AKS systems and Lepage equivalent problems

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by  Santiago Capriotti

AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS
S.CAPRIOTTI
1
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0
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Abstract. The integrable systems known as “AKS systems” admit a natu-

n ral formulation in terms of a Hamiltonian picture. The Lagrangian side of
a these systems are far less known; a version in these terms can be found in
J [4]. Thepurpose ofthese notes intoprovide anovel description ofAKSsys-

6 tems in terms of a variational system more general than usual in mechanics.
Additionally,andusingtechniques borrowed from[5],itwaspossibletobuild

] the Hamiltoniansideof this variational problem, allowingus to establish the
h equivalence withtheusualapproachtotheseintegrable systems.
p
-
h Contents
t
a
m 1. Introduction 1
2. A brief summary of AKS dynamics 2
[
2.1. Symmetries of a factorizable Lie group 2

1 2.2. AKS systems as reduced spaces 3
v 2.3. The dynamical data for AKS systems 3
2
3. Non standard variational problems 4
9
4. Lepage equivalence of variational problems 5
2
1 4.1. (Multi)hamiltonian formalism through Lepagean equivalent problems 5
. 4.2. Lepagean equivalent problems 5
1
0 4.3. Canonical Lepage equivalent of a non standard problem 7
1 5. The Lagrangianside of AKS systems 8
1 5.1. Prolongation and constraints 8
:
v 5.2. The non standard mechanics of AKS 9
i 6. Hamiltonian problem 11
X
6.1. Gotay, Nester and Hinds algorithm 11
r
6.2. Gauge fixing and reduction 15
a
References 19
1. Introduction
Any(regular)LagrangiandynamicalsystemhasaHamiltoniancounterpart. The
usualprescriptionfor the constructionofthis alternate systemrelies heavily in the
notion of Legrendre transformation. On the other side, the integrable systems
knownasAKSdynamicalsystemsaredescribedinthe Hamiltonianside,andsome
interesting Lagrangian versions has been found recently [4]. The following article
describes a kind of Lagrangian setting for the so-called AKS systems. The main
theoretical weapons used in the work are:
• The notion of Lepage equivalent variational problems [5].
1
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 2
• An alternate description of mechanics known as non standard mechanics,
described in [2].
We willsetanonstandardvariationalproblem,inspiredinthe workofFeheret al.
cited above,and use the canonicalLepageequivalent ofthis variationalproblemin
order to provide an “almost Lagrangian” version of AKS systems.
2. A brief summary of AKS dynamics
In the following section we will define what we say when we say “AKS systems”;
theschemepresentedherewasadapted,withminorchanges,fromjointworkofthe
author with H. Montani [3].
2.1. Symmetries of a factorizable Lie group. Let us begin by considering
G×g∗ as a A×B-space, if as above G×g∗ ≃ T∗G via left trivialization and we
lift the A×B-action on G given by
A×B×G→G:(a,b;g)7→agb−1.
By using the facts that the action is lifted and the symplectic form on G×g∗ is
exact, we can determine the momentum map associated to this action; then we
obtain that
J :G×g∗ →b0×a0
(2.1) (g,σ)7→ πb0 Ad♯gσ ,πa0(σ)
(cid:16) (cid:16) (cid:17) (cid:17)
where Ad♯ indicates the coadjoint action of G on g∗. Let us now define the sub-
manifold
Λµν := (g,σ)∈G×g∗ :πb0 Ad♯gσ =µ,πa0(σ)=ν
n (cid:16) (cid:17) o
for each pair µ∈b0,ν ∈a0. We have the following lemma.
Lemma 1. Let σ ∈g0,a∈A,b∈B be arbitrary elements. Then the formulas
± ∓
a·σ+ :=πb0 Ad♯aσ+
(cid:16) (cid:17)
b·σ− :=πa0 Ad♯bσ−
(cid:16) (cid:17)
defines an action of A (resp. B) on a0 (resp. b0); in fact, under the identifications
a∗ ≃b0,b∗ ≃a0
induced by the decomposition g=a⊕b these actions are nothing but the coadjoint
actions of each factor on the dual of its Lie algebras.
