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Magnetic Flows on Homogeneous Spaces!

Alexey V. Bolsinov †, Bozidar Jovanovic ‡

Dept. of Math. Sciences, Loughborough University

Loughborough, Leicestershire LE11 3TU UK, e-mail: A.Bolsinov@lboro.ac.uk

Department of Mechanics and Mathematics, Moscow State University

119992, Moscow, Russia, e-mail: bolsinov@mech.math.msu.su

and

Mathematical Institute SANU

Kneza Mihaila 35, 11000 Belgrade, Serbia, e-mail: bozaj@mi.sanu.ac.yu

February 26, 2007

Abstract

We consider magnetic geodesic flows of the normal metrics on a classof homogeneous spaces, in particular (co)adjoint orbits of compact Liegroups. We give the proof of the non-commutative integrability of flowsand show, in addition, for the case of (co)adjoint orbits, the usual Liou-ville integrability by means of analytic integrals. We also consider thepotential systems on adjoint orbits, which are generalizations of the mag-netic spherical pendulum. The complete integrability of such system isproved for an arbitrary adjoint orbit of a compact semisimple Lie group.

Contents

1 Introduction

Let Q be a smooth manifold with a Riemannian metric g = (gij). Consideran arbitrary local coordinate system x1, . . . , xn and pass from velocities xi tomomenta pj by using the standard transformation pj = gij xi. Then xi, pi

(i = 1, . . . , n) represent the local coordinate system on the cotangent bundle!MSC: 70H06, 37J35, 53D25†Supported by RFBR 05-01-00978‡Supported by the Serbian Ministry of Science, Project ”Geometry and Topology of Man-

ifolds and Integrable Dynamical Systems”.

1

T !Q and the standard symplectic form on T !Q reads ! =!

dpi " dxi. Thecorresponding canonical Poisson brackets are given by

{f, g}0 =n"

i=1

#"f

"xi

"g

"pi# "g

"xi

"f

"pi

$.

The equations of the geodesic flow have the Hamiltonian form on (T !Q,!):

df

dt= {f,H}0 $% dxi

dt=

"H

"pi,

dpi

dt= #"H

"xi, (1)

where the Hamiltonian H is

H(x, p) =12

n"

i,j=1

gijpipj =12

n"

i,j=1

gij xixj . (2)

Here gij are the coe!cients of the tensor inverse to the metric.The geodesic flow can be interpreted as the inertial motion of a particle on

Q with the kinetic energy given by (??). The motion of the particle under theinfluence of the additional magnetic field given by a closed 2-form

" ="

1"i<j"n

Fij(x)dxi " dxj ,

is described by the following equations:

dxi

dt=

"H

"pi,

dpi

dt= #"H

"xi+

n"

j=1

Fij"H

"pj. (3)

The equations (??) are Hamiltonian with respect to the ”twisted” symplecticform ! + #!", where # : T !Q & Q is the natural projection. Namely, the newPoisson bracket is given by

{f, g} = {f, g}0 +n"

i,j=1

Fij"f

"pi

"g

"pj, (4)

and the Hamiltonian equations f = {f,H} read (??).The flow (??) is called magnetic geodesic flow on the Riemannian manifold

(Q, g) with respect to the magnetic field ". For simplicity, we shall refer to (??)as a magnetic Poisson bracket and to (T !Q,! + #!") as a magnetic cotangentbundle.

Outline and Results of the Paper. Our work has been inspired by a recentpaper by Efimov [?] in which he proved the non-commutative integrability ofmagnetic geodesic flows on coadjoint orbits of compact Lie groups. We observedthat using the approach developed in [?], [?] one can extend this result to a wider

2

class of homogeneous spaces and construct, in many cases, complete algebrasof commuting integrals. Besides, this technics turned out to be useful in thetheory of integrable magnetic potential systems. Such systems on Stiefel andGrassmann manifolds of two-dimensional planes in Rn and complex projectivespaces were studied in [?, ?].

In Section 2, we recall the concept of non-commutative integrability sug-gested by Mishchenko and Fomenko [?] and its relation with the Hamiltoniangroup action established in [?]. In Section 3 we introduce a class of homoge-neous spaces G/H admitting a natural G-invariant magnetic field. Briefly, thisconstruction can be explained as follows. Let G be a compact Lie group, H itsclosed subgroup and h and g denote the Lie algebras of H and G, respectively.Suppose that a ' h is H-adjoint invariant. In particular, H ( Ga, where Ga ( Gis the G-adjoint isotropy group of a. Consider the adjoint orbit O(a) througha endowed with the standard Kirillov-Konstant symplectic form "KK (we cannaturally identify adjoint and coadjoint orbits by the use of AdG-invariant scalarproduct on g). Then we have the canonical submersion of homogeneous spaces$ : G/H & G/Ga

)= O(a) and the closed two-form " = $!"KK gives us therequired magnetic field on G/H. In particular, for H = Ga we obtain an adjointorbit with magnetic term being the Kirillov-Konstant form.

We prove the non-commutative integrability of geodesic magnetic flows of thenormal metrics (Theorem ??) on G/H and show, in addition, that for the case ofadjoint orbits one can find enough commuting analytic integrals (Theorem ??).The proof is based on recent results concerning geodesic flows on homogeneousspaces [?, ?, ?].

