Separability preserving Dirac reductions of Poisson pencils on Riemannian manifolds

25 Pages·0.27 MB·

by  Maciej Blaszak

Separability preserving Dirac reductions of
Poisson pencils on Riemannian manifolds
3
0
0 Maciej B laszak∗†
2
Institute of Physics, A. Mickiewicz University

n
Umultowska 85, 61-614 Poznan´, Poland
a
J
Krzysztof Marciniak∗‡
7
Department of Science and Technology
1
Campus Norrk¨oping, Link¨oping University

]
I 601-74 Norrk¨oping, Sweden
S
.
n February 8, 2008
i
l
n
[

1 Abstract
v
DiracdeformationofPoissonoperatorsofarbitraryrankisconsidered.
0
The question when Dirac reduction does not destroy linear Poisson pen-
2
cils is studied. A class of separability preserving Dirac reductions in the
0
1 corresponding quasi-bi-HamiltoniansystemsofBenentitypeisdiscussed.
0 Two examples of such reductions are given. This paper will appear in J.
3 Phys. A: Math. Gen.
0
/ AMS 2000 Subject Classification: 70H45,70H06,70H20,53D17,70G45
n
i
l
n 1 Introduction
:
v
i Recentlyanew(quasi)-bi-HamiltonianseparabilitytheoryofLiouvilleintegrable
X
finite dimensional systems was constructed [1]-[15] based on general properties
r
of Poissonpencils on manifolds. A natural further step within this theory is to
a
investigate admissible integrable and separable reductions of these integrable /
separable systems onto appropriate submanifolds. The most natural approach
seemstobetheonebasedontheDiractheoryofconstraineddynamics[16],[17].
The presented paper contains only some special cases of such reductions, but
eveninthesecasestheproblemisfarfrombeingtrivial. Thedifficulties wemet
duringourresearchinclinedustoreconsidertheDiracformalismfromthepoint
∗PartiallysupportedbySwedishResearchCouncilGrantNo. 629-2002-6681
†PartiallysupportedbyKBNgrantNo. 5P03B 00420
‡On leave of absence from Department of Physics, A. Mickiewicz University, Poznan´,
Poland.
1
of view of Poisson bivectors rather then from the point of view of constrained
dynamics.
The paper is organized as follows. In this introductory part we remind ba-
sic concepts of Poisson geometry and of (quasi-)bi-Hamiltonian systems. In
Section 2 we formulate the theory of Dirac reductions of Poisson brackets in
terms of Poisson bivectors. Our construction is more general than usually met
in literature since we consider reduction procedure on the whole foliation of
submanifolds. Here we also explain that our approach indicates that Dirac
classification of constraints onto those of ”first” and of ”second class” requires
further discussion as our further results show. In Section 3 we review the re-
cent results on separability theory of quasi-bi-Hamiltonian systems of Benenti
type. In section 4 we perform Dirac reduction of Poisson pencil and corre-
spondingquasi-bi-HamiltonianchainofBenentitypeontoaparticularlychosen
submanifold. Chosen constraints preserve the Liouville integrability as well as
the coordinates of separation of considered system. The main obstacle of such
a choice is that these constraints are nonexpressible in natural (original) coor-
dinates. Hence, in Section 5 we modify constraints introduced in the previous
section and obtain an equivalent reduction, that is expressible directly in origi-
nal coordinates. Finally, in Section 6 we illustrate the results by two nontrivial
examples of constrained separable dynamics.
Letus firstremindfew basicfacts fromPoissongeometry. Givenamanifold
M, a Poisson operator π on M is a mapping π : T∗M → TM that is fibre-
preserving (i.e. π|Tx∗M : Tx∗M → TxM for any x ∈ M) and such that the
inducedbracketonthe spaceC∞(M)ofallsmoothreal-valuedfunctions onM
{.,.} :C∞(M)×C∞(M)→C∞(M) , {F,G} d=ef hdF,πdGi (1)
π π
(whereh.,.iisthedualmapbetweenTMandT∗M)isskew-symmetricandsat-
isfiesJacobiidentity(thebracket(1)alwayssatisfiestheLeibnizrule{F,GH} =
π
G{F,H} +H{F,G} ). Throughoutthewholepaperthesymboldwilldenote
π π
the operator of exterior derivative. The operator π can always be interpreted
as a bivector, π ∈Λ2(M) and in a given coordinate system (x1,...,xm) on M
we have
m
∂ ∂
π = πij ∧ .
