Thus, using (2.14), we deduce that
F⊥h =
ΩLd (h)
(γ(1,0)h )
∣∣∣ γ ∈ Dc(α(h))0 } ⊆ {0} ⊕ Vhα ⊆ Fh.
The coisotropic character of F̄g is proved in a similar way. �
We also have the following result
Lemma 3.6. Let TΓF−Ld : TΓΓ→ TEΓE∗Γ (respectively, T
ΓF+Ld : TΓΓ→ TEΓE∗Γ)
be the prolongation of the Legendre transformation F−Ld : Γ → E∗Γ (respectively,
F+Ld : Γ→ E∗Γ). Then,
(TΓhF
−Ld)(Fh) = T
F−Ld(h)
E∗Γ =
(vα(h), XF−Ld(h)) ∈ T
F−Ld(h)
∣∣∣ vα(h) ∈ Dc(α(h))} ,
for h ∈Mc (respectively,
(TΓg F
+Ld)(F̄g) = T
F+Ld(g)
E∗Γ =
(vβ(g), XF+Ld(g)) ∈ T
F+Ld(g)
∣∣∣ vβ(g) ∈ Dc(β(g))} ,
for g ∈Mc).
Proof. It follows using (2.11), (2.18) (respectively, (2.12), (2.19)) and Proposition
2.2. �
Now, we may prove the following theorem.
16 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Theorem 3.7. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re-
spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism.
(ii) For every h ∈Mc (respectively, g ∈Mc)
ΓΓ)−1(ThMc) ∩ F⊥h = {0} (3.5)
(respectively, (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}).
Proof. (i) ⇒ (ii) If h ∈ Mc and (Xh, Yh) ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h then, using the
fact that F⊥h ⊆ {0} ⊕ Vhα (see the proof of Lemma 3.5), we have that Xh = 0.
Therefore,
Yh ∈ Vhα ∩ ThMc. (3.6)
Next, we will see that
(ThF−(Ld,Mc,Dc))(Yh) = 0. (3.7)
From (3.4) and (3.6), it follows that (ThF−(Ld,Mc,Dc))(Yh) is vertical with
respect to the projection τ∗Dc : D
c →M .
Thus, it is sufficient to prove that
((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = 0, for all Z ∈ Sec(τDc).
Here, Ẑ : D∗c → R is the linear function on D∗c induced by the section Z.
Now, using (3.3), we deduce that
((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = d(Ẑ ◦ i∗Dc)((F
−Ld)(h))(0, (ThF−Ld)(Yh)),
where d is the differential of the Lie algebroid τ τ
: TEΓE∗Γ → E
Consequently, if Z∗c : E∗Γ → T
EΓE∗Γ is the complete lift of Z ∈ Sec(τ), we have
that (see (2.10)),
((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = Ω(F−Ld(h))(Z∗c(F−Ld(h)),
(0, (ThF−Ld)(Yh)),
(3.8)
Ω being the canonical symplectic section associated with the Lie algebroid EΓ.
On the other hand, since Z ∈ Sec(τDc), it follows that Z∗c(F−Ld(h)) is in
F−Ld(h)
E∗Γ and, from Lemma 3.6, we conclude that there exists (X
h) ∈ Fh
such that
(TΓhF
−Ld)(X
h) = Z
∗c((F−Ld)(h)). (3.9)
Moreover, using (2.18), we obtain that
(TΓhF
−Ld)(0, Yh) = (0, (ThF−Ld)(Yh)). (3.10)
Thus, from (2.21), (3.8), (3.9) and (3.10), we deduce that
((ThF−(Ld,M,Dc))(Yh))(Ẑ) = −ΩLd(h)((0, Yh), (X
Therefore, since (0, Yh) ∈ F⊥h , it follows that (3.7) holds, which implies that Yh = 0.
This proves that (ρT
ΓΓ)−1(ThMc) ∩ F⊥h = {0}.
If F+(Ld,Mc,Dc) is a local diffeomorphism then, proceeding as above, we have
that (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc.
(ii) ⇒ (i) Suppose that h ∈Mc and that Yh is a tangent vector to Mc at h such
(ThF−(Ld,Mc,Dc))(Yh) = 0. (3.11)
DISCRETE NONHOLONOMIC MECHANICS 17
We have that (Thα)(Yh) = 0 and, thus,
(0, Yh) ∈ (ρT
ΓΓ)−1(ThMc).
We will see that (0, Yh) ∈ F⊥h , that is,
ΩLd(h)((0, Yh), (X
h)) = 0, for (X
h) ∈ Fh. (3.12)
Now, using (2.18) and (2.21), we deduce that
ΩLd(h)((0, Yh), (X
h)) = Ω(F
−Ld(h))((0, (ThF−Ld)(Yh)), (TΓhF
−Ld)(X
Therefore, from Lemma 3.6, we obtain that
ΩLd(h)((0, Yh), (X
h)) = Ω(F
−Ld(h))(0, (ThF−Ld)(Yh)), (vα(h), YF−Ld(h)))
with (vα(h), YF−Ld(h)) ∈ T
F−Ld(h)
Next, we take a section Z ∈ Sec(τDc) such that Z(α(h)) = vα(h). Then (see
(2.9)),
(vα(h), YF−Ld(h)) = Z
∗c(F−Ld(h)) + (0, Y ′F−Ld(h)),
where Y ′F−Ld(h) ∈ TF−Ld(h)E
Γ and Y
F−Ld(h)
is vertical with respect to the projection
τ∗ : E∗Γ →M .
Thus, since (see Eq. (3.7) in [23])
Ω(F−Ld(h))((0, (ThF−Ld)(Yh)), (0, Y ′F−Ld(h))) = 0,
we have that
ΩLd(h)((0, Yh), (X
h)) = −Ω(F
−Ld(h))(Z∗c(F−Ld(h)), (0, (ThF−Ld)(Yh)))
= −d(Ẑ ◦ i∗Dc)(F
−Ld(h))(0, (ThF−Ld)(Yh))
and, from (3.11), we deduce that (3.12) holds.
This proves that Yh ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h which implies that Yh = 0.
Therefore, F−(Ld,Mc,Dc) is a local diffeomorphism.
If (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0} for all g ∈ Mc then, proceeding as above, we
obtain that F+(Ld,Mc,Dc) is a local diffeomorphism. �
Now, let ρT
ΓΓ : TΓΓ→ TΓ be the anchor map of the Lie algebroid πτ : TΓΓ→ Γ.
Then, we will denote by Hh the subspace of TΓhΓ given by
Hh = (ρ
TΓΓ)−1(ThMc) ∩ Fh, for h ∈Mc.
In a similar way, for every g ∈Mc we will introduce the subspace H̄g of TΓg Γ defined
H̄g = (ρ
TΓΓ)−1(TgMc) ∩ F̄g.
On the other hand, let h be a point of Mc and G
h : (EΓ)α(h) ⊕ (EΓ)β(h) → R
be the R-bilinear map given by (2.22). We will denote by (
h the subspace of
(EΓ)β(h) defined by
b ∈ (EΓ)β(h)
∣∣ (T�(β(h))lh)(b) ∈ ThMc }
and by GLdch : (Dc)α(h)× (
h → R the restriction to (Dc)α(h)× (
h of the
R-bilinear map GLdh .
In a similar way, if g is a point of Γ we will consider the subspace (
E Γ)Mcg of
(EΓ)α(g) defined by
a ∈ (EΓ)α(g)
∣∣ (T�(α(g))(rg ◦ i))(a) ∈ TgMc }
18 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
and the restriction ḠLdcg : (
E Γ)Mcg × (Dc)β(g) → R of GLdg to the space (
E Γ)Mcg ×
(Dc)β(g).
Proposition 3.8. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) For every h ∈Mc (respectively, g ∈Mc)
ΓΓ)−1(ThMc) ∩ F⊥h = {0}
(respectively, (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}).
(ii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub-
space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle
H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener-
(iii) For every h ∈Mc (respectively, g ∈Mc){
b ∈ (
∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0}
(respectively,
a ∈ (
E Γ)Mcg
∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}).
Proof. (i) ⇒ (ii) Assume that h ∈Mc and that
ΓΓ)−1(ThMc) ∩ F⊥h = {0}. (3.13)
Let U be an open subset of Γ, with h ∈ U , and {φγ}γ=1,...,n−r a set of independent
real C∞-functions on U such that
Mc ∩ U = {h′ ∈ U | φγ(h′) = 0, for all γ } .
If d is the differential of the Lie algebroid τ̃Γ : TΓΓ → Γ then it is easy to prove
ΓΓ)−1(ThMc) =< {dφγ(h)} >0 .
Thus,
dim((ρT
ΓΓ)−1(ThMc)) ≥ n+ r. (3.14)
On the other hand, dimF⊥h = n−r. Therefore, from (3.13) and (3.14), we obtain
dim((ρT
ΓΓ)−1(ThMc)) = n+ r
TΓhΓ = (ρ
TΓΓ)−1(ThMc)⊕ F⊥h .
Consequently, using Lemma 3.5, we deduce that
Fh = Hh ⊕ F⊥h . (3.15)
This implies that dimHh = 2r. Moreover, from (3.15), we also get that
Hh ∩H⊥h ⊆ Hh ∩ F
and, since Hh ∩ F⊥h = (ρ
TΓΓ)−1(ThMc) ∩ F⊥h (see Lemma 3.5), it follows that
Hh ∩H⊥h = {0}.
Thus, we have proved that Hh is a symplectic subspace of the symplectic vector
space (TΓhΓ,ΩLd(h)).
If (ρT
ΓΓ)−1(TgMc)∩ F̄⊥g = {0}, for all g ∈Mc then, proceeding as above, we ob-
tain that H̄g is a symplectic subspace of the symplectic vector space (TΓg Γ,ΩLd(g)),
for all g ∈Mc.
(ii) ⇒ (i) Suppose that h ∈ Mc and that Hh is a symplectic subspace of the
symplectic vector space (TΓhΓ,ΩLd(h)).
DISCRETE NONHOLONOMIC MECHANICS 19
If (Xh, Yh) ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h then, using Lemma 3.5, we deduce that
(Xh, Yh) ∈ Hh.
Now, if (X ′h, Y
h) ∈ Hh then, since (Xh, Yh) ∈ F
h , we conclude that
ΩLd(h)((Xh, Yh), (X
h)) = 0.
This implies that
(Xh, Yh) ∈ Hh ∩H⊥h = {0}.
