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Stochastic chaplygin systems
Vol. 66 (2010)
REPORTS ON MATHEMATICAL PHYSICS
NO.3
STOCHASTIC CHAPLYGIN SYSTEMS SIMON HOCHGERNER Section de Mathematiques, Station 8, EPFL, CH-1015 Lausanne, Switzerland (e-mail: [email protected]) (Received February 9, 2010 -
Revised August 7, 20/0)
We mimic the stochastic Hamiltonian reduction of Lazaro-Cami and Ortega [18, 19] for the case of certain nonholonomic systems with symmetries. Using the nonholonomic connection it is shown that the drift of the stochastically perturbed n-dimensional Chaplygin ball is a certain gradient of the density of the preserved measure of the deterministic system.
Keywords: Stochastic nonholonomic systems, stochastic mechanics.
1. Introduction Imagine a ball sittmg on a rough horizontal table. Because of the roughness of the table this ball-the Chaplygin ball-cannot slip, but it can tum about the vertical axis without violating the constraints. Its geometric and gravitational center coincide but there may be an inhomogeneous mass distribution. Suppose now that the ball is subject to a Brownian noise such that there is a random jiggling in all of its angular and translational degrees of freedom. This problem is similar to the stochastic rigid body considered in [19] but upon imposing the no-slip constraints some differences can be expected. One may wonder if the stochastic Chaplygin ball will acquire a drift that makes it roll on the table or spin about its vertical axis or both? The answer to this question is given by Theorem 3.3: The drift follows a Fick's law in the following sense. The configuration space of the n-dimensional ball is Q = SO(n) x JRn-l and there are n - 1 constraints corresponding to the directions in the table JRn-l. By a symmetry reduction argument (compression) one can eliminate the JRn-l-factor and the deterministic motion of the ball can be described by the geodesic equations of the so-called nonholonomic connection V nh on SO(n). With respect to this connection one can now show that the process on SO(n) describing the balls stochastic motion is a nonholonomic diffusion separating into a drift- and a martingale-term. See Definition 3.2. Theorem 3.3 says that this Vnh-drift equals - ~ grad"? log N where N is the preserved density (3.5) of the deterministic ball and the gradient is computed with respect to the kinetic energy metric (3.4) of the ball. [385]
S.HOCHGERNER
386
In particular, when the ball is homogeneous there is no drift. Moreover, it is shown in Corollary 3.4 that the homogeneous ball's stochastic process factorizes to a Brownian motion on 'the ultimate reduced configuration space' sn-l = (Q/lRn-I)/SO(n-l). This is in analogy to the corresponding deterministic case where the motion is Hamiltonian at the ultimate reduced level (see [12].) In 3 dimensions the drift - grad"? log N does not have an angular velocity component about the vertical axis of the ball in the space frame. For dimensions n > 3 it turns out that this property is related to the hamiltonization of the deterministic system: If the inertia matrix describing the ball's mass distribution satisfies the hamiltonization condition (3.7) then the drift does not have an angular velocity component about the vertical axis in the space frame. On the other hand, the drifts angular momentum about the vertical axis is always 0, regardless of the dimension or the mass distribution. Section 2 starts by collecting some definitions and facts from stochastic differential geometry as presented in [15, 10]. Then we rehearse the basics of the stochastic Hamiltonian mechanics and their symmetries as introduced in [18, 19]. In Section 2.C these ideas are transferred to describe stochastic G-Chaplygin systems. It is noticed that the reduction of symmetries, termed compression in this context, works naturally. In Section 3 this construction is applied to the n-dimensional Chaplygin ball. First some facts about the deterministic system, such as the preserved measure and Hamiltonization, are recalled. Then the Hamiltonian construction of [18] of Brownian motion on the configuration space Q = SO(n) x lRn - 1 is reviewed. Section 3.C makes the constraint forces act on the Brownian motion according to the recipe of Section 2.C. Thus a constrained stochastic motion is obtained and the mentioned above Theorem 3.3 is found. Finally, we note that we can also treat the cases of angular jiggling only or horizontal jiggling only. The latter corresponds to the Chaplygin ball sitting on a table which undergoes a translational Brownian motion. The 3-dimensional version of this case has been considered in [20].
4
2. Stochastic Chaplygin systems and reduction of symmetries 2.A. Stochastic geometry
We state some notions from stochastic differential geometry. The references we used here are [10, 15]. See also the appendix of [18]. Let M be a manifold, let (Q, F, {Ft : t ::: O], P) be a filtered probability space, and let r : lR+ x Q ~ M be an adapted stochastic process. (We consider only continuous processes.) The process r is called a semi-martingale if for is a semi-martingale in lR for all f E COO(M), see [10, Chapter III]. The definition of a martingale in M depends upon a choice of a connection. Let V be a connection in T M ~ M. Then the Hessian of V is defined by HessV'(f)(X, Y) = XY(f) - VxY(f)
STOCHASTIC CHAPLYGIN SYSTEMS
387
for X, Y E X(M). This is bilinear in X and Y but not symmetric, in general, since HessV'(f)(X, Y) - HessV'(f)(Y, X) = -TorV'(X, Y)(f). 2.1. A semi-martingale I' : lR+ x Q -+ M is said to be a V-martingale E COO(M),
DEFINITION
if, for any
f
f
0
r, -
f
0
r,
-it
Hess'' (f)(df s, dfs)ds
is a local martingale in lR. In [10, Chapter IV] this is stated in terms of a torsionless connection but it is noted that one can also allow for connections with torsion, since it is proved ([10, (3.14)]) that J~ Hess" (f)(df s, dfs)ds depends only on the symmetric part of HessV'. The situation is similar to the notion of a V-geodesic. This also depends only on the torsionless part V - 4TorV' of V. Let M and N be manifolds, let (Q, F, {Ft : t ~ O}, P) be a filtered probability space, and let X : lR+ x Q -+ N be a semi-martingale. A Stratonovich operator S from TN to T M is a family of linear maps S(x,y) : TxN ----+ TyM
which depend smoothly on x E Nand y EM. In other words, S is a section of T* N Q9 T M -+ N x M. A Stratonovich differential equation for a semi-martingale I' : lR+ x Q -+ M is written as
sr =
S(f, X)8X.
See [10, Chapter VII] for the precise meaning of this equation as well as the existence and uniqueness (up to explosion time) of solutions. Assume now that N = lR x lRn and that X : lR+ x Q -+ lR x lRn , (t, w) 1---+ (t, Wt (w)) where W denotes n-dimensional Brownian motion. Let Xo, X I, ... ,Xn be vector fields on M and define the Stratonovich operator S(x, y) : lRn + 1 ----+ TyM, (t, wI, ... , w n ) f----+ tXo(Y) + L:w i Xi(y). Let
f
E COO(M). Then
8(f
0
f)
f
0
I' : lR+ x
Q -+
lR satisfies
= df(f).S(X, f)8X = df(f).X o(f)8t + L:df(f)·Xi(f)8W i •
Hence I' defines a diffusion in M by [15, Capter V, Thm. 1.2]. The generator of this diffusion is the second-order differential operator A given by Af = Xof
where
f
+ 4L:X iX if,
E COO(M).
DEFINITION 2.2. If M is equipped with a Riemannian metric J1, and .6. is the associated Laplacian then the diffusion is called Brownian motion in (M, J1,) if
A = 4.6..
388
S. HOCHGERNER
This definition agrees with the one given in [10, (5.16)] or [21, (8.5.18)] but differs slightly from the one in [15, Chapter V, Def. 4.2] where it is required that n = dimM.
2.B. Stochastic Hamiltonian systems
This section presents some of the concepts elaborated in [18, 19]. Let again N = ~n and consider a Poisson manifold (M, {.,.}) together with an ~n-valued Hamiltonian function h = (hi, ... , h n) : M -+ ~n. Let X : ~+ x Q -+ ~n be a semi-martingale. The associated stochastic Hamiltonian system is given by the Stratonovich equation of = S(X, r)OX where S is defined in terms of the Hamiltonian structure, that is, S(x, y) : ~n -----+ TyM, (r ', ... , x")
1-------+
LXhi (yjx',
where the Hamiltonian vector field Xhi is the vector field corresponding to the derivation {hi, .}. When (M, w) is a symplectic manifold then one uses the Hamiltonian fields defined by i (Xh)W = dh., These systems allow for a symmetry reduction analogous to classical mechanics. We state the symplectic version of this theorem of [19, Section 6]. THEOREM 2.3 (Stochastic Hamiltonian reduction). Let (M, w) be a symplectic manifold with Hamiltonian h : M -+ ~n and stochastic component X as above. Assume that (M, w, h) are invariant under the free and proper action of a Lie group G such that a coadjoint equivariant momentum map J : M -+ g* exists. Fix a level)", E g*. Then J-I()..,) is invariant under the flow of S. Moreover, S induces a Stratonovich operator SJ.. from T~n to T(J-I()"')/GJ..) and solutions of S with initial condition in J-I()..,) project to solutions of SJ... The induced operator is given by
SJ..(x, Yo) : (r ', ... .x")
1-------+
LXhA(YO)X i , I
where x
E
~n, Yo E J-I()"')/GJ.. and XhA is the Hamiltonian vector field with respect I
to the reduced symplectic form on J-I()")/GJ.. of the induced function h7.