∂x ∂x
i j
i<j
X
A function C : M → R is called Casimir function of the Poisson operator π
if for arbitrary function F : M → R we have {F,C} = 0 (or, equivalently, if
π
πdC =0). Alinearcombinationπ =π −ξπ (ξ ∈R)oftwoPoissonoperators
ξ 1 0
π and π is called Poisson pencil if the operator π is Poisson for any value
0 1 ξ
of the parameter ξ. In this case we say that π and π are compatible. Given
0 1
a Poisson pencil π = π −ξπ we can often construct a sequence of vector
ξ 1 0
fieldsY onMthathaveatwofoldHamiltonianform(asocalledbi-Hamiltonian
i
chain)
Y =π dh =π dh (2)
i 1 i 0 i+1
where h : M → R are called Hamiltonians of the chain (2) and where i is
i
2
some discrete index. This sequence of vector fields may or may not truncate
(depending on existence of Casimir functions). In the case when the Poisson
pencil π is degeneratedbut projectable into a symplectic leaf N (of dimension
ξ
2n) of π the bi-Hamiltonian chain (2) on M turns into a so called quasi-bi-
0
Hamiltonian chain on N of the form
n
θ dH =θ dH + α θ dH , i=1,...,n, H ≡0, (3)
1 i 0 i+1 ij 0 j n+1
jX6=j=i+11
where θ are projections of π onto N, the functions H are restrictions of h
i i j j
to N: H = h | , and α are some multipliers (real functions). And vice
j j N ij
versa: having a quasi-bi-Hamiltonian chain (3) on the manifold N one can
lift it to a bi-Hamiltonian chain (2) on the extended manifold M. (Quasi-)bi-
Hamiltonian chains (called also (quasi-)bi-Hamiltonian systems) possess very
interesting differential-algebraic properties and are one of key notions in the
theory of integrable systems, due to the fact that in many cases the systems
(2) and (3) are Liouville integrable [18]. Recently, much effort has been spent
in order to exploit the procedure of solving these systems by the method of
separationofvariables[2]-[15]. Inthisarticlewewillmainlyworkwiththequasi-
bi-Hamiltonian chains (3) rather than bi-Hamiltonian ones, since the pencil
θ =θ −ξθ is always non-degenerated.
ξ 1 0
2 Dirac reduction of Poisson bivectors
WebeginbyconsideringtheDiracreductionprocedureinamoregeneralsetting
that is usually met in literature. Let π be a Poissonbivector,in general degen-
erated on some manifold M. Let S =F be a submanifold in a foliation F of
0
the manifold M defined by m functionally independent functions (constraints)
ϕ :M→R, i=1,...,m.
i
F ={x∈M :ϕ (x)=s , i=1,...,m}
s i i
Thus,S is asubmanifoldofcodimensionm inM.Moreover,letZ , i=1,...,m
i
be some vectorfields transversalto F , spanninga regulardistributionZ inM
s
ofconstantdimensionm(thatisasmoothcollectionofm-dimensionalsubspaces
Z ⊂ T M at every point x in M). The word’transversal’means here that no
x x
vector field Z is at any point tangent to the submanifold F passing through
i s
this point. Hence, the tangent bundle TM splits into a direct sum
TM=TF ⊕Z
(whichmeansthatatanypointxinMwehaveT M=T F ⊕Z withssuch
x x s x
that x∈F ) and so does its dual
s
T∗M=T∗F ⊕Z∗,
3
whereT∗F istheannihilatorofZ andZ∗ istheannihilatorofTF. Thatmeans
that if α is a one form in T∗F then α(Z ) = 0 for all i = 1,...,m and if β is
i
a one-form in Z∗ then β vanishes on all vector fields tangent to F . Moreover,
s
weassumethatthe vectorfieldsZ whichspanZ arechoseninsuchawaythat
i
dϕ , i=1,...,m is a basis in Z∗ that is dualto the basis Z of the distribution
i i
Z,
hdϕ ,Z i=Z (ϕ )=δ , (4)
i j j i ij
(this is no restriction since for any distribution Z transversal to F we can
s
chooseits basis so that (4) is satisfied). Finally, let us define m vector fields X
i
on M and m2 functions ϕ :M→R on M through
ij
X =π(dϕ ), ϕ ={ϕ ,ϕ } = hdϕ ,πdϕ i=X (ϕ ). (5)
i i ij i j π i j j i
The functions ϕ define an m-dimensional skew-symmetric matrix ϕ = (ϕ ),
ij ij
i,j =1,...m. It can be easily shown that
[X ,X ]=X =π d{ϕ ,ϕ } =πdϕ , (6)
j i {ϕi,ϕj}π i j π ij
where{.,.} isaPoissonbracketdefinedbyourPoissonbivectorπand[X,Y]=
π
L Y =X(Y)−Y(X)istheLiebracket(commutator)ofthevectorfieldsX,Y.
