Vol. 48 (2001)
REPORTS ON MATHEMATICAL PHYSICS
No. 1/2
ALMOST POISSON SPACES NONHOLONOMIC SINGULAR REDUCTION*
AND
JI~DRZEJ SNIATYCKI Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada (email: sniat ~math.ucalgary.ca) (Received October 5, 2000   Revised November 30, 2000)
Dynamics of Hamiltonian systems with linear nonholonomic constraints is described by distributional Hamiltonian systems. We show that the space of orbits of a proper action of a symmetry group of a distributional Hamiltonian system is a differential space partitioned by smooth manifolds preserved by the evolution. The reduced dynamics is given by distributional Hamiltonian systems on the projections of the manifolds of the partition. It is described in terms of the almost Poisson algebra of smooth functions on the orbit space. 2000 AMS subject classification numbers: primary 70F25, secondary 58D19. K e y w o r d s : accessible sets, almost Poisson algebra, differential space, nonholonomic constraints, singular reduction, symplectic distribution.
1.
Introduction
A differential structure on a t o p o l o g i c a l space is given by a ring of flmctions which satisfy all t h e a x i o m s for a s m o o t h m a n i f o l d e x c e p t for t h e existence of charts. A differential space is a t o p o l o g i c a l s p a c e e n d o w e d w i t h a differential s t r u c t u r e [18]. A n almost P o i s s o n space is a differential space such t h a t its differential s t r u c t u r e is e n d o w e d w i t h a b r a c k e t o p e r a t i o n s a t i s f y i n g all t h e a x i o m s of a Poisson a l g e b r a e x c e p t for t h e J a c o b i identity. T h e a i m of this p a p e r is to s t u d y t h e g e o m e t r i c s t r u c t u r e of a l m o s t Poisson spaces e m e r g i n g n a t u r a l l y in t h e process of r e d u c t i o n of s y m m e t r i e s of H a m i l t o n i a n syst e m s w i t h linear n o n h o l o n o m i c c o n s t r a i n t s . A symplectic distribution is a t r i p l e t (D, H, w ) , where D is a manifold, H is a distrib u t i o n on D, a n d ~ is a skew s y m m e t r i c n o n d e g e n e r a t e b i l i n e a r form on H . If H = T D , t h e n w is a n o n d e g e n e r a t e differential 2form on D. In this case t h e p a i r (D, w ) is a n a l m o s t s y m p l e c t i c manifold. If, in a d d i t i o n , cv is closed, t h e n (D, ~ ) is a s y m p l e c t i c manifold. *Invited lecture of the XXXII Symposium on Mathematical Physics, Torufi, June 6 10, 2000.
[235]
236
Jt~DRZEJ SNIATYCKI
A distributional Hamiltonian system is a quadruplet (D, H, w, h), where (D, H, w) is a symplectic distribution and h is a smooth function on D. For every f E C ~ (D), we denote by YI the unique vector field on D with values in H such that YI_I~ = OHf,
(1)
where _1 denotes the left interior product and OHf is the restriction of df to H. We refer to Yf as the distributional Hamiltonian vector field of f . THEOREM 1. A mechanical system with linear nonholonomic constraints, defined by a distribution D on the configuration space Q, gives rise to a symplectic distribution ( D, H, w). Its dynamics is given by the distributional vector field Yh of the energy function h of the system. The proof is given in [5]. On the space C a ( D ) we define a bracket {fl, f2} = Yflf2,
(2)
where YI1 f2 is the derivative of f2 in the direction of the distributional Hamiltonian vector field Yfl o f f l " It satisfies all the properties of a Poisson bracket except for the Jacobi identity. Hence, C a ( D ) is an almost Poisson algebra and D is an almost Poisson manifold. The bracket (2) was first introduced in [16], see also [15]. Let G be a connected Lie group with Lie algebra g and a smooth and proper action •
: a x D ~ D :
(g,u)
* 4 ~ ( g , u ) = ~ 9 ( u )

g u
(3)
on D. It is a symmetry group of (D, H, w, h) if the action 4~ preserves H , w, and h. For a free and proper action of G on D, the reduction of symmetries was studied in [5]. Here we use the approach proposed by Bates [2] to generalize the results of [5] to the case of proper actions which need not be free. The reduced space of the system is the space D = D / G of orbits of G in D. It is a differential space in the sense of Sikorski [18]. Its differential structure C ~ ( D ) consists of functions f on D which pull back to smooth functions on D under the orbit m a p m
p:D+D:u~~G.u.
