Second-order constrained variational problems on Lie algebroids: applications to optimal control

0.62 MB·

by  Leonardo Colombo

SECOND-ORDER CONSTRAINED VARIATIONAL PROBLEMS ON
LIE ALGEBROIDS: APPLICATIONS TO OPTIMAL CONTROL
7 Leonardo Colombo
1 DepartmentofMathematics,UniversityofMichigan
0 530ChurchStreet,3828EastHall
2 AnnArbor,Michigan,48109,USA

n
a
J Abstract. Theaimofthisworkistostudy,fromanintrinsicandgeometricpoint
of view, second-order constrained variational problems on Lie algebroids, that is,
7 optimization problems defined by a cost function which depends on higher-order
1 derivatives of admissible curves on a Lie algebroid. Extending the classical Skin-
ner and Rusk formalism for the mechanics in the context of Lie algebroids, for
second-orderconstrainedmechanicalsystems,wederivethecorrespondingdynam-
]
ical equations. We find a symplectic Lie subalgebroid where, under some mild
h
regularity conditions, the second-order constrained variational problem, seen as a
p presymplecticHamiltoniansystem,hasauniquesolution. Westudytherelationship
- ofthisformalismwiththesecond-orderconstrainedEuler-Poincar´eandLagrange-
h Poincar´e equations, among others. Our study is applied to the optimal control of
t mechanicalsystems.
a
m
Contents
[
1. Introduction 1
1
2. Lie algebroids and admissible elements 4
v
2.1. Lie algebroids, Lie subalgebroids and Cartan calculus on Lie algebroids 4
2
2.2. E-tangent bundle to a Lie algebroid E 8
7
2.3. E-tangent bundle of the dual bundle of a Lie algebroid 10
7
4 2.4. Symplectic Lie algebroids 12
0 2.5. Admissible elements on a Lie algebroid 13
. 3. Second-order variational problems on Lie algebroids 14
1
3.1. Mechanics on Lie algebroids 14
0
3.2. Constraint algorithm for presymplectic Lie algebroids 17
7
1 3.3. Vakonomic mechanics on Lie algebroids 20
: 3.4. Second-order variational problems on Lie algebroids 22
v
4. Application to optimal control of mechanical systems 27
i
X 4.1. Optimal control problems of fully-actuated mechanical systems on Lie
algebroids 28
r
a 4.2. Optimal control problems of underactuated mechanical systems on Lie
algebroids 31
Acknowledgments 37
References 37
1. Introduction
Lie algebroids have deserved a lot of interest in recent years. Since a Lie algebroid is
a concept which unifies tangent bundles and Lie algebras, one can suspect their relation
1991 Mathematics Subject Classification. Primary: 70H25 ; Secondary: 70H30, 70H50, 37J15,
58K05,70H03,37K05.
Key words and phrases. Liealgebroids,optimalcontrol,higher-ordermechanics,higher-ordervari-
ationalproblems.
1
2 LEONARDOCOLOMBO
with mechanics. More precisely, a Lie algebroid over a manifold Q is a vector bundle
τ :E →QoverQwithaLiealgebrastructureoverthespaceΓ(τ )ofsectionsofE and
E E
an application ρ : E → TQ called anchor map satisfying some compatibility conditions
(see [51]). Examples of Lie algebroids are the tangent bundle over a manifold Q where
theLiebracketistheusualLiebracketofvectorfieldsandtheanchormapistheidentity
function; a real finite dimensional Lie algebras as vector bundles over a point, where the
anchor map is the null application; action Lie algebroids of the type pr : M ×g → M
1
wheregisaLiealgebraactinginfinitesimallyoverthemanifoldM withaLiebracketover
the space of sections induced by the Lie algebra structure and whose anchor map is the
actionofgoverM;and,theLie-AtiyahalgebroidτTQ/G :TQ/G→M(cid:99)=Q/Gassociated
with the G-principal bundle p:Q→M(cid:99)where the anchor map is induced by the tangent
application of p, Tp:TQ→TM(cid:99)[49, 51, 57, 71].
In [71] Alan Weinstein developed a generalized theory of Lagrangian mechanics on Lie
algebroids and he obtained the equations of motion using the linear Poisson structure on
the dual of the Lie algebroid and the Legendre transformation associated with a regular
Lagrangian L : E → R. In [71] also he asked about whether it is possible to develop
a formalism similar on Lie algebroids to Klein’s formalism [46] in Lagrangian mechanics.
ThistaskwasobtainedbyEduardoMart´ınezin[57](seealso[56]). Themainnotionisthat
ofprolongationofaLiealgebroidoveramappingintroducedbyHigginsandMackenziein
[51]. A more general situation, the prolongation of an anchored bundle τ : E → Q was
E
also considered by Popescu in [65, 66].