X
A very special choice of our transversalvector fields Z originates by taking
i
linear combinations of fields X with coefficients being the entries of the matrix
j
ϕ−1.
m
Z = (ϕ−1) X i=1,...,m. (7)
i ji j
j=1
X
Since the constraintfunctions ϕ arefunctionally independent, the vectorfields
i
Z in(7)willindeedbetransversaltothefoliationF. Moreover,theywillauto-
i
maticallysatisfytheorthogonalitycondition(4)asZ (ϕ )= m (ϕ−1) X (ϕ )
i j k=1 ki k j
= m (ϕ−1) ϕ =δ .
k=1 ki jk ij P
Letusnowconsiderthefollowingdeformation (modification)ofthebivector
P
π:
m
π =π− 1 X ∧Z , (8)
D 2 i i
i=1
X
where ∧ denotes the wedge product in the algebra of multivectors. This new
bi-vector π can be properly restricted to F for any s∈Rm (and thus also to
D s
S =F ), since the image of π consideredon a givenleaf F of the foliation F
0 D s
lies in TF for all s. This is the content of the following theorem.
s
Theorem 1 Suppose x ∈ F . Then for any α ∈ T∗M the vector π (α) is
s x D
tangent to F i.e. π (T∗M)⊂T F .
s D x x s
Proof. We will show, that π (dϕ ) = 0, k = 1,...,m, since it means that
D k
the constraints ϕ , i = 1,...,m are Casimirs of π (and so are then ϕ −s )
i D i i
which obviously implies the thesis of the theorem. Using the definition (8) of
4
π , the obvious fact that (X ∧ Z )dϕ = (X ⊗ Z )dϕ − (Z ⊗ X )dϕ =
D i i k i i k i i k
hdϕ ,Z iX −hdϕ ,X iZ =Z (ϕ )X −X (ϕ )Z we have
k i i k i i i k i i k i
π (dϕ )=π(dϕ )− 1 Z (ϕ )X + 1 X (ϕ )Z
D k k 2 i k i 2 i k i
i i
X X
=X − 1 δ X + 1 ϕ (ϕ−1) X
k 2 ik i 2 ki ji j
i i,j
X X
=X − 1X − 1 δ X =0
k 2 k 2 ki i
i
X
due to the fact that ϕ =−ϕ .
ij ji
Theorem 2 The bivector π in (8) with Z as in (7) satisfies the Jacobi iden-
D i
tity.
Proof. Itiseasytocheckthatouroperatorπ definesthefollowingbracket
D
on M
m
{F,G} ={F,G} − {F,ϕ } (ϕ−1) {ϕ ,G} , (9)
πD π i π ij j π
i,j=1
X
(where F,G:M→R are two arbitrary functions on M) which is just the well
known Dirac deformation [16] of the bracket {.,.} associated with π, and as
π
was shown by Dirac [17] it satisfies the Jacobi identity.
Remark 3 If C :M→R is a Casimir function of π, then it is also a Casimir
function of π , since in this case (9) yields
D
m
{F,C} ={F,C} − {F,ϕ } (ϕ−1) {ϕ ,C} =0−0=0.
πD π i π ij j π
i,j=1
X
We also know from Theorem 1 that the constraints ϕ are Casimirs of the de-
i
formed operator π . Thus, we can informally state that Dirac deformation
D
preserves all the old Casimir functions and introduces new Casimirs ϕ .
i
ItisnowpossibletorestrictourPoissonoperatorπ (orourPoissonbracket
D
{.,.} )toaPoissonoperator(bracket)onthesubmanifoldS (i.e. asymplectic
πD
leaf ϕ = ... = ϕ = 0 of π ) in a standard way (reduction to a symplectic
1 m D
leaf). Namely, for arbitrary functions f,g : S → R one defines the reduced
Dirac bracket {f,g} onS (withthecorrespondingPoissonoperatorπ )asthe
R R
restriction of the bracket of arbitrary prolongations of f and g to M, i.e.