Therefore, we have proved that (ρT
ΓΓ)−1(ThMc) ∩ F⊥h = {0}.
If H̄g ∩ H̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we obtain that
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc.
(i) ⇒ (iii) Assume that
ΓΓ)−1(ThMc) ∩ F⊥h = {0}
and that b ∈ (
h satisfies the following condition
h (a, b) = 0, ∀a ∈ (Dc)α(h).
Then, Yh = (T�(β(h))lh)(b) ∈ ThMc ∩ Vhα and (0, Yh) ∈ (ρT
ΓΓ)−1(ThMc).
Moreover, if (X ′h, Y
h) ∈ Fh, we have that
X ′h = −(T�(α(h))(rh ◦ i))(a), with a ∈ (Dc)α(h).
Thus, using (2.14) and (2.22), we deduce that
ΩLd(h)((X
h), (0, Yh)) = ΩLd(h)((X
h, 0), (0, Yh)) = G
h (a, b) = 0.
Therefore,
(0, Yh) ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h = {0},
which implies that b = 0.
If (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we
obtain that{
a ∈ (
∣∣∣ GLdcg (a, b) = 0, for all b ∈ (Dc)β(g) } = {0}.
(iii) ⇒ (i) Suppose that h ∈Mc, that{
b ∈ (
∣∣∣ GLdh (a, b) = 0, ∀a ∈ (Dc)α(h) } = {0}
and let (Xh, Yh) be an element of the set (ρT
ΓΓ)−1(ThMc) ∩ F⊥h .
Then (see the proof of Lemma 3.5), Xh = 0 and Yh ∈ ThMc∩Vhα. Consequently,
Yh = (T�(β(h)lh)(b), with b ∈ (
Now, if a ∈ (Dc)α(h), we have that
X ′h = (T�(α(h))(rh ◦ i))(a) ∈ Vhβ and (X
h, 0) ∈ Fh.
Thus, from (2.22) and since (0, Yh) ∈ F⊥h , it follows that
h (a, b) = ΩLd(h)((X
h, 0)(0, Yh)) = 0.
Therefore, b = 0 which implies that Yh = 0.
a ∈ (
E Γ)Mcg
∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}, for all g ∈ Mc, then
proceeding as above we obtain that (ρT
ΓΓ)−1(TgMc)∩F̄⊥g = {0}, for all g ∈Mc. �
Using Theorem 3.7 and Proposition 3.8, we conclude
20 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Theorem 3.9. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re-
spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism.
(ii) For every h ∈Mc (respectively, g ∈Mc)
ΓΓ)−1(ThMc) ∩ F⊥h = {0}
(respectively, (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}).
(iii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub-
space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle
H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener-
(iv) For every h ∈Mc (respectively, g ∈Mc){
b ∈ (
∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0}
(respectively,
a ∈ (
E Γ)Mcg
∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}).
3.3. Nonholonomic evolution operators and regular discrete nonholo-
nomic Lagrangian systems. First of all, we will introduce the definition of a
nonholonomic evolution operator.
Definition 3.10. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system
and Υnh : Mc →Mc be a differentiable map. Υnh is said to be a discrete nonholo-
nomic evolution operator for (Ld,Mc,Dc) if:
(i) graph(Υnh) ⊆ Γ2, that is, (g,Υnh(g)) ∈ Γ2, for all g ∈Mc and
(ii) (g,Υnh(g)) is a solution of the discrete nonholonomic equations, for all
g ∈Mc, that is,
do(Ld ◦ lg + Ld ◦ rΥnh(g) ◦ i)(�(β(g)))|Dc(β(g)) = 0, for all g ∈Mc.
Remark 3.11. If Υnh : Mc → Mc is a differentiable map then, from (3.1), (3.2)
and (3.4), we deduce that Υnh is a discrete nonholonomic evolution operator for
(Ld,Mc,Dc) if and only if
F−(Ld,Mc,Dc) ◦Υnh = F+(Ld,Mc,Dc).
Now, we will introduce the notion of a regular discrete nonholonomic Lagrangian
system.
Definition 3.12. A discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is said
to be regular if the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc)
and F+(Ld,Mc,Dc) are local diffeomorphims.
From Theorem 3.9, we deduce
Corollary 3.13. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) The system (Ld,Mc,Dc) is regular.
(ii) The following relations hold
ΓΓ)−1(ThMc) ∩ F⊥h = {0}, for all h ∈Mc,
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc.
DISCRETE NONHOLONOMIC MECHANICS 21
(iii) H and H̄ are symplectic subbundles of rank 2r of the symplectic vector
bundle (TΓMcΓ,ΩLd).
(iv) If g and h are points of Mc then the R-bilinear maps GLdch and Ḡ
g are
right and left nondegenerate, respectively.
The map GLdch (respectively, Ḡ
g ) is right nondegenerate (respectively, left non-
degenerate) if
h (a, b) = 0,∀a ∈ (Dc)α(h) ⇒ b = 0
(respectively, ḠLdcg (a, b) = 0,∀b ∈ (Dc)β(g) ⇒ a = 0).
Every solution of the discrete nonholonomic equations for a regular discrete
nonholonomic Lagrangian system determines a unique local discrete nonholonomic
evolution operator. More precisely, we may prove the following result:
Theorem 3.14. Let (Ld,Mc,Dc) be a regular discrete nonholonomic Lagrangian
system and (g0, h0) ∈Mc×Mc be a solution of the discrete nonholonomic equations
for (Ld,Mc,Dc). Then, there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0
and h0 ∈ V0, and there exists a local discrete nonholonomic evolution operator
Υ(Ld,Mc,Dc)nh : U0 ∩Mc → V0 ∩Mc such that:
(i) Υ(Ld,Mc,Dc)nh (g0) = h0;
(ii) Υ(Ld,Mc,Dc)nh is a diffeomorphism and
(iii) Υ(Ld,Mc,Dc)nh is unique, that is, if U
0 is an open subset of Γ, with g0 ∈ U ′0,
and Υnh : U ′0 ∩ Mc → Mc is a (local) discrete nonholonomic evolution
operator then
(Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc = (Υnh)|U0∩U ′0∩Mc .
Proof. From remark 3.4, we deduce that
F+(Ld,Mc,Dc)(g0) = F−(Ld,Mc,Dc)(h0) = µ0 ∈ D∗c .
Thus, we can choose two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0,
and an open subset W0 of E∗Γ such that µ0 ∈W0 and
F+(Ld,Mc,Dc) : U0 ∩Mc →W0 ∩D∗c , F
−(Ld,Mc,Dc) : V0 ∩Mc →W0 ∩D∗c
are diffeomorphisms. Therefore, from Remark 3.11, we deduce that
Υ(Ld,Mc,Dc)nh = (F
−(Ld,Mc,Dc)
−1 ◦ F+(Ld,Mc,Dc))|U0∩Mc : U0 ∩Mc → V0 ∩Mc
is a (local) discrete nonholonomic evolution operator. Moreover, it is clear that
Υ(Ld,Mc,Dc)nh (g0) = h0 and it follows that Υ
(Ld,Mc,Dc)
nh is a diffeomorphism.
Finally, if U ′0 is an open subset of Γ, with g0 ∈ U ′0, and Υnh : U ′0 ∩Mc → Mc
is another (local) discrete nonholonomic evolution operator then (Υnh)|U0∩U ′0∩Mc
is also a (local) discrete nonholonomic evolution operator. Consequently, from
Remark 3.11, we conclude that
(Υnh)|U0∩U ′0∩Mc = [F
−(Ld,Mc,Dc)−1 ◦ F+(Ld,Mc,Dc)]|U0∩U ′0∩Mc
= (Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc .
22 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
3.4. Reversible discrete nonholonomic Lagrangian systems. Let (Ld,Mc,
Dc) be a discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M .
Following the terminology used in [36] for the particular case when Γ is the pair
groupoid M ×M , we will introduce the following definition
Definition 3.15. The discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is
said to be reversible if
Ld ◦ i = Ld, i(Mc) = Mc,
i : Γ→ Γ being the inversion of the Lie groupoid Γ.
For a reversible discrete nonholonomic Lagrangian system we have the following
result:
Proposition 3.16. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys-
tem on a Lie groupoid Γ. Then, the following conditions are equivalent:
(i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) is a
local diffeomorphism.
(ii) The discrete nonholonomic Legendre transformation F+(Ld,Mc,Dc) is a
local diffeomorphism.
Proof. If h ∈Mc then, using (3.1) and the fact that Ld ◦ i = Ld, it follows that
F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ l−1h )
for v�(α(h)) ∈ (Dc)α(h). Thus, from (3.2), we obtain that
F−(Ld,Mc,Dc)(h)(v�(α(h))) = −F+(Ld,Mc,Dc)(h−1)(v�(β(h−1))).
This implies that
F+(Ld,Mc,Dc) = −F−(Ld,Mc,Dc) ◦ i.
Therefore, since the inversion is a diffeomorphism (in fact, we have that i2 = id),
we deduce the result �
Using Theorem 3.9, Definition 3.12 and Proposition 3.16, we prove the following
corollaries.
Corollary 3.17. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system
on a Lie groupoid Γ. Then, the following conditions are equivalent:
(i) The system (Ld,Mc,Dc) is regular.
(ii) For all h ∈Mc,
ΓΓ)−1(ThMc) ∩ F⊥h = {0}.
(iii) H = (ρT
ΓΓ)−1(TMc)∩F is a symplectic subbundle of the symplectic vector
bundle (TΓMcΓ,ΩLd).
(iv) The R-bilinear map GLdch : (
h ×(Dc)α(h) → R is right nondegenerate,
for all h ∈Mc.
Corollary 3.18. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system
on a Lie groupoid Γ. Then, the following conditions are equivalent:
(i) The system (Ld,Mc,Dc) is regular.
(ii) For all g ∈Mc,
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}.
(iii) H̄ = (ρT
ΓΓ)−1(TMc)∩ F̄ is a symplectic subbundle of the symplectic vector
bundle (TΓMcΓ,ΩLd).
DISCRETE NONHOLONOMIC MECHANICS 23
(iv) The R-bilinear map ḠLdcg : (Dc)β(g) × (
E Γ)Mcg → R is left nondegenerate,
for all g ∈Mc.
Next, we will prove that a reversible nonholonomic Lagrangian system is dynam-
ically reversible.
