1. Introduction - ITN - Liu

Vol. 55 (2005)

No. 3

REPORTS ON MATHEMATICAL PHYSICS

GEOMETRIC

REDUCTION

OF HAMILTONIAN

SYSTEMS*

KRZYSZTOF M A R C I N I A K t Department of Science and Technology Campus Norrk6ping, Lirlk6ping University 601-74 Norrk6ping, Sweden (e-mail: [email protected]) and MACIEJ B L A S Z A K $ Institute of Physics, A. Mickiewicz University Umultowska 85, 61-614 Poznafi, Poland (e-mail: blaszakm@ amu.edu.pl) (Received July 13, 2004 - - Revised February 17, 2005)

Given a foliation 8 of a manifold .A4, a distribution Z in A4 transversal to 8 and a Poisson bivector FI on A4, we present a geometric method of reducing this operator on the foliation 8 along the distribution Z. It encompasses the classical ideas of Dirac (Dirac reduction) and more modern theory of J. Marsden and T. Ratiu, but our method leads to formulae that allow for an explicit calculation of the reduced Poisson bracket. Moreover, we analyse the reduction of Hamiltonian systems corresponding to the bivector FI. AMS 2000 Mathematics Subject Classification: 70H45, 53D17, 70F20, 70G45 Keywords: Poisson structures, geometric reduction, constraints, Hamiltonian systems.

1.

Introduction

T h e r e d u c t i o n t h e o r y o f d y n a m i c a l s y s t e m s c o n s i s t s o f t w o b r a n c h e s : the first b r a n c h d e a l s w i t h c o n s t r a i n e d L a g r a n g i a n s y s t e m s , the s e c o n d w i t h c o n s t r a i n e d H a m i l t o n i a n s y s t e m s . In the L a g r a n g i a n a p p r o a c h o n e c o n s i d e r s s e p a r a t e l y the c a s e o f h o l o n o m i c c o n s t r a i n t s , i.e. the c o n s t r a i n t s w h i c h m a y d e p e n d o n v e l o c i t i e s , but o n l y in such a w a y that the e q u a t i o n s o f c o n s t r a i n t s c a n b e i n t e g r a t e d to e l i m i n a t e v e l o c i t i e s , a n d the n o n h o l o n o m i c case. In m a n y o f t h e s e p a p e r s o n e first c o n s i d e r s the L a g r a n g i a n f o r m u l a t i o n a n d then p a s s e s to t h e c o r r e s p o n d i n g H a m i l t o n i a n f o r m u l a t i o n (see e.g. [1]). T h e r e d u c t i o n t h e o r y in the H a m i l t o n i a n c o n t e x t h a s b e e n i n i t i a t e d b y P. A . M. D i r a c , w h o in his f a m o u s p a p e r [2] d e s c r i b e d a m e t h o d o f r e d u c i n g *Lecture given at the 36th Symposium on Mathematical Physics, Torufi, June 9-12, 2004. tPartially supported by The Swedish Research Council grant No. 624-2003-607. ."--Partiallysupported by The Swedish Institute scholarship No. 03824/2003 and by KBN Research Project 1 P03B 11127. [325]

326

K. MARC1NIAK and M. BLASZAK

a given Poisson bracket onto a submanifold given by some constraints ~p provided they were of "second class". In this approach the classical notion of holonomic constraints is usually not introduced as in this context there is no obvious division of variables between "position" and "velocity" (or "momenta"). Recently, there has been much interest in extending the theory to the case of generalized Hamiltonian systems (see e.g. [3]). The ideas of Dirac were developed in many papers, among others in [4-10] (see also the literature quoted there). A geometric meaning of this reduction procedure has been investigated in [8, 11]. In this paper we develop the ideas of [2, 8] and present a constructive, computable method of reducing (locally) a given Poisson operator 1-I to any regular submanifold 8. The idea of the method is to choose a distribution Z (not necessarily integrable) that is (i) transversal to the foliation 8, (ii) at any x c A4 it completes T:,8 to Tx.A4, (iii) it makes the operator ri Z-invariant (see definitions below), and then to deform the Poisson operator ri to a new Poisson operator rid such that its image will be tangent to the submanifold 8. This new operator rid will be always Poisson (and so its natural restriction to 8 will be Poisson). In consequence, we obtain a method of reducing a Hamiltonian system on M to a Harniltonian system on every leaf S~ of the foliation S. This reduced system strongly depends on the choice of the distribution Z. As a special case we obtain the classical Dirac reduction of Hamiltonian system. All our considerations will be local in the sense that our manifold .A4 is perhaps only an open submanifold of a larger manifold. Our construction is equivalent to the reduction method proposed by Marsden and Ratiu in [8]. However, our approach has advantages: it can be performed simultaneously on any leaf S~ of the foliation 8, it is constructive (the approach of Marsden and Ratiu requires calculations of prolongations of Z-invariant functions and as such is difficult to perform in practice) and it is formulated in the language of Poisson bivectors rather than Poisson brackets. We want to stress that this method does not require the submanifold to be given by holonomic constraints on some configuration space. In fact, we do not require our manifold to be a cotangent bundle to any configurational manifold at all, but of course our construction covers this special case as well. For example, it covers the case discussed in [1], where the authors obtain the Poisson operator on the constrained submanifold only in the case of holonomic constraints--because they simply restrict their Poisson operator ri to the constrained submanifold, and such restriction usually (apart from the holonomic case) destroys the poissonity of the operator. We have to notice that some particular versions of the proposed scheme appeared recently in [12, 13] and [14] in the context of Poisson pensils. In [15] the author applied the same setting as above but with no conclusive formulae for computing the actual deformed Poisson bracket rio and since he did not use the notion of Z-invariance, his deformed operator rio, although formally identical with our construction, was not always Poisson (it has been called there a pseudo-Poisson operator).

