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Bifurcations of relative equilibria near zero momentum in
Hamiltonian systems with spherical symmetry
James Montaldi
School of Mathematics, University of Manchester,
Manchester M13 9PL, UK
November, 2013
Abstract
For Hamiltonian systems with spherical symmetry there is a marked difference between zero and
non-zero momentum values, and amongst all relative equilibria with zero momentum there is a
marked difference between those of zero and those of non-zero angular velocity. We use tech-
niques from singularity theory to study the family of relative equilibria arising as a symmetric
Hamiltonian having a generic zero-momentum zero-velocity relative equilibrium (so a group or-
bit of equilibria with momentum equal to zero) is perturbed so that the zero-momentum relative
equilibrium no longer has zero velocity. We also analyze the stability of these nearby relative equi-
libria, and consider an application to satellites controlled by means of rotors.
MSC 2010: 70H33, 58F14, 37J20
Keywords: momentum map, symplectic reduction, bifurcations, SO(3) symmetry, relative equilibria
Contents
Introduction 2
1 Reduction and slice coordinates 3
2 The family of relative equilibria 6
Study of the ‘universal’ family G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
The energy-momentum discriminant of G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Stability of the relative equilibria 13
Zero momentum, non-zero velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Zero momentum, zero velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Example: rigid body with rotors 18
The free system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
A controlled version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Singularity theory and deformations 20
2J. MON TA LD I
Introduction
Analogous to the fact that in generic Hamiltonian systems equilibrium points form a set of isolated
points, in generic Hamiltonian systems with symmetry, for each value of the momentum the rel-
ative equilibria are isolated. It is therefore reasonable to parametrize the set of relative equilibria
by the momentum value, at least locally. As the momentum value varies, one would then expect
to see bifurcations occur, and many of these have similar descriptions to bifurcations occurring
at equilibria in generic (non-symmetric) Hamiltonian systems, such as saddle-node, pitchfork and
Hamiltonian-Hopf bifurcations (see [4] for a review). However there is one class of transition that is
due to the ‘geometry of reduction’ and which occurs as a result of the momentum passing through
a non-regular value in the dual of the Lie algebra, which for the group SO(3) means passing through
0. This type of geometric bifurcation was first investigated in [15] in the case where the angular ve-
locity is non-zero even though the angular momentum vanishes. This was extended in [16] to zero
angular velocity, where there is also an application to the dynamics of molecules.
In this paper we describe these geometric transitions in more detail for the symmetry group
SO(3). The results also apply to other compact Lie groups, where the momentum value passes
through a generic point of a reflection hyperplane in the Cartan subalgebra, but not to more de-
generate points (see Remark 1.3). There are two cases to consider, the first is the ‘generic’ one,
where the velocity at the zero-momentum relative equilibrium is non-zero (a transverse relative
equilibrium in the terminology of Patrick and Roberts [22]) and in this case the set of relative equi-
libria forms a smooth curve in the orbit space, as shown by Patrick [21], and the curve can be natu-
rally parametrized by the momentum. The other case is where the relative equilibrium in question
consists of equilibria. Although non-generic in the universe of all symmetric Hamiltonian systems,
this is the situation in systems governed by kinetic and potential energies. In this case the set of rel-
ative equilibria generically forms three smooth curves in the orbit space, as is familiar from Euler’s
equations for the rigid body.
The question we address here is how the two are related: start with a relative equilibrium p
with zero momentum and zero velocity (that is, a zero-momentum equilibrium), and perturb the
Hamiltonian so that the zero momentum relative equilibrium no longer has zero velocity. How
does the set of relative equilibria change? We find in particular that in the class of all Hamiltonian
systems with SO(3) symmetry, the zero-momentum equilibrium is of codimension 3: it would only
be seen generically in a 3-parameter family of such systems.
The most familiar example of an SO(3)-invariant system is the rigid bod, with Euler’s equa-
tions mentioned above, where the reduced picture (in (T∗SO(3))/SO(3) ≃so(3)∗≃R3) of the set
of relative equilibria consists of three lines through the origin corresponding to the three principal
axes of the body, and it follows from results in [16] that this persists when the rigid body motion is
coupled to shape deformations. Now add terms with the effect that the zero momentum relative
equilibrium is no longer an equilibrium. For most deformations, the three lines deform to three
non-intersecting curves as shown in Figure 2.2(vi), and the branches ‘reconnect’ in different ways
according to the specific deformation. This is analogous to how the two lines in the plane with
equation x y =0 break up and reconnect to form the two branches of a hyperbola with equation
x y =ε, and which half-branch connects to which depends on the sign of ε.
In the rigid body, it is well known that two of the branches are stable (Lyapunov stable) and one
is unstable (even linearly unstable). When coupled with shape oscillations, one of the Lyapunov
stable branches becomes linearly stable (elliptic) but not necessarily Lyapunov stable. The stability
type can be followed in the deformation of the Hamiltonian, and we show where the transitions of
stability occur in the deformations; one transition (between Lyapunov stable and elliptic) occurs
BIFURCATIO NS N EAR Z ERO MO MEN TUM 3
at the point of zero momentum, and the others occur at points on the other branches.
The paper is organized as follows: in Section 1we outline the approach we use for calculat-
ing relative equilibria based on the energy-Casimir method and the splitting lemma; it is the same
method used in [15] and other papers since. In Section 2we state Theorem 2.1 which uses singular-
ity theory to reduce the calculations of the geometry of the family of relative equilibria for a general
family Hof Hamiltonians, to those of a particularly simple family G, and we find the relative equi-
libria for that family. In Section 3we study the stabilities of the bifurcating relative equilibria, and
in Section 4we consider an example of a rigid body (such as a satellite) equipped with three rotors,
one parallel to each of the principal axes of inertia to find the family of relative equilibria when the
rotors are either given fixed momenta or fixed speeds of rotation.
The paper concludes with Section 5on singularity theory; this begins with a description of
Damon’s KV-equivalence, which is the singularity theoretic equivalence required for the proof of
Theorem 2.1, and then finishes with the proof of that theorem.
Acknowledgements I would like to thank J.E. Marsden and P.S. Krishnaprasad for suggesting the
example of the system of a rigid body with rotors, which is discussed in Section 4. This work was
completed during a stay at the Centre Interfacultaire Bernoulli (EPFL, Lausanne) and I would like
to thank Tudor Ratiu and the staff of the centre for organizing such a great environment, and for
the financial support during my stay.
1 Reduction and slice coordinates
Let (P,ω) be a symplectic manifold with a Hamiltonian action of G=SO(3), which throughout
we assume to be a free action. The momentum map is denoted J:P→g∗, which without loss of
generality can be assumed to be equivariant with respect to the coadjoint action on g∗, [15,18].
Since the action is free, Jis a submersion. Given an element ξ∈g=so(3) (the Lie algebra), we write
ξPfor the associated vector field on P. Finally, let H:P→Rbe a smooth G-invariant function, the
Hamiltonian.
Throughout this paper we assume we are given a relative equilibrium peof this system, with
J(pe)=0. That is, at pethere is an element ξ∈g=so(3) for which the Hamiltonian vector field
at pecoincides with ξP(pe). This is equivalent to the group orbit G·pebeing invariant under the
Hamiltonian dynamics. See for example [13] or [4] for details.
Since we are interested in existence and bifurcations of relative equilibria near pe, we describe
the local normal form for Hamiltonian actions near such a point, and then we will use the normal
from then on.
Since J(pe)=0 one has by equivariance that J(g·pe)=0 and hence dJ(ξP(pe)) =0 (for all ξ∈g).