Note 2. The symbols OA (resp. OB ) will denote the orbit in b0 (resp. a0) under
σ+ σ−
the actions defined in the previous lemma. Additionally, for each ξ ∈g∗, the form
ξ♭ ∈ g∗ is given by ξ♭ := B(ξ,·), where B(·,·) is the invariant bilinear form on g∗
by the Killing form.
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 3
2.2. AKS systems as reduced spaces. Therefore we will have that Λ =
µν
J−1(µ,ν), and taking into account the M-W reduction (see [1]) the projection
ofΛ onΛ /A ×B ispresymplectic,andthe solutioncurvesforthe dynamical
µν µν µ ν
system defined there by the invariant Hamiltonian H(g,σ) := 1σ σ♭ are closely
2
related with the solution curves of the system induced in the quotient. To work
(cid:0) (cid:1)
with these equations, let us introduce some convenient coordinates; the map
Lµν :Λµν →OµA×OνB :(g,σ)7→ πb0 Ad♯g−1µ ,πa0 Ad♯g−ν
(cid:18) (cid:18) + (cid:19) (cid:16) (cid:17)(cid:19)
7→ πb0 Ad♯g−σ ,πa0 Ad♯g−σ ,
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
where g = g g , induces a diffeomorphism on Λ /(A ×B ). If (g,ξ;σ,η) is
+ − µν µ ν
a tangent vector to G×g∗ (all the relevant bundles are left trivialized) then the
derivative of L can be written as
µν
(2.2) (Lµν)∗(cid:12)(g,σ)(g,σ;ξ,η)=(cid:18)−πb0(cid:18)ad♯ξ+Ad♯g+−1µ(cid:19),πa0(cid:16)ad♯ξ−Ad♯g−ν(cid:17)(cid:19)
= πb0 (cid:12)ad♯ξ−Ad♯g−σ +πb0 Ad♯g−η ,πa0 ad♯ξ−Ad♯g−σ +πa0 Ad♯g−η
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
if and only if g = g+g−,ξ+ = πa Adg−ξ ,ξ− = πb Adg−ξ . So the following
remarkable result is true.
(cid:0) (cid:1) (cid:0) (cid:1)
Proposition 3. Let OA ×OB be the phase space whose symplectic structure is
µ ν
ω =ω −ω , where ω are the corresponding Kirillov-Kostant symplectic struc-
µν µ ν µ,ν
tures on each orbit. If i :Λ ֒→G×g∗ is the inclusion map, then we will have
µν µν
that
i∗ ω =L∗ ω .
µν µν µν
A proof of this proposition can be found in [3].
2.3. The dynamical data for AKS systems. Therefore OA × OB with the
µ ν
symplecticstructureω issymplectomorphictothereducedspaceassociatedtothe
µν
A×B-actiondefined aboveon G×g∗. As was pointedout before,the hamiltonian
H(g,σ)= 1σ σ♭ is invariantfor this action,andthis implies thatthe solutions of
2
the dynamical system defined by such a hamiltonian on G×g∗ are in one-to-one
(cid:0) (cid:1)
correspondence with those of the dynamical system induced on OA ×OB by the
µ ν
hamiltonian H [1] defined through
µν
i∗ H=L∗ H .
µν µν µν
Let us note now that if L (g,σ)=(ω ,ω ), then Ad♯ σ =ω +ω , and so
µν 1 2 g− 1 2
♭
1
H (ω ,ω )= Ad♯ (ω +ω ) Ad♯ (ω +ω )
µν 1 2 2(cid:18) g−−1 1 2 (cid:19)"(cid:18) g−−1 1 2 (cid:19) #
1
= 2 Ad♯g−−1(ω1+ω2) Adg−−1(ω1+ω2)♭
(cid:18) (cid:19)h i
1 1
= ω ω♭ + ω ω♭ +ω ω♭ .