GEOMETRIC REDUCTION OF HAMILTONIAN SYSTEMS

327

Some basic steps of our construction have been presented in our previous paper [16]. This paper, however, was mainly devoted to completion of the above picture by its "dual" part by developing a theory of the Marsden-Ratiu type reduction of presymplectic 2-forms f2 that are (in a sense) dual to a given Poisson operator FI. The picture presented here is much clearer and moreover it is parametrization-free in the sense that we prove the main results without necessity of discussing some particular functions defining our foliation S. Moreover, in the paper [16] we focused on the Dirac case while this paper has a general character. In the end, in this paper we also consider the related reduction of Hamiltonian dynamics.

2.

Geometric reduction of Poisson bivectors

Let us consider a smooth manifold Ad of arbitrary (finite) dimension n and a foliation S of Ad consisting of the leaves S~ parametrized by v • N k (so that k • N is the codimension of every leaf $v). Consider also a regular distribution Z on A4 (that is a smooth collection of the spaces Z~ c T~Ad where v is such that x • S~) such that it completes every TxS to TxAd in the sense that

TxAd = TxS~ @ Zx

(1)

for every x in M . Here and in what follows @ denotes the direct sum of vector spaces. It means that every vector field X on the manifold .M has a unique decomposition X = X/I + X± such that for every x in .M the v e c t o r (Xii)x C TxS (XII is tangent to the leaves of the foliation S) while (X±)x • Zx (X± is contained in the distribution Z). The splitting (1) induces the following splitting of the corresponding dual space T ' M :

TSM = rS& • zS,

(2)

where T*S~ is the annihilator of Zx while the space Z~* is the annihilator of TxS~. Thus, any one-form ot on M has a unique decomposition oe = Otll -t- Or_l_ such that (oell)~ • Tx*S (all annihilates the vectors from Z) while (~±)x • Z* (c~± annihilates the vectors tangent to the foliation S). We will call Xll and o~Ei as projections of X and or, respectively, on the foliation S. Abusing notation a bit we will write that X c T S if X = XII , X C Z if X = X±, and similarly for one-forms: o~ C T*S if o~ = Otll, ot C Z* if ot = or±. Let us now suppose that our manifold Ad is equipped with a Poisson bivector lI (i.e. a bivector with vanishing Schouten bracket, see [6]). This operator induces the Poisson bracket {F, G}ri = (dF, IldG) on the algebra of smooth functions on .M, where (., .) is the dual map between T*3d and TAd. A smooth real-valued function F on Ad is called Z-invariant if the Lie derivative L z F = 0 for any vector field Z C Z. We will now adopt the following definition. DEFINITION 1. The operator rI is said to be Z-invariant if Lz {F, G}n = 0 for any pair of Z-invariant functions F and G and every vector field Z C Z.

328

K. MARCINIAK and M. BLASZAK

Notice that our definition does not necessarily mean that Lzl-I = 0 for all vector fields Z C Z , as for any pair F, G of Z-invariant functions the condition L z {F, G}n = 0 means only that the function (dF, ( L z l i ) d G ) vanishes. Thus, I7 does not have to be an invariant of the distribution Z to be Z-invariant in our meaning. Notice also that the above definition is equivalent to the statement that for any pair 0/,/~ C T*S we have (0/, (Lz17)/~) = 0 (since if F is Z-invariant then

dF C T'S). Suppose for the moment that the distribution Z is spanned by k vector fields Zi. We say, that the operator 17 is Vaisman [17] with respect to Z if for every vector field Zi there exist vector fields Wij, j = 1 . . . . k, such that k

Lzi 17 = Y~ Wij m Zj.