{f,g} ={F,G} |
R D S
where F,G : M → R are two functions such that f = F| and g = G| .
S S
Of course, an identical construction can be induced on any leaf F from the
s
foliation F. There arises, of course, a question how ’robust’ this construction
is, i.e. to what extent the obtained deformation π and the reduction π are
D R
5
independentofthechoiceofparticularfunctionsϕ thatdefineoursubmanifold
i
S. This issue will be partially addressed in next sections.
The results of this section as well as the results presented in Sections 4 and
5 suggest that the concept of the classification of constraints as being either
of ”first-class” or of ”second-class”, proposed by Dirac, should be reexamined
when one looks on the problem from the point of view of Poisson geometry.
FirstofallitisclearthattheprocedurecalledDirac reduction hastwodifferent
levels. ThefirstlevelwecallDirac deformation aswedeformaPoissonbivector
π from manifold M to another Poisson bivector π on the same manifold M.
D
A sufficient condition for the existence of π for a given set of constraints ϕ ,
D i
i = 1,...,m is a nondegeneracy of the Gram matrix ϕ. From the construction
all constraints ϕ are Casimirs of π .The second level of the construction we
i D
call Dirac restriction as we restrict a Poissonbivector π to its symplectic leaf
D
F . In the original Dirac construction it was a particular one, namely F = S.
s 0
A Poisson bivector on F is denoted by π . For the existence of the second
s R
step additional restrictions on ϕ have to be imposed. Actually, π has to
i R
be nonsingular. In a standard classification it means that we have to exclude
constraints of first-class. Let us remind that a constraint ϕ is of first class if
k
itsPoissonbracketwithallthe remainingconstantsϕ vanishesonS, thatis if
i
{ϕ ,ϕ } | =0, i=1,...,m. (10)
k i π S
Otherwise ϕ is of second-class. In general, first-class constraints make π sin-
k R
gularsothattheDiracreductionprocedurecannotbeperformed. Nevertheless,
the condition (10) seems to be too strong. In sections 4 and 5 we demonstrate
situations where constraints are of first-class but the singularity is ’removable’
andsothePoissonbivectorπ iswelldefined. Itsuggeststhattheclassification
R
of constraints given by Dirac should be reformulated in the context of Poisson
pencils and Poisson geometry.
3 Separable quasi-bi-Hamiltonian chains of Be-
nenti type
In the following section we briefly remind basic facts about separable Hamilto-
nian systems on Riemannian manifolds, which form a special class of quasi-
bi-Hamiltonian chains [3],[11], known also as the so called Benenti systems
[21],[22]. Let (Q,g) be a Riemannian manifold with covariant metric tensor
g = (g ) and with the inverse (contravariant metric tensor) g−1 = G = (Gij).
ij
Let(q1,...,qn) be somecoordinatesystemonQandlet (q1,...,qn,p ,...,p )be
1 n
the correspondingcanonicalcoordinateson the phase spaceN =T∗Q with the
associated Poisson tensor
0 I
θ = n n (11)
0 −I 0
n n
(cid:18) (cid:19)
where I is n×n unit matrix and 0 is the n×n matrix with all entries equal
n n
to zero. Let us consider the Hamiltonian E : N → R for the geodesic flow on
6
Q:
n
E =E(q,p)= Gij(q)p p . (12)
i j
i,j=1
X
As it is known, a (1,1)-type tensor B = (Bi) (or a (2,0)-type tensor B =
j
(Bij)) is called a Killing tensor with respect to g if (BG)ijp p , E = 0
i j θ0
(or (B)ijp p , E = 0). An important generalization of this notion is
i j θ0 (cid:8)P (cid:9)
formulated in the following definition.
(cid:8)P (cid:9)
Definition 4 Let L = (Li) be a second order mixed type (i.e. (1,1)-) tensor
j
n
on Q and let L : N →R be a function on N defined as L = 1 (LG)ijp p ,
2 i j
i,j=1
X
n
where LG is a (1,1) tensor with components (LG)ij = Ligkj. If
k
i,j=1
X
n
∂f
{L,E} =αE, where α= Gij p , f =Tr(L),
θ0 ∂qi j
i,j=1
X
then L is called a special conformal Killing tensor with the associated potential
f =Tr(L) [11].