The importance of Lie algebroids in mathematics is beyond doubt and in the last
years Lie algebroids has been a lot of applications in theoretical physics and other re-
lated sciences. More concretely in Classical Mechanics, Classical Field Theory and their
applications. One of the main characteristic concerning that Lie algebroids are interest-
ing in Classical Mechanics lie in the fact that there are many different situations that
can be understand in a general framework using the theory of Lie algebroids as sys-
tems with symmetries, systems over semidirect products, Hamiltonian and Lagrangian
systems, systems with constraints (nonholonomic and vakonomic) and Classical Fields
theory [1, 14, 15, 10, 16, 25, 26, 32, 47, 52, 61].
In [49] M. de Leo´n, J.C Marrero and E. Mart´ınez have developed a Hamiltonian de-
scription for the mechanics on Lie algebroids and they have shown that the dynamics is
obtainedsolvinganequationinthesamewaythaninClassicalMechanics(seealso[56]and
[71]). Moreover, they shown that the Legendre transformation leg :E →E∗ associated
L
totheLagrangianL:E →RinducesaLiealgebroidmorphismandwhentheLagrangian
is regular both formalisms are equivalent.
Marrero and collaborators also have analyzed the case of non-holonomic mechanics on
Lie algebroids [25]. In other direction, in [40] D. Iglesias, J.C. Marrero, D. Mart´ın de
Diego and D. Sosa have studied singular Lagrangian systems and vakonomic mechanics
fromthepointofviewofLiealgebroidsobtainedthroughtheapplicationofaconstrained
variational principle. They have developed a constraint algorithm for presymplectic Lie
algebroids generalizing the well know constraint algorithm of Gotay, Nester and Hinds
[36,37]andtheyalsohaveestablishedtheSkinnerandRuskformalismonLiealgebroids.
Someoftheresultsgivenareasanextensionofthisframeworkforconstrainedsecond-order
systems.
OurframeworkisbasedintheSkinner-Ruskformalismwhichcombinessimultaneously
some features of the Lagrangian and Hamiltonian classical formalisms. The idea of this
formulation was to obtain a common framework for both regular and singular dynamics,
obtaining simultaneously the Hamiltonian and Lagrangian formulations of the dynamics.
Overtheyears,however,SkinnerandRusk’sframeworkwasextendedinmanydirections:
It was originally developed for first-order autonomous mechanical systems [70], and later
generalized to non-autonomous dynamical systems [2, 24, 68], control systems [4] and,
more recently to classical field theories [12, 28].
SECOND-ORDER CONSTRAINED VARIATIONAL PROBLEMS ON LIE ALGEBROIDS 3
Briefly, in this formulation, one starts with a differentiable manifold Q as the con-
figuration space, and the Whitney sum TQ⊕T∗Q as the evolution space (with canon-
ical projections π : TQ⊕T∗Q −→ TQ and π : TQ⊕T∗Q −→ T∗Q). Define on
1 2
TQ⊕T∗Q the presymplectic 2-form Ω = π∗ω , where ω is the canonical symplectic
2 Q Q
formonT∗Q,andobservethattherankofthispresymplecticformiseverywhereequalto
2n. If the dynamical system under consideration admits a Lagrangian description, with
Lagrangian L ∈ C∞(TQ), then one can obtain a (presymplectic)-Hamiltonian represen-
tation on TQ⊕T∗Q given by the presymplectic 2-form Ω and the Hamiltonian function
H =(cid:104)π ,π (cid:105)−π∗L,where(cid:104)·,·(cid:105)denotesthenaturalpairingbetweenvectorsandcovectors
1 2 1
on Q. In this Hamiltonian system the dynamics is given by vector fields X, which are
solutions to the Hamiltonian equation iXΩ = dH. If L is regular, then there exists a
unique vector field X solution to the previous equation, which is tangent to the graph of
the Legendre map. In the singular case, it is necessary to develop a constraint algorithm
in order to find a submanifold (if it exists) where there exists a well-defined dynamical
vector field.
Recently, higher-order variational problems have been studied for their important ap-
plications in aeronautics, robotics, computer-aided design, air traffic control, trajectory
planning, and in general, problems of interpolation and approximation of curves on Rie-
mannian manifolds [6, 11, 39, 45, 50, 62, 60, 63]. There are variational principles which
involves higher-order derivatives by Gay Balmaz et.al., [29, 30, 31], (see also [48]) since
from it one can obtain the equations of motion for Lagrangians where the configuration
spaceisahigher-ordertangentbundle. Morerecently,therehavebeenaninterestinstudy
of the geometrical structures associated with higher order variational problems with the
aim of a deepest understanding of those geometric sructures [20, 23, 67, 58, 42, 43, 44] as
well the relation of higher-order mechanics and graded bundles, [8, 9, 10].