(3)

j=l

It is easy to see that this definition does not depend on the choice of basis in Z (although the vector fields Wij obviously do). If the operator li is Vaisman with respect to Z , then it is also Z-invariant, as then for any two one-forms 0/,/~ C T ' S , k

E(0/,(W j A z j ) # )

= 0,

j=l

since 0/ and /~ annihilate all the vector fields Zi. The converse statement is however not true in general. Let us now consider a Poisson operator 1-I on 3/I and define the following bivector: 1-[D (0/, r ) = 17(0/1I , 1~11) for any pair a, fl of one-forms. (4) We will often call the bivector liD a deformation of 17. This bivector occurred for example in [12] in a more restrictive context. Observe that it always exists and that it is uniquely defined once the foliation S and the distribution Z are given (it is thus a purely geometric construction). It has an important property: its image lies always in T S . LEMMA 1. I'Io(0/) C T S for any one-form 0/ on .AA, i.e. the image of l i d is tangent to the foliation S.

Proof: We have to show that (fl, FID0/) = 0 for any fl C Z*. But n

0/} = n o

0/) = n (

since /?ll = 0 for every /~ C Z*.

11,0/0 = 0

[]

Thus, the deformed bivector l i D has its image in T S and if we consider it as a mapping from one-forms to vector fields on .A4 then it can be naturally restricted to a bivector zrR~ on every leaf Sv of S by simply restricting its domain to S~, 7r Rv =

l'IDIsv

.

329

GEOMETRIC REDUCTION OF HAMILTONIAN SYSTEMS

Moreover, it induces a new bracket for functions on 34,

{F, G}rlD = riD (dF, dG) = l-I((dF)ll, (dG)ll). Of course, the bivector liD (and thus even ZrR~) in general does not have to be Poisson. However, it turns out that if li is Z-invariant then l i d (and thus every zre~) is Poisson. THEOREM 1. If li is Z-invariant then liD given by (4) is Poisson.

Proof: We will prove that the bracket {., "}riD is Poisson. Obviously, this bracket is antisymmetric and satisfies the Leibniz property. It remains to show that it also satisfies the Jacobi identity, that is {{F, G } n 9 , H } n D + cycl. = 0, for any functions F, G, H. Using the definition of rio, this condition can be written as ((d {(dF),l , l i (dG),))l I , r i ( d H ) , } + cycl. : 0.

(5)

However, for any vector field Z C Z we have (d ((dF)ll, li (dG)ll) , Z) = Z (((dF)tl, FI (dG)ll)) = Lz ((dF)ll, YI (dG)ll} = 0 due to the assumed Z-invariance of li. This means that d((dF)l I , li (dG)ll} C T*$, so that (d((dF),, rI (dG)ll))lf = d ((dF)ll , I'I (dG)H), and thus condition (5) turns out to be the Jacobi identity for I-I, which is satisfied [] since l-I is Poisson. Thus, given a foliation S on 3 4 and a transversal distribution Z on .A4 (such that (1) is satisfied) we can reduce any Poisson bivector l'I that is Z-invariant to a Poisson bivector 7rn~ on the leaf $~ of S by deforming li to riD and then by restricting l i d to S~. This construction yields the same operator Jrn~ as in the approach of Marsden and Ratiu [8]. We will however show how this construction can be easily realized in practice. REMARK 1. In case when the foliation S coincides with of 1-I we have of course that liD = I-I, since in this case function F and so {F, G}nD = l i ( ( d F ) l l , (dG)ll) = 1-I (dF, case zrR~ is the standard reduction of 1-I on its symplectic

the symplectic foliation li((dF)±) = 0 for any dG) = {F, G}n. In this leaf $~.

Let us now consider some special cases of our general situation. We firstly observe that the annihilator Z* of T S is defined as soon as the foliation g is determined (we do not need to specify a particular Z in order to define Z*). DEFINITION 2. The distribution D = I-I (Z*) (so that 50x-----li ( Z * ) )

is called

a Dirac distribution associated with the foliation ,_q. Thus, the distribution 50 is determined by ,5 and by li. A priori, two limit cases are possible here. If T 3 4 = 50 @ T $ we say that we are in the Dirac case. If 7) C T,_q we say that we are in the tangent case. In the Dirac case we have