(4)
The almost Poisson bracket on D induces an almost Poisson bracket m
__
__
C ~ ( D ) x C ~ ( D ) ~ C a ( D ) :
( f l , f 2 ) ~~ { f l , f 2 } ~
on D such that P*{fl, f 2 } ~  {P'f1, PI2} for all f l and f2 in C ~ ( D ) , where p* : C ~ ( D ) ~ C ~ ( D ) is the pullback by the orbit m a p p. Thus, C ~ (D) is an almost Poisson algebra and D is an almost Poisson space.
ALMOST POISSON SPACES AND NONHOLONOMIC SINGULAR REDUCTION
237
In principle, the geometry of a differential space D is encoded in its differential structure C ~ ( D ) . At present, we have few techniques allowing us to get at this information directly. However, we can decode it indirectly by investigating the structure induced on D by the action of G. This indirect approach is very similar to the process of singular reduction of symmetries for Hamiltonian systems [1, 4, 8, 11, 19]. The main difference is that, in addition to the known forces given by the Hamiltonian, we have the reaction forces of the constraints given by the constraint distribution. In unconstrained Hamiltonian dynamics restrictions on possible motions for Ginvariant Hamiltonians are given by constants of motion, that is, by isotropy groups and values of the m o m e n t u m map. In the presence of nonholonomic constraints, we have additional restrictions given by accessible sets of the constraint distribution. Let t H ~(t) be a curve in the orbit space D. It is an integral curve of an inner derivation corresponding to h • C a (D) if
d ] ( c ( t ) ) = {h, f}(~(t)) for all f • C ~ ( D ) . Accessible sets of the space of inner derivations of C°¢(D) are maximal sets of points in D which can be connected by piecewise integral curves of inner derivations. MAIN THEOREM.
1. For every Ginvariant function h on D, the projections to D of integral curves of Yh are integral curves of the inner derivation corresponding to h, where h = p*h. 2. Accessible sets P of the space of inner derivations of C a ( D ) are manifolds. 3. For each accessible set P, the ring C a ( P ) of smooth functions on P is an almost Poisson algebra such that the restriction map C°~(~) ~ C ~ ( P ) : f ~ 7 v = 71v is an almost Poisson algebra homomorphism. 4. Each accessible set P supports a symplectic distribution (H~, ~ ) such that { f ~ , f ' v } V = Y T ~ f ' v , where YTT is the distributional Hamiltonian vector field of
deigned in te
s of
The paper is organized as follows. Section 2 contains the statement of Stefan Sussmann theorem and its application to the decomposition of a distributional Hamiltonian system into its simple components. The decomposition of distributional Hamiltonian systems with s y m m e t r y into components determined by s y m m e t r y type is studied in Section 3. The m o m e n t u m m a p for a Hamiltonian action of a Lie group on a symplectic distribution is discussed in Section 4. In Section 5 we study singular reduction of distributional Hamiltonian systems, and complete the proof of Main Theorem. A naive example illustrating some aspects of the theory is given in Section 6. It is an expanded version of the nonholonomic oscillator treated in [3].
238
JI~DRZEJ SNIATYCKI
This paper is not a review of dynamics of Hamiltonian systems with nonholonomic constraints; such a review will be given in [9]. It is concerned only with the problem of singular reduction of symmetries of distributional Hamiltonian systems. The author is greatly indebted to Richard Cushman for illuminating discussions of problems studied here.
2.
StefanSussmann theorem
A generalized distribution on a manifold D is a subset E C TD locally spanned by smooth vector fields [22]. An accessible set of a generalized distribution E on D is a maximal subset L of D such that every pair of points of L can be joined by a piecewise integral curve of vector fields in E. THEOREM 2.