In this work, we study a geometric framework, based on the Skinner and Rusk for-
malism, for constrained second-order variational problems determined by a Lagrangian
function, playing the role of cost function in an optimal control problem, which depends
on derivatives of admissible curves on a Lie algebroid. The strategy is to apply the geo-
metric procedure described above in combination with an extension of the constraint al-
gorithm developed by Gotay, Nester and Hinds [36, 37] in the setting of Lie algebroids
[40]. Our work permits to obtain constrained second-order Euler-Lagrange equations,
Euler-Poincar´e,Lagrange-Poincar´eequationsinanunifiedframeworkandunderstandthe
geometric structures subjacent in second-order variational problems. We show how this
study can be applied to the problem of finding necessary conditions for optimality in
optimal control problems of mechanical system with symmetries, where trajectories are
parameterizedbytheadmissiblecontrolsandthenecessaryconditionsforextremalsinthe
optimal control problem are expressed using a pseudo-Hamiltonian formulation based on
the Pontryagin maximun principle.
The paper is organized as follows. In Section 2 we introduce some known notions con-
cerningLiealgebroidsthatarenecessaryforfurtherdevelopmentsinthiswork. Insection
3wewillusethenotionofLiealgebroidandprolongationofaLiealgebroiddescribedin2
to derive the Euler-Lagrange equations and Hamilton equations on Lie algebroids. Next,
after introduce the constraint algorithm for presymplectic Lie algebroids and study vako-
nomic mechanics on Lie algebroids, we study the geometric formalism for second-order
constrained variational problems using and adaptation of the classical Skinner-Rusk for-
malism for the second-order constrained systems on Lie algebroids. In section 4 we study
optimal control problems of mechanical systems defined on Lie algebroids. Optimality
conditions for the optimal control of the Elroy’s Beanie are derived. Several examples
show how to apply the techniques along all the work.
4 LEONARDOCOLOMBO
2. Lie algebroids and admissible elements
Inthissection,weintroducesomeknownnotionsanddevelopnewconceptsconcerning
Lie algebroids that are necessary for further developments in this work. We illustrate the
theory with several examples. We refer the reader to [13, 51] for more details about Lie
algebroids and their role in differential geometry.
2.1. Lie algebroids, Lie subalgebroids and Cartan calculus on Lie algebroids.
Definition 2.1. Let E be a vector bundle of rank n over a manifold M of dimension
m. A Lie algebroid structure on the vector bundle τ : E → M is a R-linear bracket
E
[[·,·]] : Γ(τ )×Γ(τ ) → Γ(τ ) on the space Γ(τ ), the C∞(M)-module of sections of E,
E E E E
and a vector bundle morphism ρ:E →TM, the anchor map, such that:
(1) The bracket [[·,·]] satisfies the Jacobi identity, that is,
[[X,[[Y,Z]]]]+[[Z,[[X,Y]]]]+[[Y,[[Z,X]]]]=0 ∀X,Y,Z ∈Γ(τ ).
E
(2) If we also denote by ρ : Γ(τ ) → X(M) the homomorphism of C∞(M)-modules
E
induced by the anchor map then
[[X,fY]]=f[[X,Y]]+ρ(X)(f)Y, for X,Y ∈Γ(τ ) and f ∈C∞(M). (1)
E
We will said that the triple (E,[[·,·]],ρ) is a Lie algebroid over M. In this context,
sections of τ , play the role of vector fields on M, and the sections of the dual bundle
E
τE∗ :E∗ →M of 1-forms on M.
Wemayconsidertwotypeofdistinguishedfunctions: givenf ∈C∞(M)onemaydefine
a function f˜on E by f˜=f ◦τ , the basic functions. And, given a section θ of the dual
E
bundleτE∗ :E∗ →M,mayberegardedasalineal function θˆonE asθˆ(e)=(cid:104)θ(τE(e)),e(cid:105)
foralle∈E. Inthissense,Γ(τ )islocallygeneratedbythedifferentialofbasicandlinear
E
functions.
If X,Y,Z ∈Γ(τ ) and f ∈C∞(M), then using the Jacobi identity we obtain that
E
[[[[X,Y]],fZ]]=f[[X,[[Y,Z]]]]+[ρ(X),ρ(Y)](f)Z. (2)
Also, from (1) it follows that
[[[[X,Y]],fZ]]=f[[[[X,Y]],Z]]+ρ[[X,Y]](f)Z. (3)
Then, using (2) and (3) and the fact that [[·,·]] is a Lie bracket we conclude that
ρ[[X,Y]]=[ρ(X),ρ(Y)],
thatis,ρ:Γ(τ )→X(M)isahomomorphismbetweentheLiealgebras(Γ(τ ),[[·,·]])and
E E
(X,[·,·]).
The algebra (cid:76)kΓ(ΛkE∗) of multisections of τE∗ plays the role of the algebra of the
differential forms and it is possible to define a differential operator as follow:
Definition 2.2. If(E,[[·,·]],ρ)isaLiealgebroidoverM,onecanbedefinethedifferential
of E, dE :Γ((cid:86)kτE∗)→Γ((cid:86)k+1τE∗), as follows;
k
dEµ(X0,...,Xk) = (cid:88)(−1)iρ(Xi)(µ(X0,...,X(cid:98)i,...,Xk))
i=0
+ (cid:88)(−1)i+jµ([[X,Y]],X0,...,X(cid:98)i,...,X(cid:98)j,...,Xk),
i<j
for µ∈Γ((cid:86)kτE∗) and X0,...,Xk ∈Γ(τE).