330

K. MARCINIAK and M. BLASZAK

a canonical choice of Z: we can choose Z = 73 ,(in this case 17 is automatically Z-invariant, since Z is spanned by the vector fields which are Hamiltonian with respect to 17). Nevertheless, we can also choose some other distribution Z # 73. In the tangent case we have no canonical choice of Z and we are free to find a distribution Z that makes FI Z-invariant. Anyway, in both cases we have many nonequivalent deformations rid (and thus projections zrR~). Generically, the distribution 73 will not be tangent to S, but it will not suffice to span TAA together with T S. Let us now suppose that the foliation 8 of M is parametrized by the set of k functionally independent real-valued functions ~oi(x ) so that its leaves have the form 8~ = {x e 34 : ~0i (x) = vi, vi c R, i = 1. . . . . k } where k is--as above--the codimension of the foliation. We will show how the above considerations can be written in the parametrization {~0i}. The one-forms dq9i constitute a basis in Z*. Then, the Dirac distribution 73 is spanned by k (possibly dependent) Hamiltonian vector fields Xi = Fide&. Let us denote by Zi a basis of Z dual to the basis {d~oi } in Z*, so that Zi(~oj)= &j. Our projections Xll and o~11 are then given by k

all

=

X

-

~ X(~oi)Zi,

i=l

( o b v i o u s l y Xll(q)i) = 0 for all i so that indeed this vector field is tangent to the leaves of S) and by k Cell = oe -- ~ ol(Zi)d{o i i=1

(and obviously OglI(Zi) = 0 for all i). Thus k rI(o~,,,/~]1) = 1[ ( o g - Y~. ol(Zi)d~oi, e i=1

k

~ ]~(aj)d(pj) j=l

k

= n(a,/~) - ~/~(zi)n(a, j=l

k

&o j) - ~ o~(Zz)rI(&0;, ~) i=l

k

-Jr- Y~. ol(Zi)t~(Zj)17(dq) i, d~oj), i,j=l so that the deformation I70 can be expressed as 1 I"I0 = rI - y~ x i A Zi -t- ~ ~ g)ijZi /k Z j , i i,j

(6)

where the functions q)ij are defined as

qgiJ = {q)i' q)J}FI = XJ (qgi) = n ( & ° e , dq)j). In the Dirac case all the vector fields Xi are transversal to the foliation S and are moreover linearly independent. It happens precisely when det(~0/j) 7~ 0 (the functions

GEOMETRIC REDUCTION OF HAMILTONIAN SYSTEMS

331

q9i are then 'second class constraints' in the terminology of Dirac). The vector fields Zi (the dual basis t o {dq)i} ) can be expressed through the vector fields Xi as k

Zi = --~_j,(~p-1)j~Xj,

i = 1. . . . . k.

j=l

Indeed,

k

k

zj(~oi) = ~(~0-1)sjXs(~0i)= ~(~o-1)sj~O~ = ~j. s=l

s=l

Moreover, in this case the deformation (6) attains the form q

1 l i D ~ l i - - 2 i=1

Xi A Zi.

(7)

This operator defines the following bracket on C~(.A4), k

{F, G}nD = [F, G}n - E

{F, q)i}n(qg-1)ij{~oj, G}I3,

(8)

i,j=l

which is just the well-known Dirac deformation [2] of the bracket {., .}n. In the tangent case all the vector fields Xi are tangent to the foliation S the deformation (6) attains the form

and

k

liD = rI - y~ xi A Zi,

(9)

i=1

and has been considered in [12, 13]. In [11] we have considered a bit more general than (6) deformation of li of the form lid = l i - Y-~i 17,-A Zi. The image of lid does not have to lie in TS. However, one can prove that it happens precisely when the vector fields 14, satisfy the following functional equation

Vi = Xi + E

VJ(~Pi)ZJ'

(10)

J

and in this case lid is Poisson. Substituting this formula into lid yields k

k

lid = l i -- E Xi A Z i -Jr" E v j ( ~ i ) z i i=1 i,j=l

A Zj,

(11)

and it can be proved (after some technical manipulations) that in this case lia = liD 1 (even though Vj(rpi) does not have to be equal to ~Pij). A natural solution of Eq. (10) in the Dirac case is given by Vi = ½Xi and in the tangent case by V/ = Xi, which turns (11) into the deformations (7) and (9), respectively. Thus, in a sense, the deformation liD is the canonical deformation in our setting. Our construction is strongly related to the construction of Marsden and Ratiu. J. Marsden and T. Ratiu presented in [8] a natural way of reducing of a given

332

K. MARCINIAK and M. BLASZAK

Poisson bracket {-, "}ri on A4 to a Poisson bracket {., "}~R on a given submanifold $~ (in our notation). Their method is non-constructive in the sense that in order to find the bracket {f, g}~R of two functions f , g : S~ ~ R one has to calculate Z[s~-invariant prolongations of these functions. Our construction is performed on the level of bivectors rather than on the level of Poisson brackets. This construction (by deformation of the bivector rI) applies directly to every leaf of the distribution $ and moreover it is constructive. At every leaf, however, both constructions are equivalent. On the other hand, we make the assumption about the transversality of the distribution Z that was not present in the original paper of Marsden and Ratiu. This assumption is however very natural since it makes all the assumptions of Poisson Reduction Theorem in [8] automatically satisfied.