It follows that the tangent space to the group orbit g·pe⊂ker dJ(pe). Let Sbe a slice to the group
orbit G·peat peinside the submanifold J−1(0). It turns out that the pull-back of the symplectic
form to Sis non-degenerate so that Sis symplectic. Since the action is free, the normal form of
Marle-Guilleimn-Sternberg states that there is a G-invariant neighbourhood Uof pewhich is G-
symplectomorphic to an invariant neighbourhood Uof the point (e,0, 0) in the symplectic space
Ywith momentum map JY:Y→g∗given by,
Y=G×g∗×S,
JY(g,ρ,v)=Coadgµ.(1.1)
4J. MON TA LD I
The G-action on Yis simply g′·(g,ρ,v)=(g′g,ρ,v). Since a neighbourhood of pein Pis diffeo-
morphic to U⊂Y, the Hamiltonian Hon Pdefines a Hamiltonian on U, which we also denote by
H. This material is standard, and can be found for instance in the book of Ortega and Ratiu [18].
Here Coad is the coadjoint action of G, and is defined by 〈Coadgµ,ξ〉=〈µ, Adg−1ξ〉, and Coadg
is often written Ad∗
g−1. For G=SO(3), if we consider µ∈R3as a column vector, then Coadgµ=gµ,
where the latter is just matrix multiplication.
In practice Scan often be interpreted as the phase space associated to ‘shape space’, so corre-
sponding to vibrational motions.
We now proceed to pass to the quotient by the free group action, obtaining
P/G≃g∗×S(1.2)
and the orbit momentum map is denoted j:g∗×S→Rand is independent of s∈S, just as the
momentum map itself is. For SO(3), j(µ,s)= kµk2for the natural coadjoint-invariant norm on g∗.
The reduced space Pµ⊂P/Gis then
Pµ=j−1(kµk2)=Oµ×S,
where Oµ⊂g∗is the coadjoint orbit through µ, which is the 2-sphere containing µif µ6= 0 and
reduces to the origin when µ=0.
Energy-Casimir method Now let Hbe a G-invariant smooth Hamiltonian on P. It descends to a
smooth function on the orbit space H:P/G→R. Write Hµfor the restriction of Hto the reduced
space Pµ; this is called the reduced Hamiltonian on Pµ. Since H(µ,s)=H(g,µ,s) (which is by
hypothesis independent of g), from now on we abuse notation and do not distinguish Hfrom H.
Relative equilibria of the Hamiltonian system are solutions of the Lagrange multiplier problem
dH−ξdJon P; moreover the Lagrange multiplier ξ∈(g∗)∗≃gcan be interpreted as the angular ve-
locity of the relative equilibrium. Equivalently, they are critical points of the reduced Hamiltonian
Hµ, for the appropriate value of µ. See for example [13] for details.
At points where µ6= 0, jis nonsingular so the critical points of Hµare solutions of the reduced
Lagrange multiplier problem
dH−λdj=0 (1.3)
for some λ∈R. Since j(µ,s)= kJ(g,µ,s)k2it follows that dj=2J·dJand comparing the two Lagrange
multiplier equations one finds that ξand λare related by ξ=2λµ, whenever µ6= 0. Note that as
µ→0 one may have λ→ ∞ so allowing ξ6= 0 with µ=0.
On the other hand, at points where µ=0, the restriction H0of Hto P0={0} ×Shas a critical
point wherever dsH=0.
Definition 1.1 A relative equilibrium at ¯
p∈Pµis said to be non-degenerate if the hessian d2Hµ(¯
p)
is non-degenerate.
We will be interested in the family of relative equilibria in a neighbourhood of a non-degenerate
relative equilibrium with zero momentum. Under this non-degeneracy assumption, it follows
from the implicit function theorem that in a neighbourhood of ¯
pe=(0, se) in P/Gwe can solve
the equation
dsH(µ,s)=0 (1.4)
uniquely for s=s(µ). In other words dsH(µ,s(µ)) ≡0, and these are the only zeros of dsH(µ,s) in a
neighbourhood of (0, se).
BIFURCATIO NS N EAR Z ERO MO MEN TUM 5
Now define the function,
h:g∗−→ R
µ7−→ H(µ,s(µ)). (1.5)
In fact this his only defined in a neighbourhood of the origin in g∗, but to save on notation we will
ignore that here and continue to write h:g∗→R.
Proposition 1.2 Assume µ6= 0. The following are equivalent:
1. x =(µ,s(µ)) ∈P/G is a relative equilibrium of the Hamiltonian system,
2. the map (h,j)is singular at µ,
3. dh(µ)−λdj(µ)=0for some λ∈R
Note that at µ=0, (1) and (2) are equivalent, while (3) might not be. Furthermore, at any
relative equilibrium, the differential dh(µ)∈(g∗)∗≃g, and can be identified with the velocity of
the relative equilibrium (which is also an element of g). Details are in [15].
PROO F: At x=(µ,s(µ)) one has dH=(dµH, 0) =(dh,0), so (1.3) is satisfied if and only if dh=λdj,
which is equivalent to (h,j) being singular at xsince jis non-singular when µ6= 0. ❒
Applications of this approach can be found in [16] (to relative equilibria of molecules) and in
[12] (to relative equilibria of point vortices).
Parametrized version We will be interested in a parametrized family Huof G-invariant Hamilto-
nians, with parameter u∈U, where Uis an open subset of Rdfor some d. Write H(z;u)=Hu(z)
for such a family. We assume His a smooth G-invariant function on P×U, where G=SO(3) acts
trivially on U. Assume 0 ∈Uand H0has a non-degenerate relative equilibrium at pe=(0, se).
The arguments of the previous paragraph can be extended to a parametrized family with no
difficulty. For example the map sobtained from solving (1.4) is now a map s:g∗×U→S, and one
defines in the same way a smooth family of functions,
h:g∗×U−→ R
(µ,u)7−→ H(µ,s(µ,u),u). (1.6)
And as in Proposition 1.2, the map (hu,j) : g∗→R2is singular at µ6= 0 if and only if the point
(µ,s(µ,u)) is a relative equilibrium of Hu.
Notice that there are two types of parameter in this problem: firstly for a given Hamiltonian
H=H(g,µ,s) (which is independent of g) there is a family of relative equilibria which is essen-
tially parametrized by µ∈g∗(an internal parameter), and secondly we consider a family of such
Hamiltonians, parametrized by an external parameter u∈U. The family of relative equilibria
parametrized by µ(i.e. those of Hµ) will then vary from one value of uto another, and the aim
of this paper is to study precisely how a particular type of family varies with an external parameter.
The particular type of family in question being one with a generic relative equilibrium with zero
momentum and zero velocity deforming to one with non-zero velocity.
Remark 1.3 The analysis in the paper is also valid for systems with compact symmetry group of
rank greater than 1 (the analysis for SU(2) is identical to the one for SO(3)), provided the action
6J. MON TA LD I
is free. The argument is briefly as follows. Let Gbe a compact Lie group of rank ℓsay, and p∈P
is such that J(p)=α∈g∗then one can find a symplectic cross section R(see [8] or [17]) which
reduces the original system to a system invariant under Gα. Let Zα✁Gαbe its centre, which is a
(submaximal) torus of dimension rsay. By reducing by Zαone obtains a system invariant under
K:=Gα/Zα, with JK(p)=0 and we have rk(K)=ℓ−r. In particular, if αis a generic point of one of
the reflection hyperplanes of the Weyl group action then r=ℓ−1, so Kis a group of rank 1, which
means it is isomorphic to either SO(3) or SU(2).
Finding a description of the geometry of the set of relative equilibria in an analogous family
near points deeper in the walls of the Weyl chamber (points fixed by Weyl group larger than Z2) re-
mains an open problem. The set near zero momentum with generic (regular) velocity is described
in [15].
2 The family of relative equilibria
The main aim of this paper is to determine the behaviour of the family of relative equilibria in the
neighbourhood of a non-transverse (relative) equilibrium when the Hamiltonian is deformed, so
making it transverse in the sense of Patrick and Roberts [22].