2 1 1 2 2 2 1 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 4
Therefore
dHµν|(ω1,ω2) πb0 ad♯ξω1 ,πa0 ad♯ζω2 =
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
=πb0 ad♯ξω1 ω1♭ +ω2♭ +πa0 ad♯ζω2 ω1♭ +ω2♭
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
and the hamiltonian vector field will be given by
(2.3) VHµν (ω1,ω2) =(cid:18)πb0(cid:18)ad♯πa(ω1♭+ω2♭)ω1(cid:19),πa0(cid:18)ad♯πb(ω1♭+ω2♭)ω2(cid:19)(cid:19).
(cid:12)
In generalthe sol(cid:12)utionof dynamical systems via reductionhas a definite direction,
opposite to the adopted by us in this discussion: That is, it is expected that the
reduced system is easiest to solve, because it involves less degrees of freedom, and
the solution of the original system is found by lifting the solution of the reduced
system. In case of AKS systems, we proceed in the reverse direction: To this end
let us note that the dynamicalsystem on G×g∗ determined by H has hamiltonian
vector field given by
(2.4) VH| = σ♭,−ad♯ σ = σ♭,0
(g,σ) σ♭
(cid:16) (cid:17) (cid:16) (cid:17)
because of the invariance condition ad♯ξ♭ = 0 for all ξ ∈ g. Then the solution in
ξ
the unreduced space passing through (g,σ) at the initial time is
t7→ gexptσ♭,σ ,
(cid:16) (cid:17)
and if this initial data verifies πb0 Ad♯gσ = µ,πa0(σ) = ν, then this curve will
belong to Λµν for all t. Therefore th(cid:16)e map(cid:17)
t7→ πb0 Ad♯(g+(t))−1µ ,πa0 Ad♯g−(t)ν ,
whereg :R→G aret(cid:16)hecu(cid:16)rvesdefinedb(cid:17)ythefa(cid:16)ctorizatio(cid:17)n(cid:17)problemg (t)g (t)=
± ± + −
gexptσ♭,issolutionforthedynamicalsystemassociatedtothevectorialfield(2.3),
which is more difficult to solve than the original system.
3. Non standard variational problems
Wewillconsiderthatavariational problem [5]isatriple(Λ,λ,I),wherep:Λ→
M is a fiber bundle on an n-dimensional manifold M, λ is an n-form on Λ, and
I ⊂Ω•(Λ)is anEDS there. A dynamicalsystemcanbe attachedto these data by
selecting an intermediate bundle
Λ→Λ →M;
1
for example, the variational problem for classical mechanics can be obtained from
the variational problem TQ×I →I,Ldt, dqi−vidt by picking up the in-
diff
termediate bundle TQ×I →Q×I →I. The new data allow us to formulate the
(cid:0) (cid:10) (cid:11) (cid:1)
variational problem of classical mechanics in the following form: To find a section
of Q×I →I such that it is an extremal of the action
S[γ]:= γ˙∗(Ldt),
ZI
where γ˙ :I →TQ×I is a section satisfying the folllowing requeriments
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 5
• It makes commutative the diagram
TQ×I
(cid:0)(cid:18)
γ˙ (cid:0)
(cid:0) τQ×id
(cid:0) ?
-
I Q×I
γ
• It is an integral section for dqi−vidt .
diff
Theintroductionofanintermediatebundle inavariationalproblem(Λ,λ,I) allow
(cid:10) (cid:11)
ustoconsidertheEDSI asaprolongation structure,andthusprovidingconditions
that associate to every section of this intermediate bundle a section of the full
bundle.
4. Lepage equivalence of variational problems
Letusintroduceamethodtoworkwithavariationalproblemofthegeneralkind
thatweareconsideringhere. Thecontentsofthefollowingsectionareadaptedfrom
the author’s work [2].