3.

Reduction of Hamiltonian dynamics

Let us begin this section by stating--in our setting--a well-known theorem about the relation between the Dirac deformation liD of li and the dynamic imposed by the constraints. Suppose thus that our manifold .M is a cotangent bundle of a Riemannian manifold Q with a covariant metric tensor g, so that .M = T*Q. Denote the corresponding contravariant metric tensor by G. Consider a Lagrangian dynamical system on T Q,

d 3L dt 30i

3L 3qi

---0,

i=l

. . . . . n,

(12)

with a potential Lagrangian function L(q, il) = lgltg~l - V(q). This of course leads to a Hamiltonian equations of motion on A4 =-*T Q, qi :

{qi,

H}ri,

,hi = {Pi, H}ri

(13)

with the Hamiltonian H = ½ p t G p + V ( q ) and with the canonical Poisson operator li. Let us now impose a physical constraint ~0(q)= 0 on our system and assume that in the beginning the coordinates of the system lie on the submanifold Q0 of Q given by ~0(q) = 0. One often makes a physical assumption here that the surface Q0 starts to act on our system with a reaction force R(q, gl) that is orthogonal to Q0 and such that the trajectories of the constrained system

d 3L dt 3Cli

3L = Ri(q, gl) Oqi

(14)

that start on Q0 remain on Q0. On the level of the phase space .M = T*Q the Hamiltonian system (13) is now subordinate to a pair of constraints: q)l(q) --= qv(q) = 0,

~02(q) ~ (V~0)t Gp = O,

where V~0 is the gradient of ~o with respect to q-variables (differential of ~o in Q; the second constraint is a consequence of the fact that the velocities c) must remain

GEOMETRIC REDUCTION OF HAMILTONIAN SYSTEMS

333

tangent to Q0) and thus modifies to

eli = {qi, H } n , THEOREM 2.

/)i = {Pi, H}n + Ri(q, p).

(15)

Eqs. (15) are Hamiltonian and can be written in the form eli = {qi, H}n o ,

/~i = {Pi, n } n D ,

(16)

where l i d = 17 -- g1 Z 2 i=1 Si A Zi is the Dirac deformation (7) of 17 given by the constraints ~oI and cp2. Thus, the response of the Lagrangian system (12) subordinated to the reaction forces R can be accounted for by the corresponding Dirac deformation of the Poisson operator I7. We will only sketch the proof.

Proof: The reaction force R can be calculated by differentiating the assumed identity ~o(q(t))=-0 twice with respect to t and eliminating the second derivatives q'/ with the help of Eqs. (14) and by using the demand that the force should be orthogonal to Q0. After some calculations we obtain that 1

R(q, p) -- (V~o)t G Vq~ ((V~°)t G V V - (ptG)H~(Gp) + A) V~o, where H~o is the Hessian of q): (H~o)ij -

A

-

(17)

°2~° Oqi Oqj and where A = A(q, p) is given by

899 ~s~qs~. ' j m PrPm, . ~l isj lf J' i r ' IJ r,ra,t,j

so that it vanishes in the Euclidean coordinates when all Christoffel symbols F~k are equal to zero. On the other hand, calculating the explicit form of (16) on the submanifold of 34 = T*Q given by the constraints q91,992 leads to the Eqs. (15) with R given by (17). [] Let us now consider a Hamiltonian vector field X = 17dH on a general Poisson manifold 34 where H is some real-valued smooth function on 34 (Hamiltonian function). We constantly assume that we have a smooth, regular foliation S on 34 and a regular distribution Z on 34 such that (1) is satisfied. The corresponding l i d defined by (4) is Poisson and has its image tangent to the foliation S, so that it can be properly restricted to every leaf Sv of S. Thus, the following definition makes sense. DEFINITION 3. We call the vector field XD = VIDdH the Hamiltonian projection of the Hamiltonian vector field X = rIdH. The vector field XD lives on every leaf of the foliation in the sense that its restriction to the leaf S~ is tangent to Sv. Moreover, on the leaf S~ it coincides with the Hamiltonian vector field zrRvdh,

liDdH]& = 7rRvdh,

334

K. M A R C I N I A K and M. B L A S Z A K

where h = HI& is the restriction of the Hamiltonian H to the leaf S~. To see this it is enough to choose a parametrization {q)i} of S and to pass to any system of coordinates of the form (x, qg). In these coordinates the bivector I-Iv has a matrix form with a nonzero upper-left block coinciding with the matrix form of zrR~ and with the remaining terms equal to zero. There is a connection between X = l i d H and its Hamiltonian projection X D = I-[DdH. THEOREM 3 (Dynamic reduction theorem). If X = FldH, XD = FIDdH and X i = Yldg) i then k X D = Xll - - ~ Z i ( H ) X i l I .