For the organizing centre of our family, we consider an invariant Hamiltonian which has an
equilibrium pewith zero momentum value J(pe)=0, and we assume the equilibrium is non-
degenerate in the sense of Definition 1.1. Since this central relative equilibrium is in fact an equi-
librium, it follows that dH0(0) =0. So after rotating the axes if necessary, the Taylor series of H0
begins,
H0(x,y,z)=ax2+by 2+c z2+O(3),
where µ=(x,y,z), and O(3) represents terms of order 3. The genericity assumption we make
throughout is that the coefficients a,b,care distinct, and we will assume a>b>c.
The set R0=R(H0) of relative equilibria near (0, 0) ∈g∗×Scoincides with the set of critical
points of (H0,j) : g∗→R2. That is,
R0=©µ∈g∗|rkF(µ)≤1ª,
where F(µ) is the Jacobian matrix at µof (H0,j), and rkF(µ) is the rank of that matrix.
The aim now is to study the geometry of the set of relative equilibria of any deformation of
H0. Consider any family of SO(3)-invariant Hamiltonian systems containing H0as above, and let
Hbe the resulting deformation of H0parametrized by u∈U, and for each u∈Uwrite hufor the
corresponding reduced Hamiltonian on g∗as constructed above—see (1.5).
To study the family R(H) of relative equilibria of any given family H, we describe a ‘universal’
3-parameter family Gof reduced Hamiltonians Gα, with α∈R3, based on a reduced Hamiltonian
G0which is just the quadratic part of the given H0. Using singularity theory techniques we show
that the family R(H) of relative equilibria for the given deformation His the inverse image under a
smooth map ϕof the family R(G) of relative equilibria of G, and then in the next section we study
the geometry of this universal family.
Thus, given H0(µ)=ax2+b y 2+c z2+O(3) as above, with a>b>c, we define
G0(µ)=ax2+b y2+c z 2
where µ=(x,y,z), and a deformation
G(x,y,z;α,β,γ)=G0(x,y,z)+αx+βy+γz; (2.1)
BIFURCATIO NS N EAR Z ERO MO MEN TUM 7
for brevity we write α=(α,β,γ) for the parameter, so we have G(µ;α).
Theorem 2.1 Let H0be an SO(3)-invariant Hamiltonian with zero linear part as above (in partic-
ular, a,b,c distinct), and let Hbe an invariant deformation of H0with parameter space U . With the
construction above, defining Gfrom H0, there exists a neighbourhood U ′of 0in U, and a smooth
map
Φ:g∗×U′−→ g∗×R3
(µ,u)7−→ (Φ1(µ,u), ϕ(u))
with ϕ(0) =0, such that (µ,u)∈R(H)if and only if Φ(µ,u)∈R(G).
In particular, if for each u ∈U′, we define Φu:g∗→g∗, by Φu(µ) :=Φ1(µ,u)then Φuis a diffeo-
morphism which identifies the set R(Hu)with the set R(Gϕ(u)).
In other words, the family Gprovides in some sense a versal deformation of H0. The precise
sense is with respect to the KV-equivalence of the map F=d(h,j), where Vis the set of 2 ×3
matrices of rank at most 1. A description of this equivalence and the proof of the theorem are
given in Section 5.
The structure of the set R(H) is therefore derived from that of R(G), and the geometry of the
latter is described in the remainder of this section. The stabilities of the relative equilibria of the
Huare discussed in Section 3.
Study of the ‘universal’ family G
Let G(x,y,z) be as given in (2.1), with a>b>cdistinct, as in the theorem above. The relative
equilibria R(G) for this family occur at points where (Gα,j) is singular, by Proposition 1.2, where
α=(α,β,γ). Then
R(G)=©(µ,α)∈R6|rk Fα(µ)≤1ª.
where Fαis the Jacobian matrix of (Gα,j):
Fα(x,y,z)=µ2ax +α2by +β2c z +γ
x y z ¶. (2.2)
Thus R(G) is given by the vanishing of the three minors of the matrix, so by the three equations
2(b−c)yz +βz−γy=0
2(c−a)zx +γx−αz=0
2(a−b)x y +αy−βx=0.
If these three minors are denoted A,Band Crespectively, then there is an algebraic relation, namely
x A +y B +zC =0 (as there is between the minors of any matrix).
When α=0, the equations become x y =y z =zx =0, whose solutions form the three coordi-
nate axes, see Figure 2.2 (i) (the colours in the figure refer to stability of different branches, which
is discussed in Section 3).
For each α=(α,β,γ)∈R3, we have the smooth map Fα:R3→Mat (2, 3) (the space of 2 ×3
matrices). Notice that the origin in R6is the only point (x,α) where F(x,α) is a matrix of rank less
than 1. Let V⊂Mat (2,3) consist of those matrices of rank at most 1, and let V◦denote its relative
8J. MON TA LD I
∆2
γ
∆1
β
α
Figure 2.1: The discriminant ∆in parameter space is the union of three planes
interior; that is, the set of matrices of rank equal to 1. So Vis the union of V◦and the zero matrix.
The set of relative equilibria for Gαis therefore
Rα=F−1
α(V).
Now, V◦is a (locally closed) submanifold of Mat (2, 3) of dimension 4 and codimension 2; in-
deed the Lie group GL(2) ×GL(3) acts by change of bases on Mat (2,3) and has three orbits cor-
responding to the rank of the matrix: the origin, V◦and Mat (2, 3) \ V. If we can show that For
Fαis transverse to V, then it follows that away from the origin R(G) is a submanifold of R6, or
respectively R(Gα) is a submanifold of g∗≃R3, in both cases of codimension 2.
Lemma 2.2 1. The map F :R6→Mat(2,3) is an invertible linear map and hence transverse to
V ; consequently R(G)is smooth (of dimension 4) except at the origin.
2. For α=(α,β,γ)∈R3, the map Fα:R3→Mat (2, 3) is transverse to V if and only if α,βand γ
are all nonzero.
This lemma tells us that the discriminant ∆=∆(G) of the family Gis the subset of the unfolding
space R3where αβγ =0, which is the union of the three coordinate planes, see Figure 2.1, and for
α6∈ ∆the set Rαof relative equilbria is a smooth 1-dimensional submanifold of g∗; that is, it is a
union of smooth curves.
PROO F: (1) This is obvious from (2.2) above.
(2) Let ˆ
x=(ˆ
x,ˆ
y,ˆ
z)∈TµR3≃R3. Then
dFα(ˆ
x)=µaˆ
x b ˆ
y c ˆ
z
ˆ
xˆ
yˆ
z¶.
Now, if A∈V◦then rk A=1 and the tangent space to Vat Ais
TAV={B∈Mat (2,3) |qB K =0}
where q∈R3is a non-zero row-vector with qA=0, and Kis a 3×2 matrix whose image spans ker A.
To see this, let A(t) be a smooth curve in Vwith tangent vector Bat A=A(0), and let q(t) and
K(t) be the corresponding row-vector and matrix. Differentiating the condition q(t)A(t)=0 gives
BIFURCATIO NS N EAR Z ERO MO MEN TUM 9
˙
qA+qB=0 and multiplying on the right by Kimplies qBK =0; this shows that TAVis a subset of
the Bwith qBK =0, and a dimension count shows they are in fact equal.
Because dFαis injective, in order to show Fαis transverse to Vone needs to find two inde-
pendent vectors ˆ
x1,ˆ
x2such that Bj:=dFα(ˆ
xj)6∈ TAV, (j=1, 2). The choice of the ˆ
xjdepends on
the point µin question. A series of straightforward calculations in different cases (x6= 0, α6= 0,. . . )
show that indeed for α6∈ ∆,Fαis transverse to Vand Fα(µ)6= 0. If on the other hand α=0 but
βγ 6= 0 then Fαfails to be transverse to Vat the point
µ=µ0, β
2(a−b),γ
2(a−c)¶.
which is therefore the singular point of F−1
α(V)—it is in fact a crossing of two components of the
curve. A similar scenario occurs if β=0, αγ 6= 0 or if γ=0, αβ 6= 0.