FromthepropertiesofLiealgebroidsitfollowsthatdE isacohomologyoperator,that
is,(dE)2 =0anddE(α∧β)=dEα∧β+(−1)kα∧dEβ,forα∈Γ(ΛkE∗)andβ ∈Γ(ΛrE∗)
(see [51] for more details).
Conversely it is possible to recover the Lie algebroid structure of E from the existence
of an exterior differential on Γ(Λ•τE∗). If f : M → R is a real smooth function, one can
define the anchor map and the Lie bracket as follows:
SECOND-ORDER CONSTRAINED VARIATIONAL PROBLEMS ON LIE ALGEBROIDS 5
(1) dEf(X)=ρ(X)f, for X ∈Γ(τ ),
E
(2) i[[X,Y]]θ=ρ(X)θ(Y)−ρ(Y)θ(X)−dEθ(X,Y)forallX,Y ∈Γ(τE)andθ∈Γ(τE∗).
Moreover, from the last equality, the section θ ∈ Γ(τE∗) is a 1-cocycle if and only if
dEθ=0, or, equivalently,
θ[[X,Y]]=ρ(X)(θ(Y))−ρ(Y)(θ(X)),
for all X,Y ∈Γ(τ ).
E
We may also define the Lie derivative with respect to a section X ∈ Γ(τ ) as the
E
operator LEX :Γ((cid:86)kτE∗)→Γ((cid:86)kτE∗) given by
LEθ=i ◦dEθ+dE◦i θ,
X X X
for θ∈Γ(ΛkτE∗). One also has the usual identities
(1) dE◦LE =LE ◦dE,
X X
(2) LEi −i LE =i ,
X Y X Y [[X,Y]]
(3) LELE −LELE =LE .
X Y Y X [[X,Y]]
Wetakelocalcoordinates(xi)onM withi=1,...,mandalocalbasis{e }ofsections
A
ofthevectorbundleτ :E →M withA=1,...,n,thenwehavethecorrespondinglocal
E
coordinates on an open subset τ−1(U) of E, (xi,yA) (U is an open subset of Q), where
E
yA(e) is the A-th coordinate of e ∈ E in the given basis i.e., every e ∈ E is expressed as
e=y1e (τ (e))+...+yne (τ (e)).
1 E n E
Such coordinates determine the local functions ρi , CC on M which contain the lo-
A AB
cal information of the Lie algebroid structure, and accordingly they are called structure
functions of the Lie algebroid. These are given by
∂
ρ(e )=ρi and [[e ,e ]]=CC e . (4)
A A∂xi A B AB C
These functions should satisfy the relations
∂ρi ∂ρi
ρj B −ρj A =ρi CC (5)
A∂xj B∂xj C AB
and
(cid:88) (cid:20)ρi ∂CBDC +CD CF (cid:21)=0, (6)
A ∂xi AF BC
cyclic(A,B,C)
which are usually called the structure equations.
If f ∈C∞(M),
∂f
dEf = ρi eA, (7)
∂xi A
where {eA} is the dual basis of {eA}. If θ∈Γ(τE∗) and θ=θCeC it follows that
(cid:18) (cid:19)
∂θ 1
dEθ= Cρi − θ CA eB∧eC. (8)
∂xi B 2 A BC
In particular,
1
dExi =ρi eA, dEeA =− CA eB∧eC.
A 2 BC
2.1.1. Examples of Lie algebroids.
Example 1. Given a finite dimensional real Lie algebra g and M = {m} be a unique
point, we consider the vector bundle τ : g → M. The sections of this bundle can be
g
identified with the elements of g and therefore we can consider as the Lie bracket the
structureoftheLiealgebrainducedbyg,anddenotedby[·,·] . SinceTM ={0}onemay
g
consider the anchor map ρ≡0. The triple (g,[·,·] ,0) is a Lie algebroid over a point.
g
Example 2. Consider a tangent bundle of a manifold M. The sections of the bundle
τ :TM →M are the set of vector fields on M. The anchor map ρ:TM →TM is the
TM
identity function and the Lie bracket defined on Γ(τ ) is induced by the Lie bracket of
TM
vector fields on M.
6 LEONARDOCOLOMBO
Example 3. Let φ : M ×G → M be an action of G on the manifold M where G is a
Liegroup. Theinducedanti-homomorphismbetweentheLiealgebrasgandX(M)bythe
action is determined by Φ : g → X(M), ξ (cid:55)→ ξ , where ξ is the infinitesimal generator
M M
of the action for ξ∈g.
The vector bundle τ : M ×g → M is a Lie algebroid over M. The anchor map
M×g
ρ:M×g→TM,isdefinedbyρ(m,ξ)=−ξ (m)andtheLiebracketofsectionsisgiven
M
by the Lie algebra structure on Γ(τ ) as
M×g
[[ξˆ,ηˆ]] (m)=(m,[ξ,η])=[(cid:91)ξ,η]
M×g
form∈M,whereξˆ(m)=(m,ξ),ηˆ(m)=(m,η)forξ,η∈g. Thetriple(M×g,ρ,[[·,·]] )
M×g
is called Action Lie algebroid.