In Section 4, we study the motion of a particle on coadjoint orbits under theinfluence of an additional potential force field. For the Lie algebra so(3), thesystem represents the magnetic spherical pendulum. The generalization of themagnetic spherical pendulum to the complex projective spaces is obtained in[?].

In Section 5, we give a representation of the system in the semi-direct prod-uct g *ad g and, following the bi-Hamiltonian approach, prove its completeintegrability for coadjoint orbits of compact semisimple Lie groups (Corollary??, Theorem ??). Various aspects of representations of (polarized) coadjointorbits in semi-direct products as magnetic cotangent bundles are studied in[?, ?, ?, ?, ?].

Let us emphasize that in the present paper we consider integrability as aqualitative phenomenon: the phase space of the system is foliated almost every-where by isotropic invariant tori with quasiperiodic dynamics. Sometimes thisproperty is not the same as the classical integrability, i.e., existence of explicitformulae for solutions. The simplest example that demonstrates the di#erencebetween two types of integrability is the magnetic geodesic flow on a compactconstant negative curvature surface. We can represent the geodesics explicitlyas projections of magnetic lines from the hyperbolic 2-plane (lines of constantgeodesic curvature), but the restriction of the flow onto energy levels H > hcr

are Anosov flows, while the restriction onto energy levels H < hcr are analyti-cally integrable, where hcr is some critical level of energy (e.g., see [?, ?]).

3

2 Integrable Systems Related to HamiltonianActions

There are a lot of examples of integrable Hamiltonian systems with n degrees offreedom that admit more than n (noncommuting) integrals. Then, under someassumptions, the n dimensional Lagrangian tori are foliated by lower dimen-sional isotropic tori. This happens in the case of the so-called non-commutativeintegrability studied by Nekhoroshev [?] and Mishchenko and Fomenko [?] (seealso [?, ?]).

Let M be a Poisson manifold and F be a Poisson subalgebra of C#(M).Suppose that in the neighborhood of a generic point x we can find exactly lindependent functions f1, . . . , fl ' F and the corank of the matrix {fi, fj} isequal to some constant r. Then numbers l and r are called di!erential dimen-sion and di!erential index of F and they are denoted by ddim F and dind F ,respectively. The algebra F is called complete if:

ddim F + dind F = dim M + corank {·, ·},

If F is any algebra of functions, then we shall say that A ( F is a completesubalgebra if

ddim A+ dind A = ddim F + dind F .

The Hamiltonian system x = XH(x) is completely integrable in the non-commutative sense if it possesses a complete algebra of first integrals F . Then(under compactness condition) M is almost everywhere foliated by (dind F #corank {·, ·})-dimensional invariant isotropic tori. Similarly as in the Liouvilletheorem, the tori are filled up with quasi-periodic trajectories.

Mishchenko and Fomenko stated the conjecture that non-commutative in-tegrable systems x = XH(x) are integrable in the usual commutative sense bymeans of integrals from A that belong to the same functional class as the originalnon-commutative algebra of integrals. The conjecture is proved in C#–smoothcase [?]. In the analytic case, when F = span R{f1, . . . , fl} is a finite-dimensionalLie algebra, the conjecture has been proved by Mishchenko and Fomenko in thesemisimple case and just recently by Sadetov [?] for arbitrary Lie algebras.

Now, let a connected compact Lie group G act on a 2n–dimensional con-nected symplectic manifold (M,!). Suppose the action is Hamiltonian with themomentum mapping $ : M & g! )= g (g! is the dual space of the Lie algebra g,we use the identification by means of AdG-invariant scalar product +·, ·, on g).

Consider the following two natural classes of functions on M . Let F1 be theset of functions in C#(M) obtained by pulling-back the algebra C#(g) by themoment map F1 = $!C#(g). Let F2 be the set of G–invariant functions inC#(M). The mapping f -& f . $ is a morphism of Poisson structures:

{f . $, g . $}(x) = {f, g}g(%), % = $(x),

where {·, ·}g is the Lie-Poisson bracket on g:

{f, g}g(%) = +%, [/f(%),/g(%)],, f, g : g & R.

4

Thus, F1 is closed under the Poisson bracket. Since G acts in a Hamiltonianway, F2 is closed under the Poisson bracket as well. The second essential fact isthat h . $ commute with any G–invariant function (the Noether theorem). Inother words: {F1,F2} = 0.

The following theorem, although it is a reformulation of some well knownfacts about the momentum mapping (e.g., see [?]), is fundamental in the con-siderations below (see [?] for more details).

Let A ( C#(g) be a Lie subalgebra and $!A = {h.$, h ' A} the pull-backof A by the momentum mapping. Then we have:

Theorem 1 (i) The algebra of functions F1 + F2 is complete:

ddim (F1 + F2) + dind (F1 + F2) = dim M.

The dimension of regular invariant isotropic tori, common level sets of functionsfrom F1 + F2, is equal to

dim Gµ # dim Gx,

for generic x ' M , µ = $(x) (Gµ and Gx denotes the isotropy groups of Gaction at µ and x).

(ii) $!A + F2 is a complete algebra on M if and only if A is a completealgebra on a generic adjoint orbit O(µ) ( $(M).

(iii) If B is complete (commutative) subalgebra of F2 and A is complete(commutative) algebra on the orbit O(µ), for generic µ ' $(M) then $!A + Bis complete (commutative) algebra on M .

Notice that instead of commutative subalgebras one usually consider setsof commuting functions. Clearly, each commutative set generates a certaincommutative subalgebra. The notions of completeness, ddim and dind for acommutative set are defined just in the same way as above.