1. Accessible sets of a generalized distribution E on a smooth manifold D are immersed submanifolds of D. 2. Accessible sets of E form a smooth foliation with singularities on D. The proof is given in [21, 22]. Since H is a distribution on D, its accessible sets are immersed submanifolds of D. For each accessible set L of H, the restriction of H to L is contained in TL. Hence, it defines a distribution HL on L. Moreover, the restriction of ~ to L gives a nondegenerate antisymmetric 2form ~ L on HL, and the restriction to L of the Hamiltonian h yields a function hL C C ~ (L). Since vector fields on D with values in H are tangent to accessible sets of H, the restriction to L of the almost Hamiltonian vector field Yh defines a vector field YhL on L, and YhLlWL = OHLhL. Hence, our nonholonomic Hamiltonian system has a decomposition
(D,H,w,h)=
U (L, HL,WL, hL), La.s.H
(5)
where L r u n s over accessible sets of H. We say that a nonholonomic Hamiltonian system (D, H, ~ , h) is simple if the distribution H on D has only one accessible set, namely D. For each accessible set L of H , the manifold L is an accessible set of HL. Hence Eq. (5) gives a decomposition of (D, H, ~ , h) into its simple components. Throughout the remainder of this paper we assume that the original nonholonomic Hamiltonian system is simple. In other words, D is the unique accessible set of H. A more detailed analysis of the case when (D, H, ~ , h) is not simple wilt be given in [9].
3.
Symmetry type
Let G be a s y m m e t r y group of a simple distributional Hamiltonian system (D, H, ~ , h). For each u E D, the isotropy group of u is
Cu= { g c G I g . u = u } .
ALMOST
POISSON
SPACES
AND NONHOLONOMIC
SINGULAR
REDUCTION
239
Since the action of G on D is proper, isotropy groups are compact. For every compact subgroup K of G, the set
DK = {u E D I G~ = K} consists of points of symmetry type K. It is a local manifold. In other words, connected components of DK are manifolds [14, 8]. Let M be a connected component of DK. It is a submanifold of D. The intersection H M = H N T M is a distribution on M and the restriction VZM of w to HM is nondegenerate [14]. For every f E C ~ ( D ) , we denote by fM the restriction of f to M, and YfM the vector field on M with values in HM given by
YL,,IWM = OHMfM.
(6)
In other words, YfM is the distributional Hamiltonian vector field of fM defined with respect to Z~M. Let Yf]M denote the restriction of Yf to points of M. PROPOSITION 1. YI]M = YfM for every Ginvariant function f E C ~ ( D ) .
Proof: Let Ct be the local 1parameter group of local diffeomorphisms of D generated by Yr. Since H , w, and f are Ginvariant, the distributional Hamiltonian vector field Yf of f is Ginvariant. Hence, Ct commutes with the action of G on D. In particular,  { g • a 1 9 .
=
= {geGlg.u=u}=G
= {9 • a l
=
~.
Therefore, if u • Dr( and M is the connected component of DK containing u, then YI(u) • T~DK = TuM. Since Yl(u) • H, it follows that Yf(u) • H N TuM = HM. Restricting Eq. (1) to vectors in HM, w e get YflM_]'I;UM = OHMfM. Taking into account Eq. ( 6 ) we obtain YIIM = YfM" [] As a consequence of Proposition 1 we have a decomposition
(D, H, ~ , h) =
U K c.s. G
U M c.c.
(M, HM, wM, hM),
(7)
DK
where K runs over compact subgroups of G and M runs over connected components of
DK. Note the essential difference between the decomposition (7) and the decomposition into simple components given by Eq. (5). The decomposition into simple components is independent of the Hamiltonian h, and depends only on the accessible sets of the distribution H. The decomposition given here depends essentially on the fact that G is a s y m m e t r y group of (D, H, w, h). Moreover, in general, the restriction HIM of H to points of M need not be contained in T M . Since HM is a distribution on M, it defines on M the structure of smooth foliation with singularities [21]. Leaves of this foliation are accessible set of HM. They are immersed submanifolds of M [21, 22].
240
Jt~DRZEJ
SNIATYCKI
Let N be be an accessible set of HM. The restriction of HM to points in N is a distribution HN on N. Let WN denote the retriction ofwM to HN, and h g the restriction of hM to N. The argument leading to decomposition (5) implies that (N, HN, WN, h N) is a distributional Hamiltonian system. Hence, we get a refinement of decomposition (7):
(D,H,w,h)=
U
U
U
K c.s. G
/tl c.c. DK
N a.s. HM
(N, HN,ZZN,hN),
(8)
where K runs over compact subgroups of G, M runs over connected components of DK, and N runs over accessible sets of HM. 4.