Proposition 3.19. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys-
tem on a Lie groupoid Γ and (g, h) be a solution of the discrete nonholonomic Euler-
Lagrange equations for (Ld,Mc,Dc). Then, (h−1, g−1) is also a solution of these
equations. In particular, if the system (Ld,Mc,Dc) is regular and Υ
(Ld,Mc,Dc)
nh is the
(local) discrete nonholonomic evolution operator for (Ld,Mc,Dc) then Υ
(Ld,Mc,Dc)
is reversible, that is,
Υ(Ld,Mc,Dc)nh ◦ i ◦Υ
(Ld,Mc,Dc)
nh = i.
Proof. Using that i(Mc) = Mc, we deduce that
(h−1, g−1) ∈ Γ2 ∩ (Mc ×Mc).
Now, suppose that β(g) = α(h) = x and that v ∈ (Dc)x. Then, since Ld ◦ i = Ld,
it follows that
do[Ld ◦ lh−1 + Ld ◦ rg−1 ◦ i](ε(x))(v) = v(Ld ◦ i ◦ rh ◦ i) + v(Ld ◦ i ◦ lg)
= v(Ld ◦ lg) + v(Ld ◦ rh ◦ i) = 0.
Thus, we conclude that (h−1, g−1) is a solution of the discrete nonholonomic
Euler-Lagrange equations for (Ld,Mc,Dc).
If the system (Ld,Mc,Dc) is regular and g ∈Mc, we have that (g,Υ
(Ld,M,Dc)
nh (g))
is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc).
Therefore, (i(Υ(Ld,M,Dc)nh (g)), i(g)) is also a solution of the dynamical equations
which implies that
Υ(Ld,M,Dc)nh (i(Υ
(Ld,M,Dc)
nh (g))) = i(g).
Remark 3.20. Proposition 3.19 was proved in [36] for the particular case when Γ
is the pair groupoid. �
3.5. Lie groupoid morphisms and reduction. Let (Φ,Φ0) be a Lie groupoid
morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′.
Denote by (E(Φ),Φ0) the corresponding morphism between the Lie algebroids
EΓ and EΓ′ of Γ and Γ′, respectively (see Section 2.2).
If Ld : Γ→ R and L′d : Γ
′ → R are discrete Lagrangians on Γ and Γ′ such that
Ld = L
d ◦ Φ
then, using Theorem 4.6 in [27], we have that
(DDELLd)(g, h)(v) = (DDELL
d)(Φ(g),Φ(h))(Ex(Φ)(v))
for (g, h) ∈ Γ2 and v ∈ (EΓ)x, where x = β(g) = α(h) ∈M.
Using this fact, we deduce the following result:
Corollary 3.21. Let (Φ,Φ0) be a Lie groupoid morphism between the Lie groupoids
Γ ⇒ M and Γ′ ⇒ M ′. Suppose that L′d : Γ
′ → R is a discrete Lagrangian on Γ′,
that (Ld = L′d ◦Φ,Mc,Dc) is a discrete nonholonomic Lagrangian system on Γ and
that (g, h) ∈ Γ2 ∩ (Mc ×Mc). Then:
24 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
(i) The pair (g, h) is a solution of the discrete nonholonomic problem (Ld,Mc,
Dc) if and only if (DDELL′d)(Φ(g),Φ(h)) vanishes over the set
(Eβ(g)Φ)((Dc)β(g)).
(ii) If (L′d,M
c) is a discrete nonholonomic Lagrangian system on Γ
′ such
that (Φ(g),Φ(h)) ∈M′c×M′c and (Eβ(g)(Φ))((Dc)β(g)) = (D′c)Φ0(β(g)) then
(g, h) is a solution for the discrete nonholonomic problem (Ld,Mc,Dc) if
and only if (Φ(g),Φ(h)) is a solution for the discrete nonholonomic problem
(L′d,M
3.6. Discrete nonholonomic Hamiltonian evolution operator. Let (Ld,Mc,
Dc) a regular discrete nonholonomic system. Assume, without the loss of gener-
ality, that the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc) :
Mc −→ D∗c and F+(Ld,Mc,Dc) : Mc −→ D∗c are global diffeomorphisms. Then,
(Ld,Mc,Dc)
nh = F
−(Ld,Mc,Dc)−1◦F+(Ld,Mc,Dc) is the discrete nonholonomic evo-
lution operator and one may define the discrete nonholonomic Hamiltonian
evolution operator, γ̃nh : D∗c → D∗c , by
γ̃nh = F+(Ld,Mc,Dc) ◦ γ
(Ld,Mc,Dc)
nh ◦ F
+(Ld,Mc,Dc)
−1 . (3.16)
From Remark 3.11, we have the following alternative definitions
γ̃nh = F−(Ld,Mc,Dc) ◦ γ
(Ld,Mc,Dc)
nh ◦ F
−(Ld,Mc,Dc)
γ̃nh = F+(Ld,Mc,Dc) ◦ F−(Ld,Mc,Dc)−1
of the discrete Hamiltonian evolution operator. The following commutative diagram
illustrates the situation
Mc Mc
(Ld,Mc,Dc)
D∗c D
F−(Ld, Mc, Dc)
F+(Ld, Mc, Dc)
F−(Ld, Mc, Dc)
F+(Ld, Mc, Dc)
γ̃nh γ̃nh
Remark 3.22. The discrete nonholonomic evolution operator is an application
from D∗c to itself. It is remarkable that D
c is also the appropriate nonholonomic
momentum space for a continuous nonholonomic system defined by a Lagrangian
L : EΓ → R and the constraint distribution Dc. Therefore, in the regular case, the
solution of the continuous nonholonomic Lagrangian system also determines a flow
from D∗c to itself. We consider that this would be a good starting point to compare
the discrete and continuous dynamics and eventually to establish a backward error
analysis for nonholonomic systems. �
3.7. The discrete nonholonomic momentum map. Let (Ld,Mc,Dc) be a reg-
ular discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M and
τ : EΓ →M be the Lie algebroid of Γ.
Suppose that g is a Lie algebra and that Ψ : g→ Sec(τ) is a R-linear map. Then,
for each x ∈M, we consider the vector subspace gx of g given by
gx = { ξ ∈ g | Ψ(ξ)(x) ∈ (Dc)x }
DISCRETE NONHOLONOMIC MECHANICS 25
and the disjoint union of these vector spaces
gDc =
We will denote by (gDc)∗ the disjoint union of the dual spaces, that is,
(gDc)∗ =
(gx)∗.
Next, we define the discrete nonholonomic momentum map Jnh : Γ →
(gDc)∗ as follows: Jnh(g) ∈ (gβ(g))∗ and
Jnh(g)(ξ) = Θ+Ld(Ψ(ξ)
(1,1))(g) =
Ψ(ξ)(g)(Ld), for g ∈ Γ and ξ ∈ gβ(g).
If ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for all x ∈ M, then we may
consider the smooth function Jnheξ : Γ→ R defined by
Jnheξ (g) = Jnh(g)(ξ̃(β(g))), ∀g ∈ Γ.
Definition 3.23. The Lagrangian Ld is said to be g-invariant with respect Ψ if
Ψ(ξ)(1,1)(Ld) =
Ψ(ξ)(Ld)−
Ψ(ξ)(Ld) = 0, ∀ξ ∈ g.
Now, we will prove the following result
Theorem 3.24. Let Υ(Ld,Mc,Dc)nh : Mc → Mc be the local discrete nonholonomic
evolution operator for the regular system (Ld,Mc,Dc). If Ld is g-invariant with
respect to Ψ : g→ Sec(τ) and ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for
all x ∈M, then
Jnheξ (Υ(Ld,Mc,Dc)nh (g))− Jnheξ (g) =
←−−−−−−−−−−−−−−−−−−−−−−−−−−
Ψ(ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g)))(Υ
(Ld,Mc,Dc)
nh (g))(Ld)
for g ∈Mc.
Proof. Using that the Lagrangian Ld is g-invariant with respect to Ψ, we have that
−−−−−−−−−−−−−−−−−−→
Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ
(Ld,Mc,Dc)
nh (g))(Ld) =
←−−−−−−−−−−−−−−−−−−
Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ
(Ld,Mc,Dc)
nh (g))(Ld).
(3.17)
Also, since (g,Υ(Ld,Mc,Dc)nh (g)) is a solution of the discrete nonholonomic equations:
←−−−−−−−
Ψ(ξ̃(β(g)))(g)(Ld) =
−−−−−−−−−−−−−−−−−−→
Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ
(Ld,Mc,Dc)
nh (g))(Ld).(3.18)
Thus, from (3.17) and (3.18), we find that
←−−−−−−−
Ψ(ξ̃(β(g))(g)(Ld) =
←−−−−−−−
Ψ(ξ̃(β(g)))(Υ(Ld,Mc,Dc)nh (g))(Ld).
26 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Therefore,
(Υ(Ld,Mc,Dc)nh (g))− J
(g) =
←−−−−−−−−−−−−−−−−−−−
ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))
(Υ(Ld,Mc,Dc)nh (g))(Ld)
←−−−−−−−−
ξ̃(β(g))
(g)(Ld)
←−−−−−−−−−−−−−−−−−−−
ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))
(Υ(Ld,Mc,Dc)nh (g))(Ld)
←−−−−−−−
Ψ(ξ̃(β(g))(Υ(Ld,Mc,Dc)nh (g))(Ld)
←−−−−−−−−−−−−−−−−−−−−−−−−−−−
ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g))
(Υ(Ld,Mc,Dc)nh (g))(Ld).
Theorem 3.24 suggests us to introduce the following definition
Definition 3.25. An element ξ ∈ g is said to be a horizontal symmetry for
the discrete nonholonomic system (Ld,Mc,Dc) and the map Ψ : g→ Sec(τ) if
Ψ(ξ)(x) ∈ (Dc)x, for all x ∈M.
Now, from Theorem 3.24, we conclude that
Corollary 3.26. If Ld is g-invariant with respect to Ψ and ξ ∈ g is a horizontal
symmetry for (Ld,Mc,Dc) and Ψ : g → Sec(τ) then Jnhξ̃ : Γ → R is a constant of
the motion for Υ(Ld,Mc,Dc)nh , that is,
◦Υ(Ld,Mc,Dc)nh = J
4. Examples
4.1. Discrete holonomic Lagrangian systems on a Lie groupoid. Let us
examine the case when the system is subjected to holonomic constraints.