4.1. (Multi)hamiltonian formalism through Lepagean equivalent prob-
lems. Wewanttoconstructahamiltonianversionforthenonstandardvariational
problem. The usualapproach[6] seems useless here,because ofthe following facts:
• The covariantmultimomentum space is a bundle in some sense dualto the
velocity space, which is a jet space.
• ThedynamicsinthemultimomentumspaceisdefinedthroughtheLegendre
transform, and it is not easy to generalize to a non standard problem this
notion.
The trick to circumvect the difficulties is to mimic the passage from Hamilton’s
principle to Hamilton-Pontryaguin principle. This is done by including the gener-
ators of the prolongationstructure in the lagrangiandensity by means of a kind of
Lagrange multipliers. This procedure will be formalized below, where the hamil-
tonianversionis defined by associatinga firstordervariationalproblemto the non
standard variational problem, whose extremals are in one to one correspondence
with its extremals. This is called canonical bivariant Lepage equivalent problem.
4.2. Lepagean equivalent problems. Here we will follow closely the exposition
of the subject in the article [5]. Before going into details, let us introduce a bit of
terminology: If Λ −π→ M is a bundle, λ ∈ Ωn(Λ) (n = dimM) and I is an EDS
π
on Λ, the symbol Λ−→M,λ,I indicates the variational problem consisting in
extremize the actio(cid:16)n (cid:17)
S[σ]= σ∗(λ)
ZM
with σ ∈ Γ(Λ) restricted to the set of integral sections of I. Furthermore, E(λ)
π
will denote the set of extremals for Λ−→M,λ,I .
The idea is to eliminate in some(cid:16)way the constr(cid:17)aints imposed by the elements
of I; intuitively, it is expected that the number of unknown increase when this is
done. The following concept captures these ingredients formally.
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 6
Definition 4 (Lepageequivalentvariationalproblem). ALepagean equivalent ofa
π
variational problem Λ−→M,λ,I is another variational problem
(cid:16) (cid:17)
Λ˜ −ρ→M,λ˜,{0}
together with a surjective subme(cid:16)rsion ν :Λ˜ →Λ (cid:17)such that
• ρ=π◦ν, and
• if γ ∈Γ Λ˜ is such that ν◦γ is an integral section of I, then
(cid:16) (cid:17) γ∗λ˜ =(ν◦γ)∗λ.
There exists a canonical way to build up a Lepage equivalent problem associ-
π
atedtoagivenvariationalproblem Λ−→M,λ,I ,thesocalledcanonicalLepage
equivalent problem. Let I be diffe(cid:16)rentially genera(cid:17)ted by the sections of a graded
subbundle I ⊂ •(T∗Λ) (this is a “constant rank” hypothesis, ensuring the exis-
tence of a bundle inthe construction,see below). Define Ialg asthe algebraicideal
in Ω•(Λ) generaVted by Γ(I), and
Ialg l :=Ialg∩Ωl(Λ).
For λ ∈ Ωn(Λ), define the a(cid:0)ffine(cid:1)subbundle Wλ ⊂ n(T∗Λ) whose fiber above
p∈Λ is
Wλ := λ| + β| :β ∈ IalgVn .
p p p
Definition 5 (Canonical (cid:12)Lepagne equivalent pro(cid:0)blem(cid:1)).oIn the previous setting,
(cid:12)
it is the triple Wλ −ρ→M,Θ˜,{0} , where ν is the canonical projection τ¯n :
Λ
n(T∗Λ)→Λ r(cid:16)estricted to Wλ, ρ:(cid:17)=π◦ν and Θ˜ is the pullback of the canonical
n-form
V Θ | :=α◦(τ¯n)
n α Λ ∗
to Wλ. The form Θ˜ will be called Cartan form of the variational problem.
Remark 1. It is worth remarking that our terminology was brought from [5],
whichis slightly differentfrom the classicaltheory as exposedin e.g. [8]. It is fully
explained in the former work how to be relate both approaches.