2.C. Stochastic G-Chaplygin systems
A (deterministic) G-Chaplygin system consists of a Riemannian configuration manifold (Q, J1), a Lie group G acting freely and properly by isometries on (Q, J1), and a horizontal space V of the principal bundle n : Q -tt Q / G. Hence V is the kernel of a connection form A : T Q -+ g. The Lagrangian of the system is the kinetic energy, i.e. L(q, v) = 4J1Q(v, v). In general, V is not J1-orthogonal to the vertical space ker T Jr. We will henceforth identify T Q = T* Q via J1. See also Section 4. Let JG : T Q -+ g* denote the standard equivariant momentum map associated to the lifted G-action on T Q. Given a G-invariant function h : T Q -+ ~ one may use Noether's theorem to conclude that the Hamiltonian vector field X h is tangent to
389
STOCHASTIC CHAPLYGIN SYSTEMS
J(;"I(O) and, moreover, is projectable for J(;"I(O) -» J(;"I(O)/G = T(QIG). This is why results such as Theorem 2.3 work. Since V is a perturbed version of J(;" 1 (0) we look for a substitute construction. Consider the horizontal space associated to the pulled back connection r" A = Ao Tr: T(TQ) ~ TQ ~ 9 of the tangent lifted G-action on TQ,
F:= ker r" A C TTQ. By assumption V is also G-invariant. Thus we can consider the restricted G-action on V and the associated connection l*r* A : TV ~ 9 where l : V ~ T Q is the inclusion. Define C:= kerl*r*A = Ft: TV C TV to be the horizontal space of the principal bundle V -» VIG = T(QIG). According to [1] we can decompose T T Q along V as
TTQIV = C EB CQ where CQ is the Q-orthogonal of C in TT QIV. This implies that the fiber-wise restriction of l*Q to C x C is nondegenerate. For z E V define the projection
P, : Tz(T Q) ------+ C,
(2.1)
where we project along C~. Moreover, for a k-form 4> on TQ we denote the fiber-wise restriction of l*4> to nkC by 4>c. For a function h : TQ ~ ~ we may thus define the vectorfield X~ on V with values in C by the formula PzXh(z) = (Qc);l(dh)~ =: X~(z)
(2.2)
where Z E V. The link to the nonholonomic system introduced above is the following: Via the Legendre transform the dynamics of the nonholonomic system (Q, V, L) can be equivalently described by the triple (T Q, QC, H. = ~ u. (p, p)) together with Eq. (2.2). Thus the Lagrange multipliers have been encoded in the two-form QC or, equivalently, in the projection P : T(TQ)IV ~ C. The idea is that a nonholonomic system is a Hamiltonian system acted upon by constraint forces. The effect of the forces is described by the projector P : T(TmIV ~ C. In addition to this structure consider now a semi-martingale X : ~+ x Q ~ ~n and a Hamiltonian function h = (hi)i : T Q ~ ~n as above. The associated stochastic nonholonomic system is given by the Stratonovich equation 8r = SC(X, r)8X where the Stratonovich operator SC : ~n X V ~ Hom(T~N, C) is given by SC(n,z): ~n ------+ Cz, (r ', ... .x") ~ '£PzXh;(Z)X i = ,£Xf;(z)x . i
(2.3)
Thus the nonholonomic Stratonovich operator arises, by applying the constraint forces, as a projection of the Hamiltonian Stratonovich operator into C. When the hi are G-invariant we refer to the collection (Q, V, h = (hi)i, X) as a stochastic G-Chaplygin system. PROPOSITION 2.4 (Compression of stochastic G-Chaplygin systems). Let (Q, V, h = (hi)i, X) be a stochastic G-Chaplygin system with QC as above. Then the
390
S. HOCHGERNER
Stratonovich operator (2.3) compresses to a Stratonovich operator Snh from TjRn to D/G = T(Q/G) which is given by Snh(n, eo) : jRn ---+ Tzo(T(Q/G)), (r ', ... , x")
1----+
LX~(Z)xi I
where zo E T(Q/G) and h? : T(Q/G) ---+ jR is the function induced on the quotient from the invariant function l*hi. Moreover, solutions of (2.3) project to solutions of snh. Proof: Everything is entirely analogous to the proof of [19, Theorem 6.7] with the only difference that now one uses Proposition 4.1 instead of the usual symplectic reduction theorem. D We think of 8f = Snh(X, f)8X as the equations of motion of the system (Q, V, h, X).
PROPOSITION 2.5 (Ito representation). Let f representation of the equation 8(f 0 f) = snh(x, f
d(f
0
r) = LX~f(r)dXi i
I
E 0
Coo (T (Q / G)). Then the Ito f)8X is
+ 1LX~~X~f(f)[dXj,dXi] i,j
I
]
where Xi = pr' 0 X. Proof: This follows exactly as in the proof [18, Proposition 2.3]. It is only necessary to notice that this proof does not depend on whether or not the nonholonomic bracket Xr: f = {g, = -{f, g}nh satisfies the Jacobi identity. The only property of the Poisson bracket which is used in [18, Proposition 2.3] is the Leibniz rule and this feature is evidently shared by the nonholonomic bracket. D
nnh
3. The stochastic Chaplygin ball 3.A. The deterministic system
For background on the Chaplygin ball we refer to [6, 8, 9, 11, 12, 17]. The configuration space of Chaplygin's n-dimensional rolling ball is Q = K x V where K = SO(n) and V = jRn-l. The no-slip constraints are given by the distribution '0= (A + pr2)-1(0) c TQ, where
A = L1Ja ® ea , where el,"" en-l is the standard basis on V and we stick to the following conventions: I T K = K x t is trivialized via left-multiplication and t is equipped with the Ad-invariant inner product (., .}.2 Let H = SO(n -1) c K be the stabilizer in K of the n-th standard vector en such that H acts on V in the natural way. We 1In Section 2.e it was the connection form A + pr2 which was called A. 2The space XL (K) of left-invariant vectorfields and the Lie algebra t will also be identified without further notice.
391
STOCHASTIC CHAPLYGIN SYSTEMS
decompose £ = f) ED f)1- with respect to the Ad-invariant inner product (.,.) on t With respect to this inner product we introduce an orthonormal system
Ya , a = 1, ... , k = dim f) and Za, a = 1, ... , n - 1, on £ such that Ya E f) and Z; right-invariant vector fields ~a and ~a(s)
f)1-. Associated to this basis we define the ~a. In the left trivialization these read
E
= Ad(s-l)Ya
and ~a(s)
= Ad(s-l)Za.
Dually we introduce the corresponding right-invariant coframe pa
= (~a,')
and fJa
= (~a, .).
The Lagrangian is the function
L : TQ = K x £ x TV ---+ R (s, u, X, x')
f-----+
~(TIu, u)
+ ~(X', x'),
where TI is the inertia matrix in body coordinates. The rolling ball with the no-slip constraint is the nonholonomic system described by the data (Q, D, L) where the equations of motion follow from the Lagrange-d'Alembert principle. However, we will not have much use for the Lagrange function below since we will only perturb the resting ball. Note also that we overload the symbol (.,.) by using it for the Euclidean inner product on V as well as for the Ad-invariant structure on t From a structural point of view the decisive feature of the Chaplygin ball is that its constraints are given by a connection prz + A : T Q ~ V on the (trivial) principal bundle V ~ K x V --» K where V acts on itself by addition. Thus D is the horizontal space of this connection. However, D is not fL-orthogonal to the vertical space of the bundle. The fact that the system is nonholonomic is reflected in the nonflatness, CurvA = dA i= O. Compression of the Chaplygin ball system yields the almost Hamiltonian system (T K, Qnh, He) where Q nh is described in Proposition 4.1, T K and T* K are identified via the induced metric fLo(UJ, uZ)s
= (TIul' uz) + (As(Ul), As(uz)) = ((TI + A;As)UI, uz),
(3.4)
and He = ~fLO(U, u)s. The metric fLo is the sum of a left invariant and a right invariant term. Thus it constitutes an L+R-system, see [16]. Note the useful formula A* A(u)
= L>~a, u)~a.
The compressed system (T K, Qnh, He) is further invariant under the lift of the left multiplication action of H on K. Physically this corresponds to rotation of the ball about the en-axis in the space frame. This is an inner symmetry and gives, by the nonholonomic Noether theorem, rise to a conserved quantity. This quantity is just the standard momentum map JH : T K ~ f)* = f), (s, u) t--+ L>~(TIU)Ya of the (co-)tangent lifted H -action on T K . The Chaplygin ball shares an important feature with Hamiltonian systems. Namely, it possesses a preserved measure [6, 11]. At the compressed level-the T K -level-the
392 density
S.HOCHGERNER
N :K
-+ lR of this measure with respect to the Liouville volume on T K is I
N(s) = (detJLo(s))-z.
(3.5)
This function plays the central role in all questions of hamiltonization of the system. Note that N is H-invariant and thus descends to a function N: K/H = sn-I -+ R For further reference we also record that
d(1ogN) = ~][JLol~a, ~a], ~b}'1b.
n
(3.6)
In [12] it is proved that Q nh can be replaced by = QK - ~L([~a, ~b], _}'1 a /\'1 b without altering the equations of motion, that is, i(X~ )Q nh = i(X~ = d'He. Now ,...... ftc Il.c
)n
the new system (T K, Q, 'He) has the same dynamics but has the advantage of being liable to reduction with respect to the internal symmetry group H: Let J H : T K -+ ~* be the momentu~ map introduced above, A E ~ * and l : J "---+ T K the inclusion. Then l*Q descends to an almost symplectic two form Q A on J// (")...)/ H A. (l*Q nh is not horizontal for the projection onto J//(A)/Hd In this way one can recover the hamiltonization of the 3-dimensional ball of [4, 5]: It is shown in [12, Proposition 4.4] that d(Nn A ) = 0 if n = 3. Thus, for n = 3, the rescaled vector field N-IX'flc is Hamiltonian with respect to (Jiil(A)/HA = TS 2 ,N n A, 'He).
Ii!, ()...)
Moreover, the homogeneous ball, TI = 1, is Hamiltonian at the Jiil(A)/HA-level for any dimension n. It is interesting to notice that none of these statements hold at the T K -level. In [13, Section 4.A] the following hamiltonization condition is proved for arbitrary dimension n: Let A = 0 E ~*. Then d(NQo) = 0 if and only if (n - 2)(/-LOI~d' [~b, ~c]) = La(/-Lol~a, [~b, ~a]8c,d - [~C> ~a]8b,d)
(3.7)
which is an algebraic condition on TI. For n = 3 this condition is trivially satisfied. In the stochastic context this condition appears in Theorem 3.3 below. 3.B. Brownian motion on the configuration space We follow [18] to construct Brownian motion on Q = K x V. Let VIL be the Levi-Civita connection of /-L«U, x'), (v, y')) = (TIu, v)
+ (x',
y').
Thus for u, vEt we have
V~v
= (V~v, 0) = (~[u, v] + ~TI-I([u, TIv] + [v, TIul), 0)
where Vi is the Levi-Civita connection of the left-invariant metric defined by (TI.,.) on K. Note that we identify t ~ t* via (.,.) whence TI: t -+ t* ~ t Let VI, •.. , V m ,
m
= dim t = ~n(n -1)
denote a basis which is orthonormal for (TI.,.). Thus VI, ..• , Vm , el, ... , en-I is a left-invariant frame on Q which is orthonormal with respect to the left-invariant
393
STOCHASTIC CHAPLYGIN SYSTEMS
metric It. (Remember that we identify T Q and T* Q via It.) Let the functions H o, Hi, Fa : TQ -+ JR be given by Ho(s, u, X, x')
= -~(Ru, I:V'~i Vi),
Hits, U, X, x')
= (Ru, Vi),
and
Fa(s, u, X, x') = (x', ea).