Let Ld : Γ → R be a discrete Lagrangian on a Lie groupoid Γ ⇒ M . Suppose
that Mc ⊆ Γ is a Lie subgroupoid of Γ over M ′ ⊆M , that is, Mc is a Lie groupoid
over M ′ with structural maps
α|Mc : Mc →M
′, β|Mc : Mc →M
′, �|M ′ : M
′ →Mc, i|Mc : Mc →Mc,
the canonical inclusions iMc : Mc −→ Γ and iM ′ : M ′ −→ M are injective immer-
sions and the pair (iMc , iM ′) is a Lie groupoid morphism. We may assume, without
the loss of generality, that M ′ = M (in other case, we will replace the Lie groupoid
Γ by the Lie subgroupoid Γ′ over M ′ defined by Γ′ = α−1(M ′) ∩ β−1(M ′)).
Then, if LMc = Ld ◦ iMc and τMc : EMc → M is the Lie algebroid of Mc, we
have that the discrete (unconstrained) Euler-Lagrange equations for the Lagrangian
function LMc are:
X (g)(LMc)−
X (h)(LMc) = 0, (g, h) ∈ (Mc)2, (4.1)
for X ∈ Sec(τMc).
We are interested in writing these equations in terms of the Lagrangian Ld
defined on the Lie groupoid Γ. From Corollary 4.7 (iii) in [27], we deduce that
(g, h) ∈ (Mc)2 is a solution of Equations 4.1 if and only if DDELLd(g, h) vanishes
over Im(Eβ(g)(iMc)). Here, E(iMc) : EMc → EΓ is the Lie algebroid morphism
induced between EMc and EΓ by the Lie groupoid morphism (iMc , Id). Therefore,
we may consider the discrete holonomic system as the discrete nonholonomic system
(Ld,Mc,Dc), where Dc = (E(iMc))(EMc) ∼= EMc .
DISCRETE NONHOLONOMIC MECHANICS 27
In the particular case, when the subgroupoid Mc is determined by the vanishing
set of n− r independent real C∞-functions φγ : Γ→ R:
Mc = { g ∈ Γ | φγ(g) = 0, for all γ } ,
then the discrete holonomic equations are equivalent to:
Y (g)(Ld)−
Y (h)(Ld) = λγd
oφγ(�(β(g)))(Y (β(g)),
φγ(g) = φγ(h) = 0,
for all Y ∈ Sec(τ), where do is the standard differential on Γ. This algorithm is
a generalization of the Shake algorithm for holonomic systems (see [10, 20, 32, 36]
for similar results on the pair groupoid Q×Q).
4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid.
Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the pair group-
oid Q×Q ⇒ Q and suppose that (q0, q1) is a point of Mc. Then, using the results
of Section 3.1, we deduce that ((q0, q1), (q1, q2)) ∈ (Q × Q)2 is a solution of the
discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc) if and only if
(D2Ld(q0, q1) +D1Ld(q1, q2))|Dc(q1) = 0,
(q1, q2) ∈Mc,
or, equivalently,
D2Ld(q0, q1) +D1Ld(q1, q2) =
j(q1),
(q1, q2) ∈Mc,
where λj are the Lagrange multipliers and {Aj} is a local basis of the annihilator
D0c . These equations were considered in [10] and [36].
Note that if (q1, q2) ∈ Γ = Q×Q then, in this particular case, GLd(q1,q2) : Tq1Q×
Tq2Q→ R is just the R-bilinear map (D2D1Ld)(q1, q2).
On the other hand, if (q1, q2) ∈Mc we have that
(TQ)Mc
(q1,q2)
vq2 ∈ Tq2Q
∣∣ (0, vq2) ∈ T(q1,q2)Mc } ,
(TQ)Mc
(q1,q2)
vq1 ∈ Tq1Q
∣∣ (vq1 , 0) ∈ T(q1,q2)Mc } .
Thus, the system (Ld,Mc,Dc) is regular if and only if for every (q1, q2) ∈ Mc
the following conditions hold:
If vq1 ∈
(TQ)Mc
(q1,q2)
〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq2 ∈ Dc(q2)
 =⇒ vq1 = 0,
If vq2 ∈
(TQ)Mc
(q1,q2)
〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq1 ∈ Dc(q1)
 =⇒ vq2 = 0.
The first condition was obtained in [36] in order to guarantee the existence
of a unique local nonholonomic evolution operator Υ(Ld,Mc,Dc)nh for the system
(Ld,Mc,Dc). However, in order to assure that Υ
(Ld,Mc,Dc)
nh is a (local) diffeomor-
phism one must assume that the second condition also holds.
28 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Example 4.1 (Discrete Nonholonomically Constrained particle). Consider
the discrete nonholonomic system determined by:
a) A discrete Lagrangian Ld : R3 × R3 → R:
Ld(x0, y0, z0, x1, y1, z1) =
x1 − x0
y1 − y0
z1 − z0
b) A constraint distribution of Q = R3,
Dc = span
,X2 =
c) A discrete constraint submanifold Mc of R3 × R3 determined by the con-
straint
φ(x0, y0, z0, x1, y1, z1) =
z1 − z0
y1 + y0
x1 − x0
The system (Ld,Mc,Dc) is a discretization of a classical continuous nonholonomic
system: the nonholonomic free particle (for a discussion on this continuous system
see, for instance, [4, 8]). Note that if E(R3×R3) ∼= TR3 is the Lie algebroid of the
pair groupoid R3 × R3 ⇒ R3 then
T(x1,y1,z1,x1,y1,z1)Mc ∩ E(R3×R3)(x1, y1, z1) = Dc(x1, y1, z1).
Since
X1 = −
X2 = −
then, the discrete nonholonomic equations are:(
x2 − 2x1 + x0
z2 − 2z1 + z0
= 0, (4.2)
y2 − 2y1 + y0
= 0, (4.3)
which together with the constraint equation determine a well posed system of dif-
ference equations.
We have that
D2D1Ld = − 1h{dx0 ∧ dx1 + dy0 ∧ dy1 + dz0 ∧ dz1}
TR3)Mc
(x0,y0,z0,x1,y1,z1)
= {a0 ∂∂x0 + b0
+ c0 ∂∂z0 ∈ T(x0,y0,z0)R
c0 = 12 (a0(y1 + y0)− b0(x1 − x0))}.
TR3)Mc
(x0,y0,z0,x1,y1,z1)
= {a1 ∂∂x1 + b1
+ c1 ∂∂z1 ∈ T(x1,y1,z1)R
c1 = 12 (a1(y1 + y0) + b1(x1 − x0))}.
Thus, if we consider the open subset of Mc defined by{
(x0, y0, z0, x1, y1, z1) ∈Mc
∣∣ 2 + y21 + y1y0 6= 0, 2 + y20 + y0y1 6= 0}
then in this subset the discrete nonholonomic system is regular.
Let Ψ : g = R2 −→ X(R3) given by Ψ(a, b) = a ∂
+ b ∂
. Then gDc =
span{Ψ(ξ̃) = X1}, where ξ̃ : R3 → R2 is defined by ξ̃(x, y, z) = (1, y). More-
over, the Lagrangian Ld is g-invariant with respect to Ψ. Therefore,
(x1, y1, z1, x2, y2, z2)− Jnhξ̃ (x0, y0, z0, x1, y1, z1)
←−−−−−−−−−
Ψ(0, y2 − y1)(x1, y1, z1, x2, y2, z2)(Ld),
DISCRETE NONHOLONOMIC MECHANICS 29
that is,(
x2 − x1
z2 − z1
x1 − x0
z1 − z0
= (y2 − y1)
z2 − z1
This equation is precisely Equation (4.2).
4.3. Discrete nonholonomic Lagrangian systems on a Lie group. Let G be
a Lie group. G is a Lie groupoid over a single point and the Lie algebra g of G is
just the Lie algebroid associated with G.
If g, h ∈ G, vh ∈ ThG and αh ∈ T ∗hG we will use the following notation:
gvh = (Thlg)(vh) ∈ TghG, vhg = (Thrg)(vh) ∈ ThgG,
gαh = (T ∗ghlg−1)(αh) ∈ T
ghG, αhg = (T
hgrg−1)(αh) ∈ T
Now, let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the Lie
group G, that is, Ld : G → R is a discrete Lagrangian, Mc is a submanifold of G
and Dc is a vector subspace of g.
If g1 ∈ Mc then (g1, g2) ∈ G × G is a solution of the discrete nonholonomic
Euler-Lagrange equations for (Ld,Mc,Dc) if and only if
g−11 dLd(g1)− dLd(g2)g
gk ∈Mc, k = 1, 2
(4.4)
where λj are the Lagrange multipliers and {µj} is a basis of the annihilator D0c of
Dc. These equations were obtained in [36] (see Theorem 3 in [36]).
Taking pk = dLd(gk)g
k , k = 1, 2 then
p2 −Ad∗g1p1 = −
λjµj ,
gk ∈Mc, k = 1, 2
(4.5)
where Ad : G × g −→ g is the adjoint action of G on g. These equations were
obtained in [14] and called discrete Euler-Poincaré-Suslov equations.
On the other hand, from (2.14), we have that
ΩLd((
−→η ,←−µ ), (−→η ′,←−µ ′)) = −→η ′(←−µ (Ld))−−→η (←−µ ′(Ld)).
Thus, if g ∈ G then, using (2.22), it follows that the R-bilinear map GLdg : g×g→ R
is given by
GLdg (ξ, η) = −
←−η (g)(
ξ (Ld)).
Therefore, the system (Ld,Mc,Dc) is regular if and only if for every g ∈ Mc the
following conditions hold:
η ∈ g/←−η (g) ∈ TgMc and ←−η (g)(
ξ (Ld)) = 0,∀ξ ∈ Dc =⇒ η = 0,
ξ ∈ g/
ξ (g) ∈ TgMc and ←−η (g)(
ξ (Ld)) = 0,∀η ∈ Dc =⇒ ξ = 0.
We illustrate this situation with two simple examples previously considered in
[14].
30 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
4.3.1. The discrete Suslov system. (See [14]) The Suslov system studies the motion
of a rigid body suspended at its centre of mass under the action of the following
nonholonomic constraint: the body angular velocity is orthogonal to some fixed
direction.
The configuration space is G = SO(3) and the elements of the Lie algebra so(3)
may be identified with R3 and represented by coordinates (ωx, ωy, ωz). Without
loss of generality, let us choose as fixed direction the third vector of the body frame
ē1, ē2, ē3. Then, the nonholonomic constraint is ωz = 0.
The discretization of this system is modelled by considering the discrete La-
grangian Ld : SO(3) −→ R defined by Ld(Ω) = 12Tr (ΩJ), where J represents the
mass matrix (a symmetric positive-definite matrix with components (Jij)1≤i,j≤3).