Example 4. Let G be a Lie group and we assume that G acts free and properly on M.
We denote by π : M → M(cid:99)= M/G the associated principal bundle. The tangent lift of
the action gives a free and proper action of G on TM and T(cid:100)M = TM/G is a quotient
manifold. The quotient vector bundle τT(cid:100)M :T(cid:100)M →M(cid:99)where τT(cid:100)M([vm])=π(m) is a Lie
algebroid over M(cid:99). The fiber of T(cid:100)M over a point π(m)∈M(cid:99)is isomorphic to TmM.
TheLiebracketisdefinedonthespaceΓ(τ )whichisisomorphictotheLiesubalgebra
T(cid:100)M
of G-invariant vector fields, that is,
Γ(τ )={X ∈X(M)|X is G-invariant}.
T(cid:100)M
Thus, the Lie bracket on T(cid:100)M is the bracket of G-invariant vector fields. The anchor map
ρ:T(cid:100)M →TM(cid:99)isgivenbyρ([vm])=Tmπ(vm).Moreover,ρisaLiealgebrahomomorpishm
satisfyingthecompatibilityconditionsincetheG-invariantvectorfieldsareπ-projectable.
This Lie algebroid is called Lie-Atiyah algebroid associated with the principal bundle
π:M →M(cid:99).
Let A : TM → g be a principal connection in the principal bundle π : M → M(cid:99) and
B :TM⊕TM →gbethecurvatureofA.Theconnectiondeterminesanisomorphismα
A
between the vector bundles T(cid:100)M → M(cid:99) and TM(cid:99)⊕(cid:101)g → M(cid:99), where (cid:101)g = (M ×g)/G is the
adjoint bundle associated with the principal bundle π:M →M(cid:99)(see [17] for example).
Wechoosealocaltrivializationoftheprincipalbundleπ:M →M(cid:99)tobeU×G,where
U is an open subset of M(cid:99). Suppose that e is the identity of G, (xi) are local coordinates
on U and {ξ } is a basis of g.
A
←−
Denote by {ξ } the corresponding left-invariant vector field on G, that is,
A
←−
ξ (g)=(T L )(ξ )
A e g A
for g∈G where L :G→G is the left-translation on G by g. If
g
(cid:18) ∂ (cid:12) (cid:19) (cid:18) ∂ (cid:12) ∂ (cid:12) (cid:19)
A (cid:12) =AA(x)ξ , B (cid:12) , (cid:12) =BA(x)ξ ,
∂xi(cid:12)(x,e) i A ∂xi(cid:12)(x,e) ∂xj(cid:12)(x,e) ij A
fori,j ∈{1,...,m}andx∈U,thenthehorizontalliftofthevectorfield ∂ isthevector
∂xi
field on π−1(U)(cid:39)U ×G given by
(cid:18) ∂ (cid:19)h ∂ ←−
= −AAξ .
∂xi ∂xi i A
Therefore, the vector fields on U ×G
∂ ←− ←−
e = −AAξ and e =ξ
i ∂xi i A B B
areG-invariantundertheactionofGoverM anddefinealocalbasis{eˆi,eˆB}onΓ(T(cid:100)M)=
Γ(τ ). The corresponding local structure functions of τ :T(cid:100)M →M(cid:99)are
TM(cid:99)⊕˜g T(cid:100)M
Ck = Cj =−Cj =Ci =0, CA =−BA, CC =−CC =cC AB,
ij iA Ai AB ij ij iA Ai AB i
CC = cC , ρj =δ , ρA =ρi =ρB =0,
AB AB i ij i A A
SECOND-ORDER CONSTRAINED VARIATIONAL PROBLEMS ON LIE ALGEBROIDS 7
being{cC }theconstantstructuresofgwithrespecttothebasis{ξ }(see[49]formore
AB A
details). That is,
[[eˆ,xˆ ]] =−BCeˆ , [[eˆ,eˆ ]] =cC ABeˆ , [[eˆ ,eˆ ]] =cC eˆ ,
i j T(cid:100)M ij C i A T(cid:100)M AB i C A B T(cid:100)M AB C
∂
ρ (eˆ)= , ρ (eˆ )=0.
T(cid:100)M i ∂xi T(cid:100)M A
The basis {eˆi,eˆB} induce local coordinates (xi,yi,y¯B) on T(cid:100)M =TM/G.
Next, we introduce the notion of Lie subalgebroid associated with a Lie algebroid.
Definition 2.3. Let (E,[[·,·]] ,ρ ) be a Lie algebroid over M and N is a submanifold of
E E
M. A Lie subalgebroid of E over N is a vector subbundle B of E over N
(cid:31)(cid:127) j (cid:47)(cid:47)
B E
τB τE
N(cid:15)(cid:15) (cid:31)(cid:127) i (cid:47)(cid:47)M(cid:15)(cid:15)
such that ρB =ρE |B:B →TN is well define and; given X,Y ∈Γ(B) and X(cid:101),Y(cid:101) ∈Γ(E)
arbitrary extensions of X,Y respectively, we have that ([[X(cid:101),Y(cid:101)]]E)|N∈Γ(B).