(3.8)
Consider the semi-martingale X : JR+ x Q ---+ JR x JRm X JRn- l, (t, w)
f-----+
(t, B/(w), ... , B;n(w), W/(w), ... , Wtn - I (w)),
where s', W a are m + n - 1 independent Brownian motions. The Stratonovich stochastic differential equation which is associated to these data is (3.9) where the Stratonovich operator from T (JR x JRm
X
JRn-l) to T K x T V is defined by
. 1)(t, b' , ... , bm ,w, I ... n ,w -I) S Ham( n,s,u,x,X
= XHo(S'
U,
x, x')t
+ I:X Hi (s, U, x, xl)b i + I:X Fa (s, U, x, xl)w a
with X H denoting the canonical Hamiltonian vector field of a function H : T Q -+ JR. Using the Ito representation of this equation [18] show that the solutions I' project via r : T Q -+ Q onto Brownian motion on Q. Using again the setting of [18] and Theorem 2.3 it is easy to see the following. Consider the V-action on Q as above. Let JVI(O) = {(s,u,x,O)} be the O-level set of the standard momentum map J v : T Q -+ V* = V of the lifted V -action on T Q. Then the Stratonovich equation (3.9) induces a Stratonovich equation ofo = So(X, fo)oX on JVI(O)/V = T K and solutions I' with initial condition in Jvl(O) project onto solutions fo. Moreover, OK 0 f o is a Brownian motion on K, where v« : T K -+ K. When we regard V as perturbed version of J l (0) we can ask how much of this observation remains true? This is the content of Section 3.C.
v
3.C. Constrained Brownian motion and compression
We now force the Brownian motion on Q to satisfy the constraints induced by V. In accordance with Section 2.C we do so by applying the constraint forces to the Stratonovich operator from (3.9). Thus we are concerned with the equation of = snh(X, f)oX where X and Ho, Hi, Fa are as above and
SC(n, z) = PzSHam(n, z) : JR x JRm X JRn-1 ---+ C, C TzV, where P was defined in (2.1) and z = (s, u, x, -A.~(u)) E V. The functions
394
S. HOCHGERNER
Ho, Hi, Fa are V -invariant and compress to functions h o, hi, fa given by hoes, u)
= Ho(s, u, x, -As(u)) = -!It((u, -As(u)), LV~Vi) = -!(rru, LV~i Vi) = -!lto(u, ltolrrLV~ivd,
hies, u) = (rru, Vi) = Ito(u, lto1rrvd, fa(s, u)
= -(As(u), ea) = -Ito(u, ltol~a(S)).
Notice that ho and hi are left-invariant while the fa are right-invariant. The compressed nonholonomic Stratonovich operator is now of the form snh(n; S, u)(t, b l, ... , b", Wi, ... , w n - l) = X~~(s, u)t + LX~~(S, u)b i
+ LXf~(s, u)w a. We think of solutions of of = snh(X, f)oX as nonholonomic diffusions. This is in analogy to [2, Chapter V] where Hamiltonian diffusions are considered in a similar manner. For a function f E CXJ(K), viewed as a function on T K via pull-back, we have
X~~(f) = -!df.ltolrrLV~i Vi, X'J!:(f) = df.lto1rrvi' and Xf~(f) = -df.ltol~a.
Let I' be the solution semi-martingale to be the projection. Then r 0 I' solves
o(r
0
of =
(3.10)
snh(X, f)oX and let r : T K ---+ K
f) = Tr.of
=
-~ltolrrLV~ivi(r
0
f)ot
+ Llt01rrVi(r 0
f)oB i - Lltol~a(r
0
f)oW a.
According to [15, Chapter V, Theorem 1.2] this means that the semi-martingale r 0 I' defines a diffusion in K whose generator is the second-order differential operator
-~ltolrrLV~ivi
+ ~L(ltolrrVi)(ltolrrVi) + ~L(ltol~a)(ltol~a)'
(3.11)
To identify the drift of the diffusion r 0 I' a connection is needed. We introduce the nonholonomic connection which is explained in [7, Section 5.1.1]:3 Let Pr" : T Q ---+ V denote the projection onto V along the It-orthogonal 'OIL of 'O. Note that 'OJ-t f= ker I'n where n : Q --tt K is the projection. Let hlA : X(K) ---+ X(Q; V), hlA(s, x)(u) = (s, u, x, -As(u)) be the horizontal lift map associated to A. Given X, Y E X(K) the nonholonomic connection is prescribed by Vnhy = TnPrJ-tVJ-t
x
(hlAy)
hIAX'
(3.12)
This connection is metric, i.e. VnhIto = 0, and its geodesic equations are exactly the equations of motion of the nonholonomic system described by (T K, Qoh, He). However, in general, Vnh will have nontrivial torsion. 3Contrary to [7] we only use the projected version of the nonholonomic connection.
STOCHASTIC CHAPLYGIN SYSTEMS
395
LEMMA 3.1. For u, VEe = XdK) we have V~h v = JLo l ([V~v
where V~v = 1[u , v]
+ A * A[u , v ])
+ 1TI-I([u , nv] + [v , nu]).
Its torsion is given by To~h(u, v) =
JLo1A*A[u , v].
Note that Vl~hv is not left-invariant any more. At the compressed level the equations of motion of the Chaplygin ball write as u' + ILo I [u, liu] = O. In 3D this corresponds to [8, Equation (3.5)]. Proof: For u, VEe we need to compute Pr/LVtu,_Au/V' -Av) = Pr/L(V~v - u.Av) = Prll(V~v - A[u, v]).
Now note that (w, X) E D/L if and only if w
o=
A(V~v
= n-1A*X. We have to solve
+ n-1A*X) -
A[u, v]
+X
for [-I A * X. The solution is found to be given by JLon- 1A * X = A* A([u , v] - V~v). Therefore, V~hv = V~v + JLo 1A * A([u , v] - V~v) = JLo l A* A[u , v] + JLolnv~v . (3.13)
o Recall that the Hessian of Vnh is defined by Hessnh(f)(X, Y) = XY(f) -
vtY(f)
for X, Y E X(K) and f E COO(K). Let f E C OO(K) and or = snh(X, r)8X. By Proposition 2.5, Eqs. (3.10) and [dB', dBi] = 8',idt , [dW a, dW b] = oa,bdt, as well as [dB', dW a] = 0 we have the Ito equation d(f
0
r) = (- ~JLOIJ[LV~i Vi
=
+ ~L(JLollIvi)(JLol][vJ + 4L (tLol~a)(tLol~a»)!(r)dt
+ L(JLo1nV,)!(r)dB' - L(JLol~a)f(r)dwa (- ~JLOIJ[LV~.Vi + ~LVn~ln .(tLo lTIv,) + ~LVn~Jr (JLol~a) /La v, /-La va I
+ 1LHessnh(JLo lnv" JLo1[vd + L(JLolnvd!(r)dB' -
+ ~LHessnh(/Lol~a, JLol~a»)f(r)dt
L(JLol~a)!(r)dWa ,
which also confirms (3.11). Having split the generator into first and purely second order part it makes sense to say what we mean by drift. DEFINITION 3.2. The vectorfield 1
-l 1f",nli
-'5Yo
JiL..v v· v, I
(-1 11") I "nnh (-1 r ) + Z1",nnh L.. - In . JLo JiV, + zL.. v /-La-I ~a JLo.,a /-La v, V
is called the drift of the diffusion r
0
r
with respect to Vnh.
396
S. HOCHGERNER
Note that r 0 I' is a y7Uh-martingale if and only if the y1Dh_drift vanishes. See also [10, Theorem (7.31)].
3.3. Let I' be a solution of the Stratonovich equation
THEOREM
of =
snh(X, f)oX
and let r : T K ---* K be the projection. (1) Then, with respect to the nonholonomic connection y7Dh introduced in (3.12), the semi-martingale r 0 I' defines a diffusion on K whose drift is the gradient -~gradJLO(logN) where N is the density function defined in (3.5). (2) The drift -~gradJLO(1ogN) is horizontal with respect to the mechanical connection, Hornech = (ker T K )JLo.L, on the principal bundle K : K _ K j H = sn-l. If n satisfies the hamiltonization condition (3.7) then the drift is also horizontal with respect to the principal bundle connection Lpa Q9 Ya : T K ---* ~ on K: K - KjH.
Item (2) means that the drift component of angular momentum about the vertical axis in the space frame vanishes, and when the hamiltonization condition holds then the same is true for the component of angular velocity about the vertical axis. For n = 3 this condition is always satisfied. Let I' be a solution of the Stratonovich equation of = r : TK ---* K and K : K - KjH = sn-l be the obvious projections. Suppose that n= 1 (i.e. the ball is homogeneous). (1) Then r 0 I' defines a martingale in K with respect to the nonholonomic connection. (2) The process K 0 r 0 I' is a Brownian motion on sn-l whose generator is ~ times the Laplacian of v, where v is the metric on sn-l induced from the left H -invariant metric /-to = ((1 + A*A).,.) on K. COROLLARY 3.4. snh(X, f)OX and let
Note that the restriction of /-to to Hor = Hornech = span{~a} equals twice the restriction of the biinvariant metric. This corollary is intuitive but nevertheless not obvious since the dynamics at the compressed level can never be described by a Hamiltonian reduction procedure. This is because A is not the mechanical connection, even if the ball is homogeneous. (Compare with [12, Corollary 4.3].) Thus it does not fall in the category of [18, 19]. For reference we note the formula v./-to(u) = v'L(~a, u}~a = L(~a, [v, u]}~a - L(~a, u}[v, ~a] = A* A[v, u] - [v, A* Au].
(3.14)
Proof of Theorem 3.3: Let f E COO(K) which we regard via pull-back as a function on T K. According to Definition 3.2 we need to show that
-/-tolnLV~.Vi I
+ Lvn~h (/-to1nVi) + LVn~l JLo .Vi JLo
~a
(/-tol~a) = -gradJLo(1ogN). (3.15)
Claim: (3.16)
397
STOCHASTIC CHAPLYGIN SYSTEMS
Indeed, we use (3.14) and the fact that
U
= L(Kv;, u)v; to see that
LVn~h (tLOIKv;) = L(KtLOIKv;, Vj)(KtLOIKv;, Vk)V~hVk ~~
J
+ L(KtLOIKv;, Vj)(Vj.(KtLo IKv;, Vk))Vk = LtLo I [tLoI Kv;, KtLo I Kv;]
+L
(KtLol Kv;, Vj) (KtLol [Vj, A* AtLol Kv;], Vk)Vk
= LtLOI[tLOIKv;, KtLo l Kv;]
+ LtLOI[tLOIKv;, A* AtLo1Kv;]
= LtLol[tLoIKv;, Kv;] = LtLol[V;, Kv;] - LtLol[tLoIA*Av;, Kv;]
where we use tLoI K= 1- tLoI A *A in the last equation. Notice that [v;, Kv;] = KV~I v;. For the gradient part of claim (3.16) we consider l l (L[tLo A* Av;, Kv;], {b) = - L(Kv;, [tLo ({a, V;}{a, {b]}
= - L({a, (Kv;, [tLol{a, {b]}V;) = L([tLol{a, {a], {b} = d(logN){b l
by (3.6). Similarly it is true that (L[tLo l A* Av;, Kv;], ~a) = L([tLo {a, {a], ~a}. Using the property [Z,. fa] = Oa,bZc - Oa,cZb with fa = [Zb, Zc] it is easy to see that L([tLol{a, {a], ~a)
Thus LtLol[tLoIA*Av;,Kv;] (3.16) follows. Claim:
=
= (tLOI{b' {c} -
(tLol{c' {b}
tLoIL(d(logN){b){b
=
= o.
grad'vrlog A") and claim
Vn~1 (tLol{a) = O. JIo (a
Again we use (3.14) to see that Vn~lr (tLol{a) = tLol[tLol{a' KtLol{a] + L(Kvj, tLo1{a) (Vj.(Kvk. tLo1{a) )Vk JIo ,a
-I[
= tLo
-Ii"
11
-Ii"]
tLo ,>a, JltLo ,>a
+ L(Kvj, tLoI {a}(Kvk. tLol[Vj, A* AtLo l {a]}Vk l l - L(Kv j, tLo {a}(Kvk. tLo [Vj, {a]}Vk = tLo-1[ tLo-I {a, 11 JltLo-Ii"] ,>a
+ tLo-I[ tLo-Ii",>a, A*AtLo-Ii"] ,>a
- tLo1[tLo1{a' {a]
=0. Now (3.16) and (3.17) imply (3.15) which shows part (1) of the assertion.