3 Magnetic Geodesic Flows

Let G be a compact connected Lie group with the Lie algebra g = TeG. Let usfix some bi-invariant metric ds2

0 on G, i.e., AdG–invariant scalar product +·, ·,on g. We can identify g! and g by +·, ·,.

Consider an arbitrary homogeneous space G/H of the Lie group G. Themetric ds2

0 induces so called normal metric on G/H. We shall denote thenormal metric also by ds2

0. By the use of ds20 we identify T !G )= TG and

T !(G/H) )= T (G/H). Let h be the Lie algebra of H and g = h + v the orthog-onal decomposition. Then v can be naturally identified with T!(e)(G/H) andT !!(e)(G/H), where & : G & G/H is the canonical projection.

Construction of the Magnetic Field. We introduce a class of homogeneousspaces G/H having a natural construction of the magnetic term, consisting ofpairs (G, H), where H have one-point adjoint orbits. Let a ' h be the H-adjoint invariant. Then H is a subgroup of the G-adjoint isotropy group Ga of

5

a. The adjoint orbit O(a) through a carries the Kirillov-Konstant symplecticform "KK . Then we have canonical submersion of homogeneous spaces

$ : G/H & G/Ga)= O(a)

and the closed two-form " = $!"KK gives us the required magnetic field onG/H.

The form " is G-invariant. From the definition of "KK (see equation (??))one can easily prove that at the point &(e), " is given by

"('1, '2)|!(e) = #+a, ['1, '2],, '1, '2 ' v )= T!(e)(G/H) (5)

Below we give another natural description of the form ".

Reduction. Consider the right action of the Lie subgroup H to G: (g, h) -&gh, g ' G, h ' H, and extend it to the right Hamiltonian action on T !G. Afteridentification h )= h!, we get the momentum mapping

% : T !G & h, %(g · ') = prh ', ' ' g.

Here prh denotes the orthogonal projection with respect to the invariant scalarproduct +·, ·,.

It is well known that the symplectic reduced space %$1(0)/H is symplecto-mophic to (T !(G/H),!), where ! is the canonical symplectic form on T !(G/H).On the other side, the reduced spaces %$1(a)/Ha, are di#eomorphic to the fibrebundles over T !(G/H) with fibres being the H-adjoint orbit through a. In par-ticular, if we deal with one-point orbit OH(a) = {a}, then the reduced space issymplectomorphic to the magnetic cotangent bundle of G/H (see [?],[?]). Notethat for connected H, a is a H-adjoint invariant if and only if a belongs to thecenter of h.

Proposition 1 Let a ' h be the H-adjoint invariant and let ( be a real pa-rameter. Then the symplectic reduced space %$1((a)/H is symplectomorphicto the magnetic cotangent bundle T !(G/H) endowed with the symplectic form! + (#!". The form " is G-invariant and at the point &(e) is given by (??).

Proof. We only need to describe the magnetic term for ( = 1. Take a principalconnection ) on the H-bundle G & G/H, that is a h-valued 1-form with theproperty that the distribution D = ker) ( TG (horizontal distribution) is H-invariant and transversal to the orbit of H-action. For example we can take) such that D is orthogonal to H-orbits with respect to the fixed bi-invariantmetric. Then )g(X) = prh(g$1 ·X) and Dg = g ·v. Define the 1-form )a = +), a,on G. The magnetic form " is the unique 2-form determined by (see Kummer[?])

d)a = &!".

Since ) is a connection and a is H-invariant, the form " is well defined. Also,since & is a submersion, " is closed but need not be exact.

6

For a vector X ' TgG we have the unique decomposition X = Xh +Xv, intothe horizontal part Xh ' g · v and the vertical part Xv ' g · h. The alternativedescription of the magnetic term is

"!(g)(X, Y ) = +a,*g(Xh, Y h),, X, Y ' T!(g)(G/H) (6)

where Xh, Y h ' TgG are horizontal lifts of X and Y and * is the curvature ofthe connection (h-valued 2-form on G).

Let Xh, Y h be arbitrary extensions of Xh, Y h to horizontal vector fields.Then

*(Xh, Y h) = #)([Xh, Y h])

where [·, ·] is the commutator of vector fields. Now, by taking the left-invariantextensions of Xh and Y h we come to the expression

*(Xh, Y h) = #prh[g$1 ·Xh, g$1 · Y h], (7)

where [·, ·] is the Lie algebra commutator.The horizontal lifts of '1, '2 ' v )= T!(e)(G/H) to g )= TeG are exactly '1

and '2 considered as elements of g. Whence, from relations (??) and (??) weget the required expression (??) for the magnetic form " at &(e). !

Note that the above magnetic cotangent bundles of the homogeneous spacesnaturally appear in the symplectic induction procedure over a point (see [?]).

The natural left G-action on T !G commutes with the right H-action andleaves the preimage %$1((a) invariant. Thus, from the well known formula forthe momentum mapping of the left G-action on T !G we get

Lemma 1 The momentum mapping

$" : T !(G/H) & g, (8)

of the natural G-action on T !(G/H) with respect to the symplectic form !+(#!"is given by

$"(g · %) = Adg(% + (a), % ' v, g · % ' T !!(g)(G/H).