The constraint submanifold Mc is determined by the constraint Tr (ΩE3) = 0
(see [14]) where
 0 0 00 0 −1
0 1 0
 , E2 =
 0 0 10 0 0
−1 0 0
 , E3 =
 0 −1 01 0 0
0 0 0
is the standard basis of so(3), the Lie algebra of SO(3).
The vector subspace Dc = span{E1, E2}. Therefore, D0c = span{E3}. Moreover,
the exponential map of SO(3) is a diffeomorphism from an open subset of Dc (which
contains the zero vector) to an open subset of Mc (which contains the identity
element I). In particular, TIMc = Dc.
On the other hand, the discrete Euler-Poincaré-Suslov equations are given
Ei(Ω1)(Ld)−
Ei(Ω2)(Ld) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2}.
After some straightforward operations, we deduce that the above equations are
equivalent to:
Tr ((EiΩ2 − Ω1Ei)J) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2}
or, considering the components Ωk = (Ω
ij ) of the elements of SO(3), we have that:(
33 − J33Ω
32 + J22Ω
−J23Ω
22 + J12Ω
13 − J13Ω
−J23Ω
33 − J22Ω
32 − J12Ω
+J33Ω
23 + J23Ω
22 + J13Ω
−J13Ω
33 + J33Ω
31 − J12Ω
+J23Ω
21 − J11Ω
13 + J13Ω
33 + J12Ω
32 + J11Ω
−J33Ω
13 − J23Ω
12 − J13Ω
Ω(1)12 = Ω
21 , Ω
12 = Ω
Moreover, since the discrete Lagrangian verifies that
Ld(Ω) =
Tr (ΩJ) =
Tr (ΩtJ) = Ld(Ω
and also the constraint satisfies Tr (ΩE3) = −Tr (Ω−1E3), then this discretization
of the Suslov system is reversible. The regularity condition in Ω ∈ SO(3) is in this
particular case:
η ∈ so(3) /Tr (E1ΩηJ) = 0, Tr (E2ΩηJ) = 0 and Tr (ΩηE3) = 0 =⇒ η = 0
It is easy to show that the system is regular in a neighborhood of the identity I.
DISCRETE NONHOLONOMIC MECHANICS 31
4.3.2. The discrete Chaplygin sleigh. (See [12, 14]) The Chaplygin sleigh system
describes the motion of a rigid body sliding on a horizontal plane. The body is
supported at three points, two of which slide freely without friction while the third
is a knife edge, a constraint that allows no motion orthogonal to this edge (see [41]).
The configuration space of this system is the group SE(2) of Euclidean motions
of R2. An element Ω ∈ SE(2) is represented by a matrix
 cos θ − sin θ xsin θ cos θ y
0 0 1
 with θ, x, y ∈ R.
Thus, (θ, x, y) are local coordinates on SE(2).
A basis of the Lie algebra se(2) ∼= R3 of SE(2) is given by
 0 −1 01 0 0
0 0 0
 , e1 =
 0 0 10 0 0
0 0 0
 , e2 =
 0 0 00 0 1
0 0 0
and we have that
[e, e1] = e2, [e, e2] = −e1, [e1, e2] = 0.
An element ξ ∈ se(2) is of the form
ξ = ω e+ v1 e1 + v2 e2
and the exponential map exp : se(2) ∼= R3 → SE(2) of SE(2) is given by
exp(ω, v1, v2) = (ω, v1
+ v2(
cosω − 1
),−v1(
cosω − 1
) + v2
), if ω 6= 0,
exp(0, v1, v2) = (0, v1, v2).
Note that the restriction of this map to the open subset U =] − π, π[×R2 ⊆ R3 ∼=
se(2) is a diffeomorphism onto the open subset exp(U) of SE(2).
A discretization of the Chaplygin sleigh may be constructed as follows:
- The discrete Lagrangian Ld : SE(2) −→ R is given by
Ld(Ω) =
Tr (ΩJΩT )− Tr (ΩJ),
where J is the matrix:
 (J/2) +ma2 mab mamab (J/2) +mb2 mb
ma mb m
(see [14]).
- The vector subspace Dc of se(2) is
Dc = span {e, e1} = { (ω, v1, v2) ∈ se(2) | v2 = 0 } .
- The constraint submanifold Mc of SE(2) is
Mc = exp(U ∩Dc). (4.6)
Thus, we have that
Mc = { (θ, x, y) ∈ SE(2) | − π < θ < π, θ 6= 0, (1− cos θ)x− y sin θ = 0 }
∪ { (0, x, 0) ∈ SE(2) | x ∈ R } .
32 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Figure 1. Submanifold Mc
From (4.6) it follows that I ∈Mc and TIMc = Dc. In fact, one may prove
T(0,x,0)Mc = span {
∂θ |(0,x,0)
∂y |(0,x,0)
∂x |(0,x,0)
for x ∈ R.
Now, the discrete Euler-Poincaré-Suslov equations are:
←−e (θ1, x1, y1)(Ld)−−→e (θ2, x2, y2)(Ld) = 0,
←−e1(θ1, x1, y1)(Ld)−−→e1(θ2, x2, y2)(Ld) = 0,
and the condition (θk, xk, yk) ∈Mc, with k ∈ {1, 2}. We rewrite these equations as
the following system of difference equations:(
−am cos θ1 − bm sin θ1 + am
+mx1 cos θ1 +my1 sin θ1
mx2 + am cos θ2
−bm sin θ2 − am
amy1 cos θ1 − amx1 sin θ1 − bmx1 cos θ1
−bmy1 sin θ1 + (a2m+ b2m+ J) sin θ1
amy2 − bmx2
+(a2m+ b2m+ J) sin θ2
together with the condition
(θk, xk, yk) ∈Mc, k ∈ {1, 2}.
On the other hand, one may prove that the discrete nonholonomic Lagrangian
system (Ld,Mc,D) is reversible.
Finally, consider a point (0, x, 0) ∈ Mc and an element η ≡ (ω, v1, v2) ∈ se(2)
such that
←−η (0, x, 0) ∈ T(0,x,0)Mc, ←−η (0, x, 0)(−→e (Ld)) = 0, ←−η (0, x, 0)(−→e1(Ld)) = 0.
Then, if we assume that a2m+ J + amx
6= 0 it follows that η = 0.
Thus, the discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is regular in a
neighborhood of the identity I.
4.4. Discrete nonholonomic Lagrangian systems on an action Lie group-
oid. Let H be a Lie group with identity element e and · : M ×H → M , (x, h) ∈
M × H 7→ xh, a right action of H on M . Thus, we may consider the action Lie
groupoid Γ = M ×H over M with structural maps given by
α̃(x, h) = x, β̃(x, h) = xh, �̃(x) = (x, e),
m̃((x, h), (xh, h′)) = (x, hh′), ĩ(x, h) = (xh, h−1).
(4.7)
DISCRETE NONHOLONOMIC MECHANICS 33
Now, let h = TeH be the Lie algebra of H and Φ : h→ X(M) the map given by
Φ(η) = ηM , for η ∈ h,
where ηM is the infinitesimal generator of the action · : M×H →M corresponding
to η. Then, Φ is a Lie algebra morphism and the corresponding action Lie algebroid
pr1 : M × h→M is just the Lie algebroid of Γ = M ×H.
We have that Sec(pr1) ∼= { η̃ : M → h | η̃ is smooth } and that the Lie algebroid
structure ([[·, ·]]Φ, ρΦ) on pr1 : M ×H →M is defined by
[[η̃, µ̃]]Φ(x) = [η̃(x), µ̃(x)]+(η̃(x))M (x)(µ̃)−(µ̃(x))M (x)(η̃), ρΦ(η̃)(x) = (η̃(x))M (x),
for η̃, µ̃ ∈ Sec(pr1) and x ∈M. Here, [·, ·] denotes the Lie bracket of h.
If (x, h) ∈ Γ = M ×H then the left-translation l(x,h) : α̃−1(xh) → α̃−1(x) and
the right-translation r(x,h) : β̃−1(x)→ β̃−1(xh) are given
l(x,h)(xh, h
′) = (x, hh′), r(x,h)(x(h
′)−1, h′) = (x(h′)−1, h′h). (4.8)
Now, if η ∈ h then η defines a constant section Cη : M → h of pr1 : M × h→M
and, using (2.4), (2.5), (4.7) and (4.8), we have that the left-invariant and the
right-invariant vector fields
C η and
C η, respectively, on M ×H are defined by
C η(x, h) = (−ηM (x),−→η (h)),
C η(x, h) = (0x,
←−η (h)), (4.9)
for (x, h) ∈ Γ = M ×H.
Note that if {ηi} is a basis of h then {Cηi} is a global basis of Sec(pr1).
On the other hand, we will denote by expΓ : EΓ = M × h → Γ = M × H the
map given by
expΓ(x, η) = (x, expH(η)), for (x, η) ∈ EΓ = M × h,
where expH : h → H is the exponential map of the Lie group H. Note that if
Φ(x,e) : R → Γ = M ×H is the integral curve of the left-invariant vector field
on Γ = M ×H such that Φ(x,e)(0) = (x, e) then (see (4.9))
expΓ(x, η) = Φ(x,e)(1).
Next, suppose that Ld : Γ = M × H → R is a Lagrangian function, Dc is a
constraint distribution such that {Xα} is a local basis of sections of the annihilator
D0c , and Mc ⊆ Γ is the discrete constraint submanifold.
For every h ∈ H (resp., x ∈ M) we will denote by Lh (resp., Lx) the real
function on M (resp., on H) given by Lh(y) = Ld(y, h) (resp., Lx(h′) = Ld(x, h′)).
A composable pair ((x, hk), (xhk, hk+1)) ∈ Γ2 ∩ (Mc × Mc) is a solution of the
discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc) if
C η(x, hk)(Ld)−
C η(xhk, hk+1)(Ld) = λαX
α(xhk)(η), for all η ∈ h,
or, in other terms (see (4.9))
{(Telhk)(η)}(Lx)− {(Terhk+1)(η)}(Lxhk) + ηM (xhk)(Lhk+1) = λαX
α(xhk)(η),
for all η ∈ h.