(3.17)
398
S.HOCHGERNER
For (2) one checks that (3.7) yields, for
~a
= [sc, Sd],
(gradJLo(1ogN), ~a} = L(fLol[Sc, Sd], ([fLoISa, Sa], ~b)~b} = L{fLol[Sc, Sd], ~b}(fLol~a, [Sa, Sb]} = (fLOI~d, [~c, ~d]}(fLol~c, [SC, td]}(n - 2)
+ (fLOI~c, [~c, ~d]}{fLoltd, [~d, ~c]}(n -
2)
=0, where we also have used that L([fLoISa, tal, ~a} = O.
0
Proof of Corollary 3.4: Part (1) is clear. Concerning part (2) let f E coo(sn-l). According to Definition 2.2 we should show that the generator (3.11) satisfies (4L(fLolvi)(fLolvJ
+ ~L(fLol~a)(fLolsa»)K*f =
~K*~v I,
where ~v is the Laplacian associated to v. Indeed, we find ~L(fLolvi)(fLolvJK* f + ~L(fLolta)(fLolta)K* f = HLtasa K* t- Now {Jz~a} is a horizontal orthonormal frame for fLo I(Hor x Hor) where Hor is the fLo-orthogonal to ker T K. Therefore, iL~aSaK* f = ~Tf'1orHessJLO(K* f) = ~K* ~v f,
where Tf'1or denotes the trace computed with respect to horizontal fields only and Hess JLO is the Hessian of the Levi-Civita connection on (K, fLo). The equation ta~aK* f = HessJLO(K* f)(ta, ~a) is justified by the observation that Vrao~a = 0 where VJLO is the Levi-Civita connection of the right-invariant metric fLo. (In fact, the 0 restriction of v oh to Hor x Hor equals the restriction of ViLa.)
In the homogeneous case the above construction yields a Brownian motion on in a manner similar to the one described in [15, Chapter V] by the notion of rolling the sphere sn-l along a Brownian motion in lRn - 1 by means of the LeviCivita connection. The difference is that [15] start from Brownian motion in lRn - 1 while we started from Brownian motion in lRm x lRn - 1 with m = dimso(n) = n(n2-1). One can recover the setting of [15] by setting (Ho, HI, ... , H m ) = 0 in (3.8). Then, with ]I = 1, we obtain a diffusion K 0 r 0 r which is driven by Brownian motion (WI, ... , W n - I) in lRn - 1 and the generator of which is given by sn-I
~L(fLol Sa)(fLo l ~a)K* f = ~K* ~v i. where f E coo(sn-l). Referring to the interpretation stated in Introduction this means that the Chaplygin ball is subject to horizontal jiggling but there is no angular jiggling. Equivalently, the ball sits on a table which undergoes a translational Brownian motion, compare [20]. Alternatively, we can set (FI , ..• , Fn - I ) = 0 in (3.8). Then there is only angular jiggling and the diffusion r 0 r is driven by (B I , ... .B'"), By (3.17) the drift remains the same as in Theorem 3.3.
399
STOCHASTIC CHAPLYGIN SYSTEMS
It seems that the notion of a stochastic nonholonomic system has been hardly investigated in the literature. We finish by asking the following questions. (1) Does an analog of Theorem 3.3 hold for general G-Chaplygin systems when there is a preserved measure? What can be said about the drift if there is no preserved measure? (2) Which is the precise relationship between G-Chaplygin systems with preserved measures and measure preserving 'Chaplygin diffusions'? Preservation of measure by diffusions is studied in [15, Chapter V]. (3) Is there a time change or Girsanov type argument to eliminate the drift in Theorem 3.3 or to make it even a Brownian motion? Is this related to the hamiltonization of the deterministic problem? Added in proof" Questions (1) and (2) are answered in the forthcoming paper [14]. This paper addresses also stochastic nonholonomic systems with constraints which are not of Chaplygin type.
4. Appendix: G-Chaplygin systems and symmetry reduction The purpose of this appendix is to shortly introduce and motivate the notion of a G-Chaplygin system and to state Proposition 4.1 which explains the symmetry reduction of such systems. This reduction is termed compression [9] to distinguish it from symplectic reduction. These concepts are closely related and compression can be viewed as a perturbed version of its symplectic counterpart. At the same time, however, there are fundamental differences; symmetries behave differently in nonholonomic mechanics and do not necessarily give rise to conserved quantities, and there need not exist a preserved measure [7, Section 5.4]; all this is related to the question of closedness of the form nnh defined in Proposition 4.1, see [1, 3, 7, 9, 12]. A nonholonomic system is a triple (Q, V, L) where Q is a configuration manifold, L : T Q -+ lR is a Lagrangian, and VeT Q is a smooth nonintegrable distribution which is supposed to be of constant rank. The equations of motion for a curve q(t) which should satisfy q' E V are then stated in terms of the Lagrange-d' Alembert principle. Suppose there is a Riemannian metric u. on Q such that we have an isomorphism T Q ;:: T* Q and assume that L is the kinetic energy Lagrangian. In this case there is also an (almost) Hamiltonian version: continue to use the symbol fL to denote the co-metric and consider the Hamiltonian Hiq, p) given by the Legendre transform of L. Since D is of constant rank there is a family of independent one-forms ¢a E n(Q) such that V is the joint kernel of these. In terms of coordinates (qi, Pi) the equations of motion are (q i )vt
aH" = -oH and p,, = - -. - ~Aa¢ a ( -a. ) 0Pi
I
oql
oq'
where the Aa are the Lagrange multp>liers to be determined from the supplementary condition that fL(¢a, p) = O. With X H := (q', p') we may thus rephrase the equations
400 as
S. HOCHGERNER
i(X~)Q =
an + L)"ar*¢a
(4.18)
where Q = -de is the canonical symplectic form on T* Q and r : T* Q ---+ Q is the footpoint projection. (The notation X~ will become clear below.) Let G be a Lie group that acts freely, properly and by isometries on the Riemannian manifold (Q, /-L). A G-Chaplygin system is a nonholonomic system (Q, L = 411 . II ~,V) that has the property that V is a principal connection on the principal bundle Q --* Q I G. Thus V is the kernel of a connection form A : T Q ---+ g. Notice that we do not require A to be the mechanical connection associated to /-L. Consider C and QC as defined in Section 2.C. Since X~ is, by construction, tangent to M and takes values in C one may now rewrite the equations of motion (4.18) in the appealing format i(X~)QC = (d'Hl
where (d'H)c is the restriction of l*d'H to C with l : V ~ T Q being the inclusion. Let /-Lo denote the induced metric on S := QI G that makes n : Q --* S a Riemannian submersion. Identify tangent and cotangent space of Q and S via their respective metrics. Consider the orbit projection map p:= TrrIV: V
--*
VIG = TS.
We may also associate a fiber-wise inverse to this mapping which is given by the horizontal lift mapping hlA associated to A. As already noted in Section 2.C, l*r* A : TV ---+ 9 defines a principal bundle connection for p, whose horizontal space is given by C. PROPOSITION 4.1 (Compression). The following are true. (1) QC descends to a nondegenerate two-form Q oh on T S. (2) Q oh = Q s - (Je ohl A , r;CurvA ). Here Q s = -des is the canonical form on T S, Jc is the momentum map of the tangent lifted G-action on T Q, Curv A E Q2(S, g) is the curvature form of A, and ts : T S ---+ S is the projection. (3) Let h : T Q ---+ ~ be G-invariant. Then the vectorfield
X~ := (QC)-l (dh)c,
where (dh)c is the restriction of l*dh to C, is p-related to the vector field X~~ on T*S defined by i(X~~)Qnh = dhi, where the compressed Hamiltonian h o : T* S = T S ---+ ~ is defined by h o := h 0 l 0 hlA , with hlA denoting the horizontal lift mapping.
In general, Q oh is an almost symplectic form, that is, it is nondegenerate and nonclosed, see [1, 9, 12].