Furthermore, if α=β=0 but γ6= 0 then there are two points where transversality fails:
µ=µ0, 0, γ
2(b−c)¶,µ=µ0, 0, γ
2(a−c)¶. (2.3)
Similarly if β=γ=0 or α=γ=0 there are two singular points, given by analogous expressions. ❒
With ∆the discriminant, let ∆0={0}, ∆1be the set consisting of the points where precisely two
of the planes intersect (ie, the union of the three axes without ∆0), and ∆2the remaining points of
∆. Then,
∆=∆0∪∆1∪∆2,
this being a disjoint union. ∆1has 6 connected components, while ∆2has 12. The geometry of
the singular set of the deformation (hα,j) (that is, the set of relative equilibria) depends on which
stratum αis in, as shown in the proof above. The descriptions are as follows (refer to Figure 2.2).
• For α6∈ ∆(so away from the discriminant: i.e. a generic deformation) the set Rαof relative
equilibria is formed of three smooth disjoint curves; see Figure 2.2 (vi). (The almost-corners
in the figure are artefacts of the projection.) Before deformation when α=0, there are 3 lines
through the origin or 6 ‘rays’, and on deformation the rays are reconnected in such a way that
opposite rays are no longer connected together. There are 8 possible ways this can be done,
corresponding to the 8 components in the complement of the discriminant (the 8 octants).
The origin must always lie on one of the 3 curves and is marked by the dot in the figure.
For such a Hamiltonian, for small values of µthere are precisely two relative equilibria, both
lying on the component passing through 0, and as kµkis increased there are two saddle-
centre bifurcations, each creating a pair of relative equilibria. These bifurcations occur at
the points closest to the origin on each of the other two components (i.e., where the sphere
Oµof the appropriate radius first touches the curve as |µ|increases from 0). See Fig. 2.3 for
an illustration of the different types of bifurcation.
• For α∈∆2—a generic point of the discriminant— two of the branches of Rαintersect at
a single singular point; see Figure 2.2 (v). The system determined by such a deformation
undergoes a pitchfork bifurcation at this singular point and a saddle-centre bifurcation on
the other branch as kµkis increased f rom 0. See Figures 2.2 (ii)–(iv).
• For α∈∆1there are two crossings in Rα: the curve through µ=0 meets both the other
curves (at different points). There are therefore two pitchfork bifurcations in this system as
10 J. MON TA LD I
b
(i) α=(0,0,0)
b
(ii) α=(α,0,0)
b
(iii) α=(0,0,γ)
b
(iv) α=(0,β,0)
b
(v) α=0, γ>β>0
b
(vi) γ>β>α>0
Figure 2.2: Deformations of R. The large dot on one of the curves in each diagram
represents the origin µ=0. (i) corresponds to α=0, (ii), (iii) and (iv) to αin three different
components of ∆1, (v) to αin one of the components of ∆2and (vi) to α6∈ ∆. The colours
of the branches refer to their stability: red for Lyapunov stable, green for elliptic and brown
for linearly unstable, see Sec. 3.
kµkis increased from 0. Since a,b,care distinct it follows from equation (2.3) that the two
bifurcations occur at different values of kµk.
Theorem 2.1 shows that the family or relative equilibria for His diffeomorphic to that from
G(or more generally can be induced from it by a map ϕon parameters). However, this does not
imply that where one has, for example, a sub-critical pitchfork so does the other—in fact it is not
straightforward to match the stability types. This is however proved in Theorem 3.3, so the conclu-
sions above are relevant to a wider class of Hand not just to G.
The energy-momentum discriminant of G
The map (hα,j) we are considering is the reduced energy-momentum map, and its singular set is
the set Rαof relative equilibria. Its discriminant is the image (hα,j)(Rα)⊂R2, and the fibres of
the reduced energy-momentum map are diffeomorphic within each connected component of the
complement of this discriminant. It is therefore useful to know what this discriminant looks like.
It is important to be aware that the KV-equivalence we use to show that a general family can be
reduced to Gby a change of coordinates, does not respect the discriminant; that is, two maps which
are equivalent in our sense do not necessarily have diffeomorphic discriminants. Bearing that in
BIFURCATIO NS N EAR Z ERO MO MEN TUM 11
Saddle-centre bifurcation
Subcritical pitchfork bifurcation Supercritical pitchfork bifurcation
Figure 2.3: Illustration of the three bifurcation types that occur
mind, the discriminants of the universal family Gare shown in Figure 2.4, and a brief description
of how they might appear for other families is given in Remark 2.3 (1) below.
For the universal family G, the discriminant of (j,G0) consists of 3 rays as in Figure 2.4 (i). On a
perturbation along the singular set ∆1of the unfolding discriminant it may look like (ii), (iii) or (iv)
(depending on the component); for a generic point of the unfolding discriminant the perturbation
may be of the form in (v), while a generic deformation is shown in (vi).
Remarks 2.3 (1) In general, if H0has higher order terms, the straight lines in Figure 2.4 may not
be straight. Moreover, in the diagrams these lines are doubly covered by the energy-momentum
map, in that each point on one of the lines corresponds to 2 distinct relative equilibria. If H0is
not an even function they would be expected to ‘open up’ and become cusps of some order. For
example, comparing Figures 2.2(i) and 2.4(i) (both with α=0) each line in the first maps 2–1 to the
corresponding ray in the second, and for a Hamiltonian which is not even, the rays in the latter
figure would become cusps.
(2) It would be natural to attempt a classification of the reduced energy-momentum maps via left-
right equivalence (A-equivalence), which consists of equivalence via diffeomorphisms in source
and target. The diffeomorphism in the source would then relate the singular sets of the two maps
and the one in the target would relate the discriminants. However, the map (j,G0) is of infinite
codimension with respect to this equivalence, because the map from the singular set Rto the
discriminant is not 1–1 (it is in fact 2–1 away from the origin as pointed out above). If we were
to add appropriate cubic terms to H0or G0, the A-codimension of the map would become finite
(though considerably higher than 3) but calculations would be harder, and would also not be valid
in settings where the higher order terms are absent, as in the example of §4.
(3) There has been no mention of what happens to the equilibrium in the family. Since an equi-
librium corresponds to a critical point of the Hamiltonian, and for α=0 the Hamiltonian (or the
reduced function h) has a non-degenerate critical point at the origin, under any perturbation this
non-degenerate critical point must persist. Though it will no longer be at the origin, it will nec-
essarily lie on one of the branches of the set of relative equilibria. However, the KV-equivalence
we use does not respect critical points of h(only critical points of hrelative to j) so we cannot use
the universal family Gto determine its location. The (unique) equilibrium will lie at some general
point of one of the curves, and not, as might first be thought, at a bifurcation point. Calculations
suggest that which curve it is on depends on the signs of a,b,c, and that in the most physical case
12 J. MON TA LD I
(i) α=0
b b
(ii) α=(α,0,0) ∈∆1
b
(ii’) α=(α,0,0) ∈∆1
b
(iii) α=(0,0,γ)∈∆1
b
(iii’) α=(0,0,γ)∈∆1
b
(iv) α=(0,β,0) ∈∆1
b
(v) α=(0,β,γ)∈∆2
b
(vi) α=(α,β,γ)6∈ ∆
Figure 2.4: Energy-momentum discriminants for the family G: the magnitude of the
momentum increases to the right in each diagram. The curves are the images of the sets of
relative equilibria for the different values of α. (ii′) is an expanded view of (ii), and similarly
(iii’) of (iii). Caveat : see Remark 2.3(1). The colours refer to the stability of the relative
equilibria, as in Figure 2.2—see also § 3.