Momentum
map
For ~ Efl, let X~ denote the vector field on D generating the action on D of exp(t() C G. We denote by ~ the set of ( E ~ such that Xe is in H and there exists ,]~ C C*~(D) satisfying X ~ A ~ = OHJ~. It has been shown in [20] that b is an ideal in g. We assunw that there exists a smooth map J : D ~ ~* such that J~ = (J [~) for every ~ E b We shall refer to J as the momentum map for the action of G on D By the nonholonomic Noether theorem, Y I J = 0 for every Ginvariant function f on D [10, 20]. Hence, the sets J  l ( a ) n N give rise to a further partition of D preserved by the motion. THEOREM 3.
Connected components P of j  l ( a ) A N are submanifolds of N.
The proof will be given in [9]. Thus, we get a refinement of the partition of D given in (8):
U
"
U
K c.s. G
AI c.c. D~,
U
N a.s. HM
U
aE~*
U
P c.c. J
I(cQNN
Let E be a generalized distribution on D spanned by distributional Hamiltonian vector fields Yf of Ginvariant functions f on D. Distributional Hanfiltonian fields have values in H. Hence, E c_ H, and accessible sets of E are contained in the corresponding accessible sets of H. PROPOSITION
2.
E[M
=
HAs n
kerdJ.
Proof: By assumption u E DK is fixed by the action CK of K oi1 D given by the restriction of ~5 to K. Hence, K acts oi1 T**D by the derived action ~K,~, : K x T~D , T~D : (g,w) H TCg(w). Let v c T , , A I A H N k e r d J . Thus, v E T ~ M = TuDK, which implies that v is invariant under action of K on T~D. Moreover, v C Hu and (dJ I v) = O. Let S,, be the slice through u for the action of G on D. We have
T¢,S,, Q T~,(G. u) = Z , D .
ALMOST POISSON SPACES AND NONHOLONOMIC SINGULAR REDUCTION
241
We can choose S~ so that
H~, = (T~Su n H~) • (H~ N ker Tp) = (TuSu N H~) • (H~ n T~(G . u) ). The second summand H~ N Tu(G • u) is spanned by the vectors fields X~ such that X ~(u) = Yj~ (u) for ~ E ~. Since v c H n ker d J, it follows that
vz(v, Yj~(u)) = (dJ~ Iv> = O.
(10)
Let c2 e T~,D be such that (~l w) = c~(v,w) for a l l w E H~. Since w~ and v are invariant with respect to the derived action of K on T~D, we may assume without loss of generality that ~ is Kinvariant. Eq. (10) implies that (~ I w) = 0 for all w E T~,(G. u). Choose a Kinvariant metric k on D. Let hor TuD be the korthogonal complement of Tu(G • u). The slice S~ can be chosen to be the image of a neighbourhood of 0 in h o r T ~ D under the exponential map exp~ : T,~D ~ D. Hence, we can extend p linearly in a neighbourhood of u in Su, and then construct a compactly supported function fl on S~ such that a*p = dfl(u), where a : S~ ~ D is the inclusion map. In particular,
Iw)
forall
weTuS~nH~.
Let f2 be the K average of fl. That is f2  v o l K
~fldk,
where dk is a Haar measure on K. By construction ~ is K invariant. Hence, dfl(u) is K invariant and df2(u) = dfl(u) = o'*(p. Let f E C ~ ( D ) be a Ginvariant extension of f2 and YS the distributional Hamiltonian vector field of f. For each w C TuSu n H~,
w)  (as(u) I w) = Since f is Ginvariant, w(Ys(u), w) = (df(u) lw) = 0 for all w E H u n Tu(G. p). Taking into account Eq. (10) we see that w ( Y s ( u ) , w ) = w ( v , w ) for all w E H~. Hence, v = Y$(u). This implies that TuM n H n k e r d J C_ Eu. For u E DK, let a = J(u). Let M be the connected component of DK containing u, N the accessible set of HM containing u, and P the accessible set of E through u. Then p C_ j  l ( ~ ) A N . Since YSJ = 0, for every Ginvariant function f on D, integral curves of vector fields in E are contained in the level sets of J. Hence, E C_ ker dJ. Proposition 1 implies that the restriction ElM of E to points of M is contained in T M . Hence, ElM C_ H n T M . Therefore, ElM = T M n H n k e r d J = HM N ker dJ. [] THEOREM 4. For each u E D, the accessible set of E through u is the connected component of j  1 (c~) O N containing u. The proof will be given in [9].
