Persistence of stationary motion under explicit symmetry breaking perturbation
PPersistence of stationary motion under explicit symmetrybreaking perturbation
Marine Fontaine and
James Montaldi
Abstract
Explicit symmetry breaking occurs when a dynamical system having a certain symme-try group is perturbed to a system which has strictly less symmetry. We give a geometricapproach to study this phenomenon in the setting of hamiltonian systems. We provide amethod for determining the equilibria and relative equilibria that persist after a symmetrybreaking perturbation. In particular a lower bound for the number of each is found, interms of the equivariant Lyusternik-Schnirelmann category of the group orbit.
Keywords:
Symmetry breaking, hamiltonian systems, Lie group actions
1. Introduction
When we talk about symmetries, we either refer to the symmetry of a physical law (dynamicalequations) or the symmetry of a physical state (solution of these equations). The symmetry or symmetry group of a physical law (or a physical state) is defined to be the group oftransformations which leave these equations (or this solution) invariant. Explicit symme-try breaking is defined as a process of perturbing symmetric dynamical equations so thatthe resulting equations have a lower symmetry group. When the system is not hamiltonianinteresting results are obtained showing for example that periodic solutions of an unperturbeddynamical system can become heteroclinic cycles under a perturbation that breaks the sym-metry [CL04, GL01, LR92].In this paper we focus on dynamical systems which are hamiltonian. We address the ques-tion whether equilibria or relative equilibria persist under a symmetry breaking perturbation.Some aspects of explicit symmetry breaking phenomena for hamiltonian systems havebeen studied by several authors [ACZE87, GMO04, GBT10] to cite just a few, but exceptfor [GMO04] their results do not overlap with ours (for [GMO04] see further below). In thecase when none of the symmetries are broken the question of persistence of relative equilibriais raised in [Mon97] for compact symmetry groups and further developed in [LS98, Wul03].The persistence in those papers was for perturbing the momentum value rather than theHamiltonian function, but the arguments also apply if the Hamiltonian is perturbed in amanner that preserves the symmetry.In applications explicit symmetry breaking phenomena appear in various ways. As ex-plained for example in [BC03] terms can be introduced artificially in the equations of motionin order to match with theoretical or experimental observations. Besides quantization pro-cesses might also be a cause for the appearance of such terms which are the so-called quantumanomalies [FS13, MS17]. In this case the terms are not artificially introduced but they appearafter a renormalization procedure. 1 a r X i v : . [ m a t h . D S ] J un he problem is as follows. Phase spaces of hamiltonian systems are symplectic manifoldsand the symmetries of such systems are encoded into Lie group actions on those manifolds. Asymplectic manifold is a smooth manifold M equipped with a non-degenerate closed two-form ω . A (proper) smooth action of a Lie group G on M is symplectic if it preserves ω . Animportant class of symplectic group actions on symplectic manifolds are the Hamiltonian actions, which are those actions to which there is associated a Noether conserved quantityexpressed in term of a momentum map Φ G : M → g ∗ , where g ∗ is the dual of the Liealgebra of G . This notion generalizes the notion of angular momentum in classical mechanics,when the phase space is (a product of copies of) T ∗ R , acted on by the group of rotations SO (3) . By a hamiltonian (proper) G -manifold we mean a quadruple ( M, ω, G, Φ G ) asdescribed above, where Φ G : M → g ∗ is equivariant with respect to the coadjoint actionAd ∗ : ( g, µ ) ∈ G × g ∗ (cid:55)→ Ad ∗ g − µ ∈ g ∗ of G on the dual Lie algebra.The dynamics is governed by a Hamiltonian h which is a G -invariant smooth real-valuedfunction defined on M . The ring of such functions is denoted by C ∞ ( M ) G . The non-degeneracy of ω implies that, associated to any Hamiltonian h ∈ C ∞ ( M ) G , there is a uniquevector field X h defined by ι X h ω = − d h . Since the action of G on M is symplectic and h is G -invariant, the integral curve ϕ t ( m ) of X h starting at m ∈ M satisfies ϕ t ( g · m ) = g · ϕ t ( m ) for all g ∈ G . The resulting Hamiltonian equations dd t ϕ t ( m ) = X h ( ϕ t ( m )) (1.1)are thus G -equivariant and we say that G is the symmetry group of (1.1). We study theeffect of a small hamiltonian perturbation of these equations, which is invariant with respectto a subgroup of G . Definition . Let h ∈ C ∞ ( M ) G and H ⊂ G be a closed subgroup. An H -pertubation of h is a family of functions h λ ∈ C ∞ ( M ) H , with λ in a neighbourhood of 0 in R such that themap ( m, λ ) ∈ M × R (cid:55)→ h λ ( m ) ∈ R is smooth and h = h .We focus on specific solutions of (1.1), namely equilibria (fixed points under the dynam-ics) and relative equilibria (group orbits fixed under the dynamics). Under a specific non-degeneracy condition on a (relative) equilibrium of the unperturbed Hamiltonian h there is achance that this (relative) equilibrium persists under an H -perturbation.Section 2 is devoted to the question of persistence of equilibria. The required non-degeneracy condition on an equilibrium m ∈ M of h is a particular case of the Morse-Bottcondition when the critical manifold of h is the group orbit G · m (cf. Definition 2.1). We showthat at least a certain number of H -orbits of equilibria persist under a small H -perturbationin a tubular neighbourhood of G · m (cf. Theorem 2.2 and Corollary 2.5). This number is thepositive integer Cat H ( G/G m ) , which is the H -equivariant Lyusternik-Schnirelmann categoryof the group orbit. At the end of the section we present applications of this result including theproblem of an ellipse-shaped planar rigid body moving in a planar irrotational, incompressiblefluid with zero vorticity and zero circulation around the body.Extending Theorem 2.2 and Corollary 2.5 to the case of relative equilibria is more subtlebecause we must take into account the conservation of momentum and the correspondingnon-degeneracy condition takes into account the ambient symplectic structure. This questionis treated in Section 4. Whereas equilibria are critical points of the Hamiltonian function h ,relative equilibria are critical points of the restriction of this same function to a level set of2he momentum map, the problem being that as the group changes, so do these level sets. Let m ∈ M be one of those critical points. The element ξ ∈ g playing the role of a Lagrangemultiplier is called the velocity of m , which is in general not unique when the action is notfree. For that reason we refer to a relative equilibrium as a pair ( m, ξ ) ∈ M × g . We denotethe underlying Lagrange function associated to ξ by h ξ = h − φ ξG , where φ ξG ( m ) := (cid:104) Φ G ( m ) , ξ (cid:105) is a G ξ -invariant function on M . It is called the augmented Hamiltonian .A standard definition says that a relative equilibrium ( m, ξ ) of h is non-degenerate if theHessian of h ξ at m is a non-singular quadratic form when restricted to some symplectic sub-space N ⊂ T m M , called the symplectic slice at m . If the perturbations h λ are invariant withrespect to the full symmetry group G , this notion of non-degeneracy is enough to guarantee thepersistence of a relative equilibrium. This is no longer the case if h λ has a smaller symmetrygroup than does h and we require a stronger non-degeneracy condition on the relative equilib-rium (Definition 4.2). In [GMO04] a step in that direction is taken, when the symmetry groupis a torus that breaks into a subtorus. In addition, the group actions in consideration are as-sumed to be free. We extend their result to non-free actions and non-abelian symmetry groups.A necessary condition for a relative equilibrium of h to persist under an H -perturbation isthat the velocity ξ belongs to h , the Lie algebra of H . If the non-degeneracy condition on ( m, ξ ) ∈ M × h holds, and modulo some technicalities, the least number of H µ -orbits of relativeequilibria with velocity close to ξ , which persist under a small H -perturbation in some neigh-bourhood of G µ · m in Φ − H ( α ) , is the positive integer Cat H µ ( G µ /G m ) . This is the content ofTheorem 4.5 and Corollary 4.6. We illustrate this result for the spherical pendulum on S , asa perturbation of the geodesic flow; an example of symmetry breaking from SO (4) to SO (3) . Acknowledgments.