Returningtoourmainconcern,itcanbeprovedthatthecanonicalLepageequiv-
π
alent is a Lepagean equivalent problem of Λ−→M,λ,I . Now, the extremals of
some variational problem has, in general,(cid:16)nothing to do (cid:17)with the extremals of its
Lepageanequivalentproblem,soitisnecessarytointroducethefollowingdefinition.
Definition 6 (Covariant and contravariant Lepage equivalent problems). We say
that a Lepageanequivalentproblem Wλ −ρ→M,Θ˜,{0} for the variationalprob-
lem Λ−π→M,λ,I iscovariant ifν(cid:16)◦γ ∈E(λ)forallγ(cid:17)∈E Θ˜ ;onthecontrary,
it is(cid:16)called contrava(cid:17)riant if every σ ∈ E(λ) is the projection(cid:16)of(cid:17)some extremal in
E Θ˜ through ν. A Lepagean equivalent problem is bivariant if and only if it is
bo(cid:16)th(cid:17)covariantand contravariant.
Thereexistsafundamentalrelationbetweentheextremalsofavariationalprob-
lem and the extremals of its canonical Lepage equivalent.
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 7
Theorem 7. The canonical Lepage equivalent is covariant.
For a proof, see [5]. The contravariantnature of a Lepage equivalent problem is
more subtle to deal with; in fact, it must be verified in each case separately.
4.3. CanonicalLepageequivalentofanonstandardproblem. Wewillapply
these considerations to our problem. The important thing to note is that, if a
variationalproblemhasacovariantandcontravariantLepageanequivalentproblem,
thenthe latter canbe consideredas a kindofHamilton-Pontryaguin’sprinciple for
thegivenvariationalproblem;infact,itisshownbelowthatthecanonicalLepagean
equivalentproblemassociatedtothevariationalproblemunderlyingtheHamilton’s
principle gives rise to the classical Hamilton-Pontryaguin’s principle. This will be
our starting point for assigning a multisymplectic space to the variationalproblem
we aredealing with. The bivarianceensuresus that everyextremalhasbeen taken
intoaccountinthenewsetting. Otherwise,namely,fornoncontravariantcanonical
Lepage equivalent problems, some extremals for the original variational problem
could be lost in the process.
Soletussupposethatwehavethenonstandardproblemdefinedbythefollowing
data
Λ→Λ →M,
1
I ⊂Ω•(Λ),

S[σ]:= (prσ)∗(λ).
M
IfI inthisnonstandardproblemhastRherequiredregularity(i.e. a“constantrank”
hypothesis,see reference[2]), thenthe variationalproblem(Λ→M,λ,I) will have
a canonical Lepage equivalent problem W˜λ →M,Θ˜,{0} , and we can apply the
scheme described above. In order to ca(cid:16)rry out this task l(cid:17)ocally, let us suppose as
above that the fibers of the bundle I on an open set U ⊂Λ can be written as
I| =R α1 ,··· ,α1 ,α2 ,··· ,α2 ,··· ,αp| ,··· ,αp , γ ∈U
γ 1 γ k1 γ 1 γ k2 γ 1 γ kp γ
(cid:28) (cid:12) (cid:29)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
so that the prol(cid:12)ongation str(cid:12)uctur(cid:12)e I is (differ(cid:12)entially) generated by(cid:12)
I = αj :1≤i≤p,1≤j ≤k
i i
diff
D E
onU,where αj,··· ,αj =I(j) =:I∩Ωj(U)andαj ∈Z (Λ)foralli,j. Then
1 kj diff i 1
ifU0 :=π(UD)⊂M,wecaEndefineforeach1≤l≤pthenumbersml :=dim(M)−l
and the n-form on Λ˜ :=U × p [ ml(T∗U )]⊕kl will reads
U U0 l=1 0
L V
p kl
λ˜ := αl ∧βj −λ
 j ml
l=1 j=1
X X
 
where βm11,··· ,βmk11,··· ,βm1p,··· ,βmkpp denotessectionsof ml(T∗U0); thesub-
script i(cid:16)n these sections thus indicates th(cid:17)eir degree (in the exterior algebra sense).