242 5.
JI~DRZEJ SNIATYCKI Reduction
Let t ~+ u(t) be an integral curve of Yh and t ~+ ~(t) = p(u(t)) its projection to D. For every f E C°°(D),
d f(u(t)) = (Yhf)(u(t)) = {h, f}(u(t)).
(11)
If h is Ginvariant, then h = p ' h , where h E Coo(D). Restricting Eq. (11) to Ginvariant functions f = p*f we get
d ] = ({ h , f g} ~ ( g ( t() ) fort all) f E) Coo(D).
(12)
Hence t ~~ g(t) is an integral curve of the inner derivation of C ~ ( 9 ) corresponding to h. This proves part 1 of Main Theorem. For each connected component M of DK, its projection M = p(M) to D is a manifold [14, 8]. By Proposition 1, for each Ginvariant function h o n D , the restriction of Yh to M is equal to YhM, where hM is the restriction of h to M. Since h is Ginvariant, the vector field YhM projects to a vector field YhM on M. In other words,
TpM o YhM = YhM o PM,
(13)
where tiM : M ~ M is given by the restriction to M of the orbit m a p p : D ~ D. Let t ~ u(t) be an integral curve of Yh contained in M. Then its projection t ~7(t) = p(u(t)) to D is contained in M and it is an integral curve of Y h i . Eq. (12) yields {h, f } ~ ( g ( t ) ) = d f ( g ( t ) )
= (YhMf)(g(t))
for all
] E Coo(D).
(14)
Eq. (14) implies that inner derivations corresponding to functions h E Coo(D), when restricted to points of M, are given by smooth vector fields YhA~ o n M. Hence, accessible sets of inner derivations of Coo(D) coincide with accessible sets of generalized distributions on manifolds M C_ D. From the Stefan Sussmann Theorem we conclude that accessible sets of inner derivations of Coo(D) are manifolds. This proves part 2 of Main Theorem. Let P be an accessible set of inner derivations of C ~ ( D ) contained in M . Let f v E C ~ (P). Since P is an immersed submanifold of M , for each g E P there is a neighbourhood U of g in P, open in the manifold topology of P, such that the restriction f T I u of f T to U can be extended to a function f s / E C ~ ( M ) . Since M is closed in D, every Ginvariant smooth function on M can be extended to a smooth Ginvariant flmction on D. Hence, f ~ can be extended to a flmction f C Coo (D). For every g E P, and f g , f ' g E Coo(P), let f, f ' E Coo(D) be extensions to D of f p , f ' T restricted to an appropriate neighbourhood of g. Eq. (14) implies t h a t
{f,
= (gsA,?v)
(15)
ALMOST POISSON SPACES AND NONHOLONOMIC SINGULAR REDUCTION
243
where fM is the restriction of f = p * f to M, depends only on the functions ] p and f ' ~ . Hence, we can define an almost Poisson bracket {., .}p on C°°(P) such that { f p , f ' p } T ( ~ ) = {f, f ' } ~ ( ~ )
for all
~ E P.
(16)
Clearly, the restriction map
coo( )
coo(P): S
is an almost Poisson algebra homomorphism. This proves part 3 of Main Theorem. For each connected component M of DK, the distribution HM also projects to a distribution Tp(HM) on M. However, the form "6uM will not push forward to a form on Tp(HM) unless it annihilates the vectors tangent to the fibres of the projection map
pM :M* M. PROPOSITION 3.
ElM = {w E HM [ w(w,v) = 0
V v E HMNkerT(pM)}.