2.1.2. Examples of Lie subalgebroids.
Example 5. LetE beaLiealgebroidoverM.GivenasubmanifoldN ofM,ifB =E |
N
∩(ρ | )−1(TN) exists as a vector bundle, it will be a Lie subalgebroid of E over N, and
N
will be called Lie algebroid restriction of E to N (see [51]).
Example 6. Let N be a submanifold of M. Then, TN is a Lie subalgebroid of TM.
Now, let F be a completely integrable distribution on a manifold M. F equipped with
the bracket of vector fields is a Lie algebroid over M since τ | : F → M is a vector
E F
bundleandifF isafoliation,(Γ(F),[·,·])isaLiealgebra. Theanchormapistheinclusion
i :F →TM (i is a Lie algebroid monomorphism).
F F
Moreover, if N is a submanifold of M and F is a foliation on N, then F is a Lie
N N
subalgebroid of the Lie algebroid τ :TM →M.
TM
Example 7. Let g be a Lie algebra and h be a Lie subalgebra. If we consider the Lie
algebroid induced by g and h over a point, then h is a Lie subalgebroid of g.
Example 8. Let M ×g→M be an action Lie algebroid and let N be a submanifold of
M.LethbeaLiesubalgebraofgsuchthattheinfinitesimalgeneratorsoftheelementsof
h are tangent to N; that is, the application
h→X(N)
ξ(cid:55)→ξ
N
is well defined. Thus, the action Lie algebroid N ×h → N is a Lie subalgebroid of
M ×g→M.
Example 9. Suppose that the Lie group G acts free and properly on M. Let π : M →
M/G = M(cid:99)be the associated G−principal bundle. Let N be a G−invariant submanifold
of M and F be a G−invariant foliation over N. We may consider the vector bundle
N
F(cid:100)N = FN/G → N/G = N(cid:98) and endow it with a Lie algebroid structure. The sections of
F(cid:100)N are
Γ(F(cid:98)N)={X ∈X(N)|X is G-invariant and X(q)∈FN(q),∀q∈N}.
The standard bracket of vector fields on N induces a Lie algebra structure on Γ(F(cid:98)N).
The anchor map is the canonical inclusion of F(cid:98)N on TN(cid:98) and F(cid:98)N is a Lie subalgebroid of
T(cid:100)M →M(cid:99).
8 LEONARDOCOLOMBO
2.2. E-tangent bundle to a Lie algebroid E. We consider the prolongation over the
canonical projection of the Lie algebroid E over M, that is,
TτEE = (cid:91) (E × T E)= (cid:91){(e(cid:48),v )∈E×T E |ρ(e(cid:48))=(T τ )(v )},
ρ TτE e e e e E e
e∈E e∈E
where Tτ :TE →TM is the tangent map to τ .
E E
Infact,TτEE isaLiealgebroidofrank2noverE whereτ(1) :TτEE →E isthevector
E
bundle projection, τE(1)(b,ve) = τTE(ve) = e, and the anchor map is ρ1 : TτEE → TE is
given by the projection over the second factor. The bracket of sections of this new Lie
algebroid will be denoted by [[·,·]] (See [57] for more details).
τ(1)
E
If we denote by (e,e(cid:48),ve) an element (e(cid:48),ve) ∈ TτEE where e ∈ E and where v is
tangent; we rewrite the definition for the prolongation of the Lie algebroid as the subset
of E×E×TE by
TτEE ={(e,e(cid:48),v )∈E×E×TE |ρ(e(cid:48))=(Tτ )(v ),v ∈T E and τ (e)=τ (e(cid:48))}.
e E e e e E E
Thus, if (e,e(cid:48),ve)∈TτEE; then ρ1(e,e(cid:48),ve)=(e,ve)∈TeE, and τE(1)(e,e(cid:48),ve)=e∈E.
Next,weintroducetwocanonicaloperationsthatwehaveonaLiealgebroidE. Thefirst
oneisobtainedusingtheLiealgebroidstructureofE andthesecondoneisaconsequence
ofE beingavectorbundle. Ononehand,iff ∈C∞(M)wewilldenotebyfcthecomplete
lift toE off definedbyfc(e)=ρ(e)(f)foralle∈E. LetX beasectionofE thenthere
exists a unique vector field Xc on E, the complete lift of X, satisfying the two following
conditions:
(1) Xc is τ -projectable on ρ(X) and
E
(cid:91)
(2) Xc(αˆ)=LEα,
X
for every α ∈ Γ(τE∗) (see [33]). Here, if β ∈ Γ(τE∗) then βˆ is the linear function on E
defined by
βˆ(e)=(cid:104)β(τ (e)),e(cid:105), for all e∈E.