BIFURCATIO NS N EAR Z ERO MO MEN TUM 13
where all are positive, it lies on the branch that contains the point with zero momentum, and more-
over it lies on the Lyapunov stable side of the zero momentum point. But this has not been proved
in general.
3 Stability of the relative equilibria
There are well-developed methods for proving the Lyapunov stability of relative equilibria, based
on Dirichlet’s criterion for the stability of an equilibrium. Since we are assuming the original ac-
tion of SO(3) is free, the reduced spaces are smooth manifolds—indeed, we have seen they are of
the form Oµ×S—and the Dirichlet criterion for reduced Lyapunov stability (that is, Lyapunov sta-
bility on the reduced space) is that the Hessian of the reduced Hamiltonian hµshould be positive
or negative definite at the (relative) equilibrium in question. Under this hypothesis, the relative
equilibrium in the full phase space Pis G-Lyapunov stable [13], and it is shown by Lerman and
Singer [11], based on work of Patrick [19], that moreover it is Lyapunov stable relative to the pos-
sibly smaller group Gµ, where µis its momentum value, see also [18]. In [15,17] it is shown that
if the Hessian is definite (in which case the relative equilibrium is extremal) then on each nearby
reduced space there is also an extremal (hence stable) relative equilibrium.
If the momentum µis non-zero, then the coadjoint orbits form a smooth foliation near µ, and
hence so do the reduced spaces near (µ,s). Furthermore, if the reduced Hamiltonian has a non-
degenerate critical point at (µ,s) then it does so for nearby reduced spaces as well (as observed
essentially by Arnold [1, Appendix 2], at least in the case S=0), and if one Hessian is definite so are
all nearby ones.
On the other hand, if pis a non-degenerate relative equilibrium with momentum µ=0, then
the local structure depends on the value of the velocity ξ. If ξ6= 0 the situation was first studied
by Patrick [20], also by the author [15] and again in more detail by Patrick [21], where he considers
the eigenvalues of the linear approximations to the flow at such relative equilibria. We begin by
considering the stability of nearby relative equilibria in this case, and afterwards we consider the
case where both µand ξare zero.
Returning to the decomposition (1.2), and the Hamiltonian H(µ,s), the Poisson Hamiltonian
system on P/G≃g∗×Sis,
½˙
µ= −coadDµHµ,
˙
s=JDsH.(3.1)
Here Jis the usual symplectic/Poisson structure matrix on S. Linearizing these equations at the
origin gives
µ˙
s
˙
ν¶=Lµs
ν¶,
with
L=µJD2
sH C
0−coadξ¶
and where ξ=DνH(0,0) ∈g, and C=J D2
sνH(0,0) which is a linear map g→S. The spectrum of
−coadξis equal to ©0,±i|ξ|ª. More details can be found in [21,22].
Write L0=JD2
sH(0,0). If the spectra of L0and of −coadξare disjoint, then a change of coor-
dinates (or choice of symplectic slice) can be chosen to eliminate the matrix C. If on the other
hand, the spectra are not disjoint, one says there is a rotation-vibration resonance, and the matrix
Ccontributes a nilpotent term to the linear system. See Remark 3.2(c) below for further comments.
Recall that an infinitesimally symplectic matrix is said to be,
14 J. MON TA LD I
•spectrally stable if its spectrum is pure imaginary,
•linearly stable or elliptic if it is spectrally stable with zero niloptent part,
•strongly stable if it lies in the interior of the set of linearly stable matrices,
•linearly unstable if it has an eigenvalue with non-zero real part.
An equilibrium is said to be spectrally stable, elliptic, strongly stable or linearly unstable if the
linear part of the Hamiltonian vector field has the corresponding property. Note that if an equi-
librium point is linearly unstable then it is also nonlinearly unstable. In any continuous family
of (relative) equilibria, no matter how it is parametrized, the transitions from one stability type
to another occur only at points where the spectrum has a double eigenvalue. This could be at
zero, in the transition between spectrally stable and unstable, and where the Hessian matrix of the
Hamiltonian becomes degenerate, or a double imaginary eigenvalue (with mixed sign) resulting
usually in a Hamiltonian-Hopf bifurcation and a change again from spectrally stable to unstable.
The difference between linear stability and Lyapunov stability (in the full nonlinear system) lies
with the ‘Krein sign’ of the linear vector field, which is a question of whether the quadratic part of
the Hamiltonian is positive definite or not—this is Dirichlet ’s criterion for (Lyapunov) stability. If
the Hamiltonian is definite (positive or negative) then the equilibrium is strongly stable, and if a
pair of complex conjugate eigenvalues crosses the origin but remains on the imaginary axis, then
there is a transition from Lyapunov stable to elliptic. See [4] for more details.
Zero momentum, non-zero velocity
We now present the first stability theorem appropriate for relative equilibria with zero momentum
but non-zero velocity.
Theorem 3.1 Suppose G =SO(3) acts freely on Pas before, with equivariant momentum map
J. Suppose that p0∈P0(zero momentum) is a non-degenerate relative equilibrium with non-zero
angular velocity ξ∈g. Then,
1. there is a neighbourhood of p0in P/G such that the relative equilibria in the neighbourhood
form a smooth curve through p0intersecting each nearby reduced space in precisely 2 points;
2. if the Hessian of the reduced Hamiltonian d2H0(p0)is definite then on one side of p0on the
curve the relative equilibrium will be Lyapunov stable, and if there is no rotation-vibration
resonance then on the other it will be elliptic.
3. if p0is strongly stable and there is no rotation-vibration resonance then throughout a neigh-
bourhood of p0on the curve the equilibrium will be elliptic;
4. if p0is linearly unstable, then throughout the curve (in a neighbourhood of p0), the relative
equilibria will all be linearly unstable.
Remarks 3.2 (a) The transition between Lyapunov stable and elliptic relative equilibria described
in part (2) can be seen in Figure 2.2 in all cases except α=0, where the black dot represents the
point µ=0.
(b) In the case that P0is just a point (so S=0), the relative equilibria will be Lyapunov stable
throughout the curve; see the example of the rigid body with rotors described in §4.
BIFURCATIO NS N EAR Z ERO MO MEN TUM 15
(c) The rotation-vibration resonance was introduced in [21]. If there is a rotation-vibration reso-
nance and C6= 0 then it might be expected to see a singular Hamiltonian-Hopf bifurcation along
the curve of relative equilibria (singular because it occurs at µ=0, where the dimension of the
reduced space changes). This possibility was also suggested at the very end of [15], and to the au-
thor’s knowledge has not yet been investigated as a bifurcation. An example of this phenomenon
can be found in systems of point vortices—see [10, §9].
PROO F: (1) Except for the intersection with reduced spaces, this is proved in [20]. Here we sum-
marize the argument using the constructions described here in Section 1. We define the function
h:g∗→Ras in (1.5), and we have dh(0) =ξ6= 0. Now hµis the restriction of hto the sphere through
µand it follows that for small µthere are precisely two critical points of hµ, and as µvaries these
form a smooth curve through µ=0.
(2, 3) For each small, non-zero value of µ, the function hµhas an isolated minimum and an isolated
maximum on the sphere Oµ, and no other critical points. Suppose for (2) that D2
sH(p0) is positive
definite (the argument for negative definite being similar), then the point which is a local minimum
of hµwill be Lyapunov stable, because at that point the reduced Hamiltonian on Pµwill also have
a non-degenerate local minimum.