Proof: Let OM be the algebra consisting of ~ E ~ such that the oneparameter groups exp(t~) preserve M. Then, HM A ker T(pM) is spanned by distributional Hamiltonian vector fields Yje, where ~ E I~M. For w E HM, cv(w, Yde) = wAdJ~ = 0 for all E [}M if and only if w E HM MkerdJ. However, by Proposition 2, HM Mker d J = ElM, which completes the proof. [] Let H ~ = TpM(EIM). It follows from Proposition 3 that there exists a unique skew symmetric z ~ on H ~ such that its pullback by PM agrees with the restriction of sv to ElM. It has been proved in [5] that w ~ is nondegenerate. Hence, (M, H~7 , w ~ ) is a symplectic distribution. PROPOSITION 4. For every Ginvariant h E C°°(D), the vector field given by (13) satisfies the equation
Yh M
on
M
Proof: For every Ginvariant h E Coo(D), Proposition 1 ensures that ]"hiM = Yh,~,. Since Yh^, has values in E l M ~ we can restrict equation YhMAV~M = C~HMhM to ElM obtaining Yh~,i.JgiYitl[EIM : OE[Mh M. (18) However, H ~ = TpM(E]M), (SVM)[EIM = p ~ z a ~ , and hM = PMh~. Pushing equation (18) forward by PM : M * M we get Eq. (17). [] COROLLARY. Y h M is the distributional Hamiltonian vector field of h ~ E C°°(M) relative to the symplectic distribution (M, H ~ , cv~).
244
Jt~DRZEJ SNIATYCKI
We can decompose the distributional Hamiltonian system (M, H ~ , c ~ , h~/) into its simple components (M, H~, wZ3, h~)
=
U
(P, HN, wp,
(19)
hp),
a.s. H~3
where P are accessible sets of H ~ . Since ~ = TpM(E]M), accessible__sets of H ~ are accessible sets of inner derivations of C ~ ( D ) which are contained in M. Hence each simple component (P, H~, wp, hp) on the righthand side of Eq. (19) is a distributional Hamiltonian system. Moreover, Eqs. (15), (16) and Corollary imply that, for every f_z and f ' p in C ~ ( P ) , {fv, f'g}~ = Yf~f'P, where Yf~ is the restriction of Y f ~ to P. This completes the proof of part 4 of Main Theorem. It should be noted that our proof of Main Theorem is independent of Theorems 3 and 4. If we take into account these theorems, we obtain a generalization to sympleetic distributions of results of [19] and [4] proved for sympleetie manifolds. 6.
An example On ]I~4 with coordinates (x, y, z, q) consider a point particle with energy h ~(x 1 "2 + ~)2 + ~2 + q2) + l~( x 2 q _ y 2 + q2)
subject to a nonholonomic constraint [condition] 0 = ( x2 + y2 + 1)~, which corresponds to a distribution D = span {Ox, Oy, Oz + (x 2 + y2 + 1)0q} on ]R4. Since [Ox, [Ox, Oz + (x 2 + y2 + 1)0q] = 20q, Chow's theorem, [7], implies that IR4 is an accessible set of D. Every vector in D is of the form
u = dCOx+ ~/cgy+ ~[0~ + (x 2 + y2 + 1)0q]. Hence, we can use (x, y, z, q, 2, ~), 5) as coordinates in D. However, it is more convenient to describe D in terms of its inclusion in R s ~ T R 4. A vector field on If(8 is tangent to D if it preserves the equality 0 = (x 2 + y2 + 1)~. Hence,
T D = span {05 + 2xi:O4, Ou + 2y~.04, Oz, Oq, 05, 09, Oe + (x 2 + y2 + 1)04 }
]D"
The distribution H is given by H = span{0x + 2x~OO, Oy + 2y~.Oq, O~ + (x 2 + y2 + 1)0q,
05, 09, {1+ (x2 + y2 + l)2}l[Oe + (x2 + y2 + l)Oo]}lD . An alternative description of H is H = ker{(x 2 + y2 + 1)dz  dq} A ker{(x 2 + y2 + 1)d~ + 2x~'dx + 2y~,dy  dgt}lD.
(20)
ALMOST POISSON SPACES AND NONHOLONOMIC SINGULAR REDUCTION
245
Tile symplectic form w on H is the restriction of the symplectic form
w = dx A d~ + dy A dy + dz A d~ + dq A do
(21)
on R s to vectors in H . It is easy to check t h a t the vector fields in Eq. (20) fornl a symplectic fl'ame of H . T h e distributional H a m i l t o n i a n vector field of a s m o o t h function f oi1 D is the unique vector field Yf in H satisfying the equation
Y/J~
= df+A{(x 2+y2+l)dzdq} + #{(x 2 + y2 + 1)di + 2x~dx + 2y~dy  dO}
for some L a g r a n g e multipliers A and #. Consider tile action of G = T x R on R 4 given by rotations in the (x, y)plane and translations along the zaxis. Its prolongation to TIR 4 preserves D. Hence, it restricts to an action (P: G x D ~ D such t h a t , if (x', y', z', q', ~', ~)', :i') = ¢ ( ( e it, s), (x, y, z, q, J:, ~), ~)), then x ~ = xcostysint,
J=xsint+ycost,
z~=z+s,
2 ~ = 2cost~sint,
/)~=2sint+ycost,
~=~.