E
We may introduce the complete lift Xc of a section X ∈ Γ(τ ) as the sections of τ(1) :
E E
TτEE →E given by
Xc(e)=(X(τ (e)),Xc(e)) (9)
E
for all e∈E (see [57]).
Given a section X ∈ Γ(τ ) we define the vertical lift as the vector field Xv ∈ X(E)
E
given by
Xv(e)=X(τ (e))v, for e∈E,
E e
wherev :E →T E forq=τ (e)isthecanonicalisomorphismbetweenthevectorspaces
e q e q E
E and T E .
q e q
Finally we may introduce the vertical lift Xv of a section X ∈ Γ(τ ) as a section of
E
τ(1) given by
E
Xv(e)=(0,Xv(e)) for e∈E.
With these definitions we have the properties (see [33] and [57])
[Xc,Yc]=[[X,Y]]c, [Xc,Yv]=[[X,Y]]v, [Xv,Yv]=0 (10)
for all X,Y ∈Γ(τ ).
E
If (xi) are local coordinates on an open subset U of M and {e } is a basis of sections
A
ofτ thenwehaveinducedcoordinates(xi,yA)onE. Fromthebasis{e }wemaydefine
E A
a local basis {e(1),e(2)} of sections of τ(1) given by
A A E
(cid:18) ∂ (cid:12) (cid:19) (cid:18) ∂ (cid:12) (cid:19)
e(1)(e)= e,e (τ (e)),ρi (cid:12) , e(2)(e)= e,0, (cid:12) ,
A A A A∂xi(cid:12)e A ∂yA(cid:12)e
for e∈(τ )−1(U) with U an open subset of M (see [49] for more details).
E
From this basis we have that the structure of Lie algebroid is determined by
SECOND-ORDER CONSTRAINED VARIATIONAL PROBLEMS ON LIE ALGEBROIDS 9
(cid:18) ∂ (cid:12) (cid:19) (cid:18) ∂ (cid:12) (cid:19)
ρ (e(1)(e))= e,ρi (cid:12) , ρ (e(2)(e))= e, (cid:12)
1 A A∂xi(cid:12)e 1 A ∂yA(cid:12)e
[[e(1),e(1)]] =CC e(1),
A B τ(1) AB C
E
[[e(1),e(2)]] =[[e(2),e(2)]] =0,
A B τ(1) A B τ(1)
E E
for all A, B and C; where CC are the structure functions of E determined by the Lie
AB
bracket [[·,·]] with respect to the basis {e }.
A
Using {e(1),e(2)} one may introduce local coordinates (xi,yA;zA,vA) on E. If V is a
A A
section of τ(1), locally it is determined by
E
V(x,y)=(xi,yA,zA(x,y),vA(x,y));
therefore the expression of V in terms of the basis {e(1),e(2)} is V =zAe(1)+vAe(2) and
A A A A
the vector field ρ (V)∈X(E) has the expression
1
∂ (cid:12) ∂ (cid:12)
ρ (V)=ρi zA(x,y) (cid:12) +vA(x,y) (cid:12) .
1 A ∂xi(cid:12)(x,y) ∂yA(cid:12)(x,y)
Moreover, if {eA ,eA } denotes the dual basis of {e(1),e(2)},
(1) (2) A A
dTτEEF(xi,yA)=ρi ∂F eA + ∂F eA ,
A∂xi (1) ∂yA (2)
dTτEEeC =−1CC eA ∧eB , dTτEEeC =0.
(1) 2 AB (1) (1) (2)
Example 10. In the case of E = TM one may identify TτEE with TTM with the
standard Lie algebroid structure over TM.
Example 11. Let g be a real Lie algebra of finite dimension. g is a Lie algebroid over a
singlepointM ={q}.Theanchormapofgiszeroconstantfunction,andfromtheanchor
map we deduce that
Tτgg={(ξ ,ξ ,v )∈g×Tg}(cid:39)g×g×g(cid:39)3g.
1 2 ξ1
The vector bundle projection τ(1) : 3g → g is given by τ(1)(ξ ,ξ ,ξ ) = ξ with anchor
g g 1 2 3 1
map ρ (ξ ,ξ ,ξ )=(ξ ,ξ )(cid:39)v ∈T g.
1 1 2 3 1 3 ξ1 ξ1
Let {e } be a basis of the Lie algebra g, this basis induces local coordinates yA on g,
A
that is, ξ=yAe . Also, this basis induces a basis of sections of τ(1) as
A g
(cid:18) (cid:19)
∂
e(1)(ξ)=(ξ,e ,0), e(2)(ξ)= ξ,0, .
A A A ∂yA
Moreover
(cid:18) (cid:19)
∂
ρ (e(1))(ξ)=(ξ,0), ρ (e(2))(ξ)= ξ, .