On the other hand, for the local maximum νof hµ, and for both critical points in case (3), the
function Hµwill have a critical point at p=(ν,s(ν)) for which the Hessian is indefinite. However,
the spectrum of Lµ=LTpPµwill be a perturbation of the union of the spectra of L0and coadξ
excluding the zero, and by hypothesis these are pure imaginary and disjoint. Moreover, in (2) the
assumption that d2H0(p) is positive definite implies that L0is strongly stable. Thus in both (2) and
(3) any sufficiently small perturbation of it has purely imaginary eigenvalues. It follows that the
spectrum of Lµwill also be pure imaginary for µsufficiently small.
(4) This follows a similar argument. The spectrum of Lµwill contain perturbations of the spec-
trum of L0, and as the latter has a non-zero real part, so will the former. ❒
Zero momentum, zero velocity
We now turn to the unfolding of the relative equilibrium pwith µ=0 and ξ=0, as in Theorem 2.1.
Now, the equivalence relation used for the theorem only respects the set of relative equilibria; it
does not respect dynamics, nor even the level sets of the energy-momentum map so one cannot a
priori deduce the stability of the bifurcating relative equilibria from studying the normal form G.
On the other hand, stability only changes (along a branch of relative equilibria) through one of the
two scenarios as described above.
Recall that the unfolding discriminant ∆of Gconsists of three planes in R3, that ∆2denotes
the open strata (regular points of ∆) and ∆1the points where two planes intersect; that is, ∆is
the disjoint union of ∆2,∆1and the origin (see Figure 2.1). It follows from Theorem 2.1 that the
unfolding discriminant of any other family based on H0is pulled back from this ∆by some smooth
map Φ. The following theorem justifies the pictures and stabilities shown in Fig. 2.2.
Theorem 3.3 Let Hbe a family of SO(3)-invariant Hamiltonians, with parameter u ∈U , and
with H0having a non-degenerate relative equilibrium at p0∈P0with D2
sH0(p0)positive definite,
and with DνH0(p0)=0(so the velocity is zero). Assume moreover that D2
νH0(p0)is a quadratic form
whose three eigenvalues (with respect to an SO(3)-invariant inner product on so(3)∗) are distinct.
(This is the setting of Theorem 2.1.) Let ϕ:U→R3be the map given by Theorem 2.1, inducing H
from G, and write ∆H=φ−1(∆).
16 J. MON TA LD I
1. There is a neighbourhood of p0in P/G in which the set of relative equilibria for H0consists of
three curves; along one of these the relative equilibria are Lyapunov stable, along another they
are elliptic and along the third they are linearly unstable.
2. There is a neighbourhood M of the origin in so(3)∗such that, for u ∈U with ϕ(u)6= 0,
(i) if kµkis sufficiently small there are two relative equilibria as in Theorem 3.1, of which one
is Lyapunov stable and the other elliptic;
(ii) if the Hamiltonian is analytic then for u 6∈ ∆H, as kµkis increased, there are two saddle-
centre bi furcations, one producing a Lyapunov stable relative equilibr ium ( RE) and an unsta-
ble RE, while the other produces an elliptic RE and an unstable RE;
(iii) for ϕ(u)∈∆2, as kµkis increased, again one of the RE persists, and the other undergoes a
supercritical1pitchfork bifurcation; there is also a saddle-centre bifurcation. Whether it is the
Lyapunov stable or elliptic R E that bifurcates depends on the connected component of ∆2;
(iv) for ϕ(u)∈∆1, as kµkis increased, one RE persists, while the other undergoes two successive
supercritical pitchfork bifurcations. Which persists and which bifurcates will depend on the
component of ∆1.
In Figures 2.2 and 2.4, (i) corresponds to part (1) of the theorem, (ii)–(iv) to part 2(iv), (v) to part
2(iii) and (vi) to part 2(ii). Note that the theorem does not imply that all these cases arise for every
family: it is certainly possible that the image of ϕis contained in ∆1, for example.
PROO F: (1) The existence of the three curves is part of the calculations in § 2. And the stability part
is proved in [16, Section 2.5].
(2) (i) For u6= 0 we have DνHu(0) 6= 0, consequently the relative equilibrium has non-zero velocity
(ξ6= 0), and since ξis continuous function of u, it follows that for usufficiently small there is no
rotation-vibration resonance, so this follows from Theorem 3.1.
(ii) For each perturbation Hu, with u6∈ ∆the set Rconsists of three disjoint curves, as described
in § 2. By Lemma 3.5 below, the restriction of the function jto the curve Rαhas at most 4 criti-
cal points in a fixed neighbourhood of the origin (for sufficiently small values of α). Now in this
fixed neighbourhood of the origin, jis increasing and on each curve reaches its maximum at the
ends (for sufficiently small values of α). The function therefore has at least one minimum on each
branch, and in general an odd number of critical points (counting multiplicity) on each branch.
The only way that is compatible with the upper bound of 4 is that there is a single non-degenerate
critical point of jon each branch, and so 3 in all. Now one of the branches passes through the ori-
gin, where jreaches its minimum value of 0, while on the other two branches, the minimum will
be a point where the branch is tangent to the momentum sphere (coadjoint orbit), so producing
a saddle-centre bifurcation point. These will be the two points of bifurcation mentioned in the
theorem.
There remains to show that the saddle-centre bifurcations involve the creation of critical points
with the stated stability properties. This is true for the model family Gby direct calculation. Now
consider a 1-parameter family of systems perturbing Gto H. By the multiplicity argument above,
no other critical points are introduced, so the index of each critical point is the same for Gas the
corresponding one for H, and the stability type depends only on the index.
(iii), (iv). Here the proof is analogous to part (ii), but more straightforward as the pitchfork bifurca-
tions will correspond to points where the map d(h,j) is not transverse to V, a property preserved by
1after a supercritical pitchfork bifurcation, a stable RE becomes two stable REs and one unstable one, see Fig. 2.3
BIFURCATIO NS N EAR Z ERO MO MEN TUM 17
the equivalence we use in the proof above of Theorem 2.1, so is clearly preserved by the diffeomor-
phism. If the pitchfork bifurcations were transcritical rather than sub- or super-critical then this
would involve extra critical points which we know from the lemma below cannot happen. The type
of pitchfork and stability properties is the same in the family Has for Gby the homotopy argument
given above.
That there are no other bifurcations or loss of stability, except those involving an eigenvalue
becoming zero, follows because there is no rotation-vibration resonance, so the spectra from the
rotation part and the shape part are disjoint, and so can be continued with no extra multiple eigen-
values occurring. ❒
Remarks 3.4 (i) If the hypothesis that D2
sH0is positive definite is replaced with that of L0=JD2
sH0
being strongly stable, then the conclusions of the theorem are the same, but with Lyapunov stable
replaced by elliptic throughout. If on the other hand, L0is linearly unstable, then all the existence
and bifurcation statements are the same except that all the R E are likewise linearly unstable.
(ii) In the proof of 2(ii) we need to assume the Hamiltonian is analytic in order to use methods of
commutative algebra to estimate the number of critical points (see Lemma 3.5 below); it would be
surprising if this were an essential hypothesis.
In the proof above we used the following lemma, which we prove using some commutative
algebra based on ideas of Bruce and Roberts [3].
Lemma 3.5 There is a neighbourhood U1of the origin in so(3)∗and a neighbourhoodU2of 0such
that for all α∈U2the restriction of the function jto Rαhas at most 4 critical points in U1, counting
multiplicity.
PROO F: Let fbe a smooth function on a manifold M. The restriction of fto a submanifold Xhas a
critical point at x∈Xif the graph of the differential 1-form dfintersects the conormal variety N∗X
at a point over x(this is all in the cotangent bundle T∗M). The conormal variety is the bundle over
X, given by
©(x,λ)∈T∗M|x∈X,λ∈(TxX)◦ª.
Here (TxX)◦is the annihilator of the tangent space TxX. The total space of this bundle has the
same dimension as the ambient manifold M. The multiplicity of the critical point is equal to the
intersection number of the graph and the conormal bundle. This multiplicity can be defined using
modules of vector fields tangent to X, or using the sum of the ideals defining the graph of dfand
the conormal bundle of X.