We would like to thank Luis García-Naranjo for suggesting the exampleof the D-rigid body submerged in a fluid, also the editor and referees for suggestions improvingthe exposition. This work forms part of the first author’s Ph.D. thesis [Fon18a] from theUniversity of Manchester. It was partially funded by the project “symplectic techniques indifferential geometry” within the Excellence of Science program of the F.R.S.-FNRS and FWO.
2. Symmetry breaking for equilibria
The aim of this section is to give a lower bound for the number of H -orbits of equilibria thatpersist under a small H -perturbation of some G -invariant Hamiltonian h . Since equilibria ofthe hamiltonian vector field are the same as critical points of the Hamiltonian function, westate the main theorem in terms of critical points of smooth functions. The persistence resultwill require a non-degeneracy condition which we now recall. Definition . A G -nondegenerate critical point of an invariant function h ∈ C ∞ ( M ) G isa point m ∈ M such that(i) d h ( m ) = 0 ,(ii) if N is any subspace of T m M complementary to g · m , the restriction D N h ( m ) of theHessian D h ( m ) to N × N is non-singular. In other words, the Hessian is non-singularin the directions normal to the group orbit.
Remark. (i) We also say an equilibrium of an invariant Hamiltonian is G -nondegenerateunder the same conditions. 3ii) If m ∈ M is a G -nondegenerate equilibrium of h then so is any p ∈ G · m , by G -invariance. For this reason, the tangent space T p ( G · m ) is contained in ker (cid:0) D h ( p ) (cid:1) for any p ∈ G · m . Definition 2.1 is a particular case of Morse-Bott non-degeneracy when G · m is the critical manifold of h (cf. [Bot54]). Note that Condition (ii) implies that thecritical manifold G · m is isolated in the sense that there exists a tubular neighbourhoodof G · m that does not contain any other critical points of h .(iii) This definition is also valid in infinite dimensional Hilbert spaces, provided the non-degeneracy is interpreted as saying that the linear map N → N ∗ given by v → D N h ( m )( v ) is invertible (has bounded inverse). We say that a closed subgroup H ⊂ G is co-compact (in G ) if the left multiplication of H on G is co-compact, i.e. the orbit space H \ G under this action is compact. Theorem . Let G be a Lie group acting properly on a manifold M and let H ⊂ G be a closedco-compact subgroup. Assume that h λ ∈ C ∞ ( M ) H is an H -pertubation of some h ∈ C ∞ ( M ) G in the sense of Definition 1.1, and that m ∈ M is a G -nondegenerate equilibrium of h .Then there is a G -invariant neighbourhood U ⊂ M of m such that, if λ is sufficiently small,there exists a function f λ ∈ C ∞ ( G/G m ) H whose critical points are in one-to-one correspon-dence with those of h λ in U .Proof — Let m ∈ M be a G -nondegenerate equilibrium of h whose stabilizer is denoted by K := G m . Let N ⊂ T m M be a K -invariant vector subspace complementary to g · m in T m M .Recall that (see e.g., [OR04]) the natural action of the direct product G × K on G × N isgiven by ( h, k ) · ( g, ν ) = ( hgk − , k · ν ) . The Palais Tube Theorem then states that there is a G -invariant neighbourhood U ⊂ M of m and a K -invariant neighbourhood N ε of in N such that the associated bundle G × K N ε ⊂ G × K N is a local model for U and the only critical points of h in U are on G · m . In thatmodel the point m reads [( e, and the H -pertubation is identified with a family also denoted h λ : G × K N ε → R . Let ρ : G × N ε → G × K N ε be the orbit map for the K -action on G × N ε .We define the lift of h λ by (cid:101) h λ := h λ ◦ ρ : G × N ε → R . The critical points of (cid:101) h λ are then the inverse image under ρ of those of h λ . We may thus workwith (cid:101) h λ instead of h λ .By assumption the lift (cid:101) h is G × K -invariant and (cid:101) h λ is H × K -invariant. Since ( e, ∈ G ×N ε is a G -nondegenerate critical point of (cid:101) h , d (cid:101) h ( e,
0) = 0 and D N (cid:101) h ( e, is non-singular . (2.1)In particular the map d N (cid:101) h : G × N ε → N ∗ satisfies d N (cid:101) h ( e,
0) = 0 and its derivative with respect to the N -variables, evaluated at ( e, ,is invertible. 4e now wish to apply the implicit function theorem globally and uniformly along theorbit G × { } . The implicit function theorem implies the existence, for each g ∈ G , of aneighbourhood V g × W g of (0 , g ) in R × G such that, for each ( λ, g (cid:48) ) ∈ V g × W g , there is aunique φ gλ ( g (cid:48) ) ∈ N ε satisfying d N (cid:101) h λ ( g (cid:48) , φ gλ ( g (cid:48) )) = 0 . (2.2)for all g (cid:48) ∈ W g . By H × K -invariance of (cid:101) h λ and hence of d N (cid:101) h λ , we can choose W g to be H × K -invariant. Note that when we refer to H or K individually we are thinking of them assubgroups of H × K . This procedure defines an H -invariant smooth function φ g : V g × W g −→ N ε ( λ, g (cid:48) ) (cid:55)−→ φ gλ ( g (cid:48) ) . Since the W g are H -invariant open subsets of G , and H \ G is compact, it follows that we canextract a finite subcover of { W g } , call this { W g , . . . , W g n } . Now let V = V g ∩ · · · ∩ V g n . Byrestricting the above maps to λ ∈ V , we obtain a globally (in G ) defined map φ : V × G −→ N ε satisfying d N (cid:101) h λ ( g, φ λ ( g )) = 0 , for all ( λ, g ) ∈ G × V (this gives the global and uniformapplication of the implicit function theorem we required). Moreover, φ is H -invariant and K -equivariant: φ λ ( hgk − ) = k · φ λ ( g ) . (2.3)Now define a family f λ of functions on G by f λ ( g ) = (cid:101) h λ ( g, φ λ ( g )) . (2.4)We claim that f λ is H × K -invariant and has a critical point at g if and only if (cid:101) h λ has a criticalpoint at ( g, φ λ ( g )) ∈ G × N ε . The H × K -invariance implies that f λ passes down to a smooth H -invariant function on G/K , as required.To check the critical point property, note that, with ( g, w ) = ( g, φ λ ( g )) , d f λ ( g ) = d G (cid:101) h λ ( g, w ) + d N (cid:101) h λ ( g, w )d φ λ ( g ) . However, for λ ∈ V , w = φ λ ( g ) if and only if (2.2) holds, with φ in place of φ g . Thus f λ hasa critical point at g if and only if (cid:101) h λ has a critical point at ( g, φ λ ( g )) .The invariance properties of f λ follows from those of φ λ given in (2.3) above. (cid:4) Before progressing to give a lower bound for the number of critical points of the perturbation,we recall an important concept in the calculus of variations. In their original paper [LS47],Lyusternik and Schnirelmann introduce a numerical homotopy invariant of a topological space M that they denote Cat ( M ) . They define it to be the least number of open subsets of M , whoseinclusion is nullhomotopic, that are required to cover M . They show that if M is a closed (i.e.compact without boundary) smooth manifold, then any smooth function f on M has at leastCat ( M ) critical points. The equivariant analogue Cat G ( M ) when G is a compact Lie groupis obtained in [Fad85, Mar89] and the extension to proper Lie group actions in [ALQ01]. The5 quivariant Lyusternik-Schnirelmann category , denoted Cat G ( M ) , is the least numberof G -invariant open subsets of M , contractible by mean of a G -homotopy onto a G -orbit,that are required to cover M . The extension of the result of [LS47] to proper G -manifolds(possibly infinite dimensional) requires a certain compactness condition, called the orbitwisePalais-Smale condition (OPS) [ALQ01]. In our applications M is finite dimensional andthe orbit space M/G is compact, and in this case the OPS condition is automatic.