1 A 1 A ∂yA
The basis {e(1),e(2)} induces adapted coordinates (yA,zA,vA) in 3g and therefore a
A A
section Y on Γ(τ(1)) is written as Y(ξ) = zA(ξ)e(1) +vA(ξ)e(2). Thus, the vector field
g (cid:12) A A
ρ (Y)∈ghastheexpressionρ (Y)=vA(ξ) ∂ (cid:12) . Finally,theLiealgebroidstructureon
1 1 ∂yA(cid:12)ξ
τ(1) is determined by the Lie bracket [[(ξ,ξ˜),(η,η˜)]]=([ξ,η],0).
g
Example 12. We consider a Lie algebra g acting on a manifold M, that is, we have a
Lie algebra homomorphism g → X(M) mapping every element ξ of g to a vector field
ξ on M. Then we can consider the action Lie algebroid E =M ×g. Identifying TE =
M
TM×Tg=TM×2g, an element of the prolongation Lie algebroid to E over the bundle
projection is of the form (a,b,v )=((x,ξ),(x,η),(v ,ξ,χ)) where x∈M, v ∈T M and
a x x x
10 LEONARDOCOLOMBO
(ξ,η,χ)∈3g. The condition Tτ (v)=ρ(b) implies that v =−η (x). Therefore we can
g x M
identify the prolongation Lie algebroid with M ×g×g×g with projection onto the first
two factors (x,ξ) and anchor map ρ (x,ξ,η,χ)=(−η (x),ξ,χ). Given a base {e } of g
1 M A
the basis {e(1),e(2)} of sections of TτM×g(M ×g) is given by
A A
e(1)(x,ξ)=(x,ξ,e ,0), e(2)(x,ξ)=(x,ξ,0,e ).
A A A A
Moreover,
ρ (e(1)(x,ξ))=(x,−(e ) (x),ξ,0), ρ (e(2)(x,ξ))=(x,0,ξ,e ).
1 A A M 2 A A
Finally, the Lie bracket of two constant sections is given by [[(ξ,ξ˜),(η,η˜)]]=([ξ,η],0).
Example13. LetusdescribetheE-tangentbundletoE inthecaseofE beinganAtiyah
algebroid induced by a trivial principal G−bundle π :G×M →M. In such case, by left
trivialization we get the Atiyah algebroid, the vector bundle τ : g×TM → TM. If
g×TM
X ∈X(M) and ξ ∈g then we may consider the section Xξ :M →g×TM of the Atiyah
algebroid by
Xξ(q)=(ξ,X(q)) for q∈M.
Moreover, in this sense
[[Xξ,Yξ]] =([X,Y] ,[ξ,η] ), ρ(Xξ)=X.
g×TM TM g
Ontheotherhand,if(ξ,vq)∈g×TqM,thenthefiberofTτg×TM(g×TM)over(ξ,vq)
is
(cid:26)
Tτg×TM(g×TM)= ((η,u ),(η˜,X ))∈g×T M ×g×T (TM)
(ξ,vq) q vq q vq
(cid:27)
such that u =T τ (X ) .
q vq g×TM vq
ThisimpliesthatTτg×TM(g×TM)maybeidentifiedwiththespace2g×T (TM).Thus,
(ξ,vq) vq
the Lie algebroid Tτg×TM(g×TM) may be identified with the vector bundle g×2g×
TTM →g×TM whose vector bundle projection is
(ξ,((η,η˜),X ))(cid:55)→(ξ,v )
vq q
for (ξ,((η,η˜),X ))∈g×2g×TTM. Therefore, if (η,η˜)∈2g and X ∈X(TM) then one
vq
may consider the section ((η,η˜),X) given by
((η,η˜),X)(ξ,v )=(ξ,((η,η˜),X(v ))) for (ξ,v )∈g×T M.
q q q q
Moreover,
[[((η,η˜),X),((ξ,ξ˜),Y)]] =(([η,ξ] ,0),[X,Y] ),
τ(1) g TM
g×TM
and the anchor map ρ :g×2g×TTM →g×g×TTM is defined as
1
ρ (ξ,((η,η˜),X))=((ξ,η˜),X).
1
2.3. E-tangent bundle of the dual bundle of a Lie algebroid. Let (E,[[ , ]],ρ) be
a Lie algebroid of rank n over a manifold of dimension m. Consider the projection of the
dual E∗ of E over M, τE∗ :E∗ →M, and define the prolongation TτE∗E of E over τE∗;
that is,
TτE∗E = (cid:91) {(e,vµ)∈E×TµE∗ |ρ(e)=TτE∗(vµ)}.
µ∈E∗
TτE∗EisaLiealgebroidoverE∗ofrank2nwithvectorbundleprojectionτE(1∗) :TτE∗E →
E∗ given by τE(1∗)(e,vµ)=µ, for (e,vµ)∈TτE∗E.
Asbefore,ifwenowdenoteby(µ,e,vµ)anelement(e,vµ)∈TτE∗E whereµ∈E∗,we
rewritethedefinitionoftheprolongationLiealgebroidasthesubsetofE∗×E×TE∗ by
TτE∗E ={(µ,e,vµ)∈E∗×E×TE∗ |ρ(e)=(TτE∗)(vµ),vµ ∈TµE∗ and τE∗(µ)=τE(e)}.