V
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 8
It can be shown in the examples below that, in many important cases, the Euler-
Lagrange equations associated to the non standard problem defined by the data
id
Λ˜ −→Λ˜ →M,
U U
0⊂Ω• Λ˜ ,
U
S[σ]:=(cid:16) ((cid:17)prσ)∗λ˜,
M
has a family of solutions which is isoRmorphic to the family of solutions of the
previous system; this means that in these cases the canonical Lepage equivalent
problem is also contravariant. The Euler-Lagrange eqs for an stationary section
σ ∈Γ Λ˜ are
U
(cid:16) (cid:17)
(4.1) σ∗ Vydλ˜ =0, ∀V ∈Γ VΛ˜
U
(cid:16) (cid:17) (cid:16) (cid:17)
because there are no conditions for admisibility of variations; these sections are
then integral sections for the EDS
(4.2) I := Vydλ˜ :V ∈Γ VΛ˜ .
U
diff
D (cid:16) (cid:17)E
We call this EDS the (local version of) Hamilton-Cartan EDS.
5. The Lagrangian side of AKS systems
5.1. Prolongation and constraints. The purpose of this section is to use our
description of dynamical systems as a way of thinking about AKS systems. In
order to motivate the essential ideas, let us consider the free dynamical system on
R described by the lagragianI×TR×R
1
L| := v2−v w dt
(t;q,v;w) 2 0
(cid:20) (cid:21)
and the prolongationstructure
θ| :=dq−(v−w)dt.
(t;q,v;w)
This means that the allowed variations must keep invariant the equation dδq −
(δv−δw)dt=0; if we vary the curve γ :t7→(t;q(t),v(t);w(t)), the variation of
the associated action will be (by neglecting boundary terms)
δS = γ∗(vδv−v δw)dt
0
ZI
= γ∗[v(dδq+δwdt)−v δw]
0
ZI
= γ∗(δqdv+(v−v )δwdt).
0
ZI
So the Euler-Lagrange equations for this dynamical system will be
v˙ =0,
(v =v0.
This oversimplificatedexampleclearlyshowsafundamental characteristicinvolved
with this kind of modifications in the prolongation structure: The ability of intro-
ducing constraints on the velocities.
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 9
On the other hand, for G a factorizable Lie group G = AB, and taking into ac-
countG×g∗ withthecanonicalsymplecticstructure(usinglefttrivialization). The
assumed factorization induces the decompositions
g=a⊕b
g∗ =a0⊕b0
of the Lie algebra and its dual, where (·)0 indicates annihilator. Let πa0,πb0 be
the projections associated with the decomposition on g∗; additionally, let us take
µ∈b0,ν ∈a0. Then we can define the submanifold
Λ˜µν := (g,ζ)∈G×g∗ :πb0 Ad∗g−1ζ =µ,πa0(ζ)=ν .
Wewillseelaterthat(cid:8)thisisafirstclasssub(cid:0)manifold(cid:1),andthattheker(cid:9)neloftheform
induced by the canonical form on it by pullback is generated by the infinitesimal
generators of the action of A ×B on G given by
µ ν
(a,b)·g :=agb−1
lifted to the cotangent bundle. Here A (resp. B ) are the isotropy groups of µ
µ ν
(resp. ν) for the actions
a·φ:=πb0(Ad∗a−1φ), ∀φ∈b0,
b·ψ :=πa0(Ad∗b−1ψ), ∀ψ ∈a0.
Quotient out by this action we can obtain a symplectic manifold, whose equations
of motion are nothing but the equations of an AKS system.