When Xis singular, the conormal bundle is replaced by the so-called logarithmic characteris-
tic variety LC−(X) which is essentially the union over the (logarithmic) strata of Xof the closure
of the conormal bundle to each stratum, see [3] for details. Under certain algebraic conditions
(namely LC−(X) should be Cohen-Macaulay), and providing everything is complex analytic, the
multiplicity is preserved in deformations of the function, and without this algebraic condition the
multiplicity is upper semicontinuous [3, Proposition 5.11].
We need to extend this by allowing the variety to deform as well as the function, but the semi-
continuity is a general algebraic property, regardless of how the data deforms (provided the dimen-
sions are constant).
To return to our setting, first consider the central case with α=0, and neglect the higher order
terms in the Hamiltonian, so we are in the setting of § 2, and consider everything complex. The
variety R0consists of the three axes in C3, and the function j=1
2(x2+y2+z2). We are interested in
18 J. MON TA LD I
critical points of jrestricted to R0(and later to Rα). This clearly has a single critical point, namely
the origin, so we need to understand its multiplicity.
The variety LC−(R0)⊂T∗C3≃C6consists of a 3-dimensional subspace for each of the axes,
and a further one for the stratum {0}. Explicitly, if we use the coordinates (k,ℓ,m) in the dual space
(to form T∗C3) then the union of the four 3-dimensional subspaces is,
LC−(R0)={x=y=m=0} ∪{y=z=k=0} ∪{z=x=ℓ=0} ∪{x=y=z=0}.
The ideal of this variety is 〈x y ,yz ,z x ,x k ,yℓ,z m〉. Now the graph of djis the set {k=x,ℓ=y,m=
z}, and a calulation shows that the algebraic intersection number is 4 (in fact the variety LC−(R0)
is Cohen-Macualay so the algebraic and geometric multiplicities coincide). That is, jhas a critical
point of multiplicity 4 at the origin.
Now we wish to deform the set R0to Rα(which is smooth as α6∈ ∆), and take the new conormal
variety but defined with the higher order terms of Hincluded and at the same time add in the
higher order terms to H.
Under this deformation, the multiplicity cannot increase (it remains constant if the whole fam-
ily of conormal varieties is Cohn-Macaulay [3], but this turns out not to be the case here, explaining
why the 4 ultimately drops to 3 in the course of the proof of Theorem 3.3). It follows that in the de-
formed setting there are at most 4 (complex) critical points, and therefore at most 4 real ones, as
claimed. ❒
4 Example: rigid body with rotors
We give an application to the system consisting of a free rigid body with three freely rotating rotors
attached so that their respective axes lie along the three principal axes of the body [9,2,14]. The
configuration space for this system is the Lie group G=SO(3) ×T3, where T3=S1×S1×S1which
acts by rotation of the three rotors. A matrix A∈SO(3) corresponds to the attitude of the rigid body,
while the components of θ=(θ1,θ2,θ3)∈T3are the angles of rotation of the three rotors, relative
to the body.
The Lagrangian of this system is given by the kinetic energy, which in a principal basis is
L=1
2ωT(I−Ir)ω+1
2(ω+˙
θ)TIr(ω+˙
θ).
Here Iris the diagonal matrix whose entries are the respective moments of inertia of the rotors
about their axes, Iis the inertia tensor of the rigid body with the rotors locked to the body and
ω∈R3≃so(3) is the angular velocity vector in the body; we assume I−Iris invertible. For details
see Sec. 3 of [2], where ωis denoted Ωand ˙
θis denoted Ωr.
The corresponding momenta are therefore,
µ=∂L/∂ω=Iω+Ir˙
θ,
σ=∂L/∂˙
θ=Ir(ω+˙
θ). (4.1)
Here µ∈so(3)∗is the angular momentum in the body, and σ∈t∗=R3is the gyrostatic momentum
(µis denoted mand σis denoted ℓin [2]). The Hamiltonian is then
H=1
2(µ−σ)T(I−Ir)−1(µ−σ)+1
2σTI−1
rσ(4.2)
The momentum map for the G-action is J(A,θ,µ,σ)=(Aµ,σ). Indeed, Aµis the angular momen-
tum of the body in space, and σis the conserved quantity due to the T3-symmetry of the system,
the gyrostatic momentum.
BIFURCATIO NS N EAR Z ERO MO MEN TUM 19
The free system
As a first step to analyzing the system as it is (with no external constraints), we reduce by the free
T-action putting σconstant. This gives the reduced Hamiltonian on T∗SO(3),
Hσ(A,µ)=1
2(µ−σ)T(I−Ir)−1(µ−σ)
(the other term in (4.2) is now a constant so can be ignored). When σ=0, H0is the usual rigid body
Hamiltonian 1
2µT(I−Ir)−1µwhich is homogeneous of degree 2, as is G0Theorem 2.1.
Varying σgives a straightforward example of the family G. Indeed if (I−Ir)−1=diag[a,b,c] and
σ= −(α,β,γ) then Hσhere is precisely Guin Theorem 2.1 with u= −σ, so it does not depend on
Theorems 2.1 or 3.3, just on the calculations of Section 2.
For the stability, let µ=(x,y,z) and (I−Ir)−1=diag[a,b,c] with a>b>c. Then with σ=0
the x- and z-axes consist of stable relative equilibria, and the y-axis of linearly unstable relative
equilibria, as for the ordinary free rigid body. Note that here the T3-reduced system is a phase
space of dimension 6 so the symplectic slice at µ=0 reduces to 0 and we are in the situation of
Remark 3.2(b), so all elliptic RE are in fact Lyapunov stable.
Now suppose we consider σ=(σ1,0,0) with σ16= 0. In other words we have ‘activated’ the rotor
along the principal direction of lowest moment of inertia (although looking at (4.1) shows the idea
of ‘activation’ is not entirely accurate) This will give a family of relative equilibria as in Fig. 2.2(ii)
(and with all green curves being made red). This means that if the satellite is given a small angular
momentum, there are two relative equilibria, both rotating about the axis with the activated rotor,
and both are stable. As the angular momentum is increased, one of these (the one of lower energy)
will undergo a supercritical pitchfork bifurcation—see Fig. 2.3—so that the rotation about the axis
becomes unstable, while there are two new relative equilibria, rotating about axes initially close to
the given axis. As the angular momentum increases further, the unstable RE stabilizes again, and
two new unstable RE appear.
A similar scenario occurs if we activate the rotor along the axis of greatest moment of inertia,
except here it is the RE with greater energy (for given angular momentum) that loses stability in a
supercritical bifurcation, before stabilizing again.
If instead the rotor along the middle axis is activated, there are again two stable RE and as kµk
is increased, they both lose stability in supercritical pitchfork bifurcations; which one occurs first
depends on the relative values of a,b,c(if a−b=b−cthen they occur simultaneously).
The reader is invited to supply the storyline if two or three of the rotors are activated, following
Fig. 2.2. But in every case, it should be noted that R E with sufficiently small angular momentum µ
are always stable if σ6= 0, which is the setting of Theorem 3.1.
Note that the energy-momentum discriminants in Figure 2.4 are in fact accurate for this sys-
tem, as the Hamiltonian is of degree 2.
A controlled version
Now suppose the rotors are used as control mechanisms, and their angular velocities relative to
the body can be fixed. That is, put ˙
θi=ui, fixed (a constraint). The Lagrangian is then
L=1
2ωT(I−Ir)ω+1
2(ω+u)TIr(ω+u),
where u∈R3is constant. The corresponding Hamiltonian, with variables A,µis
H(A,µ)=1
2µTI−1µ−µTI−1α
20 J. MON TA LD I
where α=Ir˙
θ(a constant vector whose components are the angular momenta of the spinning
rotors).