Theorem . If a proper G -manifold M and a function f ∈ C ∞ ( M ) G satisfy condition (OPS), then f has at leastCat G ( M ) group orbits of critical points. As a corollary of Theorem 2.2 we obtain:
Corollary . Under the assumptions of Theorem 2.2 the number of H -orbits of criti-cal points of h that persist near G · m under a small H -perturbation is bounded below byCat H ( G/G m ) .Proof — If λ is sufficiently small, Theorem 2.2 implies that the H -orbits of critical points of h λ in some neighbourhood of G · m are in one-to-one correspondence with those of a function f λ ∈ C ∞ ( G/K ) H where K := G m . Since H \ G is compact it follows that so too is H \ G/K .We may therefore apply Theorem 2.3 and conclude that the number of H -orbits of criticalpoints of h λ is at least Cat H ( G/K ) near G · m . (cid:4) Remark.
If one knows a priori that the critical points of f λ are all H -nondegenerate, thena (usually better) lower bound can be given using equivariant Morse theory, see for example[AB83, Hin84].The corollary can be reformulated in the hamiltonian setting. Corollary . Suppose h ∈ C ∞ ( M ) G is a Hamiltonian definedon a symplectic proper G -manifold ( M, ω ) and has a G -nondegenerate equilibrium m . Thenthe number of H -orbits of equilibria that persist near G · m under a small H -perturbation isbounded below by Cat H ( G/G m ) .Example . Think of the cylinder M = S × R as embedded in R with coordinates ( θ, z ) andendow it with the standard symplectic form ω = d θ ∧ d z . The Lie group G = O (2) acts on M by R ϕ · ( θ, z ) = ( θ + ϕ, z ) , if R ϕ ∈ O (2) is a rotation of angle ϕ ; and by r α · ( θ, z ) = (2 α − θ, z ) , if r α ∈ O (2) is a reflection about the line forming an angle α with the x -axis in R . The actionof G on M is hamiltonian with momentum map Φ G : ( θ, z ) ∈ M (cid:55)→ z ∈ R . Consider the -parameter family h λ : S × R → R defined by h λ ( θ, z ) = z + λ cos( nθ ) . Then h = h is G -invariant and m = (0 , is a G -nondegenerate equilibrium of h whosestabilizer is G m = (cid:104) r (cid:105) . The perturbation h λ is invariant under H = D n , where D n is thedihedral group of order n . In fact, the full symmetry group is D n × Z since Z acts onthe z -component by changing its sign. However such an action is not symplectic. Since thisdiscrete part does not contribute in the further application, we do not take it into account.The perturbed Hamiltonian h λ has n critical points whose coordinates are ( πn k, for k =0 , . . . , n − , which form a regular n -gone as shown in Figure 1 for the case n = 3 . Since G/G m = O (2) / (cid:104) r (cid:105) is topologically a circle, we find Cat H ( G/G m ) = 2 (cf. [Mar89] Corollary . ). There are thus two H -orbits of equilibria of h which persist, each of them being aregular n -gone (cf. Figure 2). 6 igure 1: When n = 3 , h has a G -orbit (redcircle) consisting of G -nondegenerate equilib-ria, on which the six equilibria of h λ lie. (cid:0)(cid:0)(cid:0)(cid:0) (cid:0) (cid:0) D -orbitof equilibriaof h λ Circle of equilibria of h Figure 2:
At the level of coordinate z = 0 ,the six equilibria of h λ form two different D -orbits. One orbit is stable and one is unstable. We apply the result of Corollary 2.5 to the problem of a planar rigid body B of mass m moving in a planar irrotational, incompressible fluid with zero vorticity and zero circulationaround the body. The motion is governed by Kirchhoff equations [Kir77]. Classical treatmentsof the problem can be found in [Lam93] and [MT60]. The configuration space of the body-fluid system is a submanifold Q of the product SE (2) × Emb vol (cid:0) F , R (cid:1) , where SE (2) is thespecial Euclidean group describing the motion of the body, and Emb vol (cid:0) F , R (cid:1) is the spaceof volume-preserving embeddings of the fluid reference space F in R . The symmetry groupof this system is the direct product of SE (2) (group of uniform body-fluid translations androtations) and the particle relabeling symmetry group (volume-preserving diffeomorphisms of F ). Since these actions commute, the system can be reduced by the process of symplecticreduction by stages [MMO + T ∗ Q in a hamiltonian fashion.The associated momentum map has two components corresponding to the vorticity and thecirculation. The reduction at zero momentum corresponds to a fluid with zero circulation andzero vorticity. In this case, the symplectic reduced space is identified with T ∗ SE (2) , endowedwith the canonical symplectic form and the SE (2) -invariant reduced Hamiltonian is the sumof the kinetic energy of the body-fluid system by the addition of the so-called “added masses”,and the kinetic energy of the body. Those added masses depend only on the body’s shapeand not on the mass distribution. The reader is refered to [KMRMH05] and [VKM10] fordetails. Since SE (2) acts symplectically on T ∗ SE (2) , the dynamics can be reduced a secondtime using Poisson reduction and thereby the reduced motion is governed by the Kirchhoffequations that are the Lie-Poisson equations on the dual Lie algebra se (2) ∗ .For the sake of simplicity we will assume that the body B is shaped as an ellipse with semi-axes of length A > B > . We will use the formulae and follow the notations of [FGNV13].At the center of mass of B we attach a frame { E , E } that is aligned with the symmetry axesof the body. Its position is related at any time to a fixed space frame { e , e } by an element7f SE (2) . An element of the Lie algebra ξ ∈ se (2) is identified with a vector ( ˙ θ, v , v ) ∈ R (2.5)where ˙ θ ∈ R is the angular velocity of B and ( v , v ) T ∈ R is the linear velocity of its centerof mass, expressed in the body’s frame. In this setting the body has kinetic energy T B = 12 ξ · I B ξ (2.6)with I B := diag ( I B , m, m ) , where I B is the moment of inertia of the body about its center ofmass. The kinetic energy of the fluid is given by T F = 12 ξ · I F ξ (2.7)where I F = ρπ diag (( A − B ) , B , A ) is the tensor of added masses, and ρ is the fluid density.In the absence of external forces, the Lagrangian of the body-fluid system L : T SE (2) → R is given by L = T B + T F . It defines a Riemannian metric on SE (2) with respect to whichthe motion of the body B is geodesic. Since L does not depend on the group variables, it is SE (2) -invariant and can thus be reduced to the function (cid:96) : se (2) → R given by (cid:96) ( ξ ) = 12 ξ · ( I B + I F ) ξ (2.8)with ξ as in (2.5). An element ν of the dual Lie algebra se (2) ∗ is identified with a one by threematrix ( x, α , α ) . The dual pairing (cid:104)· , ·(cid:105) between se (2) ∗ and se (2) is thus given by (cid:104) ν, ξ (cid:105) := ( x, α , α )( ˙ θ, v , v ) T = x ˙ θ + α v + α v . (2.9)We perform the Legendre transform F L : ξ ∈ se (2) (cid:55)→ (( I B + I F ) ξ ) T ∈ se (2) ∗ to obtain thereduced Hamiltonian h : se (2) ∗ → R defined by h ( ν ) = 12 ν · ( I B + I F ) − ν T . The Lie-Poisson equations on se (2) ∗ that describe the motion of the body-fluid system are ˙ ν = ad ∗ δhδν ν. (2.10)where ad ∗ ξ ν is identified with ( α v − α v , ˙ θα , − ˙ θα ) . This problem turns out to exhibitsymmetry breaking phenomena from different points of view:(i) One point of view consists in looking at the body B without the fluid ( ρ = 0 ). Addingthe fluid amounts to seeing the fluid density ρ as a “parameter”. The O (2) -symmetry ofthe kinetic reduced Hamiltonian breaks into a D -symmetry, where D is the symmetrygroup of an ellipse.(ii) On the other hand we can consider the original system as being a circular planar rigidbody ( A = B ) in a fluid and the symmetry can be broken by deforming the body into anelliptical shaped body. This case exhibits the same pattern of symmetry breaking from O (2) to the subgroup D . 8hese two approaches are the same from a group theoretical point of view. Contrary toExample 2.6, the Hamiltonian in consideration will not be perturbed by adding some potentialenergy. In this case, there is no potential energy involved, only the metric is perturbed givingrise to a modified kinetic energy. Let us now discuss the two cases mentioned above.