qt=q,
Clearly, this action is a s y m m e t r y of the distributional Hanfiltonian s y s t e m (D, H, vz, It). T h e Lic algebra g of G is isomorphic to ll~2. Let ~ E ~ generate the action of ~P on R 4. T h e corrcsponding f u n d a m e n t a l vector field is x~ = yG
+ z G  90~ + 20y.
Since yOx + xOy = y(Ox + 2Xi0q) + x(cgy + 2y~0q), it follows t h a t X ~ has vMues in H. Moreover, X~w
= OH(x9  y~),
which implies t h a t X~ = Yj, where J = xy  yx is a constant of motion. Similarly, we denote by ~ the element of ~ generating the action of ~ on R 4. T h e corresponding f u n d a m e n t a l vector field X ~ = c3~ is not in H . Hence, [} = span{~}. T h e group G has c o m p a c t subgroups: the circle ~r and the trivial s u b g r o u p {e}, where e is the identity in G. T h e set of points in D of s y m m e t r y type T is
Dv = {(O,O,z,q,O,O,~,~) l (z,q, 5 ) E I~3}.
(22)
It is connected. T h e intersection of H with T D v is HDv : s p a n { O z + Oq, O~ + Cgq}. HD v is an involutive distribution on DT with integral manifolds G:
{(z,q,~) ~ 3 1 z  q : p : c o n s t } .
T h e restriction of HD v to points of Np is HN, = T N w Further, the restriction of w, given by Eq. (21), to T N , gives a symplectic form a~N,, on Np. Hence, tile distributional
246
Jt~DRZEJ SNIATYCKI
H a m i l t o n i a n s y s t e m (NB, HNp,"SJNp,hN,) is a H a m i l t o n i a n s y s t e m (Np, wm~, hN~). T h e set D is the c o m p l e m e n t of Dv in D. It is a connected open dense submanifold of D. T h e decomposition (8), applied to our example, is
Each Np is a manifold contained in the zero level of the m o m e n t u m m a p J = xy  y2. Hence, we need only to consider the intersections D~ N j  1 (a) as c~ runs over II~. If a ~ 0, then D~ N J  l ( a ) is a connected submanifold of D~. T h e intersection J  ~ ( 0 ) N D~ has 2 connected c o m p o n e n t s P+, which are submanifolds of D~, see [8]. This confirms the assertions of T h e o r e m 3. Moreover, decomposition (9) reads
(23) A function f E C~(D) is Ginvariant if it depends on J = x y  y2, r = x 2 + y2, s  22 + y2, q and ~. [17]. T h e distributional H a m i l t o n i a n vector fields of these functions are
Yj = yOx + xOy  flO.~+ ~Oy, Y~
=
 2xO~ 
2yOy,
Y8 = 25r(Ox+ 2xkO0) + 2~)(Oy + 2y~Oq) 4(xkk + 4 y y k ) ( x 2 + y2 + 1)[0~ + (x 2 + y2 + 1)0q], ~
Ya = Y~ =
(x 2 + y2 + 1)2 + 1 (x 2 + y2 + 1) l+(x 2+y2+1)2
[0e + (x 2 + y2 + 1)0~],
1 l+(x 2+y2+1)2[0~+(x
2+y2+1)0q]
2k(x 2 + y2 + 1) ~ (x 2 + y2 + 1)2 + 1 (xO~ + yOy). These vector fields are independent in De, and their flows preserve J. Hence, dim E[ D,, = 5, and accessible sets of E[D~ are D~ N J  l ( a ) , for a ¢ 0, P+, and P_ when (~ = 0. At points in D T , the vector fields Yj, Y, and Y8 vanish identically, and E[D.~ = HIDe;. Hence, accessible sets of E[D.r are the manifolds Np. This confirms the s t a t e m e n t of T h e o r e m 4 in our case. We can now describe the reduced phase spaces P a p p e a r i n g in the decomposition (19). P + and fi~ are the spaces of orbits of G in the manifolds P+, and D~ N J  l ( a ) , respectively. For fi E P + , or g c P ~ , H a = Tp(Hu), where u C p  l ( ~ ) . Since dim P:~ = dim P~ = 6, dim G = 2, dim E~ = 5, and d i m ( E ~ A ker Tp) = 1, it follows t h a t dim P~: =