4.4.1. The discrete Veselova system. As a concrete example of a nonholonomic
system on a transformation Lie groupoid we consider a discretization of the Veselova
system (see [44]). In the continuous theory [9], the configuration manifold is the
transformation Lie algebroid pr1 : S2 × so(3)→ S2 with Lagrangian
Lc(γ, ω) =
ω · Iω −mglγ · e,
34 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
where S2 is the unit sphere in R3, ω ∈ R3 ' so(3) is the angular velocity, γ is the
direction opposite to the gravity and e is a unit vector in the direction from the
fixed point to the center of mass, all them expressed in a frame fixed to the body.
The constants m, g and l are respectively the mass of the body, the strength of
the gravitational acceleration and the distance from the fixed point to the center
of mass. The matrix I is the inertia tensor of the body. Moreover, the constraint
subbundle Dc → S2 is given by
γ ∈ S2 7→ Dc(γ) =
ω ∈ R3 ' so(3)
∣∣ γ · ω = 0} .
Note that the section φ : S2 → S2 × so(3)∗, (x, y, z) 7→ ((x, y, z), xe1 + ye2 + ze3),
where {e1, e2, e3} is the canonical basis of R3 and {e1, e2, e3} is the dual basis, is a
global basis for D0c .
If ω ∈ so(3) and ω̂ is the skew-symmetric matrix of order 3 such that ω̂v = ω×v
then the Lagrangian function Lc may be expressed as follows
Lc(γ, ω) =
Tr(ω̂IIω̂T )−mg l γ · e,
where II = 1
Tr(I)I3×3− I. Here, I3×3 is the identity matrix. Thus, we may define
a discrete Lagrangian Ld : Γ = S2 × SO(3)→ R for the system by (see [27])
Ld(γ,Ω) = −
Tr(IIΩ)− hmg l γ · e.
On the other hand, we consider the open subset of SO(3)
V = {Ω ∈ SO(3) | Tr Ω 6= ±1 }
and the real function ψ : S2 × V → R given by
ψ(γ,Ω) = γ · (Ω̂− ΩT ).
One may check that the critical points of ψ are
(γ,Ω) ∈ S2 × V
∣∣ Ωγ − γ = 0} .
Thus, the subset Mc of Γ = S2 × SO(3) defined by
(γ,Ω) ∈ (S2 × V )− Cψ
∣∣∣ γ · (Ω̂− ΩT ) = 0} ,
is a submanifold of Γ of codimension one. Mc is the discrete constraint submanifold.
We have that the map expΓ : S
2× so(3)→ S2×SO(3) is a diffeomorphism from
an open subset of Dc, which contains the zero section, to an open subset of Mc,
which contains the subset of Γ given by
�̃(S2) = {(γ, e) ∈ S2 × SO(3)}.
So, it follows that
(Dc)(γ) = T(γ,e)Mc ∩ EΓ(γ), for γ ∈ S2.
Following the computations of [27] we get the nonholonomic discrete Euler-Lagrange
equations, for ((γk,Ωk), (γkΩk,Ωk+1)) ∈ Γ2
Mk+1 − ΩTkMkΩk +mglh
2( ̂γk+1 × e) = λγ̂k+1,
γk( ̂Ωk − ΩTk ) = 0, γk+1(
̂Ωk+1 − ΩTk+1) = 0,
where M = ΩII − IIΩT . Therefore, in terms of the axial vector Π in R3 defined by
Π̂ = M , we can write the equations in the form
Πk+1 = ΩTk Πk −mglh
2γk+1 × e + λγk+1,
γk( ̂Ωk − ΩTk ) = 0, γk+1(
̂Ωk+1 − ΩTk+1) = 0.
DISCRETE NONHOLONOMIC MECHANICS 35
Note that, using the expression of an arbitrary element of SO(3) in terms of the
Euler angles (see Chapter 15 of [31]), we deduce that the discrete constraint sub-
manifold Mc is reversible, that is, i(Mc) = Mc. However, the discrete nonholonomic
Lagrangian system (Ld,Mc,Dc) is not reversible. In fact, it is easy to prove that
Ld ◦ i 6= Ld.
On the other hand, if γ ∈ S2 and ξ, η ∈ R3 ∼= so(3) then it follows that
C ξ(γ, I3)(
C η(Ld)) = −ξ · Iη.
Consequently, the nonholonomic system (Ld,Mc,Dc) is regular in a neighborhood
(in Mc) of the submanifold �̃(S2).
4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie group-
oid. Let p : Q → M = Q/G be a principal G-bundle and choose a local trivial-
ization G× U , where U is an open subset of M . Then, one may identify the open
subset (p−1(U) × p−1(U))/G ' ((G × U) × (G × U))/G of the Atiyah groupoid
(Q×Q)/G with the product manifold (U ×U)×G. Indeed, it is easy to prove that
the map
((G× U)× (G× U))/G→ (U × U)×G,
[((g, x), (g′, y))]→ ((x, y), g−1g′)),
is bijective. Thus, the restriction to ((G × U) × (G × U))/G of the Lie groupoid
structure on (Q × Q)/G induces a Lie groupoid structure in (U × U) × G with
source, target and identity section given by
α : (U × U)×G→ U ; ((x, y), g)→ x,
β : (U × U)×G→ U ; ((x, y), g)→ y,
� : U → (U × U)×G; x→ ((x, x), e),
and with multiplication m : ((U × U) × G)2 → (U × U) × G and inversion i :
(U × U)×G→ (U × U)×G defined by
m(((x, y), g), ((y, z), h)) = ((x, z), gh),
i((x, y), g) = ((y, x), g−1).
(4.10)
The Lie algebroid A((U×U)×G) may be identified with the vector bundle TU×g→
U . Thus, the fibre over the point x ∈ U is the vector space TxU × g. Therefore, a
section of A((U ×U)×G) is a pair (X, ξ̃), where X is a vector field on U and ξ̃ is a
map from U on g. The space Sec(A((U × U)×G)) is generated by sections of the
form (X, 0) and (0, Cξ), with X ∈ X(U), ξ ∈ g and Cξ : U → g being the constant
map Cξ(x) = ξ, for all x ∈ U (see [27] for more details).
Now, suppose that Ld : (U ×U)×G→ R is a Lagrangian function, Dc a vector
subbundle of TU×g and Mc a constraint submanifold on (U×U)×G. Take a basis
of sections {Y α} of the annihilator Doc . Then, the discrete nonholonomic equations
are ←−−−−−
(Xα, η̃α)((x, y), gk)(Ld)−
−−−−−→
(Xα, η̃α)((y, z), gk+1)(Ld) = 0,
with (Xα, η̃α) : U → TU × g a basis of the space Sec(τDc) and (((x, y), gk), ((y, z),
gk+1)) ∈ (Mc×Mc)∩ ((U ×U)×G)2. The above equations may be also written as
(X, 0)((x, y), gk)(Ld)−
(X, 0)((y, z), gk+1)(Ld) = λαY α(y)(X(y)),←−−−−
(0, Cξ)((x, y), gk)(Ld)−
−−−−→
(0, Cξ)((y, z), gk+1)(Ld) = λαY α(y)(Cξ(y)),
with X ∈ X(U), ξ ∈ g and (((x, y), gk), ((y, z), gk+1)) ∈ (Mc×Mc)∩((U×U)×G)2.
An equivalent expression of these equations is
D2Ld((x, y), gk) +D1Ld((y, z), gk+1) = λαµα(y),
pk+1(y, z) = Ad∗gkpk(x, y)− λαη̃
α(y),
(4.11)
36 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
where pk(x̄, ȳ) = d(r∗gkL(x̄,ȳ, ))(e) for (x̄, ȳ) ∈ U × U and we write Y
α ≡ (µα, η̃α),
µα being a 1-form on U and η̃α : U → g∗ a smooth map.
4.5.1. A discretization of the equations of motion of a rolling ball without sliding on
a rotating table with constant angular velocity. A (homogeneous) sphere of radius
r > 0, mass m and inertia about any axis I rolls without sliding on a horizontal
table which rotates with constant angular velocity Ω about a vertical axis through
one of its points. Apart from the constant gravitational force, no other external
forces are assumed to act on the sphere (see [41]).
The configuration space for the continuous system isQ = R2×SO(3) and we shall
use the notation (x, y;R) to represent a typical point in Q. Then, the nonholonomic
constraints are
Tr(ṘRTE2) = −Ωy,
Tr(ṘRTE1) = Ωx,
where {E1, E2, E3} is the standard basis of so(3).
The matrix ṘRT is skew symmetric, therefore we may write
ṘRT =
 0 −w3 w2w3 0 −w1
−w2 w1 0
where (w1, w2, w3) represents the angular velocity vector of the sphere measured
with respect to the inertial frame. Then, we may rewrite the constraints in the
usual form:
ẋ− rw2 = −Ωy,
ẏ + rw1 = Ωx.
The Lagrangian for the rolling ball is:
Lc(x, y;R, ẋ, ẏ; Ṙ) =
m(ẋ2 + ẏ2) +
I Tr(ṘRT (ṘRT )T )
m(ẋ2 + ẏ2) +
I(ω21 + ω
2 + ω
Moreover, it is clear that Q = R2 × SO(3) is the total space of a trivial princi-
pal SO(3)-bundle over R2 and the bundle projection φ : Q → M = R2 is just the
canonical projection on the first factor. Therefore, we may consider the correspond-
ing Atiyah algebroid E′ = TQ/SO(3) over M = R2. We will identify the tangent
bundle to SO(3) with so(3)× SO(3) by using right translation.
Under this identification between T (SO(3)) and so(3)×SO(3) the tangent action
of SO(3) on T (SO(3)) ∼= so(3)× SO(3) is the trivial action
(so(3)× SO(3))× SO(3)→ so(3)× SO(3), ((ω,R), S) 7→ (ω,RS). (4.12)
Thus, the Atiyah algebroid TQ/SO(3) is isomorphic to the product manifold
TR2×so(3) and the vector bundle projection is τR2 ◦pr1, where pr1 : TR2×so(3)→
TR2 and τR2 : TR2 → R2 are the canonical projections.
A section of E′ = TQ/SO(3) ∼= TR2 × so(3) → R2 is a pair (X,u), where X is
a vector field on R2 and u : R2 → so(3) is a smooth map. Therefore, a global basis
of sections of TR2 × so(3)→ R2 is
s′1 = (
, 0), s′2 = (
, 0),
s′3 = (0, E1), s
4 = (0, E2), s
5 = (0, E3).
DISCRETE NONHOLONOMIC MECHANICS 37
The anchor map ρ′ : E′ = TQ/SO(3) ∼= TR2 × so(3) → TR2 is the projection
over the first factor and if [[·, ·]]′ is the Lie bracket on the space Sec(E′ = TQ/SO(3))
then the only non-zero fundamental Lie brackets are
[[s′3, s
′ = s′5, [[s
′ = s′3, [[s
′ = s′4.