Magnetic Geodesic Flows of Normal Metrics. The Hamiltonian functionof the geodesic flow of the normal metric ds2

0 is simply given by

H0 =12+$0,$0,. (9)

Now, let F"1 be the algebra of all analytic, polynomial in momenta, functions

of the formF"

1 = {p . $", p ' R[g]}and F2 be the algebra of all analytic, polynomial in momenta, G–invariantfunctions on T !(G/H). Then

{F"1 ,F2}" = 0,

7

where {·, ·}" are magnetic Poisson bracket with respect to ! + (#!".Consider the Hamiltonian H" = 1

2 +$",$", ' F"1 . We have

H"(g · %) =12+Adg %, Adg %,+ (+Adg %, Adg a,+ (2

12+Adg a,Adg a,

= H0(g · %) + (212+a, a, = H0(g · %) + const,

where we used that % ' v is orthogonal to a ' h. Thus, we see that Hamiltonianflows of H0 and H" coincides: f = {f,H0}" = {f,H"}". Since H" belongs to F"

1

its commutes with F2. On the other side, as a composition of the momentummapping with an invariant polynomial, the function H" is also G-invariant andcommutes with F"

1 . Hence {H0,F"1 + F2}" = 0.

From the above consideration and Theorem ?? we get the following result.Let, as before, G be a compact Lie group and H ( G be a closed subgroup suchthat AdH a = a for a certain element a ' h ( g.

Theorem 2 The magnetic geodesic flow of the normal metrics ds20 on the ho-

mogeneous space G/H with respect to the closed 2-form (" given by (??) iscompletely integrable in the non-commutative sense. The complete algebra offirst integrals is F"

1 + F2.

Remark 1 The magnetic geodesic flow can be seen also as the reduction of thegeodesic flow of the bi-invariant metric ds2

0 from the invariant subspace %$1((a)to %$1((a)/H )= T !(G/H). With the above notation, this means that eachmagnetic geodesic line on G/H is the projection of a certain geodesic +(t) ( G

+(t) = g0 · exp((' + (a)t), t ' R,

where ' ' o and g0 is the initial position. In such a way, the integrability ofthe magnetic geodesic flow can be also studied from the point of view of thesymplectic reduction (see [?, ?]). More precisely, the reduction of the normalgeodesic system from T !G to the Poisson manifold (T !G)/H is completely inte-grable in non-commutative sense (see Zung [?]). Since the symplectic leaves in(T !G)/H are Marsden-Weinstein reduced spaces, it appears that the symmetryreduction for a generic value of the momentum map % yields a system which isintegrable in the non-commutative sense. The interpretation of these reducedsystems in terms of the Yang-Mills analogue of the Lorentz force is well-known(e.g., see [?]). The magnetic bundle %$1((a)/H, in general, corresponds to asingular leaf in (T !G)/H and in this case some complementary work has to begiven to prove the integrability.

Magnetic Geodesic Flows on Adjoint Orbits. Consider the (co)adjointaction of G and the G-orbit O(a) through an element a ' g. In what follows, weshall use the representation of the adjoint orbit as a homogeneous space G/H,where H = Ga is the isotropy group of a. Since G is a compact connected Liegroup, Ga is also connected (e.g, see [?], page 259). We have

ann(a) = {' ' g, [', a] = 0} = TeGa.

8

By definition, the Kirillov-Konstant symplectic form "KK on G/Ga is aG–invariant form, given at the point &(e) ' G/Ga by

"KK('1, '2)|!(e) = #+a, ['1, '2],, '1, '2 ' ann(a)% = [a, g], (10)

where '1, '2 are considered as tangent vectors to the orbit at &(e). It followsfrom Proposition ?? that the Kirillov-Konstant form can be seen as a magneticform obtained after right symplectic Ga-reduction of T !G as well.

From Theorem ?? we recover the Efimov result [?] (see also [?]):

Corollary 1 Let G be a compact Lie group and a ' g. The magnetic geodesicflows of normal metric ds2

0 on the (co)adjoint orbit O(a) = G/Ga with respect tothe magnetic form ("KK is completely integrable in the non-commutative sense.

Remark 2 For a generic % ' ann(a)% we have equality dim G#+"a = dim G# forall ( ' R (see [?, ?]). It implies that the dimension of the regular invariant toridoes not depend on ( and is equal to

dind (F"1 + F2) = dim G# # dim(Ga)#, (11)

for a generic element % ' ann(a)% (see [?]). Therefore, the influence of themagnetic fields ("KK , ( ' R reflects as a deformation of the foliation of thephase space (T !O(a),! + (#!"KK) by invariant isotropic tori. As the magneticfield increases, the magnetic geodesic lines become more curved.

Example 1 On the unit round sphere (see the next section), the magnetic geodesiclines are circles on the sphere. It can be easily proved that for the motion withunit velocity, the radius of the circles is equal to r" = arctg( 1

|"| ). As |(| tends toinfinity, r" tends to zero, and as ( tends to zero, then r" tends to !

2 .

Commutative Integrability. To prove the commutative integrability, weuse the argument shift method developed by Mischenko and Fomenko [?] as ageneralization of Manakov’s construction [?].