(18)

i=1

Proof: A direct calculation yields k

XD = liDdH

= X - ~

k

(Zi(H)Xi

- Xi(H)Zi)

i=1 k

/

k

i=1

\

j=l

the last equality due to X i ( H ) = q)ji = Xi((Dj) it yields (18).

.3¢_ ~ ( i ) i j Z j ( H ) Z i i,j=l

\

/

(dH, r l d q ) i ) = - { & o i , r l d H ) = - X ( q o i ) .

Since []

k

Observe that the difference between XD and XII is the term Y~ Zi(H)Xil[ that i=1

is tangent to the foliation S, as it should be. Since for the Dirac c a s e Xill = 0 (by definition) we have C O R O L L A R Y 1. In the Dirac case Xo -----XII, so that in this case the Hamiltonian projection is just the natural projection (in the sense of direct sum) along the distribution Z.

The term XII in XD has a well-known physical interpretation: it describes the evolution of the system X = FIdH imposed with the constraints given by ~0i. The physical meaning of the second term in XD is not clear for us, although it should represent some additional force (friction) acting on the system and tangent to the constraints. The authors will be grateful for any hints in this matter. We are now in position to discuss the degeneracy of l i d using the above dynamic reduction theorem (Theorem 3). Let us first discuss the Dirac case. PROPOSITION 1. Consider the Dirac deformation l i d given by (7) of a Poisson operator FI on M. Suppose that the real-valued functions ci, i = 1 . . . . . s on Ad are such that they span the kernel of the operator 1-I (i.e. are Casimir functions of I7 in the sense that 1-Idci = O) and such that the functions {ci, @j } constitute a functionally independent set. Then

GEOMETRIC REDUCTION OF HAM1LTONIAN SYSTEMS

335

(i) the constraints ftgi and the 'old' Casimirs ci are all Casimirs of liD, (ii) any Casimir of l i d must be of the f o r m C(cl . . . . . cs, qg~. . . . qgk).

Proof: The proof of (i) is just a calculation. To prove (ii), let us complete the functionally independent functions {ci, ~oj} to a coordinate system {ci, ~oj, x~} on .A4. Suppose that a function C - - C ( c , ~o, x) is a Casimir of l i d given by (7), i.e. that l i D d C = 0. Then, according to Theorem 3, (I-IdC)[i = 0, i.e. FIdC C Z . In the Dirac case the distribution Z is spanned by the vector fields Xi so that there must exist f u n c t i o n s ot i s u c h that I-IdC

1"I dC - Z~-I °lid~°i

= Z~=I °liXi = Z~=I °~ilId~i = II ( Z k l olid~i ).

= 0 or

de

= Ei=I

°lidq)i -'~ ~-~i=1 fli dci

Thus,

w h i c h p r o v e s (ii). []

Thus, we can state that the Dirac deformation (7) preserves all old Casimir functions and introduces new Casimirs ~oi and that no other Casimirs arise in this process. The situation in the general case is more complicated, since the Casimirs of 1-I does not have to survive the general deformation (4) (or (6)) and moreover since new Casimirs, different from ~oi can arise. We can merely state that in the general case the function C is a Casimir of I-ID if. and only if the vector field Y = l i d C satisfies the relation k

Y, = ~ z i ( c ) x i , i=1

which for the Dirac case degenerates to the already discussed condition I111= 0.

4. Example Let us conclude this article by an example. Consider the so-called first Newton representation of the seventh-order stationary flow of the KdV hierarchy [18-20]. It is the following system of second-order Newton equations

ql,tt

=

--10ql 2 + 4q2,

qz,tt = --16qlq2 + 10ql 3 + 4q3, q3,tt = --20qlq3

-

8q22 + 30qZq2 -- 15q 4

(where the subscript , t denotes differentiation with respect to the evolution parameter t). By putting Pl = q3,t, P2 = qz.t, P3 = q~,t it can be written in a Hamiltonian form d d t (ql, q2, q3, Pl, P2, P3) w = X = I-IdH,

E0, I

where FI is the canonical Poisson operator on the space AA = {(ql, q2, q3, pl, p2,

FI=

-/3

0

'

P3)},

336

K. MARCINIAK and M. BLASZAK

and with the Hamiltonian 1 2 H = PiP3 + gP2 + 10q12q3 - 4q2q3 + 8qlq2 2 - 10q13q2 -k 3ql 5.