Moreover, the Lagrangian function Lc = T and the constraint functions are
SO(3)-invariant. Consequently, Lc induces a Lagrangian function L′c on E
TQ/SO(3)
L′c(x, y, ẋ, ẏ;ω) =
m(ẋ2 + ẏ2) +
I Tr(ωωT ),
m(ẋ2 + ẏ2)−
I Tr(ω2),
where (x, y, ẋ, ẏ) are the standard coordinates on TR2 and ω ∈ so(3). The con-
straint functions defined on E′ = TQ/SO(3) are:
ẋ+ r
Tr(ωE2) = −Ωy,
ẏ − r
Tr(ωE1) = Ωx.
(4.13)
We have a nonholonomic system on the Atiyah algebroid E′ = TQ/SO(3) ∼= TR2×
so(3). This kind of systems was recently analyzed by J. Cortés et al [9] (in particular,
this example was carefully studied).
Eqs. (4.13) define an affine subbundle of the vector bundle E′ ∼= TR2× so(3)→
R2 which is modelled over the vector subbundle D′c generated by the sections
D′c = {s
5, rs
1 + s
4, rs
2 − s
Our objective is to discretize this example directly on the Atiyah algebroid. The
Atiyah groupoid is now identified to R2 × R2 × SO(3) ⇒ R2. We may construct
the discrete Lagrangian by
L′d(x0, y0, x1, y1;W1) = L
c(x0, y0,
x1 − x0
y1 − y0
; (logW1)/h)
where log : SO(3) −→ so(3) is the (local)-inverse of the exponential map exp :
so(3) −→ SO(3). For simplicity instead of this procedure we use the following
approximation:
logW1/h ≈
W1 − I3×3
where I3×3 is the identity matrix.
L′d(x0, y0, x1, y1;W1) = L
c(x0, y0,
x1 − x0
y1 − y0
W1 − I3×3
x1 − x0
y1 − y0
(2h)2
Tr(I3×3 −W1)
Eliminating constants, we may consider as discrete Lagrangian
L′d =
x1 − x0
y1 − y0
Tr(W1)
The discrete constraint submanifold M′c of R
2 × R2 × SO(3) is determined
by the constraints:
x1 − x0
Tr(W1E2) = −Ω
y1 + y0
y1 − y0
Tr(W1E1) = Ω
x1 + x0
38 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
We have that the system (L′d,M
c) is not reversible. Note that the Lagrangian
function L′d is reversible. However, the constraint submanifold M
c is not reversible.
The discrete nonholonomic Euler-Lagrange equations for the system (L′d, ,M
D′c) are:
s′5(x0, y0, x1, y1;W1)(L
s′5(x1, y1, x2, y2;W2)(L
d) = 0
←−−−−−−
(rs′1 + s
4)(x0, y0, x1, y1;W1)(L
−−−−−−→
(rs′1 + s
4)(x1, y1, x2, y2;W2)(L
d) = 0
←−−−−−−
(rs′2 − s
3)(x0, y0, x1, y1;W1)(L
−−−−−−→
(rs′2 − s
3)(x1, y1, x2, y2;W2)(L
d) = 0
with the constraints defining Mc.
On the other hand, the vector fields ←−s ′5,
−→s ′5,
←−−−−−
rs′1 + s
−−−−−→
rs′1 + s
←−−−−−
rs′2 − s′3 and−−−−−→
rs′2 − s′3 on (R2 × R2)× SO(3) are given by
←−s ′5 = ((0, 0),
E 3),
−→s ′5 = ((0, 0),
E 3),←−−−−−
rs′1 + s
4 = ((0, r
E 2),
−−−−−→
rs′1 + s
4 = ((−r
, 0),
E 2),←−−−−−
rs′2 − s′3 = ((0, r
E 1),
−−−−−→
rs′2 − s′3 = ((0,−r
E 1),
where
E i (respectively,
E i) is the left-invariant (respectively, right-invariant) vec-
tor field on SO(3) induced by Ei ∈ so(3), for i ∈ {1, 2, 3}. Thus, we deduce the
following system of equations:
Tr ((W1 −W2)E3) = 0,
x2 − 2x1 + x0
Tr ((W1 −W2)E2) = 0,
y2 − 2y1 + y0
Tr ((W1 −W2)E1) = 0,
x2 − x1
Tr(W2E2) + Ω
y2 + y1
y2 − y1
Tr(W2E1)− Ω
x2 + x1
where (x0, x1, y0, y1;W1) are known. Simplifying we obtain the following system of
equations:
x2 − 2x1 + x0
I +mr2
y2 − y0
= 0 (4.14)
y2 − 2y1 + y0
I +mr2
x2 − x0
= 0 (4.15)
Tr ((W1 −W2)E3) = 0 (4.16)
x2 − x1
Tr(W2E2) + Ω
y2 + y1
= 0, (4.17)
y2 − y1
Tr(W2E1)− Ω
x2 + x1
= 0. (4.18)
Now, consider the open subset U of R2 × R2 × SO(3)
U = (R2 × R2)× {W ∈ SO(3) | W − Tr(W )I3×3 is regular } .
Then, using Corollary 3.13 (iv), we deduce that the discrete nonholonomic La-
grangian system (L′d,M
c) is regular in the open subset U
′ of M′c given by
U ′ = U ∩M′c.
DISCRETE NONHOLONOMIC MECHANICS 39
If we denote by uk = (xk+1 − xk)/h and vk = (yk+1 − yk)/h, k ∈ N then from
Equations (4.14) and (4.15) we deduce that(
4 + α2h2
4− α2h2 −4αh
4αh 4− α2h2
or in other terms
x(k + 2) =
8x(k + 1) + (α2h2 − 4)x(k)− 4αh(y(k + 1)− y(k))
α2h2 + 4
y(k + 2) =
8y(k + 1) + (α2h2 − 4)y(k) + 4α(x(k + 1)− x(k))
α2h2 + 4
where α = IΩ
I+mr2
. Since A ∈ SO(2), the discrete nonholonomic model predicts that
the point of contact of the ball will sweep out a circle on the table in agreement
with the continuous model. Figure 2 shows the excellent behaviour of the proposed
numerical method
Figure 2. Orbits for the discrete nonholonomic equations of mo-
tion (left) and a standard numerical method (right) (initial condi-
tions x(0) = 0.99, y(0) = 1, x(1) = 1, y(1) = 0.99 and h = 0.01
after 20000 steps).
4.6. Discrete Chaplygin systems. Now, we present the theory for a particu-
lar (but typical) example of discrete nonholonomic systems: discrete Chaplygin
systems. This kind of systems was considered in the case of the pair groupoid in
[10].
For any groupoid Γ ⇒ M , the map χ : Γ → M × M , g 7→ (α(g), β(g)) is a
morphism over M from Γ to the pair groupoid M ×M (usually called the anchor
of Γ). The induced morphism of Lie algebroids is precisely the anchor ρ : EΓ → TM
of EΓ (the Lie algebroid of Γ).
Definition 4.2. A discrete Chaplygin system on the groupoid Γ is a discrete
nonholonomic problem (Ld,Mc,Dc) such that
- (Ld,Mc,Dc) is a regular discrete nonholonomic Lagrangian system;
- χMc = χ ◦ iMc : Mc −→M ×M is a diffeomorphism;
- ρ ◦ iDc : Dc −→ TM is an isomorphism of vector bundles.
Denote by L̃d : M ×M −→ R the discrete Lagrangian defined by L̃d = Ld ◦
iMc ◦ (χMc)−1.
In the following, we want to express the dynamics on M×M , by finding relations
between de dynamics defined by the nonholonomic system on Γ and M ×M .
40 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
From our hypothesis, for any vector field Y ∈ X(M) there exists a unique section
X ∈ Sec(τDc) such that ρ ◦ iDc ◦X = Y .
Now, using (2.4), (2.5) and (2.6), it follows that
X (g)) = −Y (α(g)) and Tgβ(
X (g)) = Y (β(g))
with some abuse of notation. In other words,
Tgχ(X
(1,0)(g)) = Y (1,0)(α(g), β(g)) and Tgχ(X
(0,1)(g)) = Y (0,1)(α(g), β(g))
for g ∈ Mc, where Tχ : TΓΓ ∼= V β ⊕Γ V α → TM×M (M ×M) ∼= T (M ×M) is the
prolongation of the morphism χ given by
(Tgχ)(Xg, Yg) = ((Tgα)(Xg), (Tgβ)(Yg)),
for g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ∼= Vgβ ⊕ Vgα.
Since χMc is a diffeomorphism, there exists a unique X
g ∈ TgMc (respectively,
X̄ ′g ∈ TgMc) such that
(TgχMc)(X
g) = Y
(1,0)(α(g), β(g)) = (−Y (α(g)), 0β(g))
(respectively, (TgχMc)(X̄
g) = Y
(0,1)(α(g), β(g)) = (0α(g), Y (β(g)))) for all g ∈Mc.
Thus,
X ′g ∈ TgMc ∩ Vgβ,
X (g)−X ′g = Z ′g ∈ Vgα ∩ Vgβ,
X̄ ′g ∈ TgMc ∩ Vgα,
X (g)− X̄ ′g = Z̄ ′g ∈ Vgα ∩ Vgβ,
for all g ∈Mc.
Now, if (g, h) ∈ Γ2 ∩ (Mc ×Mc) then
X (g)(Ld)−
X (h)(Ld) = X̄
g(Ld) + Z̄
g(Ld)−X
h(Ld)− Z
h(Ld)
Y (α(g), β(g))(L̃d)−
Y (α(h), β(h))(L̃d)
+Z̄ ′g(Ld)− Z
h(Ld).
Therefore, if we use the following notation
(α(g), β(g)) = (x, y), (α(h), β(h)) = (y, z)
F+Y (x, y) = −Z̄
(x,y)
(Ld), F
Y (y, z) = Z
(y,z)
(Ld),
X (g)(Ld)−
X (h)(Ld) =
Y (x, y)(L̃d)−
Y (y, z)(L̃d)
−F+Y (x, y) + F
Y (y, z).
In conclusion, we have proved that (g, h) is a solution of the discrete nonholonomic
Euler-Lagrange equations for the system (Ld,Mc,Dc) if and only if ((x, y), (y, z))
is a solution of the reduced equations
Y (x, y)(L̃d)−
Y (y, z)(L̃d) = F
Y (x, y)− F
Y (y, z), Y ∈ X(M).