STOCHASTIC CHAPLYGIN SYSTEMS
401
REFERENCES [1] [2] [3] [4]
[5] [6]
[7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
L. Bates and J. Sniatycki: Nonholonomic reduction, Rep. Math. Phys. 32, No. I (1993) 99-115. J.-M. Bismut: Mecanique Aleatoire, LN in Math., vol. 866, Springer 1981. A. M. Bloch: Nonholonomic Mechanics and Control, Springer 2003. A. V. Borisov and I. S. Mamaev: Chaplygin's ball rolling problem is Hamiltonian, Mathematical Notes 70 (2001), 793-795. A. V. Borisov and I. S. Mamaev, Hamiltonizatian of nonholonomic systems, (2005), arXiv:nlinl0509036vl. S. A. Chaplygin, On a ball's rolling on a horizontal plane, Regul. Chaotic Dyn. 7 (2002), 131-148; Translation of original in: Mathematical collection of the Moscow Mathematical Society 24 (1903), 139-168 (Russian). J. Cortes Monforte: Geometric control and numerical aspects of non-holonomic systems, LN in Math. 1793, Springer 2002. I. J. Duisterrnaat, Chaplygin's sphere, arXiv:mathl0409019vl. K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios: Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in "The breath of Symplectic and Poisson Geometry", Progress in Mathematics 232 (2004), 75-120. M. Emery: Stochastic Calculus in Manifolds, Universitext, Springer 1989. Y. N. Fedorov and V. V. Kozlov: Various aspects of n-dimensional rigid body dynamics, Amer. Math. Soc. Transl. Ser. (2) 168 (1995), 141-171. S. Hochgerner and L. Garcia-Naranjo: G-Chaplygin systems with internal symmetries, Truncation, and an (almost) symplectic view of Chaplygin's ball, J. Geom. Mech. 1, No. I (2009), 35-53. S. Hochgerner: Chaplygin systems associated to Cartan decompositions of semi-simple Lie groups, Diff. Geom. Appl. 28 (4) (2010), 436-453. S. Hochgerner and T. Ratiu: Non-holonomic diffusions, in preparation. N. Ikeda and S. Watanabe: Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Company, Kodansha, 2nd Ed., 1989. B. Jovanovic: LR and L+R systems, J. Phys. A: Math. Theor. 42, No. 22 (2009). B. Jovanovic: Hamiltonization and integrability of the Chaplygin sphere in IR n , arXiv:math-phl0902.4397v I. J.-A. Lazaro-Cami and I.-P. Ortega: Stochastic Hamiltonian dynamical systems, Rep. Math. Phys. 61 (2008), 65-122. J.-A. Lazaro-Cami and J.-P. Ortega: Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations, arXiv:0705.3156v2, 2008. N. K. Moshchuk and I. N. Sinitsyn: On stochastic non-holonomic systems, PMM USSR 54, No.2 (1990), 174-182. B. Oksendal: Stochastic Differential Equations, Universitext, Springer, 2007.
REPORTS ON MATHEMATICAL PHYSICS
NO.3
STOCHASTIC CHAPLYGIN SYSTEMS SIMON HOCHGERNER Section de Mathematiques, Station 8, EPFL, CH-1015 Lausanne, Switzerland (e-mail: [email protected]) (Received February 9, 2010 -
Revised August 7, 20/0)
We mimic the stochastic Hamiltonian reduction of Lazaro-Cami and Ortega [18, 19] for the case of certain nonholonomic systems with symmetries. Using the nonholonomic connection it is shown that the drift of the stochastically perturbed n-dimensional Chaplygin ball is a certain gradient of the density of the preserved measure of the deterministic system.
Keywords: Stochastic nonholonomic systems, stochastic mechanics.
1. Introduction Imagine a ball sittmg on a rough horizontal table. Because of the roughness of the table this ball-the Chaplygin ball-cannot slip, but it can tum about the vertical axis without violating the constraints. Its geometric and gravitational center coincide but there may be an inhomogeneous mass distribution. Suppose now that the ball is subject to a Brownian noise such that there is a random jiggling in all of its angular and translational degrees of freedom. This problem is similar to the stochastic rigid body considered in [19] but upon imposing the no-slip constraints some differences can be expected. One may wonder if the stochastic Chaplygin ball will acquire a drift that makes it roll on the table or spin about its vertical axis or both? The answer to this question is given by Theorem 3.3: The drift follows a Fick's law in the following sense. The configuration space of the n-dimensional ball is Q = SO(n) x JRn-l and there are n - 1 constraints corresponding to the directions in the table JRn-l. By a symmetry reduction argument (compression) one can eliminate the JRn-l-factor and the deterministic motion of the ball can be described by the geodesic equations of the so-called nonholonomic connection V nh on SO(n). With respect to this connection one can now show that the process on SO(n) describing the balls stochastic motion is a nonholonomic diffusion separating into a drift- and a martingale-term. See Definition 3.2. Theorem 3.3 says that this Vnh-drift equals - ~ grad"? log N where N is the preserved density (3.5) of the deterministic ball and the gradient is computed with respect to the kinetic energy metric (3.4) of the ball. [385]
S.HOCHGERNER
386
In particular, when the ball is homogeneous there is no drift. Moreover, it is shown in Corollary 3.4 that the homogeneous ball's stochastic process factorizes to a Brownian motion on 'the ultimate reduced configuration space' sn-l = (Q/lRn-I)/SO(n-l). This is in analogy to the corresponding deterministic case where the motion is Hamiltonian at the ultimate reduced level (see [12].) In 3 dimensions the drift - grad"? log N does not have an angular velocity component about the vertical axis of the ball in the space frame. For dimensions n > 3 it turns out that this property is related to the hamiltonization of the deterministic system: If the inertia matrix describing the ball's mass distribution satisfies the hamiltonization condition (3.7) then the drift does not have an angular velocity component about the vertical axis in the space frame. On the other hand, the drifts angular momentum about the vertical axis is always 0, regardless of the dimension or the mass distribution. Section 2 starts by collecting some definitions and facts from stochastic differential geometry as presented in [15, 10]. Then we rehearse the basics of the stochastic Hamiltonian mechanics and their symmetries as introduced in [18, 19]. In Section 2.C these ideas are transferred to describe stochastic G-Chaplygin systems. It is noticed that the reduction of symmetries, termed compression in this context, works naturally. In Section 3 this construction is applied to the n-dimensional Chaplygin ball. First some facts about the deterministic system, such as the preserved measure and Hamiltonization, are recalled. Then the Hamiltonian construction of [18] of Brownian motion on the configuration space Q = SO(n) x lRn - 1 is reviewed. Section 3.C makes the constraint forces act on the Brownian motion according to the recipe of Section 2.C. Thus a constrained stochastic motion is obtained and the mentioned above Theorem 3.3 is found. Finally, we note that we can also treat the cases of angular jiggling only or horizontal jiggling only. The latter corresponds to the Chaplygin ball sitting on a table which undergoes a translational Brownian motion. The 3-dimensional version of this case has been considered in [20].
4
2. Stochastic Chaplygin systems and reduction of symmetries 2.A. Stochastic geometry
We state some notions from stochastic differential geometry. The references we used here are [10, 15]. See also the appendix of [18]. Let M be a manifold, let (Q, F, {Ft : t ::: O], P) be a filtered probability space, and let r : lR+ x Q ~ M be an adapted stochastic process. (We consider only continuous processes.) The process r is called a semi-martingale if for is a semi-martingale in lR for all f E COO(M), see [10, Chapter III]. The definition of a martingale in M depends upon a choice of a connection. Let V be a connection in T M ~ M. Then the Hessian of V is defined by HessV'(f)(X, Y) = XY(f) - VxY(f)
STOCHASTIC CHAPLYGIN SYSTEMS
387
for X, Y E X(M). This is bilinear in X and Y but not symmetric, in general, since HessV'(f)(X, Y) - HessV'(f)(Y, X) = -TorV'(X, Y)(f). 2.1. A semi-martingale I' : lR+ x Q -+ M is said to be a V-martingale E COO(M),
DEFINITION
if, for any
f
f
0
r, -
f
0
r,
-it
Hess'' (f)(df s, dfs)ds
is a local martingale in lR. In [10, Chapter IV] this is stated in terms of a torsionless connection but it is noted that one can also allow for connections with torsion, since it is proved ([10, (3.14)]) that J~ Hess" (f)(df s, dfs)ds depends only on the symmetric part of HessV'. The situation is similar to the notion of a V-geodesic. This also depends only on the torsionless part V - 4TorV' of V. Let M and N be manifolds, let (Q, F, {Ft : t ~ O}, P) be a filtered probability space, and let X : lR+ x Q -+ N be a semi-martingale. A Stratonovich operator S from TN to T M is a family of linear maps S(x,y) : TxN ----+ TyM
which depend smoothly on x E Nand y EM. In other words, S is a section of T* N Q9 T M -+ N x M. A Stratonovich differential equation for a semi-martingale I' : lR+ x Q -+ M is written as
sr =
S(f, X)8X.
See [10, Chapter VII] for the precise meaning of this equation as well as the existence and uniqueness (up to explosion time) of solutions. Assume now that N = lR x lRn and that X : lR+ x Q -+ lR x lRn , (t, w) 1---+ (t, Wt (w)) where W denotes n-dimensional Brownian motion. Let Xo, X I, ... ,Xn be vector fields on M and define the Stratonovich operator S(x, y) : lRn + 1 ----+ TyM, (t, wI, ... , w n ) f----+ tXo(Y) + L:w i Xi(y). Let
f
E COO(M). Then
8(f
0
f)
f
0
I' : lR+ x
Q -+
lR satisfies
= df(f).S(X, f)8X = df(f).X o(f)8t + L:df(f)·Xi(f)8W i •
Hence I' defines a diffusion in M by [15, Capter V, Thm. 1.2]. The generator of this diffusion is the second-order differential operator A given by Af = Xof
where
f
+ 4L:X iX if,
E COO(M).
DEFINITION 2.2. If M is equipped with a Riemannian metric J1, and .6. is the associated Laplacian then the diffusion is called Brownian motion in (M, J1,) if
A = 4.6..
388
S. HOCHGERNER
This definition agrees with the one given in [10, (5.16)] or [21, (8.5.18)] but differs slightly from the one in [15, Chapter V, Def. 4.2] where it is required that n = dimM.
2.B. Stochastic Hamiltonian systems
This section presents some of the concepts elaborated in [18, 19]. Let again N = ~n and consider a Poisson manifold (M, {.,.}) together with an ~n-valued Hamiltonian function h = (hi, ... , h n) : M -+ ~n. Let X : ~+ x Q -+ ~n be a semi-martingale. The associated stochastic Hamiltonian system is given by the Stratonovich equation of = S(X, r)OX where S is defined in terms of the Hamiltonian structure, that is, S(x, y) : ~n -----+ TyM, (r ', ... , x")
1-------+
LXhi (yjx',
where the Hamiltonian vector field Xhi is the vector field corresponding to the derivation {hi, .}. When (M, w) is a symplectic manifold then one uses the Hamiltonian fields defined by i (Xh)W = dh., These systems allow for a symmetry reduction analogous to classical mechanics. We state the symplectic version of this theorem of [19, Section 6]. THEOREM 2.3 (Stochastic Hamiltonian reduction). Let (M, w) be a symplectic manifold with Hamiltonian h : M -+ ~n and stochastic component X as above. Assume that (M, w, h) are invariant under the free and proper action of a Lie group G such that a coadjoint equivariant momentum map J : M -+ g* exists. Fix a level)", E g*. Then J-I()..,) is invariant under the flow of S. Moreover, S induces a Stratonovich operator SJ.. from T~n to T(J-I()"')/GJ..) and solutions of S with initial condition in J-I()..,) project to solutions of SJ... The induced operator is given by
SJ..(x, Yo) : (r ', ... .x")
1-------+
LXhA(YO)X i , I
where x
E
~n, Yo E J-I()"')/GJ.. and XhA is the Hamiltonian vector field with respect I
to the reduced symplectic form on J-I()")/GJ.. of the induced function h7.