5.2. The non standard mechanics of AKS. This section is inspired in the
article of Fèher et al. [4]. Let us consider as velocity space the bundle
Λ:=TG⊕a ⊕b
G G
where,asbefore,wesupposethatGadmitsadecompositionG=AB andthefiber
bundle on G denoted by a and b are simply G×a and G×b; a point in this
G G
space (with the left trivialization for TG) will be (g,J;α,β). There we will define
the 1−form g−valued
θ|(g,J;α,β) := λ|g− J −Adg−1α−β dt,
(hereλistheleftMaurer-Cartanform)d(cid:0)efiningtheprolon(cid:1)gationstructure,andwe
will consider the lagrangian
1
L | := B(J,J)−µ(α)−ν(β) dt
µν (g,J;α,β) 2
(cid:20) (cid:21)
where µ ∈ b0,ν ∈ a0 and, as pointed out before, (·)0 is the annihilator of the
corresponding subalgebra.
The Euler-Lagrangeequations can be obtained performing variations of the action
S [γ]:= γ∗(L dt)
µν µν
ZI
where γ : t 7→ (t;g(t),J(t);α(t),β(t)) represents a curve in Λ. Concretely, a
variation of such a curve is a vector field
δγ :(t;g,J;α,β)7→(0;g,ξ,J,δJ;α,δα,β,δβ)
c
AKS SYSTEMS AND LEPAGE EQUIVALENT PROBLEMS 10
for certain funtions ξ : Λ → g,δJ : Λ → g,δα : Λ → a,δβ : Λ → b (we are
considering here TG left trivialized as before) such that
γ∗ Lcθ =0.
δγ
(cid:16) (cid:17)
Fromthis lastconditionwewillobtainthatonγ the mapsdescribingthe variation
must obey the relation
(5.1) dξ+[λ,ξ]− δJ −Adg−1(δα)− Adg−1α,ξ −δβ dt=0.
Therefore (cid:0) (cid:2) (cid:3) (cid:1)
δS [γ]= γ∗ δγ·L dt
µν µν
ZI (cid:16) (cid:17)
where
c
δγ·L =B(δJdt,J)−[µ(δα)+ν(δβ)]dt
µν
=B dξ+ λ+ Adg−1α dt,ξ ,J +
c
(cid:0)+ B A(cid:2)dg−1(cid:0)δα+δβ(cid:1),J −(cid:3)µ(δ(cid:1)α)−ν(δβ) dt.
Each φ ∈ g∗ and κ ∈ Ω1(cid:2)(Λ,(cid:0)g) allow us to de(cid:1)fine the 1−form g(cid:3)∗−valued ad♯φ
κ
through
ad♯φ (ξ):=φ([ξ,κ]), ∀ξ ∈g;
κ
letus indicate withthe(cid:16)symbo(cid:17)lξ♭ the formB(ξ,·), andletus denote by Ad♯ :G→
Hom(g∗) the coadjoint representation,defined through
Ad♯gφ (ξ)=φ Adg−1ξ
for all ξ ∈ g,φ ∈ g∗,g ∈ G.(cid:16) With(cid:17)this not(cid:0)ation, th(cid:1)e element of g∗ obtained by
contracting a vector v tangent to Λ with the 1−form satisfy the equation
ad♯φ (v)=ad♯ φ,
κ κ(v)
(cid:16) (cid:17)
where the element in the right hand side is the infinitesimal generator for the
coadjoint action associated to κ(v)∈g in the point φ∈g∗. Then the variations in
δα,δβ will give us the equations
πb0 Ad♯gJ♭−µ =0,πa0 J♭−ν =0
(cid:16) (cid:17) (cid:16) (cid:17)
respectively, and the variation along TG (compatible with the restriction (5.1))
yields to
dJ♭+ad♯ J♭ =0.
(λ+Adg−1αdt)
(cid:16) (cid:17)
The solutionsofthe Euler-Lagrangeequationsareintegralsections ofI×Λ forthe
EDS generated by
dJ♭−ad♯ J♭ ,
βdt
λ− J −Adg(cid:0)−1α(cid:1)−β dt,
πb0 (cid:0)Ad♯gJ♭ −µ, (cid:1)
πa0(cid:16)J♭ −ν(cid:17)
satisfyingthe independence cond(cid:0)itio(cid:1)ndt6=0;herewewillassumethatad♯ζdtζ♭ =0
for all ζ ∈g.