Let R[g]G be the algebra of AdG-invariant polynomials on g. Then thepolynomials

Ac = {p(·+ ,c), , ' R, p ' R[g]G} (12)

obtained from the invariants by shifting the argument are in involution withrespect to the Lie-Poisson bracket [?]. Furthermore, for every adjoint orbit ing, one can find c ' g, such that Ac is a complete commutative set of functionson this orbit. For regular orbits it is shown in [?]. For singular orbits there areseveral di#erent proofs, see [?, ?, ?]. Thus, the argument shift method allowsus to construct a complete commutative subalgebra in F"

1 .The G-invariant, polynomial in momenta functions on T !(G/H) are in one-

to-one correspondence with AdH -invariant polynomials on v, via their restric-tions to T !! (e)(G/H) )= v. Within this identification, from (??), (??) and

9

Thimm’s formula for ( = 0 [?], the magnetic Poisson bracket {·, ·}" on T !(G/H)corresponds to the following bracket on R[v]H

{f, g}"v(%) = #+% + (a, [/f(%),/g(%)],, f, g ' R[v]H , (13)

where R[v]H denotes the algebra of AdH -invariant polynomials on v.It is interesting that

&$1,$2 = ,1{·, ·}0v + ,2{·, ·}a

v, ,21 + ,2

2 0= 0 (14)

is a pencil of the compatible Poisson brackets on R[v]H . Here

{f, g}av(%) = #+a, [/f(%),/g(%)],, f, g ' R[v]H . (15)

By the use of the pencil (??) and the completeness criterion derived in [?],for the case of adjoint orbits, i.e, when H = Ga and v = ann(a)% one canconclude that the collection of Casimir functions of all the brackets &1,$, , ' R:

Ba = {p$a(%) = p(% + ,a), , ' R, p ' R[g]G, % ' ann(a)%} (16)

is a complete commutative subset in R[ann(a)%]Ga with respect to the canonicalbracket &1,0. If a is regular in g then Ga is a maximal torus. In this casethe completeness of Ba can be easily verified (e.g., see [?, ?]). A nontrivialgeneralization to singular orbits of classical groups is done in [?, ?] and byMykytyuk and Panasyuk for a general case [?]. Namely, it is proved that all(complexified) brackets &$1,$2 have the same corank in a generic point % 'ann(a)%, equal to (??). It follows from Theorem 1.1 [?] that (??) is a completecommutative algebra with respect to each Poisson bracket (??) as well.

Whence, according Theorem ??, we get the following statement

Theorem 3 The magnetic geodesic flows of the normal metric ds20 on the orbit

O(a) = G/Ga, with respect to the magnetic field ("KK is completely integrablein the commutative sense, by means of analytic, polynomial in momenta firstintegrals $!" (Ac)+Ba, where Ac and Ba are given by (??) and (??), respectively.

The commutative integrability of the magnetic flows on the complex projec-tive spaces is proved by Efimov in [?] (since the complex projective spaces aresymmetric spaces, in this case the algebra of F2 is commutative).

The integrals Ba can be used for deforming the normal metric to a cer-tain class of G-invariant metrics on O(a) with completely integrable magneticgeodesic flows. Theorem ?? is announced in [?], where one can find the explicitdescription of the deformed flows within the standard representation of the orbitO(a), as a submanifold of g.

Remark 3 If H is a subgroup of the isotropy group Ga, then the rank of thebracket &0,1, in general, is smaller then the rank of the other brackets from thepencil. Then Theorem 1.1 [?] implies that Casimir functions of the brackets&1,$, , ' R do not form a complete set with respect to the magnetic bracket(??). In order to get complete commutative algebra in (R[v]H ,&1,"), one has tofind enought additional commutative functions among Casimirs of &0,1.

10

4 Magnetic Pendulum on Adjoint Orbits

From now on, we shall consider the orbit O(a) realised as a submanifold of g.In this representation, the geometry of Hamiltonian flows on T !O(a) is studiedby Bloch, Brockett and Crouch [?], while G-invariant magnetic geodesic flowsare studied in [?].

The tangent space at x = Adg(a) is simply the orthogonal complement toann(x). Consider the cotangent bundle T !O(a) as a submanifold of g1 g:

T !O(a) = {(x, p) |x = Adg(a), p ' ann(x)%},

with the paring between p ' T !xO(a) )= ann(x)% and % ' TxO(a) given byp(%) = +p, %,. Then the canonical symplectic form ! on T !O(a) can be seen asa restriction of the canonical linear symplectic form of the ambient space g1 g:!dim g

i=1 dpi " dxi, where pi, xi are coordinates of p and x with respect to somebase of g.

Let ' ' g and x = Adg(a). Since 'x = dds Adexp(s%)(x)|s=0 = [', x], the

momentum mapping of the Hamiltonian G-action

g · (x, p) = (Adg x,Adg p)

on (T !O(a),!) is given by the relation +$0(x, p), ', = +p, 'x, = +p, [', x],. Thatis

$0(x, p) = [x, p].

Therefore, the momentum mapping (??), for ( 0= 0, and normal metric Hamil-tonian read

$"(x, p) = [x, p] + (x,

H0(x, p) =12+[x, p], [x, p],.