2.C. Stochastic G-Chaplygin systems
A (deterministic) G-Chaplygin system consists of a Riemannian configuration manifold (Q, J1), a Lie group G acting freely and properly by isometries on (Q, J1), and a horizontal space V of the principal bundle n : Q -tt Q / G. Hence V is the kernel of a connection form A : T Q -+ g. The Lagrangian of the system is the kinetic energy, i.e. L(q, v) = 4J1Q(v, v). In general, V is not J1-orthogonal to the vertical space ker T Jr. We will henceforth identify T Q = T* Q via J1. See also Section 4. Let JG : T Q -+ g* denote the standard equivariant momentum map associated to the lifted G-action on T Q. Given a G-invariant function h : T Q -+ ~ one may use Noether's theorem to conclude that the Hamiltonian vector field X h is tangent to
389
STOCHASTIC CHAPLYGIN SYSTEMS
J(;"I(O) and, moreover, is projectable for J(;"I(O) -» J(;"I(O)/G = T(QIG). This is why results such as Theorem 2.3 work. Since V is a perturbed version of J(;" 1 (0) we look for a substitute construction. Consider the horizontal space associated to the pulled back connection r" A = Ao Tr: T(TQ) ~ TQ ~ 9 of the tangent lifted G-action on TQ,
F:= ker r" A C TTQ. By assumption V is also G-invariant. Thus we can consider the restricted G-action on V and the associated connection l*r* A : TV ~ 9 where l : V ~ T Q is the inclusion. Define C:= kerl*r*A = Ft: TV C TV to be the horizontal space of the principal bundle V -» VIG = T(QIG). According to [1] we can decompose T T Q along V as
TTQIV = C EB CQ where CQ is the Q-orthogonal of C in TT QIV. This implies that the fiber-wise restriction of l*Q to C x C is nondegenerate. For z E V define the projection
P, : Tz(T Q) ------+ C,
(2.1)
where we project along C~. Moreover, for a k-form 4> on TQ we denote the fiber-wise restriction of l*4> to nkC by 4>c. For a function h : TQ ~ ~ we may thus define the vectorfield X~ on V with values in C by the formula PzXh(z) = (Qc);l(dh)~ =: X~(z)
(2.2)
where Z E V. The link to the nonholonomic system introduced above is the following: Via the Legendre transform the dynamics of the nonholonomic system (Q, V, L) can be equivalently described by the triple (T Q, QC, H. = ~ u. (p, p)) together with Eq. (2.2). Thus the Lagrange multipliers have been encoded in the two-form QC or, equivalently, in the projection P : T(TQ)IV ~ C. The idea is that a nonholonomic system is a Hamiltonian system acted upon by constraint forces. The effect of the forces is described by the projector P : T(TmIV ~ C. In addition to this structure consider now a semi-martingale X : ~+ x Q ~ ~n and a Hamiltonian function h = (hi)i : T Q ~ ~n as above. The associated stochastic nonholonomic system is given by the Stratonovich equation 8r = SC(X, r)8X where the Stratonovich operator SC : ~n X V ~ Hom(T~N, C) is given by SC(n,z): ~n ------+ Cz, (r ', ... .x") ~ '£PzXh;(Z)X i = ,£Xf;(z)x . i
(2.3)
Thus the nonholonomic Stratonovich operator arises, by applying the constraint forces, as a projection of the Hamiltonian Stratonovich operator into C. When the hi are G-invariant we refer to the collection (Q, V, h = (hi)i, X) as a stochastic G-Chaplygin system. PROPOSITION 2.4 (Compression of stochastic G-Chaplygin systems). Let (Q, V, h = (hi)i, X) be a stochastic G-Chaplygin system with QC as above. Then the
390
S. HOCHGERNER
Stratonovich operator (2.3) compresses to a Stratonovich operator Snh from TjRn to D/G = T(Q/G) which is given by Snh(n, eo) : jRn ---+ Tzo(T(Q/G)), (r ', ... , x")
1----+
LX~(Z)xi I
where zo E T(Q/G) and h? : T(Q/G) ---+ jR is the function induced on the quotient from the invariant function l*hi. Moreover, solutions of (2.3) project to solutions of snh. Proof: Everything is entirely analogous to the proof of [19, Theorem 6.7] with the only difference that now one uses Proposition 4.1 instead of the usual symplectic reduction theorem. D We think of 8f = Snh(X, f)8X as the equations of motion of the system (Q, V, h, X).
PROPOSITION 2.5 (Ito representation). Let f representation of the equation 8(f 0 f) = snh(x, f
d(f
0
r) = LX~f(r)dXi i
I
E 0
Coo (T (Q / G)). Then the Ito f)8X is
+ 1LX~~X~f(f)[dXj,dXi] i,j
I
]
where Xi = pr' 0 X. Proof: This follows exactly as in the proof [18, Proposition 2.3]. It is only necessary to notice that this proof does not depend on whether or not the nonholonomic bracket Xr: f = {g, = -{f, g}nh satisfies the Jacobi identity. The only property of the Poisson bracket which is used in [18, Proposition 2.3] is the Leibniz rule and this feature is evidently shared by the nonholonomic bracket. D
nnh
3. The stochastic Chaplygin ball 3.A. The deterministic system
For background on the Chaplygin ball we refer to [6, 8, 9, 11, 12, 17]. The configuration space of Chaplygin's n-dimensional rolling ball is Q = K x V where K = SO(n) and V = jRn-l. The no-slip constraints are given by the distribution '0= (A + pr2)-1(0) c TQ, where
A = L1Ja ® ea , where el,"" en-l is the standard basis on V and we stick to the following conventions: I T K = K x t is trivialized via left-multiplication and t is equipped with the Ad-invariant inner product (., .}.2 Let H = SO(n -1) c K be the stabilizer in K of the n-th standard vector en such that H acts on V in the natural way. We 1In Section 2.e it was the connection form A + pr2 which was called A. 2The space XL (K) of left-invariant vectorfields and the Lie algebra t will also be identified without further notice.
391
STOCHASTIC CHAPLYGIN SYSTEMS
decompose £ = f) ED f)1- with respect to the Ad-invariant inner product (.,.) on t With respect to this inner product we introduce an orthonormal system
Ya , a = 1, ... , k = dim f) and Za, a = 1, ... , n - 1, on £ such that Ya E f) and Z; right-invariant vector fields ~a and ~a(s)
f)1-. Associated to this basis we define the ~a. In the left trivialization these read
E
= Ad(s-l)Ya
and ~a(s)
= Ad(s-l)Za.
Dually we introduce the corresponding right-invariant coframe pa
= (~a,')
and fJa
= (~a, .).
The Lagrangian is the function
L : TQ = K x £ x TV ---+ R (s, u, X, x')
f-----+
~(TIu, u)
+ ~(X', x'),
where TI is the inertia matrix in body coordinates. The rolling ball with the no-slip constraint is the nonholonomic system described by the data (Q, D, L) where the equations of motion follow from the Lagrange-d'Alembert principle. However, we will not have much use for the Lagrange function below since we will only perturb the resting ball. Note also that we overload the symbol (.,.) by using it for the Euclidean inner product on V as well as for the Ad-invariant structure on t From a structural point of view the decisive feature of the Chaplygin ball is that its constraints are given by a connection prz + A : T Q ~ V on the (trivial) principal bundle V ~ K x V --» K where V acts on itself by addition. Thus D is the horizontal space of this connection. However, D is not fL-orthogonal to the vertical space of the bundle. The fact that the system is nonholonomic is reflected in the nonflatness, CurvA = dA i= O. Compression of the Chaplygin ball system yields the almost Hamiltonian system (T K, Qnh, He) where Q nh is described in Proposition 4.1, T K and T* K are identified via the induced metric fLo(UJ, uZ)s
= (TIul' uz) + (As(Ul), As(uz)) = ((TI + A;As)UI, uz),
(3.4)
and He = ~fLO(U, u)s. The metric fLo is the sum of a left invariant and a right invariant term. Thus it constitutes an L+R-system, see [16]. Note the useful formula A* A(u)
= L>~a, u)~a.
The compressed system (T K, Qnh, He) is further invariant under the lift of the left multiplication action of H on K. Physically this corresponds to rotation of the ball about the en-axis in the space frame. This is an inner symmetry and gives, by the nonholonomic Noether theorem, rise to a conserved quantity. This quantity is just the standard momentum map JH : T K ~ f)* = f), (s, u) t--+ L>~(TIU)Ya of the (co-)tangent lifted H -action on T K . The Chaplygin ball shares an important feature with Hamiltonian systems. Namely, it possesses a preserved measure [6, 11]. At the compressed level-the T K -level-the
392 density
S.HOCHGERNER
N :K
-+ lR of this measure with respect to the Liouville volume on T K is I
N(s) = (detJLo(s))-z.
(3.5)
This function plays the central role in all questions of hamiltonization of the system. Note that N is H-invariant and thus descends to a function N: K/H = sn-I -+ R For further reference we also record that
d(1ogN) = ~][JLol~a, ~a], ~b}'1b.
n
(3.6)
In [12] it is proved that Q nh can be replaced by = QK - ~L([~a, ~b], _}'1 a /\'1 b without altering the equations of motion, that is, i(X~ )Q nh = i(X~ = d'He. Now ,...... ftc Il.c
)n
the new system (T K, Q, 'He) has the same dynamics but has the advantage of being liable to reduction with respect to the internal symmetry group H: Let J H : T K -+ ~* be the momentu~ map introduced above, A E ~ * and l : J "---+ T K the inclusion. Then l*Q descends to an almost symplectic two form Q A on J// (")...)/ H A. (l*Q nh is not horizontal for the projection onto J//(A)/Hd In this way one can recover the hamiltonization of the 3-dimensional ball of [4, 5]: It is shown in [12, Proposition 4.4] that d(Nn A ) = 0 if n = 3. Thus, for n = 3, the rescaled vector field N-IX'flc is Hamiltonian with respect to (Jiil(A)/HA = TS 2 ,N n A, 'He).
Ii!, ()...)
Moreover, the homogeneous ball, TI = 1, is Hamiltonian at the Jiil(A)/HA-level for any dimension n. It is interesting to notice that none of these statements hold at the T K -level. In [13, Section 4.A] the following hamiltonization condition is proved for arbitrary dimension n: Let A = 0 E ~*. Then d(NQo) = 0 if and only if (n - 2)(/-LOI~d' [~b, ~c]) = La(/-Lol~a, [~b, ~a]8c,d - [~C> ~a]8b,d)
(3.7)
which is an algebraic condition on TI. For n = 3 this condition is trivially satisfied. In the stochastic context this condition appears in Theorem 3.3 below. 3.B. Brownian motion on the configuration space We follow [18] to construct Brownian motion on Q = K x V. Let VIL be the Levi-Civita connection of /-L«U, x'), (v, y')) = (TIu, v)
+ (x',
y').