The importance of this notion lies in the fact that on manifolds with tensor
L the geodesic flows are separable. There exist, in this case, n constants of
motion, quadratic in momenta, of the form
n n
E = Aijp p = (K G)ijp p , r =1,...,n, (13)
r r i j r i j
i,j=1 i,j=1
X X
n
(K G)ij = (K )iGkj
r r k
k=1
X
whereA andK areKillingtensorsoftype(2,0)and(1,1),respectively. More-
r r
over, all the Killing tensors K are given by the following ’cofactor’ formula
r
n−1
cof(ξI −L)= K ξi, (14)
n−i
i=1
X
where cof(A) stands for the matrix of cofactors, so that cof(A)A = (detA)I.
NoticethatK =I,henceA =GandE ≡E.SincethetensorsK areKilling,
1 1 1 r
with a common set of eigenfunctions, the functions E satisfy {E ,E } = 0
r s r θ0
and thus they constitute a system of n constants of motion in involution with
respecttothePoissonstructureθ . So,foragivenmetrictensorg,theexistence
0
of a special conformal Killing tensor L is a sufficient condition for the geodesic
flow on N to be a Liouville integrable Hamiltonian system.
7
The special conformal Killing tensor L can be lifted from Q to a (1,1)-type
tensor on N =T∗Q where it takes the form
L 0 ∂ ∂
N = n , Fi = (Lp) − (pTL) . (15)
F LT j ∂qi j ∂qj i
(cid:18) (cid:19)
The lifted (1,1)tensorN is calleda recursion operator. An importantproperty
ofN isthatwhenitactsonthe canonicalPoissontensorθ itproducesanother
0
Poissontensor
0 L
θ =Nθ = ,
1 0 −LT F
(cid:18) (cid:19)
compatiblewiththe canonicalone(actuallyθ iscompatible withNkθ forany
0 0
integer k).
It is now possible to show that the geodesic Hamiltonians E satisfy on
r
N =T∗Q the set of relations
θ dE =θ dE +ρ θ dE , E =0, r =1,...,n, (16)
0 r+1 1 r r 0 1 n+1
where the functions ρ (q) are coefficients of the characteristic polynomial of
r
L (i.e. minimal polynomial of N), which is a special case of the quasi-bi-
Hamiltonian chain (3) [1].
ItturnsoutthatwiththetensorLwecan(generically)associateacoordinate
system on N in which the flows associated with all the functions E separate.
r
Namely,let(λ1(q),...,λn(q))bendistinct,functionallyindependenteigenvalues
ofL,i.e. solutionsofthecharacteristicequation det(ξI−L)=0. Solvingthese
relations with respect to q we get the transformation λ→q
qi =α (λ), i=1,...,n. (17)
i
The remaining part of the transformation to the separation coordinates can
be obtained as a canonical transformation reconstructed from the generating
function W(p,λ) = p α (λ) in the standard way by solving the implicit
i i i
relations µ = ∂W(p,λ) with respect to p obtaining p = β (λ,µ). In the (λ,µ)
i ∂λi P i i i
coordinates, known as the Darboux-Nijenhuis coordinates (DN), the tensor L
isdiagonalL=diag(λ1,...,λn)≡Λ , while the Hamiltonians(13)ofourquasi-
n
bi-Hamiltonian chain attain the form [3]
n ∂ρ f (λi)µ2
E (λ,µ)=− r i i r =1,...,n,
r ∂λi ∆
i
i=1
X
where
∆ = (λi−λk),
i
k=1,...n,k6=i
Y
ρ (λ) are symmetric polynomials (Vi´ete polynomials) defined by the relation
r
n
det(ξI −Λ)=(ξ−λ1)(ξ−λ2)...(ξ−λn)= ρ ξr, (18)
r
r=0
X
8
and where f are arbitrary smooth functions of one real argument.
i
ItturnsoutthatthereexistsasequenceofgenericseparablepotentialsV(k),
r
k ∈ Z, which can be added to geodesic Hamiltonians E such that the new
r
Hamiltonians
H (q,p)=E (q,p)+V(k)(q), r =1,...,n, (19)
r r r
are still separable in the same coordinates (λ,µ). These generic potentials are
given by some recursion relations [4],[10]. The Hamiltonians H : N → R in
r
(19) satisfy the following quasi-bi-Hamiltonian chain
θ dH =θ dH +ρ θ dH , H =0, r =1,...,n. (20)
0 r+1 1 r r 0 1 n+1
In the DN coordinates the Hamiltonians H attain the form [3]
r
n ∂ρ f (λi)µ2+γ (λi)
H (λ,µ)=− r i i i , r =1,...,n, (21)
r ∂λi ∆
i
i=1
X
wherepotentialsV(k) enterH asγ (λi)=(λi)n+k−1. From(21)itimmediately
r r i
follows that in (λ,µ) variables the contravariant metric tensor G and all the
Killing tensors K are diagonal
r
f (λi) ∂ρ
Gij = i δij, (K )i =− rδi.