ALMOST POISSON SPACES AND NONHOLONOMIC SINGULAR REDUCTION
247
d i m P ~ = 4, and d i m H ~ = 4. Hence, H p + = T P + , and wT± defines a differential 2form wp± on P + . Similarly, Hp, = T P ~ and ~vp.~ defines a 2form w~, on P~. Under the orbit m a p p which we denote diffeomorphic to form wT. Hence,
: D P. Np. the
* D map, all the manifolds Np have the same projection to D, Since every Gorbit intersects Np only once, it follows t h a t P is Thus, P is a 2dimensional manifold endowed with a symplectic projection to D of decomposition (23) is
Examples often have more structure than the general theory they illustrate. In this example, the s y m m e t r y group G acts on the configuration space R 4. The action of G on D C T]I(4 is obtained by prolongation of its the action on the base space. Hence, following [6], one could study the splitting of distributional Hamiltonian vector fields of invariant functions into the corresponding reduced equations and the m o m e n t u m equations. However, in the theory developed here, the action of G on D need not be the prolongation of its action on the configuration space. Hence, in this general setting, there is no natural way of defining the the splitting of distributional Hamiltonian vector fields of invariant functions into the corresponding reduced equations and the m o m e n t u m equations. REFERENCES [1] J. M. Arms, R. Cushman and M. J. Gotay: A Universal Reduction Procedure for Hamiltonian Group Actions, in The Geometry of Hamiltonian Systems, T. S. Ratiu (ed.), 3151, Birkh~user, Boston 1991. [2] L. Bates: Rep. Math. Phys., 42 (1998), 231247. [3] L. Bates and R. Cushman: Rep. Math. Phys. 44 (1999), 2935. [4] L. Bates and E. Lerman: Pacific J. Math. 191 (1997), 201229. [5] L. Bates and J. Sniatycki: Rep. Math. Phys. 32(1993), 99115. [6] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. Murray: Arch. Ration. Mech. Anal. 136 (1996), 2199. [7] W. L. Chow: Math. Ann. 117 (1939), 98115. [8] R. Cushman and L. Bates: Global Aspects of Classical Integrable Systems, Birkh~user, Basel 1997. [9] R. Cushman, J. J. Duistermaat and J. Sniatycki: Chaplygin and the Geometry of Nonholonomic Constraints, in preparation. [10] R. Cushman, D. Kemppainen, J. Sniatycki and L. Bates: Rep. Math. Phys. 36 (1995), 275268. [11] R. Cushman and R. Sjamaar: On Singular Reduction of Hamiltonian Systems, in Symplectic Geometry and Mathematical Physics, P. Donato (ed.), 114128, Birkh~iuser, Boston 1991. [12] R. Cushman and J. Sniatycki: Canad. J. Math. (to appear). [13] J. J. Duistermaat and J. A. C. Kolk: Lie Groups, Springer, New York 1999. [14] V. Guillemin and S. Sternberg: Symplectic Techniques in Physics, University Press, Cambridge 1984.
248 [15] [16] [17] [18] [19] [20] [21] [22]
JI~DRZEJ SNIATYCKI Wang Sang Koon and J. E. Marsden: Rep. Math. Phys. 42 (1998), 101134. A. J. van der Schaft and B. Maschke: Rep. Math. Phys. 34 (1994) 225 233. G. Schwarz: Topology 14 (1975), 6368. R. Sikorski: Introduction to Differential Geometry, PWN, Warszawa 1972 (in Polish). R. Sjamaar and E. Lerman: Ann. Math. 134 (1991), 375 422. J. Sniatycki: Rep. Math. Phys. 42 (1998), 523. P. Stefan: Proc. London Math. Soc. 29 (1974), 699713. H. Sussmann: Trans. Amer. Math. Soc. 180 (1973), 171 188.