(i) The unperturbed system on the Poisson reduced space se (2) ∗ is governed by the Hamil-tonian h ( ν ) = 12 ν · I ν = 12 (cid:18) x I B + α + α m (cid:19) (2.11)where ν := ( x, α , α ) and I := I − B . The Hamiltonian is invariant with respect to thegroup G = O (2) . In particular, for each c ∈ R , the level sets h ( ν ) = c describe spheroidsin R . Note that the full symmetry group is in fact O (2) × Z since Z acts on the x -component by swapping the sign. However this discrete part does not contribute toour analysis.Adding a fluid to the system amounts to look at the variation of the dimensionlessparameter λ = dρ where d := A − B m > is fixed . This gives rise to the perturbed Hamiltonian h λ ( ν ) = ν · I λ ν with I λ = diag (cid:18) I B + λc , m + λc , m + λc (cid:19) . (2.12)where c = m dπ , c = π ( A − md )4 d and c = π ( B + md )4 d are fixed constants encoding thedatas of the system. The perturbed Hamiltonian reads h λ ( ν ) = 12 (cid:18) x I B + λc + α m + λc + α m + λc (cid:19) (2.13)and has symmetry H = D , the dihedral group of order four: recall that the group D is isomorphic to Z × Z , and acts here by changing the signs of α and α .This perturbation coincides with h when λ = 0 and the function ( λ, ν ) (cid:55)→ h λ ( ν ) issmooth. Therefore, h λ is an H -pertubation of h . The symmetry is broken because thefluid influences the motion of the body if it is elliptical. If the body is circular ( A = B ),or if it moves in the vacuum, its center of mass would move at constant velocity and itwould rotate at constant angular speed.(ii) We carry out another kind of perturbation: rather than perturbing the rigid body motionby adding a fluid to the system, we start with a circular planar rigid body ( A = B )in a fluid and break the symmetry by changing the body shape into an ellipse. Theunperturbed Hamiltonian is given by h ( ν ) = 12 ν · I ν = 12 (cid:18) x I B + α + α m + d (cid:19) (2.14)where d = ρπB , ν := ( x, α , α ) , I := ( I B + I F ) − and A = B in the definition of I F .The Hamiltonian is invariant with respect to G = O (2) . For each c ∈ R , the level sets h ( ν ) = c also describe spheroids in R . 9e perturb the body shape by setting λ = A − B B where B > is fixed and A ≥ B > varies. This gives rise to the perturbed Hamiltonian h λ ( ν ) = ν · I λ ν with I λ = diag (cid:18) I B + λ d , m + d , m + ( λ + 1) d (cid:19) (2.15)where d = ρπB . The perturbed Hamiltonian is thus given by h λ ( ν ) = 12 (cid:18) x I B + λ d + α m + d + α m + ( λ + 1) d (cid:19) (2.16)and is again symmetric with respect to the action of H = D . In this case, if there wasno fluid ( ρ = d = 0) , no symmetries would have been broken.Since the reduced motion is governed by the Lie-Poisson equations (2.10), it is constrainedto the coadjoint orbits of SE (2) . As shown in [MR99] (Chapter . ), almost all of them arecylinders (the singular orbits consist of points on the vertical dashed line in Figure 3). In bothcases, the level sets of h λ are ellipsoids and those of h = h are spheroids. Their intersectionswith a coadjoint orbit are shown in Figure 3. In particular, the circle of equilibria of h (in redin Figure 3) breaks into four fixed points of h λ , two of which are connected by four heterocliniccycles. Figure 3:
The flow lines are given by intersecting the levelsets of h λ (the ellipsoids) and the coadjoint orbits. On theleft hand side, we see the flow lines of h on a coadjoint orbit.On the right hand side, the flow has been perturbed. Let us go back to the first case we discussed above with h λ as in (2.13). We will applyCorollary 2.5 to predict the existence of the four fixed points that persist (cf. Figure 3). TheFréchet derivative of h λ is δh λ δν = (cid:18) xI B + λc , α m + λc , α m + λc (cid:19) . (2.17)Therefore, the Lie-Poisson equations (2.10) reduces to ˙ x = λ ( c − c )( m + λc )( m + λc ) α α ˙ α = xα m + λc ˙ α = − xα m + λc (2.18)10etting λ = 0 in (2.18), we see that the fixed points of h = h are either of the form (0 , α , α ) with ( α , α ) ∈ ( R ) ∗ , or of the form ( x, , which correspond to points on the singularcoadjoint orbit.Let µ := (0 , α , α ) with α + α = 1 be a fixed point of the unperturbed hamiltonian h .The isotropy subgroup of µ is G µ = (cid:104) r ϑ (cid:105) where r ϑ is a reflection in the plane. The quotient G/G µ = O (2) / (cid:104) r ϑ (cid:105) is topologically a circle yielding Cat D ( S ) = 2 . The four fixed pointsappearing in Figure 3 are the two H -orbits that persist.
3. Symplectic preliminaries
For our principal result in Section 4 below, the symplectic geometry is crucial, and we relystrongly on the symplectic local model for a hamiltonian proper G -manifold near a grouporbit.The Symplectic Tube Theorem is used to study the local dynamics and the local geometryof a hamiltonian proper G -manifold ( M, ω, G, Φ G ) . It states essentially that every m ∈ M admits a G -invariant neighbourhood, which is G -equivariantly symplectomorphic to a neigh-bourhood of the zero section of a symplectic associated bundle. This construction providestractable semi-global coordinates for M near G -orbits. Those coordinates are sometimes re-ferred as slice coordinates . This theorem appears in [GS84, Mar85] for symplectic Lie groupactions with equivariant momentum maps. Its extension to general symplectic Lie group ac-tions can be found in [OR04, BL97]. See also [Sch07, PROSD08] for the particular case where M is a cotangent bundle.We briefly recall the construction underlying the Symplectic Tube Theorem. The readeris referred to [OR04] or [CB15] for details. Let m ∈ M with momentum µ = Φ G ( m ) . Denoteby G m and G µ the stabilizers of m and µ respectively and by g m and g µ their respective Liealgebras. The stabilizer G m is compact by properness of the action of G on M . We can thussplit g µ and g into a direct sum of G m -invariant subspaces g µ = g m ⊕ m and g = g m ⊕ m ⊕ n . We denote by g · m the tangent space at m of G · m . Elements of g · m are vectors of the form x M ( m ) := ddt (cid:12)(cid:12)(cid:12) t =0 exp ( tx ) · m , where x ∈ g and exp : g → G is the group exponential. Thetangent space T m M can be decomposed into a direct sum of four G m -invariant subspaces T m M = T ⊕ T ⊕ N ⊕ N , (3.1)known as the Witt-Artin decomposition, defined as follows:(i) T := ker ( D Φ G ( m )) ∩ g · m = g µ · m .(ii) T := n · m which is a symplectic vector subspace of ( T m M, ω ( m )) .(iii) N is a choice of G m -invariant complement to T in ker ( D Φ G ( m )) . It is a symplecticsubspace of ( T m M, ω ( m )) and is called the symplectic slice . The linear action of G m on N is hamiltonian with momentum map Φ N : N → g ∗ m given by (cid:104) Φ N ( ν ) , x (cid:105) = ω ( x N ( ν ) , ν ) for every ν ∈ N and x ∈ g m .11iv) N is a G m -invariant Lagrangian complement to T in the symplectic orthogonal ( T ⊕ N ) ω ( m ) . There is an isomorphism f : N → m ∗ given by (cid:104) f ( w ) , y (cid:105) = ω ( m ) ( y M ( m ) , w ) for every w ∈ N and y ∈ m .Since N is a G m -invariant subspace, there is a well-defined action of G m on the product G × m ∗ × N given by k · ( g, ρ, ν ) = ( gk − , Ad ∗ k − ρ, k · ν ) . (3.2)This action is free and proper by freeness and properness of the action on the G -factor. Theorbit space Y is thus a smooth manifold whose points are equivalence classes of the form [( g, ρ, ν )] . The group G acts smoothly and properly on Y , by left multiplication on the G -factor. Let m ∗ ε ⊂ m ∗ and ( N ) ε ⊂ N be G m -invariant neighbourhoods of zero in m ∗ and N ,respectively. Then Y ε := G × G m ( m ∗ ε × ( N ) ε ) (3.3)is a neighbourhood of the zero section in Y . It comes with a symplectic form ω Y ε if it is chosensmall enough [OR04] (Proposition . . ). Theorem . Let ( M, ω, G, Φ G ) be a hamiltonian proper G -manifold. Let m ∈ M with momentum µ = Φ G ( m ) and let ( Y ε , ω Y ε ) as in (3.3) . Thenthere is a G -invariant neighbourhood U ⊂ M of m and a G -equivariant symplectomorphism ϕ : ( Y ε , ω Y ε ) → ( U, ω (cid:12)(cid:12) U ) such that ϕ ([ e, , m . We call the triplet ( ϕ, Y ε , U ) a symplectic G -tube at m and we also say that ( Y ε , ω Y ε ) is a symplectic local model for ( U, ω (cid:12)(cid:12) U ) . Moreover, the momentum map Φ G : M → g ∗ can beexpressed in terms of the slice coordinates: Theorem . Let ( M, ω, G, Φ G ) be a hamiltonianproper G -manifold and let ( ϕ, Y ε , U ) be a symplectic G -tube at m ∈ M . Then the G -action on Y ε is hamiltonian with associated momentum map (cid:101) Φ G : Y ε → g ∗ defined by (cid:101) Φ G ([ g, ρ, ν ]) = Ad ∗ g − (Φ G ( m ) + ρ + Φ N ( ν )) . (3.4) If G is connected, (3.4) coincides with Φ G (cid:12)(cid:12) U when pulled back along ϕ − .