Note that the above equations are the standard forced discrete Euler-Lagrange
equations (see [32]).
4.6.1. The discrete two wheeled planar mobile robot. We now consider a discrete
version of the two-wheeled planar mobile robot [8, 9]. The position and orientation
of the robot is determined, with respect a fixed cartesian reference, by an element
Ω = (θ, x, y) ∈ SE(2), that is, a matrix
 cos θ − sin θ xsin θ cos θ y
0 0 1
DISCRETE NONHOLONOMIC MECHANICS 41
Moreover, the different positions of the two wheels are described by elements
(φ, ψ) ∈ T2. Therefore, the configuration space is SE(2) × T2. The system is
subjected to three nonholonomic constraints: one constraint induced by the condi-
tion of no lateral sliding of the robot and the other two by the rolling conditions of
both wheels.
It is well known that this system is SE(2)-invariant and then the system may
be described as a nonholonomic system on the Lie algebroid se(2)×TT2 → T2 (see
[9]). In this case, the Lagrangian is
Jω2 +m(v1)2 +m(v2)2 + 2m0lωv
2 + J2φ̇
2 + J2ψ̇
Tr(ξJξT ) +
φ̇2 +
where
ξ = ω e+ v1 e1 + v
2 e2 =
 0 −ω v1ω 0 v2
0 0 0
 and J =
 J/2 0 m0l0 J/2 0
m0l 0 m
Here, m = m0 + 2m1, where m0 is the mass of the robot without the two wheels,
m1 the mass of each wheel, J its the moment of inertia with respect to the vertical
axis, J2 the axial moments of inertia of the wheels and l the distance between the
center of mass of the robot and the intersection point of the horizontal symmetry
axis of the robot and the horizontal line connecting the centers of the two wheels.
The nonholonomic constraints are
v1 + R
φ̇+ R
ψ̇ = 0,
v2 = 0,
ω + R
φ̇− R
ψ̇ = 0,
(4.19)
determining a submanifold M of se(2) × TT2, where R is the radius of the two
wheels and 2c the lateral length of the robot.
In order to discretize the above nonholonomic system, we consider the Atiyah
groupoid Γ = SE(2)×(T2×T2) ⇒ T2. The Lie algebroid of SE(2)×(T2×T2) ⇒ T2
is TT2 × se(2)→ T2. Then:
- The discrete Lagrangian Ld : SE(2)× (T2 × T2)→ R is given by:
Ld(Ωk, φk, ψk, φk+1, ψk+1) = 12h2 Tr ((Ωk − I3×3)J(Ωk − I3×3)
(∆φk)
(∆ψk)
where I3×3 is the identity matrix, ∆φk = φk+1 − φk, ∆ψk = ψk+1 − ψk
 cos θk − sin θk xksin θk cos θk yk
0 0 1
We obtain that
mx2k +my
k − 2lm0xk(1− cos θk)
+2J(1− cos θk) + 2lm0yk sin θk) +
(∆φk)2
(∆ψk)2
- The constraint vector subbundle of se(2)×TT2 is generated by the sections:{
, s2 =
42 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
- The continuous constraints of the two-wheeled planar robot are written in
matrix form (see 4.19):
 0 −ω v1ω 0 v2
0 0 0
 0 R2c φ̇− R2c ψ̇ −R2 φ̇− R2 ψ̇− R
φ̇+ R
ψ̇ 0 0
0 0 0
We discretize the previous constraints using the exponential on SE(2)
(see Section 4.3.2) and discretizing the velocities on the right hand side
cos( R2c∆φk− R2c∆ψk) sin( R2c∆φk− R2c∆ψk) −c
∆φk+∆ψk
∆φk−∆ψk
sin( R2c∆φk− R2c∆ψk)
− sin( R2c∆φk− R2c∆ψk) cos( R2c∆φk− R2c∆ψk) c
∆φk+∆ψk
∆φk−∆ψk
(1−cos( R2c∆φk− R2c∆ψk))
0 0 1
if ∆φk 6= ∆ψk and
 1 0 −R∆φk0 1 0
0 0 1
if ∆φk = ∆ψk.
Therefore, the constraint submanifold Mc is defined as
θk = −
∆φk +
∆ψk (4.20)
xk = −c
∆φk + ∆ψk
∆φk −∆ψk
∆φk −
(4.21)
yk = c
∆φk + ∆ψk
∆φk −∆ψk
1− cos
∆φk −
(4.22)
if ∆φk 6= ∆ψk and θk = 0, xk = −R∆φk and yk = 0 if ∆φk = ∆ψk.
We have that the discrete nonholonomic system (Ld,Mc,Dc) is reversible. More-
over, if �Γ : T2 → SE(2) × (T2 × T2) is the identity section of the Lie groupoid
Γ = SE(2)× (T2 × T2) then it is clear that
�Γ(T2) = {I3×3} ×∆T2×T2 ⊆Mc.
Here, ∆T2×T2 is the diagonal in T2 × T2. In addition, the system (Ld,Mc,Dc) is
regular in a neighborhood U of the submanifold �Γ(T2) = {I3×3} ×∆T2×T2 in Mc.
Note that
T(I3×3,φ1,ψ1,φ1,ψ1)Mc ∩ EΓ(φ1, ψ1) = Dc(φ1, ψ1),
for (φ1, ψ1) ∈ T2, where EΓ = se(2)× TT2 is the Lie algebroid of the Lie groupoid
Γ = SE(2)× (T2 × T2).
On the other hand, it is easy to show that the system (Ld, U,Dc) is a discrete
Chaplygin system.
The reduced Lagrangian on T2 × T2 is
L̃d =


(mc2(
∆φk + ∆ψk
∆φk −∆ψk
)2(1− cos(
∆φk −
∆ψk))
+J(1− cos(
∆φk −
∆ψk))) +
(∆φk)2
(∆ψk)2
if ∆φk 6= ∆ψk
(J1 +
(∆φk)2
, if ∆φk = ∆ψk
The discrete nonholonomic equations are:
(Ω1,φ1,ψ1,φ2,ψ2)
(Ld)−−→s1
(Ω2,φ2,ψ2,φ3,ψ3)
(Ld) = 0
(Ω1,φ1,ψ1,φ2,ψ2)
(Ld)−−→s2
(Ω2φ2,ψ2,φ3,ψ3)
(Ld) = 0
DISCRETE NONHOLONOMIC MECHANICS 43
These equations in coordinates are:
2J1(φ3 − 2φ2 + φ1) = lRm0(cos θ2 + cos θ1) +
(sin θ2 − sin θ1)
R cos θ1
(lm0y1 + cmx1) +
R sin θ1
(lm0x1 − cmy1)
(cmx2 + lm0(y2 − 2c)) (4.23)
2J1(ψ3 − 2ψ2 + ψ1) = lRm0(cos θ2 + cos θ1)−
(sin θ2 − sin θ1)
R cos θ1
(lm0y1 − cmx1)−
R sin θ1
(lm0x1 + cmy1)
(cmx2 − lm0(y2 + 2c)) (4.24)
Substituting constraints (4.20), (4.21) and (4.22) in Equations (4.23) and (4.24)
we obtain a set of equations of the type 0 = f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) and 0 =
f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) which are the reduced equations of the Chaplygin system.
5. Conclusions and Future Work
In this paper we have elucidated the geometrical framework for nonholonomic
discrete Mechanics on Lie groupoids. We have proposed discrete nonholonomic
equations that are general enough to produce practical integrators for continuous
nonholonomic systems (reduced or not). The geometric properties related with
these equations have been completely studied and the applicability of these devel-
opments has been stated in several interesting examples.
Of course, much work remains to be done to clarify the nature of discrete non-
holonomic mechanics. Many of this future work was stated in [36] and, in particular,
we emphasize:
- a complete backward error analysis which explain the very good energy
behavior showed in examples or the preservation of a discrete energy (see
[14]);
- related with the previous question, the construction of a discrete exact
model for a continuous nonholonomic system (see [17, 32, 36]);
- to study discrete nonholonomic systems which preserve a volume form on
the constraint surface mimicking the continuous case (see, for instance,
[13, 46] for this last case);
- to analyze the discrete hamiltonian framework and the construction of
integrators depending on different discretizations;
- and the construction of a discrete nonholonomic connection in the case of
Atiyah groupoids (see [21, 27]).
Related with some of the previous questions, in the conclusions of the paper of R.
McLachlan and M. Perlmutter [36], the authors raise the question of the possibility
of the definition of generalized constraint forces dependent on all the points qk−1,
qk and qk+1 (instead of just qk) for the case of the pair groupoid. We think that
the discrete nonholonomic Euler-Lagrange equations can be generalized to consider
this case of general constraint forces that, moreover, are closest to the continuous
model (see [25, 36]).
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D. Iglesias: Instituto de Matemáticas y F́ısica Fundamental, Consejo Superior de
Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain
E-mail address: iglesias@imaff.cfmac.csic.es
Juan C. Marrero: Departamento de Matemática Fundamental, Facultad de Matemá-
ticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain
E-mail address: jcmarrer@ull.es
D. Mart́ın de Diego: Instituto de Matemáticas y F́ısica Fundamental, Consejo Supe-
rior de Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain
E-mail address: d.martin@imaff.cfmac.csic.es
Eduardo Mart́ınez: Departamento de Matemática Aplicada, Facultad de Ciencias,
Universidad de Zaragoza, 50009 Zaragoza, Spain
E-mail address: emf@unizar.es
	1. Introduction
	2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids
	2.1. Lie algebroids
	2.2. Lie groupoids
	2.3. Discrete Unconstrained Lagrangian Systems
	3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids
	3.1. Discrete Generalized Hölder's principle
	3.2. Discrete Nonholonomic Legendre transformations
	3.3. Nonholonomic evolution operators and regular discrete nonholonomic Lagrangian systems
	3.4. Reversible discrete nonholonomic Lagrangian systems
	3.5. Lie groupoid morphisms and reduction
	3.6.  Discrete nonholonomic Hamiltonian evolution operator
	3.7. The discrete nonholonomic momentum map
	4. Examples
	4.1. Discrete holonomic Lagrangian systems on a Lie groupoid
	4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid
	4.3. Discrete nonholonomic Lagrangian systems on a Lie group
	4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid
	4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid
	4.6. Discrete Chaplygin systems
	5. Conclusions and Future Work
	References