∆ r j ∂λi j
i
Moreover, in the (λ,µ) coordinates the recursion operator and the tensor θ
1
attain the form
Λ 0 0 Λ
N = n n , θ = n n
0 Λ 1 −Λ 0
n n n n
(cid:18) (cid:19) (cid:18) (cid:19)
while θ remains in the form (11) since the transformation (q,p) → (λ,µ) is
0
canonical.
The quasi-bi-Hamiltonian chain (20) on N can easily be lifted to a bi-
Hamitonian chain (2) on the extended manifold M = T∗Q × R = N × R
[11].
Having suchacomplete picture ofseparablequasi-bi-Hamiltonianchainson
Riemannian manifolds, one can ask a question: what kind of holonomic con-
straintscanbeimposedonconsideredsystemssothatthisquasi-bi-Hamiltonian
separability schema is preserved? The simplest admissible case of such con-
straints will be considered in the next sections.
4 Reduction λn = 0
LetusconsideraparticlemovinginourRiemannianmanifoldQequipped with
the coordinates (q1,...,qn). Suppose that this particle is subordinated to some
holonomic constraints on Q defined by the set of relations
ϕ (q)=0, k =1,...,r (22)
k
9
that define some submanifold of Q. The velocity v = vi ∂ of this particle
i ∂qi
must then remain tangent to this submanifold so that
P
n
∂ϕ
0=hdϕ ,vi= kvi.
k ∂qi
i=1
X
and thus in our coordinates vi = Gijp the motion of the particle in the
j j
phase space N = T∗Q is constrained not only by the r relations (22) but also
P
by the r relations
n
∂ϕ (q)
ϕ (q,p)≡ Gij(q) k p =0, k=1,...,r. (23)
r+k ∂qi j
i,j=1
X
that are nothing else than the lift of (22) to N. We will call the constraints
(23) a g-consequence of the constraints (22), as they are natural differential
consequences of (22) at given metric tensor g. The constraints (22)-(23) define
a submanifold S of N of dimension n−2r.
Let us now consider our quasi-bi-Hamiltonian chain (20) in N. We would
like to know what types of holonomic constraints (22) on Q do not destroy the
separability of the constrained chain? This is a complicated question and due
to its nature it is most convenient (for a moment) to consider it directly in
our separation coordinates (λ,µ). Thus, in this section we will analyze only
a very special choice of the functions ϕ in (22). Namely, we put r = 1 (so
k
that n−2r = n−2 which corresponds to m = 2 in Section 2) and define the
corresponding function ϕ′ in (λ,µ) variables as
1
ϕ′(λ)=λn (24)
1
Since the metric tensor G = (Gij) has in (λ,µ) coordinates the diagonal form
G=diag f (λi)/∆ , i=1,...,nandsincetheequation(23)hasaninvariant
i i
form(i.e. in(λ,µ) coordinatesit has the same form with q andp replacedby λ
(cid:0) (cid:1)
and µ respectively) the g-consequence of (24) reads as
f (λn)
ϕ′(λ,µ)= n µ (25)
2 ∆ n
n
(we use ′ here since soon we will modify the constraints (24)-(25) to a simpler
form). These two constraints define a subset S′ = {ϕ′ =0,ϕ′ =0} of N. We
1 2
will however restrict us to a submanifold S = {λn =0,µ =0} neglecting the
n
term f (λn)/∆ in ϕ′ (for ∆ 6=0 S ⊂S′ and in case of the systems when f
n n 2 n n
nevervanishesSandS′coincide). Thereasonforthisisthatvariouscalculations
with the help of these new constraints λn =0,µ =0 are simpler. Let us now
n
perform the Dirac reduction of our Poisson operators θ and θ
0 1
0 I 0 Λ
θ = n n , θ = n n (26)
0 −I 0 1 −Λ 0
n n n n
(cid:18) (cid:19) (cid:18) (cid:19)
10