4. Symmetry breaking for relative equilibria
In this section, we extend Theorem 2.2 and Corollary 2.5 to the case of relative equilibriawhich is more subtle for two reasons: firstly we must take into account the conservation ofmomentum, and secondly for a non-zero velocity the so-called augmented Hamiltonian nolonger has symmetry G .We start by briefly recalling some standard facts about relative equilibria (see [Mar92] fordetails). Given a hamiltonian proper G -manifold ( M, ω, G, Φ G ) , a relative equilibrium of aHamiltonian h ∈ C ∞ ( M ) G is a pair ( m, ξ ) ∈ M × g such that X h ( m ) = ξ M ( m ) . Equivalently, if ( m, ξ ) is a relative equilibrium of h , then m is a critical point of the augmented Hamiltonian h ξ := h − φ ξG ∈ C ∞ ( M ) G ξ where φ ξG ( m ) := (cid:104) Φ G ( m ) , ξ (cid:105) , which is a G ξ -invariant function which depends linearly on ξ . Astandard fact about relative equilibria is that the velocity ξ and the momentum µ = Φ G ( m ) ξ ∈ g µ . Note that, if the isotropy group G m is non trivial and ( m, ξ ) is a relativeequilibrium of h , then ( m, ξ + η ) is also a relative equilibrium of h , for any η ∈ g m . Moreover if ( m, ξ ) is a relative equilibrium of h then so is ( g · m, Ad g ξ ) for every g ∈ G . In general a relativeequilibrium is said to be non-degenerate if the Hessian D h ξ ( m ) is a non-singular quadraticform, when restricted to the symplectic slice N at m relative to the G -action. However, thisdefinition of non-degeneracy is not enough to guarantee that a relative equilibrium of some h ∈ C ∞ ( M ) G persists under an H -perturbation. For that reason, we need a stronger versionof non-degeneracy. Let H be a closed subgroup of G . The dual of the inclusion of Lie algebras i h : h (cid:44) → g is theprojection i ∗ h : g ∗ → h ∗ and is given by i ∗ h ( µ ) = µ (cid:12)(cid:12) h , which is the restriction of the linear form µ to the Lie subalgebra h . The action of H on M is still both symplectic and Hamiltonian. Amomentum map for this action is given by Φ H = i ∗ h ◦ Φ G : M → h ∗ and is called the inducedmomentum map for the H -action. Proposition . Consider the decomposition of T m M as in (3.1) , and define thesubspace M := { z M ( m ) + w ∈ T ⊕ N | − ad ∗ z µ + f ( w ) ∈ h ◦ } where f denotes the isomorphism between N and m ∗ , and h ◦ is the annihilator of h in g ∗ .Then ker ( D Φ H ( m )) = ker ( D Φ G ( m )) ⊕ M . Proof —
It is clear from the definitions that there is an inclusion of subspaces ker ( D Φ G ( m )) ⊂ ker ( D Φ H ( m )) . (4.1)Let ( ϕ, G × G m ( m ∗ ε × ( N ) ε ) , U ) be a symplectic G -tube at m as in Theorem 3.1. Linearising ϕ − at m yields a linear symplectomorphism T m ϕ − : T ⊕ T ⊕ N ⊕ N → T ϕ − ( m ) ( G × G m ( m ∗ × N )) . For x + y ∈ g m ⊕ m and z ∈ n we have T m ϕ − · (( x + y ) M ( m ) + z M ( m ) + w + ν ) = T ( e, , ρ · ( x + y + z, f ( w ) , ν ) where ρ : G × m ∗ × N → G × G m ( m ∗ × N ) is the orbit map. By definition, the subspace ker ( D Φ H ( m )) consists of the elements (( x + y ) M ( m ) + z M ( m ) + w + ν ) ∈ T ⊕ T ⊕ N ⊕ N satisfying D (Φ H (cid:12)(cid:12) U ◦ ϕ ◦ ρ )( e, , · ( x + y + z, f ( w ) , ν ) = 0 . Equivalently ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Φ H (cid:12)(cid:12) U ◦ ϕ ([(exp( t ( x + y + z )) , tf ( w ) , tν )])= ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 i ∗ h (cid:16) Ad ∗ exp( − t ( x + y + z )) ( µ + tf ( w ) + Φ N ( tν )) + C (cid:17) , C ∈ R = i ∗ h ( − ad ∗ z µ + f ( w )) where the normal form for the momentum map is given by Theorem 3.2. As required − ad ∗ z µ + f ( w ) ∈ h ◦ since the kernel of i ∗ h is equal to h ◦ . Note that we do not need the assumption ofTheorem 3.2 that G is connected because the statement only depends on the differential. (cid:4) .2 Non-degeneracy condition and regularity condition We now state a stronger version of non-degeneracy of a relative equilibrium.
Definition . Let ( M, ω, G, Φ G ) be a hamiltonian proper G -manifold with H ⊂ G be a closedsubgroup, and Φ H : M → h ∗ be the induced momentum map. Setting α := Φ H ( m ) , a relativeequilibrium ( m, ξ ) ∈ M × g of h ∈ C ∞ ( M ) G is said to be α -nondenegerate if D h ξ ( m ) is anon-singular quadratic form on N ⊕ M with M as in Proposition 4.1.Definition 4.2 only depends on α and not on the underlying Witt-Artin decomposition of T m M . If G is non-abelian, the space M might have an non-trivial intersection with g · m .This intersection is the subspace q · m ⊂ g · m where q is an H m -invariant complement to g µ in the “symplectic orthogonal” h ⊥ µ := (cid:110) x ∈ g | x M ( m ) ∈ ( h · m ) ω ( m ) (cid:111) . The non-singularity of D h ξ ( m ) along g · m depends only on that of D φ ξG ( m ) which hassymmetry group G ξ . The condition is a consequence of the following lemma which is provedin Section 5. Lemma . Let ( M, ω, G, Φ G ) be a hamiltonian proper G -manifold. Let m ∈ M with mo-mentum µ = Φ G ( m ) and an element ξ ∈ g µ . If g is semi-simple then the Hessian D φ ξG ( m ) restricted to g · m is singular precisely along ( g ξ + g µ ) · m . Therefore if an equilibrium ( m, ξ ) ∈ M × g of some h ∈ C ∞ ( M ) G with momentum µ =Φ G ( m ) is α -nondegenerate in the sense of Definition 4.2, then g ξ has trivial intersection with q . In Theorem 4.5 we show that a number of orbits of relative equilibria of h persist under H -perturbations. Such relative equilibria must have their velocity ξ in h µ . We assume anadditional regularity assumption g µ ⊂ g ξ (R)This says essentially that µ needs to be more regular than ξ (cf. [Fon18a] Definition . . for more details). In particular if condition (R) is satisfied then the null-space of the Hessianreferred to in Lemma 4.3 above is equal to g ξ · m . Example . In this example we show when condition (R) holds for g = so (4) and subalgebras h isomorphic to so (3) . The Lie algebra g is identified with the set of pairs ( x, a ) ∈ R × R with Lie bracket [( x, a ) , ( y, b )] = ( x × y + a × b, x × b + a × y ) . (4.2)The dual Lie algebra g ∗ consists of pairs ( χ, ρ ) ∈ R × R which satisfy (cid:104) ( χ, ρ ) , ( x, a ) (cid:105) = χ · x + ρ · a. The linearized coadjoint action of g on g ∗ is given byad ∗ ( x,a ) ( χ, ρ ) = ( χ × x + ρ × a, χ × a + ρ × x ) . (4.3) Lie subalgebras isomorphic to so (3) . Elements of so (3) are identified with vectors x ∈ R . Weconsider two inequivalent Lie subalgebras of h ⊂ g isomorphic to so (3) , namely14i) The Lie algebra of rotations in R denoted so (3) r = (cid:8) ( x, ∈ R | x ∈ R (cid:9) with Liebracket [( x, , ( y, x × y, . (ii) The diagonal elements denoted so (3) d = (cid:8)(cid:0) x , x (cid:1) ∈ R | x ∈ R (cid:9) with Lie bracket [( x , x , ( y , y x × y , x × y . Regularity condition.