Thus for u, vEt we have
V~v
= (V~v, 0) = (~[u, v] + ~TI-I([u, TIv] + [v, TIul), 0)
where Vi is the Levi-Civita connection of the left-invariant metric defined by (TI.,.) on K. Note that we identify t ~ t* via (.,.) whence TI: t -+ t* ~ t Let VI, •.. , V m ,
m
= dim t = ~n(n -1)
denote a basis which is orthonormal for (TI.,.). Thus VI, ..• , Vm , el, ... , en-I is a left-invariant frame on Q which is orthonormal with respect to the left-invariant
393
STOCHASTIC CHAPLYGIN SYSTEMS
metric It. (Remember that we identify T Q and T* Q via It.) Let the functions H o, Hi, Fa : TQ -+ JR be given by Ho(s, u, X, x')
= -~(Ru, I:V'~i Vi),
Hits, U, X, x')
= (Ru, Vi),
and
Fa(s, u, X, x') = (x', ea).
(3.8)
Consider the semi-martingale X : JR+ x Q ---+ JR x JRm X JRn- l, (t, w)
f-----+
(t, B/(w), ... , B;n(w), W/(w), ... , Wtn - I (w)),
where s', W a are m + n - 1 independent Brownian motions. The Stratonovich stochastic differential equation which is associated to these data is (3.9) where the Stratonovich operator from T (JR x JRm
X
JRn-l) to T K x T V is defined by
. 1)(t, b' , ... , bm ,w, I ... n ,w -I) S Ham( n,s,u,x,X
= XHo(S'
U,
x, x')t
+ I:X Hi (s, U, x, xl)b i + I:X Fa (s, U, x, xl)w a
with X H denoting the canonical Hamiltonian vector field of a function H : T Q -+ JR. Using the Ito representation of this equation [18] show that the solutions I' project via r : T Q -+ Q onto Brownian motion on Q. Using again the setting of [18] and Theorem 2.3 it is easy to see the following. Consider the V-action on Q as above. Let JVI(O) = {(s,u,x,O)} be the O-level set of the standard momentum map J v : T Q -+ V* = V of the lifted V -action on T Q. Then the Stratonovich equation (3.9) induces a Stratonovich equation ofo = So(X, fo)oX on JVI(O)/V = T K and solutions I' with initial condition in Jvl(O) project onto solutions fo. Moreover, OK 0 f o is a Brownian motion on K, where v« : T K -+ K. When we regard V as perturbed version of J l (0) we can ask how much of this observation remains true? This is the content of Section 3.C.
v
3.C. Constrained Brownian motion and compression
We now force the Brownian motion on Q to satisfy the constraints induced by V. In accordance with Section 2.C we do so by applying the constraint forces to the Stratonovich operator from (3.9). Thus we are concerned with the equation of = snh(X, f)oX where X and Ho, Hi, Fa are as above and
SC(n, z) = PzSHam(n, z) : JR x JRm X JRn-1 ---+ C, C TzV, where P was defined in (2.1) and z = (s, u, x, -A.~(u)) E V. The functions
394
S. HOCHGERNER
Ho, Hi, Fa are V -invariant and compress to functions h o, hi, fa given by hoes, u)
= Ho(s, u, x, -As(u)) = -!It((u, -As(u)), LV~Vi) = -!(rru, LV~i Vi) = -!lto(u, ltolrrLV~ivd,
hies, u) = (rru, Vi) = Ito(u, lto1rrvd, fa(s, u)
= -(As(u), ea) = -Ito(u, ltol~a(S)).
Notice that ho and hi are left-invariant while the fa are right-invariant. The compressed nonholonomic Stratonovich operator is now of the form snh(n; S, u)(t, b l, ... , b", Wi, ... , w n - l) = X~~(s, u)t + LX~~(S, u)b i
+ LXf~(s, u)w a. We think of solutions of of = snh(X, f)oX as nonholonomic diffusions. This is in analogy to [2, Chapter V] where Hamiltonian diffusions are considered in a similar manner. For a function f E CXJ(K), viewed as a function on T K via pull-back, we have
X~~(f) = -!df.ltolrrLV~i Vi, X'J!:(f) = df.lto1rrvi' and Xf~(f) = -df.ltol~a.
Let I' be the solution semi-martingale to be the projection. Then r 0 I' solves
o(r
0
of =
(3.10)
snh(X, f)oX and let r : T K ---+ K
f) = Tr.of
=
-~ltolrrLV~ivi(r
0
f)ot
+ Llt01rrVi(r 0
f)oB i - Lltol~a(r
0
f)oW a.
According to [15, Chapter V, Theorem 1.2] this means that the semi-martingale r 0 I' defines a diffusion in K whose generator is the second-order differential operator
-~ltolrrLV~ivi
+ ~L(ltolrrVi)(ltolrrVi) + ~L(ltol~a)(ltol~a)'
(3.11)
To identify the drift of the diffusion r 0 I' a connection is needed. We introduce the nonholonomic connection which is explained in [7, Section 5.1.1]:3 Let Pr" : T Q ---+ V denote the projection onto V along the It-orthogonal 'OIL of 'O. Note that 'OJ-t f= ker I'n where n : Q --tt K is the projection. Let hlA : X(K) ---+ X(Q; V), hlA(s, x)(u) = (s, u, x, -As(u)) be the horizontal lift map associated to A. Given X, Y E X(K) the nonholonomic connection is prescribed by Vnhy = TnPrJ-tVJ-t
x
(hlAy)
hIAX'
(3.12)
This connection is metric, i.e. VnhIto = 0, and its geodesic equations are exactly the equations of motion of the nonholonomic system described by (T K, Qoh, He). However, in general, Vnh will have nontrivial torsion. 3Contrary to [7] we only use the projected version of the nonholonomic connection.
STOCHASTIC CHAPLYGIN SYSTEMS
395
LEMMA 3.1. For u, VEe = XdK) we have V~h v = JLo l ([V~v
where V~v = 1[u , v]
+ A * A[u , v ])
+ 1TI-I([u , nv] + [v , nu]).
Its torsion is given by To~h(u, v) =
JLo1A*A[u , v].
Note that Vl~hv is not left-invariant any more. At the compressed level the equations of motion of the Chaplygin ball write as u' + ILo I [u, liu] = O. In 3D this corresponds to [8, Equation (3.5)]. Proof: For u, VEe we need to compute Pr/LVtu,_Au/V' -Av) = Pr/L(V~v - u.Av) = Prll(V~v - A[u, v]).
Now note that (w, X) E D/L if and only if w
o=
A(V~v
= n-1A*X. We have to solve
+ n-1A*X) -
A[u, v]
+X
for [-I A * X. The solution is found to be given by JLon- 1A * X = A* A([u , v] - V~v). Therefore, V~hv = V~v + JLo 1A * A([u , v] - V~v) = JLo l A* A[u , v] + JLolnv~v . (3.13)
o Recall that the Hessian of Vnh is defined by Hessnh(f)(X, Y) = XY(f) -
vtY(f)
for X, Y E X(K) and f E COO(K). Let f E C OO(K) and or = snh(X, r)8X. By Proposition 2.5, Eqs. (3.10) and [dB', dBi] = 8',idt , [dW a, dW b] = oa,bdt, as well as [dB', dW a] = 0 we have the Ito equation d(f
0
r) = (- ~JLOIJ[LV~i Vi
=
+ ~L(JLollIvi)(JLol][vJ + 4L (tLol~a)(tLol~a»)!(r)dt
+ L(JLo1nV,)!(r)dB' - L(JLol~a)f(r)dwa (- ~JLOIJ[LV~.Vi + ~LVn~ln .(tLo lTIv,) + ~LVn~Jr (JLol~a) /La v, /-La va I
+ 1LHessnh(JLo lnv" JLo1[vd + L(JLolnvd!(r)dB' -
+ ~LHessnh(/Lol~a, JLol~a»)f(r)dt
L(JLol~a)!(r)dWa ,
which also confirms (3.11). Having split the generator into first and purely second order part it makes sense to say what we mean by drift. DEFINITION 3.2. The vectorfield 1
-l 1f",nli
-'5Yo
JiL..v v· v, I
(-1 11") I "nnh (-1 r ) + Z1",nnh L.. - In . JLo JiV, + zL.. v /-La-I ~a JLo.,a /-La v, V
is called the drift of the diffusion r
0
r
with respect to Vnh.
396
S. HOCHGERNER
Note that r 0 I' is a y7Uh-martingale if and only if the y1Dh_drift vanishes. See also [10, Theorem (7.31)].
3.3. Let I' be a solution of the Stratonovich equation
THEOREM
of =
snh(X, f)oX
and let r : T K ---* K be the projection. (1) Then, with respect to the nonholonomic connection y7Dh introduced in (3.12), the semi-martingale r 0 I' defines a diffusion on K whose drift is the gradient -~gradJLO(logN) where N is the density function defined in (3.5). (2) The drift -~gradJLO(1ogN) is horizontal with respect to the mechanical connection, Hornech = (ker T K )JLo.L, on the principal bundle K : K _ K j H = sn-l. If n satisfies the hamiltonization condition (3.7) then the drift is also horizontal with respect to the principal bundle connection Lpa Q9 Ya : T K ---* ~ on K: K - KjH.
Item (2) means that the drift component of angular momentum about the vertical axis in the space frame vanishes, and when the hamiltonization condition holds then the same is true for the component of angular velocity about the vertical axis. For n = 3 this condition is always satisfied. Let I' be a solution of the Stratonovich equation of = r : TK ---* K and K : K - KjH = sn-l be the obvious projections. Suppose that n= 1 (i.e. the ball is homogeneous). (1) Then r 0 I' defines a martingale in K with respect to the nonholonomic connection. (2) The process K 0 r 0 I' is a Brownian motion on sn-l whose generator is ~ times the Laplacian of v, where v is the metric on sn-l induced from the left H -invariant metric /-to = ((1 + A*A).,.) on K. COROLLARY 3.4. snh(X, f)OX and let
Note that the restriction of /-to to Hor = Hornech = span{~a} equals twice the restriction of the biinvariant metric. This corollary is intuitive but nevertheless not obvious since the dynamics at the compressed level can never be described by a Hamiltonian reduction procedure. This is because A is not the mechanical connection, even if the ball is homogeneous. (Compare with [12, Corollary 4.3].) Thus it does not fall in the category of [18, 19]. For reference we note the formula v./-to(u) = v'L(~a, u}~a = L(~a, [v, u]}~a - L(~a, u}[v, ~a] = A* A[v, u] - [v, A* Au].