Magnetic Spherical Pendulum. As an example, consider the Lie groupSO(3). The Lie algebra so(3) is isomorphic to the Euclidean space R3 withbracket operation being the standard vector product. The adjoint orbits arespheres +-x, -x, = const. Let us consider the unit sphere S2 and its cotangentbundle T !S2 realized as a submanifold of R6:

T !S2 = {(-x, -p) ' R6 |.1 = +-x, -x, = 1, .2 = +-x, -p, = 0}. (17)

The momentum mapping of the natural SO(3) action on (T !S2,!) is $0 =-x1-p and the Hamiltonian of the normal metric ds2

0 reduces to H = 12 +-p, -p,. This

is the kinetic energy of the unit mass particle motion on the sphere. Addingthe magnetic term (#!"KK to the canonical form represents the influence ofthe magnetic monopole with the force equal to (-x 1 -p. It is well known thattwo famous integrable potential systems on the sphere remain integrable after

11

including the magnetic monopole, namely the spherical pendulum and the Neu-mann system. Let us consider the spherical pendulum. Then the Hamiltonianof the system becomes:

H =12+-p, -p, # /+-b, -x,, (18)

where -b is a constant unit vector. The equations of the motion of the particlewith the energy (??) under the influence of the magnetic force (-x 1 -p, in theredundant variables (x, p) are

d

dt-x = p,

d

dt-p = /-b + (-x1 -p + ,-x,

where the reaction force ,-x is determined from the condition that the trajectory(-x(t), -p(t)) satisfies the constraints .1 = 1, .2 = 0. The system is completelyintegrable due to the linear first integral f = +-b, -x1 -p + (-x,.

Generalization to Adjoint Orbits. The natural generalization of the mag-netic spherical pendulum to the orbit O(a) is the natural mechanical systemwith the kinetic energy given by the normal metric ds2

0 and the potential func-tion V (x) = #+b, x,, i.e., with Hamiltonian

H(x, p) =12+[x, p], [x, p], # +b, x,,

under the influence of the magnetic force field given by ("KK , where "KK isthe standard Kirillov-Kostant form. Similar systems on the complex projectivespaces are studied in [?].

Proposition 2 The equations of the magnetic pendulum, in redundant vari-ables (x, p), are given by

x = [x, [p, x]], (19)p = [p, [p, x]] + ([x, p] + b# prann(x) b, (20)

Proof. The equations of the geodesic flow of the normal metric are derived in[?] (see also [?]). The flow is given by the equations:

x = [x, [p, x]], (21)p = [p, [p, x]]. (22)

The magnetic and potential forces have no influence to the equation (??),while the second equation takes the form

p = [p, [p, x]] + b + ' + &, (23)

where the Lagrange multiplier & ' ann(x) is determined from the condition thatthe trajectory (x(t), p(t)) belongs to T !O(a). On the other side, the magnetic

12

force ' is equal to ([x, p]. For example, one can get this relation consider-ing the magnetic geodesic flow of the normal metric (i.e., V (x) 2 0) and theconservation of the shifted momentum map

$" = [x, p] + [x, p] + (x = [x, [p, [p, x]] + ' + &] +[[x, [p, x]], p] + ([x, [p, x]] = [x,' + ([p, x]] = 0.

In order to find &, take the (local) base e1(x), . . . , er(x), of ann(x) ([ei(x), x] =0), which is orthonormal at x = x. Then & =

!ri=1 ,iei(x). The Lagrange mul-

tipliers ,i are determined from the conditions

d

dt+p, ei(x), = +p, ei(x),+ +p, ei(x), = 0, i = 1, . . . , r (24)

From the identity [ei(x), x] 2 0 we have [ei(x), x] + [ei(x), x] = [ei(x), x] +[ei(x), [x, [p, x]]]. On the other side, the Jacobi identity gives [ei(x), [x, [p, x]]] =[[[p, x], ei(x)], x]. Therefore

ei(x) + [[p, x], ei(x)] ' ann(x), i = 1, . . . , r. (25)

Finally, combining (??), (??) and (??), we get ,i = #+b, ei(x),, i.e,

& = #prann(x) b.

This proves (??). !

5 Integrability of Magnetic Pendulum

By the use of the momentum mapping $", the equations of motion of the mag-netic pendulum on the orbit O(a), can be rewritten in the symmetric form

x = [$", x], (26)$" = [x, b]. (27)

This system can be naturally understood via another representation of (T !O(a),!+(#!"KK), as a coadjoint orbit in the dual space of semidirect product g*ad g.

Realization of T !O(a) in (g *ad g)!. To prove the complete integrabilitybelow, we shall introduce the semi direct product g*ad g in a slightly unusualway, by the use of contraction of Lie algebras.

From now on we suppose that G is a compact semisimple Lie group. Thenfor +·, ·, we can take the Killing form multiplied by #1. Let gC = g 3 C. ThengC is a semisimple complex Lie algebra. Denote by g0 the real semisimple Liealgebra obtained from gC:

g0 = g* ig, i2 = #1.

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Then dim g0 = dimR gC = 2 dimC gC = 2dim g and

rankC gC = rank g = r, rank g0 = 2 rank g = 2r.

Let p1, . . . , pr be the set of basic homogeneous invariant polynomials ong considered as complex invariant polynomials on gC. Then their real andimaginary parts form a set of basic polynomial invariants on g0.

The real algebra g0 has the symmetric pair decomposition: g0 = g + ig:

[g, g] ( g, [g, ig] ( ig, [ig, ig] ( g,

and one can consider the contraction of g0: the real Lie algebra g& with thesame linear space as g0 and the Lie bracket defined by

['1 + i%1, '2 + i%2]& = ['1, '2] + i[%1, '2] + i['1, %2], 'i, %i ' g.

It is clear that g& is the semidirect product g*ad ig, where the second term igis considered as a commutative subalgebra.