Given a fixed momentum µ := ( χ, ρ ) ∈ g ∗ , the stabilizer Lie subalgebrais g µ = { ( x, a ) ∈ g | χ × x + ρ × a = 0 and χ × a + ρ × x = 0 } by (4.3). We show below whether condition (R) is satisfied for our two different choices of Liesubalgebra.(i) Let h = so (3) r with inclusion map i h : x ∈ h (cid:55)→ ( x, ∈ g . To compute the dual of thisinclusion i ∗ h : g ∗ → h ∗ , we take ( χ, ρ ) ∈ g ∗ and x ∈ h and we compute (cid:104) i ∗ h ( χ, ρ ) , x (cid:105) = (cid:104) ( χ, ρ ) , i h ( x ) (cid:105) = (cid:104) ( χ, ρ ) , ( x, (cid:105) = χ · x. Then i ∗ h (( χ, ρ )) = χ ∈ h ∗ . The symplectic orthogonal is h ⊥ µ = { ( x, a ) ∈ g | χ × x + ρ × a = 0 } . Since the velocity ξ ∈ h must commute with µ , it has to belong to the subspace h µ = g µ ∩ h . Using equation(4.3), h µ = { ( x, ∈ so (3) r | χ × x = 0 and ρ × x = 0 } . There are three cases to consider:(a) If χ = ρ = 0 then g µ = g and h µ = h . We choose ξ = ( y, ∈ h where y ∈ R isarbitrary. Using (4.2) we get g ξ = { ( λ y, λ y ) ∈ g | λ , λ ∈ R } and clearly (R) does not hold.(b) If χ and ρ are not collinear, h µ = { (0 , } . In this case, the only available velocityis ξ = 0 and thus g ξ = g . In particular (R) holds.(c) If µ = ( χ, ρ ) is such that χ = sρ for some s ∈ R , we choose ξ (cid:54) = 0 of the form ξ := ( λχ, ∈ h µ for some λ ∈ R and thus g ξ = { ( x, a ) ∈ g | x × χ = 0 and a × χ = 0 } . Note that in particular, g ξ ⊂ g µ . To see whether g µ ⊂ g ξ , pick an element ( x, a ) ∈ g µ . By definition, it satisfies x × χ = ρ × a and χ × a = x × ρ. (4.4)Using (4.4) and the fact that χ = sρ we get, x × χ = s ( x × ρ ) = s ( χ × a ) = s ( ρ × a ) = s ( x × χ ) . igure 4: Condition (R) holds as long as µ is awayform the red dashed lines which represent subspacesof codimension three in R . Similarly a × χ = s ( a × ρ ) = s ( χ × x ) = s ( ρ × x ) = s ( a × χ ) . Therefore, ( x, a ) ∈ g ξ as long as s (cid:54) = 1 ; that is, (R) holds for such ξ as long as µ (cid:54) = ( χ, ± χ ) (see Figure 4).(ii) Let h = so (3) d with inclusion map i h : x ∈ h (cid:55)→ (cid:16) x , x (cid:17) ∈ g . To compute the dual of this inclusion i ∗ h : g ∗ → h ∗ , we take ( χ, ρ ) ∈ g ∗ and x ∈ h andwe compute (cid:104) i ∗ h ( χ, ρ ) , x (cid:105) = (cid:104) ( χ, ρ ) , i h ( x ) (cid:105) = (cid:104) ( χ, ρ ) , (cid:16) x , x (cid:17) (cid:105) = χ + ρ · x, Then i ∗ h (( χ, ρ )) = χ + ρ ∈ h ∗ . Set µ := ( χ, ρ ) ∈ g ∗ and α := i ∗ h ( µ ) = χ + ρ ∈ h ∗ . Using Equation (4.3) we get h µ = (cid:110)(cid:16) x , x (cid:17) ∈ so (3) d | α × x = 0 (cid:111) . We thus choose a velocity of the form ξ := ( λα, λα ) ∈ h µ for some λ ∈ R . By (4.2) the stabilizer Lie algebra of ξ is g ξ = { ( x, a ) ∈ g | x × α + a × α = 0 } . (4.5)In particular, g µ ⊂ g ξ and (R) is satisfied for any choice of µ .16 .3 Persistence of relative equilibria We are now ready to state the corresponding version of Theorem 2.2 for relative equilibria.The proof follows the same steps as Theorem 2.2. For that reason some details have beenskipped.
Theorem . Let ( M, ω, G, Φ G ) be a hamiltonian proper G -manifold. Assume that h ∈ C ∞ ( M ) G has a relative equilibrium ( m, ξ ) ∈ M × h with momentum µ = Φ G ( m ) . Let α ∈ h ∗ be the restriction of µ to h . We assume that(i) α is a regular value of Φ H ,(ii) ( m, ξ ) is α -nondegenerate and assumption (R) is satisfied,(iii) H µ ⊂ G µ is co-compact.Then there is a G µ -invariant neighbourhood U ⊂ Φ − H ( α ) of m and a neighbourhood V ⊂ R × h of (0 , ξ ) such that, for each ( λ, η ) ∈ V , there is a function f ηλ ∈ C ∞ ( G µ /G m ) H µ ,depending smoothly on ( λ, η ) ∈ V , whose critical points are in one-to-one correspondence withthose of h ηλ in U .Proof — The proof is similar to that of Theorem 2.2, except that h ξ is not G -invariant, butonly G ξ (or thanks to condition (R) G µ -invariant), and the extra ingredient is to control thedifference between G and G µ -nondegeneracy, and between the level sets of Φ G and Φ H .Let ( m, ξ ) ∈ M × h be an α -nondegenerate relative equilibrium h , where α is the restrictionof the momentum µ = Φ G ( m ) to h . In particular ξ ∈ h µ . Set K := G m and fix a Witt-Artindecomposition (3.1) relative to the G -action. Note that N ⊕ T is a symplectic slice at m relative to the G µ -action on M [Fon18b]. Define Y = G µ × K ( m ∗ × ( N ⊕ T )) By the Symplectic Tube Theorem 3.1 a sufficiently small neighbourhood Y ε ⊂ Y of the zerosection is equipped with a symplectic form ω Y ε and there is a G µ -invariant neighbourhood U ⊂ M of m and a G µ -equivariant symplectomorphism ϕ : ( Y ε , ω Y ε ) −→ (cid:16) U , ω (cid:12)(cid:12) U (cid:17) with ϕ ([ e, m . We define N = { ( ρ, ν + z M ( m )) ∈ m ∗ × ( N ⊕ T ) | − ad ∗ z µ + ρ ∈ h ◦ } . By Proposition 4.1, N is isomorphic to N ⊕ M , a K -vector space complementary to g µ · m in ker ( D Φ H ( m )) . Let N ε ⊂ N be a K -invariant neighbourhood of such that G µ × K N ε ⊂ Y ε .We thus define U ⊂ U by U := ϕ ( G µ × K N ε ) (4.6)which, by Proposition 4.1, is a G µ -invariant neighbourhood of m in Φ − H ( α ) .Let h λ ∈ C ∞ ( M ) H be an H -perturbation of h with augmented Hamiltonian h ξλ . We searchfor critical points of h ξλ in U . Using the local model (4.6) this reads h ξλ : G µ × K N ε → R .