Consider also a foliation S given by a pair of constraints

q)l

=

q3 + qlq2,

q)2 = Pl -q- qlP2 q- q2P3,

where q92 is the so-called G-consequence of ~01 (i.e. a lift from the configuration space {(ql, q2, q3)} to 3/l) with respect to the antidiagonal metric tensor G [18]. The x?ector fields Xi = rid~o i have the form 0 0 -X1 = -q2-~p 1 -- ql Op2

0 Op3 ,

0 X2 = ~

0 0 0 0 + ql ~q2 + q2 ~q3 - P20pl - - - P 3 Op2 --

and they are transversal to S, so that we have the Dirac case. Thus, the distribution Z = 79 = Sp{Xi} makes ri Z-invarianc The basis in Z that is dual to {dq)i} is q)121X1 ' where q)12 ~- {q)t, q)2}Fl = 2q2+q 2. The Dirac deformation rid given by (7) attains in the adopted coordinate system {(ql, q2, ~01, (/92, P2, P3)} the form 0 0 0 0 --ql --1 Z1 ~-" --' Xrpl2 ~-1 2, Z 2 - -

riD-

2q2 + q2

0

0

0

0

2q2

--ql

0

0

0

0

0

0

0

0

0

0

0

0

ql

--2q2

0

0

0

P3

1

ql

0

0

--P3

0

It has, as it should, two Casimirs qh, (P2. We can now easily restrict rid to the operator zrR~ on S~. If we parametrize S~ with the coordinates {(ql, q2, p2, P3)} (the constraints q)r, (f12 are constant on every S~) then

7"( Rv - -

0

0

--ql

--1

1

0

0

2q2

--ql

2q2 + q]

ql

--2q2

0

P3

1

ql

--P3

0

which, in accordance with the theory, is nondegenerate. Observe that this expression actually does not depend on the choice of the leaf S~ in the foliation S. The Hamiltonian projection XD = rlDdH attains on every leaf Sv the form

zr R~dhR~

--

1 ( 0_~1 -[0 2q2 + q2 Oll a2 -8q2

-'[- f l 2 ~ p

02

At- f13

0)

,

where the functions oti, fii are some rather complicated polynomial functions of coordinates and the parameters vi of the leaf $~.

337

GEOMETRIC REDUCTION OF HAMILTONIAN SYSTEMS

Let us now choose another distribution Z for which rl is also Z-invariant. Since the operator Yl has a very simple form, any pair of constant fields will span a distribution Z that mak,s YI Z-invariant, since then the Vaisman condition (3) 0 Oq 03 + ~ P0l / (observe that is trivially satisfied. Thus, let us take Z = Sp { ~0 + Op3, this distribution is integrable). We have now to change the basis in Z to a new basis {Z1, Z2} such that the condition Zi(q)j) = ~ij is satisfied. A simple calculation yields

0 Z1 =

1

--q2

-~q3 -1- 1 ~ - - ~ 2

1

q~Pl

'

(~

q2

Z 2 - - .1 - -

1

0) Op3

"

Now, the general deformation (6) defined by the above distribution attains in the coordinates {(ql, q2, ~Ol, ~02, P2, P 3 ) } the form

1--IO _

-0

0

0

0

0

-1

0

0

0

0

q2 - 1

-ql

1

0

0

0

0

0

0

q2 -- 1

0

0

0

0

0

0

0

1 - q2

0

0

0

P3 - qa

1

q~

0

0

ql - P3

0

and thus the restricted operator re R~ to the leaf $~ parametrized with the coordinates {(ql, q2, P2, P3)} is m

0

0

0

-1

1

0

0

q: - 1

-ql

q2 -- 1

0

1 - q2

0

7r Rv - - _ _

1

ql

ql

-

P3 P3

--

ql

0

m

which is again nondegenerate. Again, this expression does not depend on the choice of the leaf S~ in the foliation S. In the end, consider yet another distribution that makes I-I Z-invariant, namely 0 + 0p3' 0 OPl ~ ] " The appropriate basis of Z is given by Z = Sp ~q3 Z1