Now, identify g!& with g& and g!0 with g0 by means of nondegenerate scalarproduct

('1 + i%1, '2 + i%2) = +'1, '2, # +%1, %2,, (28)

which is propontional to the Killing form of g0. Then the di#erential of a smoothfunction f on g0 (or g&) is /f |%+i# = /%f # i/#f and the Lie-Poisson bracketson g0 and g& become

{f, g}g0(' + i%) = (' + i%, [/%f # i/#f,/%g # i/#g])= +', [/%f,/%g]# [/#f,/#g],+ +%, [/%f,/#g] + [/#f,/%g],,

{f, g}g! (' + i%) = (' + i%, [/%f # i/#f,/%g # i/#g]&)= +', [/%f,/%g],+ +%, [/%f,/#g] + [/#f,/%g],.

Note that a generic symplectic leaf (coadjoint orbit) in (g&, {·, ·}g! ) has thesame dimension as the orbit in g0, that is 2 dim g# 2r (see [?]).

It is well known that the cotangent bundle to the orbit of the linear repre-sentation of a Lie group to the vector space can be seen as a coadjoint orbit inthe dual space of the semidirect of the group with the vector space (e.g., see[?]). The similar statement holds for the magnetic cotangent bundles:

Proposition 3 The mapping (" : T !O(a) & (g*ad ig)! given by

("(x, p) = $"(x, p) + ix (29)

is a symplectomorphism between T !O(a) endowed with the twisted symplecticform ! + (#!"KK and the coadjoint orbit of the element (a + ia in (g*ad ig)!endowed with the canonical Kirillov-Konstant symplectic form.

Novikov and Schmeltzer have constucted such mapping for the orbits ofcoadjoint representation of the three-dimensional Euclidean space motion group

14

which are symplectomorphic to the megnetic cotangent bundles of the sphere(see [?]). This problem is further developed in [?, ?, ?, ?].

Proof. We use the following general statement. Suppose that we have a sym-plctic manifold M endowed with a transitive Hamiltonian action of a certainLie group K. Consider the corresponding momentum mapping ( : M & k!.Then from the standard properties of a momentum mapping it follows that

1) the image of ( is a single coadjoint orbit O ( k!,2) ( : M & O is a symplectic covering.In our case we just need to describe the transitive Hamiltonian action of the

semidirect product G1Ad ig on the cotangent bundle T !O(a). Such an actionexists and is very natural. Indeed, let us consider first the standard actionof G on the cotangent bundle T !O(a). As we saw above, the correspondingmomentum mapping is exactly $". Now we extend this action by adding thefollowing action of the vector space g:

% · (x, p) = (x, p + prann(x)!(%)), % ' g, (30)

where x ' O, p ' T !xO(a) = ann(x)%. Whence the action of the whole semidi-rect product G1Ad ig on T !O(a) is given by:

(g, i%) · (x, p) = (Adg x,Adg p + prann(Adg x)!(%)), (g, i%) ' G1Ad ig.

It is easy to verify that this formula defines an action and this action isHamiltonian. We know this for the first component, and the action of thesecond one (??) is generated by translations along the Hamiltonian vector fieldsof the functions

H#(x) = #+x, %,.

(Notice that this flow is the same for all structures ! + (#!"KK , ( ' R).Thus the Hamiltonian with respect to ! + (#!"KK corresponding to an

element ' + i% ' g*ad ig takes the form:

H%+i#(x, p) = H%(x, p) + H#(x, p)= +$"(x, p), ', # +x, %, = ($"(x, p) + ix, ' + i%)

This implies that the formula for the momentum mapping is given by (??),where we use the natural identification of (g*ad ig)! and g*ad ig by means ofthe scalar product (??).

Since the above action is obviously transitive, we conclude that (" is asymplectic covering over a certain coadjoint orbit. It is not hard to verify that(" is one-to-one with the image, and therefore is a global symplectomorphism,as required. !

Compatible Poisson Brackets and Integrability. Since the Hamiltonianflow

f = {f, h}g!

15

of h(' + i%) = 12 +', ', # +b, %, is given by

d

dt(' + i%) = [%, b] + i[', %], (31)

from Proposition ?? we reobtain the equations (??), (??).The flow (??) is completely integrable. This is related to the general con-

struction of integrable systems by the use of symmetric pair decompositions ofLie algebras and compatibility of Poisson brackets {·, ·}g0 , {·, ·}g! and {·, ·}ib,where

{f, g}ib(' + i%) = (ib, [/%f # i/#f,/%g # i/#g])= +b, [/%f,/#g] + [/#f,/%g],

(Reyman [?], see also [?]).Let A be the algebra of linear functions on ann(b), lifted to the linear func-

tions on g&:A = {fµ(' + i%) = +µ, ',, µ ' ann(b)}

and let

B = {Re(pj(,'+i(%+,2b)), Im(pj(,'+i(%+,2b)), , ' R, j = 1, . . . , r}, (32)

where pj are basic invariant polynomials of the Lie algebra gC. Then B iscommutative, and A + B is a complete (non-commutative) set of integrals of(??) (see [?] or Theorem 1.5 in [?]).

From Proposition ?? we get

Corollary 2 The magnetic pendulum system (??), (??) is completely integrableon a generic orbit O(a). The complete set of integrals is

(!" (A + B) = {f($"(x, p) + ix), f ' A + B}.