17s in the proof of Theorem 2.2 the critical points of h ξλ in G µ × K N ε lift to those of (cid:101) h ξλ := ρ ∗ h ξλ : G µ × N ε → R where ρ : G µ × N ε → G µ × K N ε is the orbit map. We may thus work with (cid:101) h ξλ instead of h ξλ .We define as usual the (left) action of the direct product G µ × K on G µ × N ε by ( h, k ) · ( g, ν ) = ( hgk − , k · ν ) . As G µ ⊂ G ξ by the (R) assumption, the lift (cid:101) h ξ is G µ × K -invariant whereas (cid:101) h ξλ is only H µ × K -invariant. By α -nondegeneracy of ( m, ξ ) and because ϕ ([ e, m : d (cid:101) h ξ ( e,
0) = 0 and D N (cid:101) h ξ ( e, is non-singular . We use the Implicit Function Theorem and the compactness of H µ \ G µ to get an H µ -invariantsmooth function φ ηλ : G µ → N ε , depending on parameters ( λ, η ) taken in a neighbourhood V ⊂ R × h of (0 , ξ ) , satisfying d N (cid:101) h ηλ ( g, φ ηλ ( g )) = 0 for every g ∈ G µ . For every fixed parameters ( λ, η ) ∈ V , the H µ × K -invariance of (cid:101) h ηλ allows us to define afamily of H µ × K -invariant functions f ηλ by f ηλ ( g ) := (cid:101) h ηλ ( g, φ ηλ ( g )) . Hence the implicit function f ηλ has a critical point at g if and only if (cid:101) h ηλ has a critical pointat ( g, φ ηλ ( g )) ∈ G × N ε . Then for ( λ, η ) ∈ V the critical points of h ηλ in U are in one-to-onecorrespondence with those of the function f ηλ . (cid:4) Corollary . Under the assumptions of Theorem 4.5 thenumber of H µ -orbits of relative equilibria of h with velocity close to ξ , that persist under a small H -perturbation in a neighbourhood of G µ · m in Φ − H ( α ) , is bounded below by Cat H µ ( G µ /G m ) .Proof — We apply Theorem 2.3 to f ηλ ∈ C ∞ ( G µ /G m ) H µ and we obtain that the number of H µ -orbits of critical points of f ηλ is bounded below by Cat H µ ( G µ /G m ) . In other words, aslong as ( λ, η ) ∈ V , the number of H µ -orbits of relative equilibria of h λ with velocity η in aneighbourhood of G µ · m in Φ − H ( α ) is at least Cat H µ ( G µ /G m ) . (cid:4) Example . As a first application, we recover the result of [GMO04] forcompact abelian groups and free actions. Let ( M, ω, T n , Φ T n ) be a hamiltonian T n -manifoldwhere T n is an n -dimensional torus acting freely on M and let T r be a subtorus of T n . Assume h ∈ C ∞ ( M ) T n has an α -nondegenerate relative equilibrium ( m, ξ ) ∈ M × t r with momentum µ = Φ T n ( m ) and where α = µ (cid:12)(cid:12) t r . As T n and T r are abelian, condition (R) always hold.Hence any T r -perturbation h λ with λ small enough has at least Cat T r ( T n ) T r -orbit of relativeequilibria with velocity closed to ξ in a neighbourhood of T n · m in Φ − T r ( α ) . Since T n actsfreely on T r by left multiplication,Cat T r ( T n ) = Cat ( T n / T r ) = Cat ( T n − r ) . Hence Cat T r ( T n ) = ( n − r ) + 1 . xample S ) . As an application of Corollary 4.6, we consider thecase of a spherical pendulum on S , whose Hamiltonian is viewed as a perturbation of the freeHamiltonian on S . Endow R with the standard inner product (cid:104)· , ·(cid:105) and let e , e , e , e bethe standard basis. The phase space is ( T ∗ S , ω, G, Φ G ) where G = SO (4) acts on T ∗ S = (cid:8) ( x, y ) ∈ S × R | (cid:104) x, y (cid:105) = 0 (cid:9) by matrix multiplication A · ( x, y ) = ( Ax, Ay ) . The associated momentum map Φ G : T ∗ S → (cid:86) ( R ) is given by Φ G ( x, y ) = y ∧ x. Let H = SO (3) ⊂ SO (4) be the rotations about the e -axis with Lie algebra h = so (3) r as defined in Example 4.4. The Hamiltonian of the spherical pendulum h λ ( x, y ) = 12 (cid:107) y (cid:107) + λ (cid:104) x, e (cid:105) (4.7)is an H -perturbation of the free Hamiltonian h ( x, y ) = (cid:107) y (cid:107) . By definition, the relativeequilibria of (4.7) are pairs (( x, y ) , ξ ) ∈ T ∗ S × h such that d h λ ( x, y ) = d φ ξH ( x, y ) (4.8)where φ ξH ( x, y ) := − Tr (( y ∧ x ) ξ ) = (cid:104) ξx, y (cid:105) . Solving (4.8) is a straightforward calculation. The result is summarized as follows:
Lemma . With the notation of Example 4.4 we fix ξ = ( w, ∈ h for some w ∈ R anddenote by p V ( x ) the projection of x on V = span ( e , e , e ) . The relative equilibria (( x, y ) , ξ ) ∈ T ∗ S × h of (4.7) satisfy the equations(i) (cid:104) x, e (cid:105) = − λ (cid:107) w (cid:107) − (ii) (cid:107) p V ( x ) (cid:107) = 1 − (cid:104) x, e (cid:105) and w · p V ( x ) = 0 (dot product in R )(iii) y = ξx satisfies p V ( y ) = w × p V ( x ) and (cid:104) y, e (cid:105) = 0 . When λ = 0 (4.7) is the free Hamiltonian h ( x, y ) = (cid:107) y (cid:107) on T ∗ S . The integral curvesof the corresponding hamiltonian vector field project to the great circles on S . We fix ξ =( w, ∈ h with w = (0 , , T . The relative equilibria (( x, y ) , ξ ) of h are such that (cid:104) x, e (cid:105) = 0 and p V ( x ) lies on the unit sphere in the hyperplane orthogonal to the line [ w ] , and y is tangentto this sphere. In particular the pair ( m, ξ ) with m = ( x, y ) = ( e , e ) is a relative equilibriumof h . Its momentum is µ = Φ G ( e , e ) = ξ and its projection on h ∗ is α = w T = (cid:0) (cid:1) .The stabilizer G µ is a copy of SO (3) in SO (4) and the orbit G µ · m is the unit sphere S ⊂ S lying on the hyperplane of equation (cid:104) x, e (cid:105) = 0 .We want to find the relative equilibria (( (cid:101) x, (cid:101) y ) , η ) of the perturbed Hamiltonian (4.7) whichlie on Φ − H ( α ) where Φ H ( (cid:101) x, (cid:101) y ) = ( p V ( (cid:101) x ) × p V ( (cid:101) y )) T is the induced momentum map. Writing η = ( u, ∈ h for some u ∈ R , those relativeequilibria satisfy the equation (cid:107) p V ( (cid:101) x ) (cid:107) u − ( p V ( (cid:101) x ) · u ) p V ( (cid:101) x ) = w (4.9)19ith w = (0 , , T , as fixed earlier. In addition they satisfy the equations of Lemma 4.9,which require in particular that p V ( (cid:101) x ) · u = 0 . Replacing in (4.9) we obtain u = (cid:107) p V ( (cid:101) x ) (cid:107) − w .From Lemma 4.9 we get (cid:104) x, e (cid:105) = − λ (cid:107) p V ( (cid:101) x ) (cid:107) and (cid:107) p V ( (cid:101) x ) (cid:107) + λ (cid:107) p V ( (cid:101) x ) (cid:107) = 1 (4.10)Setting t = (cid:107) p V ( (cid:101) x ) (cid:107) in (4.10), we obtain the equation of an algebraic curve λ t + t − t > . For a fixed λ there is exactly one solution representing the square of the radius r ( λ ) of thesphere on which p V ( (cid:101) x ) lies. This sphere is an H µ -orbit of relative equilibria of (4.7). Since r (0) = 1 , it lies in a neighbourhood of the orbit G µ · m in Φ − H ( α ) . Furthermore η is such that u = r ( λ ) − w and thus η is close to ξ in h . We also see from (4.10) that λ must be chosensmall enough such that λ < (cid:107) u (cid:107) < r ( λ ) − < c. where c is some constant coming from the fact that r ( λ ) is bounded below.We conclude that for λ small enough, h λ has exactly one H µ -orbit of relative equilibria ina neighbourhood of G µ · m in Φ − H ( α ) with velocity close to ξ . For this example, we verify theassumptions of Theorem 4.5. We have G µ = H = SO (3) and the stabilizer G m is an SO (2) ,as it is the subgroup of rotations in SO (4) which preserve both axis e and e . The quotient G µ /G m is thus a unit sphere S and H µ = G µ ∩ H = SO (3) . Furthermore, as µ = ξ , theassumption (R) is satisfied, as well as the other assumptions of Theorem 4.5. As expected, wehave Cat H µ ( G µ /G m ) = Cat SO (3) ( S ) = 1 .