(3.14)
Proof of Theorem 3.3: Let f E COO(K) which we regard via pull-back as a function on T K. According to Definition 3.2 we need to show that
-/-tolnLV~.Vi I
+ Lvn~h (/-to1nVi) + LVn~l JLo .Vi JLo
~a
(/-tol~a) = -gradJLo(1ogN). (3.15)
Claim: (3.16)
397
STOCHASTIC CHAPLYGIN SYSTEMS
Indeed, we use (3.14) and the fact that
U
= L(Kv;, u)v; to see that
LVn~h (tLOIKv;) = L(KtLOIKv;, Vj)(KtLOIKv;, Vk)V~hVk ~~
J
+ L(KtLOIKv;, Vj)(Vj.(KtLo IKv;, Vk))Vk = LtLo I [tLoI Kv;, KtLo I Kv;]
+L
(KtLol Kv;, Vj) (KtLol [Vj, A* AtLol Kv;], Vk)Vk
= LtLOI[tLOIKv;, KtLo l Kv;]
+ LtLOI[tLOIKv;, A* AtLo1Kv;]
= LtLol[tLoIKv;, Kv;] = LtLol[V;, Kv;] - LtLol[tLoIA*Av;, Kv;]
where we use tLoI K= 1- tLoI A *A in the last equation. Notice that [v;, Kv;] = KV~I v;. For the gradient part of claim (3.16) we consider l l (L[tLo A* Av;, Kv;], {b) = - L(Kv;, [tLo ({a, V;}{a, {b]}
= - L({a, (Kv;, [tLol{a, {b]}V;) = L([tLol{a, {a], {b} = d(logN){b l
by (3.6). Similarly it is true that (L[tLo l A* Av;, Kv;], ~a) = L([tLo {a, {a], ~a}. Using the property [Z,. fa] = Oa,bZc - Oa,cZb with fa = [Zb, Zc] it is easy to see that L([tLol{a, {a], ~a)
Thus LtLol[tLoIA*Av;,Kv;] (3.16) follows. Claim:
=
= (tLOI{b' {c} -
(tLol{c' {b}
tLoIL(d(logN){b){b
=
= o.
grad'vrlog A") and claim
Vn~1 (tLol{a) = O. JIo (a
Again we use (3.14) to see that Vn~lr (tLol{a) = tLol[tLol{a' KtLol{a] + L(Kvj, tLo1{a) (Vj.(Kvk. tLo1{a) )Vk JIo ,a
-I[
= tLo
-Ii"
11
-Ii"]
tLo ,>a, JltLo ,>a
+ L(Kvj, tLoI {a}(Kvk. tLol[Vj, A* AtLo l {a]}Vk l l - L(Kv j, tLo {a}(Kvk. tLo [Vj, {a]}Vk = tLo-1[ tLo-I {a, 11 JltLo-Ii"] ,>a
+ tLo-I[ tLo-Ii",>a, A*AtLo-Ii"] ,>a
- tLo1[tLo1{a' {a]
=0. Now (3.16) and (3.17) imply (3.15) which shows part (1) of the assertion.
(3.17)
398
S.HOCHGERNER
For (2) one checks that (3.7) yields, for
~a
= [sc, Sd],
(gradJLo(1ogN), ~a} = L(fLol[Sc, Sd], ([fLoISa, Sa], ~b)~b} = L{fLol[Sc, Sd], ~b}(fLol~a, [Sa, Sb]} = (fLOI~d, [~c, ~d]}(fLol~c, [SC, td]}(n - 2)
+ (fLOI~c, [~c, ~d]}{fLoltd, [~d, ~c]}(n -
2)
=0, where we also have used that L([fLoISa, tal, ~a} = O.
0
Proof of Corollary 3.4: Part (1) is clear. Concerning part (2) let f E coo(sn-l). According to Definition 2.2 we should show that the generator (3.11) satisfies (4L(fLolvi)(fLolvJ
+ ~L(fLol~a)(fLolsa»)K*f =
~K*~v I,
where ~v is the Laplacian associated to v. Indeed, we find ~L(fLolvi)(fLolvJK* f + ~L(fLolta)(fLolta)K* f = HLtasa K* t- Now {Jz~a} is a horizontal orthonormal frame for fLo I(Hor x Hor) where Hor is the fLo-orthogonal to ker T K. Therefore, iL~aSaK* f = ~Tf'1orHessJLO(K* f) = ~K* ~v f,
where Tf'1or denotes the trace computed with respect to horizontal fields only and Hess JLO is the Hessian of the Levi-Civita connection on (K, fLo). The equation ta~aK* f = HessJLO(K* f)(ta, ~a) is justified by the observation that Vrao~a = 0 where VJLO is the Levi-Civita connection of the right-invariant metric fLo. (In fact, the 0 restriction of v oh to Hor x Hor equals the restriction of ViLa.)
In the homogeneous case the above construction yields a Brownian motion on in a manner similar to the one described in [15, Chapter V] by the notion of rolling the sphere sn-l along a Brownian motion in lRn - 1 by means of the LeviCivita connection. The difference is that [15] start from Brownian motion in lRn - 1 while we started from Brownian motion in lRm x lRn - 1 with m = dimso(n) = n(n2-1). One can recover the setting of [15] by setting (Ho, HI, ... , H m ) = 0 in (3.8). Then, with ]I = 1, we obtain a diffusion K 0 r 0 r which is driven by Brownian motion (WI, ... , W n - I) in lRn - 1 and the generator of which is given by sn-I
~L(fLol Sa)(fLo l ~a)K* f = ~K* ~v i. where f E coo(sn-l). Referring to the interpretation stated in Introduction this means that the Chaplygin ball is subject to horizontal jiggling but there is no angular jiggling. Equivalently, the ball sits on a table which undergoes a translational Brownian motion, compare [20]. Alternatively, we can set (FI , ..• , Fn - I ) = 0 in (3.8). Then there is only angular jiggling and the diffusion r 0 r is driven by (B I , ... .B'"), By (3.17) the drift remains the same as in Theorem 3.3.
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STOCHASTIC CHAPLYGIN SYSTEMS
It seems that the notion of a stochastic nonholonomic system has been hardly investigated in the literature. We finish by asking the following questions. (1) Does an analog of Theorem 3.3 hold for general G-Chaplygin systems when there is a preserved measure? What can be said about the drift if there is no preserved measure? (2) Which is the precise relationship between G-Chaplygin systems with preserved measures and measure preserving 'Chaplygin diffusions'? Preservation of measure by diffusions is studied in [15, Chapter V]. (3) Is there a time change or Girsanov type argument to eliminate the drift in Theorem 3.3 or to make it even a Brownian motion? Is this related to the hamiltonization of the deterministic problem? Added in proof" Questions (1) and (2) are answered in the forthcoming paper [14]. This paper addresses also stochastic nonholonomic systems with constraints which are not of Chaplygin type.
4. Appendix: G-Chaplygin systems and symmetry reduction The purpose of this appendix is to shortly introduce and motivate the notion of a G-Chaplygin system and to state Proposition 4.1 which explains the symmetry reduction of such systems. This reduction is termed compression [9] to distinguish it from symplectic reduction. These concepts are closely related and compression can be viewed as a perturbed version of its symplectic counterpart. At the same time, however, there are fundamental differences; symmetries behave differently in nonholonomic mechanics and do not necessarily give rise to conserved quantities, and there need not exist a preserved measure [7, Section 5.4]; all this is related to the question of closedness of the form nnh defined in Proposition 4.1, see [1, 3, 7, 9, 12]. A nonholonomic system is a triple (Q, V, L) where Q is a configuration manifold, L : T Q -+ lR is a Lagrangian, and VeT Q is a smooth nonintegrable distribution which is supposed to be of constant rank. The equations of motion for a curve q(t) which should satisfy q' E V are then stated in terms of the Lagrange-d' Alembert principle. Suppose there is a Riemannian metric u. on Q such that we have an isomorphism T Q ;:: T* Q and assume that L is the kinetic energy Lagrangian. In this case there is also an (almost) Hamiltonian version: continue to use the symbol fL to denote the co-metric and consider the Hamiltonian Hiq, p) given by the Legendre transform of L. Since D is of constant rank there is a family of independent one-forms ¢a E n(Q) such that V is the joint kernel of these. In terms of coordinates (qi, Pi) the equations of motion are (q i )vt
aH" = -oH and p,, = - -. - ~Aa¢ a ( -a. ) 0Pi
I
oql
oq'
where the Aa are the Lagrange multp>liers to be determined from the supplementary condition that fL(¢a, p) = O. With X H := (q', p') we may thus rephrase the equations
400 as
S. HOCHGERNER
i(X~)Q =
an + L)"ar*¢a
(4.18)
where Q = -de is the canonical symplectic form on T* Q and r : T* Q ---+ Q is the footpoint projection. (The notation X~ will become clear below.) Let G be a Lie group that acts freely, properly and by isometries on the Riemannian manifold (Q, /-L). A G-Chaplygin system is a nonholonomic system (Q, L = 411 . II ~,V) that has the property that V is a principal connection on the principal bundle Q --* Q I G. Thus V is the kernel of a connection form A : T Q ---+ g. Notice that we do not require A to be the mechanical connection associated to /-L. Consider C and QC as defined in Section 2.C. Since X~ is, by construction, tangent to M and takes values in C one may now rewrite the equations of motion (4.18) in the appealing format i(X~)QC = (d'Hl
where (d'H)c is the restriction of l*d'H to C with l : V ~ T Q being the inclusion. Let /-Lo denote the induced metric on S := QI G that makes n : Q --* S a Riemannian submersion. Identify tangent and cotangent space of Q and S via their respective metrics. Consider the orbit projection map p:= TrrIV: V
--*
VIG = TS.
We may also associate a fiber-wise inverse to this mapping which is given by the horizontal lift mapping hlA associated to A. As already noted in Section 2.C, l*r* A : TV ---+ 9 defines a principal bundle connection for p, whose horizontal space is given by C. PROPOSITION 4.1 (Compression). The following are true. (1) QC descends to a nondegenerate two-form Q oh on T S. (2) Q oh = Q s - (Je ohl A , r;CurvA ). Here Q s = -des is the canonical form on T S, Jc is the momentum map of the tangent lifted G-action on T Q, Curv A E Q2(S, g) is the curvature form of A, and ts : T S ---+ S is the projection. (3) Let h : T Q ---+ ~ be G-invariant. Then the vectorfield
X~ := (QC)-l (dh)c,
where (dh)c is the restriction of l*dh to C, is p-related to the vector field X~~ on T*S defined by i(X~~)Qnh = dhi, where the compressed Hamiltonian h o : T* S = T S ---+ ~ is defined by h o := h 0 l 0 hlA , with hlA denoting the horizontal lift mapping.
In general, Q oh is an almost symplectic form, that is, it is nondegenerate and nonclosed, see [1, 9, 12].
STOCHASTIC CHAPLYGIN SYSTEMS
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