Note that we can always construct a complete commutative subalgebra inA (more precisely, in the symmetric algebra of A, i.e. the polynomial algebragenerated by linear functions fµ ' A). Thus we have complete commutativeintegrability. In particular, if b is a regular element of g, then A is commutativeand contained in B.

The integrability of the system (??) on the whole phase space g& do notimplies directly the integrability on singular orbits. As usual (see, for instance [?,?, ?]), to prove the completeness of integrals on singular orbits, some additionalanalysis has to be done.

Let O(a) be an arbitrary orbit.

Theorem 4 For a generic regular element b ' g, the magnetic pendulum sys-tem on (T !O(a),! + (#!"KK), described by equations (??), (??), is completelyintegrable. The complete commutative set of integrals is

(!"B = {Re(pj(,[x, p] + (x + i(x + ,2b)), Im(pj(,[x, p] + (x + i(x + ,2b))}.

16

Proof. (!"B is complete on (T !O(a),! + (#!"KK) if and only if B is completeon the coadjoint orbit ("(T !O(a)).

Consider the pencil of compatible Poisson structures

&$1,$2 = ,1{·, ·}g! + ,2 ({·, ·}g0 + {·, ·}ib) , ,1,,2 ' R, ,21 + ,2

2 0= 0.

The functions (??) are Casimir functions for &$1,$2 , where ,1 + ,2 0= 0,,2 0= 0, , =

%,2/(,2 + ,1) [?]. Since we deal with analytic functions, we

only need to prove the completeness of B at one point in ("(T !O(a)). Also,the completeness of B for one element b implies the completeness for a genericb ' g.

Consider the point (a + ia ' ("(T !O(a)) and take an arbitrary regularelement b ' ann(a). According Theorem 1.1 [?], B is complete at (a + ia withrespect to the Poisson bracket &1,0 = {·, ·}g! if and only if

(A1) rank&$1,$2 = 2dim g# 2r, for all (,1,,2) 0= (1, 0).(A2) dim{' + i% ' ker &1,0 |&0,1(' + i%, ker &1,0) = 0} = 2r.

Here the Poisson brackets &$1,$2 are taken at the point (a + ia and they areconsidered as skew-symmetric bilinear forms on g&. Furthermore, all objects areassumed to be (again) complexified.

Since b is a regular element in g, it follows that ib is a regular element of thesemisimple Lie algebra g0:

anng0(ib) = anng(b) + i anng(b).

(anng(b) is a maximal commutative subalgebra of g.)The condition (A1), for ,1 + ,2 0= 0 is equivalent to the regularity of the

elements ,(a+ ia+,2ib, , ' C, , 0= 0, in the Lie algebra g0. This follows easilyfrom the regularity of ib.

Now, consider the skew-symmetric form &$1,1. We have

&$1,1('1 # i%1, '2 # i%2) = #((a + ia, ['1 # i%1, '2 # i%2]&)+((a + ia, ['1 # i%1, '2 # i%2])+(ib, ['1 # i%1, '2 # i%2])

= #+(a, [%1, %2],+ +b, ['1, %2] + [%1, '2],

Therefore, ' # i% ' ker &$1,1 if and only if

[b, %] = 0, [', b]# [%, (a] = 0.

The first equation yields % ' anng(b). On the other hand, since b ' anng(a),we have [anng(b), a] = 0. Thus, the second equation reduces to [', b] = 0, i.e.,' ' anng(b) and

dim ker&$1,1 = 2r.

It remains to verify (A2). We have

&1,0('1 # i%1, '2 # i%2) = ((a + ia, ['1 # i%1, '2 # i%2]&)= +(a, ['1, '2],+ +a, ['1, %2] + [%1, '2],

17

Similarly as above ' # i% belongs to ker &1,0 if and only if [', a] = 0, [%, a] = 0,i.e.,

ker &1,0 = anng(a) + i anng(a).

We need to find the dimension of the space

K = {' # i% ' ker &1,0 |&0,1(' # i%, ker &1,0) = 0}= {' # i% ' ker &1,0 | ((a + ia + ib, [' # i%, anng(a) + i anng(a)]) = 0}= {' # i% ' ker &1,0 | (ib, [' # i%, anng(a) + i anng(a)]) = 0}

Whence, K consists of those elements in anng(a)+ i anng(a) which commutewith ib. But, since ib is a regular element of g0, we find dim K = 2r. Thetheorem is proved. !

Remark 4 From Corollary ??, by taking b = 0, we get complete integrabilityof the magnetic geodesic flows of normal metrics on regular orbits O(a). It isinteresting that in this case we have

F"1 = (!"A,

and (!"B coincides with the set of commuting G-invariant functions on T !O(a)obtained by shifting of argument (??):

Ba = (!"B.

In this sense, remarkably, the shifting of argument method [?] can be seen as aparticular case of the method of contraction of Lie algebras [?, ?]. By modifyingthe proof of Theorem ??, it would be possible to give a proof of the completenessof Ba + F0

1 on singular orbits O(a), di#erent from those given in [?, ?].

Remark 5 As it follows from Reyman and Semenov-Tian-Shanski [?] (or bystraightforward computation) the equations (??), (??) are equivalent to theL-A pair

L(,) = [L(,), A(,)]

with a spectral parameter ,, where L(,) = ,$" + i(x + ,2b), A(,) = $" + i,b.The integrals (!"B are exactly the integrals arising from the L-A representationof the system.

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