5. Proof of Lemma 4.3
This section is devoted to the proof of Lemma 4.3 where we assume that g is semi-simple.Further details are available in [Fon18a]. For each ξ ∈ g a momentum map Φ G : M → g ∗ defines a smooth function φ ξG : M → R depending linearly on ξφ ξG ( m ) := (cid:104) Φ G ( m ) , ξ (cid:105) . Assume that ( m, ξ ) ∈ M × g is a relative equilibrium of some Hamiltonian h ∈ C ∞ ( M ) G with momentum µ = Φ G ( m ) . By definition of a relative equilibrium, ξ and µ commute i.e.ad ∗ ξ µ = 0 . We would like to describe the space of degeneracy of the Hessian D φ ξG ( m ) alongthe orbit g · m . A straightforward calculation yields D φ ξG ( m ) ( y M ( m ) , x M ( m )) = (cid:104) Φ G ( m ) , [ x, [ y, ξ ]] (cid:105) . (5.1)Set µ = Φ G ( m ) and note that the Jacobi identity of the Lie bracket and the fact thatad ∗ ξ µ = 0 imply that (cid:104) µ, [ x, [ y, ξ ]] (cid:105) = (cid:104) µ, [ y, [ x, ξ ]] (cid:105) , reflecting the symmetric property of theHessian. The degeneracy space of D φ ξG ( m ) along g · m consists of the elements y ∈ g suchthat (cid:104) µ, [ y, [ x, ξ ]] (cid:105) = 0 for all x ∈ g . (5.2)20ince µ and ξ commute, we can fix a maximal commutative Lie algebra t ⊂ g such that ξ ∈ t and µ ∈ t ∗ . We complexify both of them g C = C ⊗ R g and t C = C ⊗ R t with extended Lie bracket [ · , · ] C . After this step the velocity and momentum read ξ = 1 ⊗ R ξ and µ = 1 ⊗ R µ and there respective stabilizer subalgebras are g ξ := { x ∈ g C | [ x, ξ ] C = 0 } and g µ := { x ∈ g C | ad ∗ x µ = 0 } . Consider the Cartan Lie subalgebra h = t C . Since ξ ∈ h and µ ∈ h ∗ , it is clear that h is asubspace of both g ξ and g µ . We thus write g ξ = h ⊕ (cid:77) β ∈ S f g β and g µ = h ⊕ (cid:77) α ∈ D f g α (5.3)for some finite subsets S f and D f of the root space R with the property: α ∈ S f ( resp. D f ) = ⇒ − α ∈ S f ( resp. D f ) . Definition . ξ (resp. µ ) is regular if S f = ∅ (resp. D f = ∅ ).Since g C is semi-simple, the Killing form κ induces an isomorphism κ (cid:93) : h ∗ → h . Let t µ ∈ h be the image of µ by this isomorphism and let O t µ be the adjoint orbit of t µ . There is anidentification T t µ O t µ = (cid:88) α ∈R\ D f g α . The problem stated in (5.2), after complexification of the Lie algebra g , reduces to find all the y ∈ g C satisfying κ ([ y ∗ , t µ ] C , [ x, ξ ] C ) = 0 for all x ∈ g C . (5.4)Let { H , . . . , H k } ∪ { X α | α ∈ R} be a Weyl-Chevalley basis of g C , where the H i ’s form a basisof h . Let y ∈ g C be an arbitrary element and let y ∗ = − ¯ y . With respect to the Weyl-Chevalleybasis, this element is expressed as y ∗ = k (cid:88) i =1 a i H i + (cid:88) α ∈R µ α X α for some unique a i , µ α ∈ C . (5.5)Hence [ y ∗ , t µ ] C = [ k (cid:88) i =1 a i H i + (cid:88) α ∈R µ α X α , t µ ] C = (cid:88) α ∈R µ α [ X α , t µ ] C as t µ ∈ h = − (cid:88) α ∈R µ α α ( t µ ) X α = − (cid:88) α ∈R\ D f µ α α ( t µ ) X α [ y ∗ , t µ ] C ∈ T t µ O t µ . Similarly (5.3) allows us to write an element [ x, ξ ] C ∈ T ξ O ξ as [ x, ξ ] C = (cid:88) β ∈R\ S f λ β X β with λ β ∈ C . Solving (5.4) is equivalent to solve (cid:88) α ∈R\ D f (cid:88) β ∈R\ S f µ α λ β α ( t µ ) κ ( X α , X β ) = 0 for any λ β ∈ C . Using the fact that the g α ’s appearing in the root decomposition are mutually orthogonal withrespect to κ (except for those corresponding to the same root with opposite sign), we get (cid:88) α ∈R\ D f (cid:88) β ∈R\ S f µ α λ β α ( t µ ) κ ( X α , X β )= (cid:88) α,β ∈R\ ( D f ∪ S f ) µ α λ β α ( t µ ) κ ( X α , X β )= (cid:88) α ∈R\ ( D f ∪ S f ) µ α λ α α ( t µ ) κ ( X α , X α )+ (cid:88) α ∈R\ ( D f ∪ S f ) µ α λ − α α ( t µ ) κ ( X α , X − α )= (cid:88) α ∈R\ ( D f ∪ S f ) µ α α ( t µ ) ( λ α κ ( X α , X α ) + λ − α κ ( X α , X − α )) . This is true for any λ α ∈ C if and only if µ α = 0 for all α ∈ R \ ( D f ∪ S f ) as such roots satisfy α ( t µ ) (cid:54) = 0 and both κ ( X α , X α ) and κ ( X α , X − α ) do not vanish. We conclude that y ∈ g C fulfils(5.4) for all x ∈ g C if and only if y ∗ decomposes as y ∗ = k (cid:88) i =1 a i H i + (cid:88) α ∈ D f ∪ S f µ α X α . (5.6)Therefore, y ∗ ∈ h ⊕ (cid:77) α ∈ D f ∪ S f g α = g ξ + g µ . In particular this shows that the degeneracy set of the Hessian D Φ G ( m ) along g · m belongsto g ξ + g µ , by considering only the elements y ∈ g C which are real. This proves the lemmabecause the other inclusion is clear. 22 eferences [ACZE87] A. Ambrosetti, V. Coti Zelati, and I. Ekeland. Symmetry breaking in Hamilto-nian systems. J. Differential Equations , 67(2):165–184, 1987.[AB83] M. F. Atiyah, and R. Bott. The Yang-Mills equations over Riemann surfaces.
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