0 _

__

Oq3

0 ~

q2opl

+

0 --, Op3

Z 2 --

0

Opl

,

so that we have Zi(~0j)= 8ij. The general deformation (6) defined by the above

338

K. MARCINIAK and M. BLASZAK

distribution

yields i n t h e c o o r d i n a t e s { ( q l , q2, gOl, q92, P2, P3)}

YI D - -

-0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

q2- 1

0

0

0

0

0

0

0

-1

0

0

0

ql

0

0

0

0

-ql

0

so that the restricted operator Jr n~ to the leaf S~ (again parametrized with the coordinates {(ql, q2, P2, P3)}) is

Jr Rv - - - q2 -- 1

-0

0

0

01

0

0

1

0

0

-1

0

ql

0

0

-ql

0

and is degenerate this time. Thus, given a foliation S, by choosing different distributions Z we can obtain several different Hamiltonian projections of our original Hamiltonian system, not only just the well-known Dirac reduction. 5.

Conclusions

In this article we formulated a comprehensive, geometric picture of what is known as Dirac reduction and Marsden-Ratiu reduction of a Poisson operator I7 on a foliation S not related with the symplectic foliation of H. As a consequence, we obtain a geometric method of reducing of any Hamiltonian system to a Hamiltonian system on the foliation S. Any such reduction depends merely on the choice of the distribution Z along which the reduction takes place. Thus, we can reduce Hamiltonian systems on .A4 to Hamiltonian systems on $ in many nonequivalent ways. However, the procedure of finding appropriate distributions Z (i.e. those that make YI Z-invariant) is nontrivial and nonalgorithmic. REFERENCES [1] A. J. van der Schaft and B. M. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys. 34 (1994), no. 2, 225-233. [2] P. A. M. Dirac, Generalized Hamiltonian Dynamics, Can. J. Math. 2 (1950), 129-148. [3] G. Blankenstein and A. J. van der Schaft: Symmetry and reduction in implicit generalized Hamiltonian systems, Rep. Math. Phys. 47 (2001), 57-100. [4] J. Sniatycki: Dirac brackets in geometric dynamics, Ann. h~st. Henri Poincarg Sect. A (N.S,) 20 (1974), 365-372.

GEOMETRIC REDUCTION OF HAMILTONIAN SYSTEMS

339

[5] A. Lichnerowicz: Vari6t6 symplectique et dynamique associ6e ~ une sous-vari6t6, C.R. Acad. Sci. Par& Ser. A-B 280 (1975), A523-A527. [6] A. Lichnerowicz: Les vari6t6s de Poisson et leurs alg6bres de Lie associ6es, J. Diff. Geom. 12 (1977), 253-300. [7] J. E. Marsden and A. Weinstein: Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130. [8] Marsden and T. Ratiu: Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), 161-169. [9] J. Grabowski, G. Landi, G. Marmo and G. Vilasi, Generalized reduction procedure: symplectic and poisson formalism, Fortschr. Phys. 42 (1994), 393427. [10] M. Flato, A. Lichnerowicz and D. Sternheimer: Deformations of Poisson brackets, Dirac brackets and applications, J. Math. Phys. 17 (1976), 1754-1762. [11] K. Marciniak and M. Blaszak: Dirac reduction revisited, J. Nonlin. Math. Phys. 10 (2003), 4514-463, or arXiv.org/nlin.SI/0303014. [12] L. Degiovanni and G. Magnano: Tri-Hamiltonian vector fields, spectral curves and separation coordinates, Rev. Math. Phys. 14 (2002), 115-1163. [13] G. Falqui and M. Pedroni: Separation of variables for Bi-Hamiltonian systems, Math. Phys. Anal. Geom. 6 (2003), 139-179. [14] G. Falqui and M. Pedroni: On a Poisson reduction for Gel'fand-Zakharevich manifolds, Rep. Math. Phys. 50 (2002), 3954-407. [15] Ch.-M. Marie: Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys. 42 (1998), 211-229. [16] M. Blaszak and K. Marciniak: Dirac reduction of dual Poisson-presymplectic pairs, J. Phys. A: Math. Gen. 37 (2004), 5173-5187. [17] I. Vaisman: Lectures on the Geometry of Poisson Manifolds, Progress in Math., Birkh~tuser 1994. [18] M. Blaszak and K. Marciniak: Separability preserving Dirac reductions of Poisson pencils on Riemannian manifolds, J. Phys. A: Math. Gen. 36 (2003), 1337-1356. [19] S. Ranch-Wojciechowski, K. Marciniak and M. Blaszak: Two Newton decompositions of stationary flows of KdV and Harry Dym hierarchies, Physica A 233 (1996), 307-330. [20] M. B~aszak: On separability of bi-Hamiltonian chain with degenerated Poisson structures, J. Math. Phys. 39 (1998), 3213.