The three components of αgive three coefficients in place of ˙
θ, which unfold the singularity
occurring when α=0, and provided the three principal moments of inertia of the rigid body are
distinct, we obtain the same unfolding as described above, and again the Hamiltonian is of degree
2 so the energy-momentum discriminants shown in Fig. 2.4 are accurate.
5 Singularity theory and deformations
Given a reduced hamiltonian hon so(3)∗, the set R⊂so(3)∗of relative equilibria coincides with
the set of critical points of the energy-Casimir map
(j,h)=¡1
2(x2+y2+z2), h(x,y,z)¢.
Thus Ris the set where the rank of the Jacobian matrix,
F(x,y,z)=d(j,h)=·x y z
hxhyhz¸(5.1)
is at most 1. Let V⊂Mat(2, 3) consist of all 2 ×3 matrices of rank at most 1. Then R=F−1(V).
Singularity theory gives a technique for deciding which deformations of F−1(V) can arise by
perturbing F, and the appropriate equivalence relation on Fis called KV-equivalence and was
introduced by Damon [6]. We recall this briefly before continuing with the proof.
Let F,G:X→Y, and let V⊂Y(everything in sight should be considered as germs). Then F
and Gare said to be KV-equivalent if there is a diffeomorphism ψof Xand a diffeomorphism Ψof
X×Ypreserving X×Vand of the form Ψ(x,y)=(ψ(x), ψ1(x,y)), such that
Ψ(x,F(x)) =(ψ(x), G(ψ(x)));
that is, Ψmaps the graph of Fto the graph of G. It follows in particular that ψ(F−1(V)) =G−1(V).
(If Vis just a point, then this reduces to ordinary K-equivalence.)
There are several rings and modules we need to consider. For F:X→Y, let EXand EYbe
the rings of germs at 0 of C∞functions on Xand on Yrespectively. Similarly, θXand θYare the
modules over EXand EYof germs of vector fields on Xand Yrespectively. For V⊂Ywe write θV
for the submodule of θYconsisting of vector fields tangent to V(often denoted Derlog(V) in the
singularity theory literature). And finally one writes θ(F) for the EX-module of ‘vector fields along
F’, meaning sections of the pull back of T Y to Xvia F, or more prosaically if Xand Yare linear
spaces, θ(F) is the EXmodule of all germs at 0 of maps X→Y
In our setting, X=R3,Y=Mat(2,3), and V⊂Yis the set of matrices of rank at most 1. Now,
there is a natural action of GL(2) ×GL(3) on Mat(2,3) given by (A,B)·M=AM B−1, and of course
this action preserves V; indeed, Vconsists of just two orbits of this action: the origin and the set of
matrices of rank equal to 1. The infinitesimal version of this action gives a map gl(2) ×gl(3) →θY,
whose image therefore lies in θV. Write θ′
V⊂θVfor the EY-module generated by these vector fields.
(It seems likely that θ′
V=θV, though for the computations we will see that in fact θ′
Vsuffices.)
Now, dim(gl(3) ×gl(2)) =13, but the element (I,−I) acts trivially, so that θ′
Vhas 12 generators;
they are vector fields such as
µa11 a12 a13
0 0 0 ¶,µ000
a11 a12 a13 ¶,µ0a11 0
0a21 0¶, .. .
BIFURCATIO NS N EAR Z ERO MO MEN TUM 21
Here the matrix (ui j ) refers to the vector field Pi,jui j ∂
∂ai j . In words, the generators are obtained by
taking a single row (ai1ai2ai3)of Aand placing it in either row with 0s in the other row (there
are 4 such vector fields), and then taking a single column µa1j
a2j¶and placing it in any column and
filling the remaining 2 columns with zeros (9 such vector fields).
Given any EY-module θof vector fields on Y, one defines two Kθtangent spaces of a map
F:X→Yto be the EX-submodules of θ(F), the Kθ-tangent space
TKθ·F=t F (mXθX)+F∗θ,
and the extended Kθ-tangent space
TKθ,e·F=t F (θX)+F∗θ.
Here
•t F (θX) means the EX-module generated by the partial derivatives of F, and so t F (mXθX) is
the maximal ideal mXtimes t F (θX), and
•F∗θ=EX{v◦F|v∈θ}, the EX-module generated by the vector fields in θcomposed with F.
The ordinary tangent space is used for finite determinacy properties while the extended one is
used for versal deformations.
Now consider the map Fdefined in (5.1). The partial derivatives of Flead to
t F (θX)=EX½µ 1 0 0
hxx hx y hxz ¶,µ0 1 0
hyx hy y hy z ¶,µ0 0 1
hzx hz y hz z ¶¾,
where subscripts refer to partial derivatives. The second term in TKV·Fcontains 12 generators
such as µhxhyhz
0 0 0 ¶,µx y z
0 0 0 ¶,µ0 0 y
0 0 hy¶.
Now apply this to the function h(x,y)=1
2(ax2+by 2+c z2), with a,b,cdistinct. We obtain
TKV·F=mXθ(F), and
TKV,e·F=mXθ(F)+R½µ1 0 0
0 0 0¶,µ0 1 0
0 0 0¶,µ0 0 1
0 0 0¶¾,
where mXis the maximal ideal of functions on X=R3that vanish at 0, so mX=〈x,y,z〉.
Here we have used θ′
Vrather than θVand a priori the expression above is for the corresponding
module TK′
V·F. However, the fact that the vector fields tangent to Vall vanish at the origin in
Mat(3,2) implies there are no other elements of TKV·F, so that for this function hone has indeed
that TK′
V·F=TKV·F, and similarly for the extended tangent spaces.
We are now in a position to prove Theorem 2.1.
PROO F OF TH EOREM 2.1: Since KV-equivalence is one of Damon’s geometric subgroups of K, it
follows that the usual finite determinacy and versal deformation theorems hold [5]. In particular,
with the family Gas in the statement of the theorem, one has that
θ(F)=TKV,e·F+R·½∂G
∂α,∂G
∂β ,∂G
∂γ ¾.
22 J. MON TA LD I
Consequently, by Damon’s theorems, the family Gα(with α=(α,β,γ)) is a KV-versal deformation
of F, which is what is required for the theorem.
Furthermore, mXθ(F)⊂TKV·Fimplies that Fis 1-determined w.r.t. KV-equivalence. It fol-
lows that if h(x,y,z) has the same 2-jet as gthen the map Fassociated to Gand Hhave the same
1-jet so are KV-equivalent; consequently gis in this sense 2-determined. ❒
Remark 5.1 The vector field constructions of singularity theory will all produce diffeomorphisms
whose linear part at the origin is the identity. However, allowing more general diffeomorphisms,
one can show further that His equivalent to the version of Gwith coefficients a=1, b=0, c= −1
say. Indeed, one can be obtained from the other by row operations on the matrix in (5.1).
Other singularities We considered above the KV-equivalence arising from a quadratic Hamilto-
nian at the origin. To understand the equivalence better, two further examples are worth consid-
ering:
(1) If dh(0,0, 0) 6= 0, we have h(x,y,z)=ax +by +c z + · ·· with (a,b,c)6= (0,0, 0). In this case
TKV·F=θ(F), so his ‘stable’ in the appropriate sense: any sufficiently small deformation
h′of hgives rise to a map F′which is KV-equivalent to F, so having diffeomorphic sets of
relative equilibria (this is not surprising: we know it is just a non-singular curve through the
origin).
(2) At points away from 0 in R3,KV-equivalence to Fis more familiar: choose local coordinates
so that j(x,y,z)=z(this is possible in a neighbourhood of a point with dj6= 0), then locally
F=µ0 0 1
hxhyhz¶.
Thus F(x,y,z)∈Vif and only if hx=hy=0. That is, this approach is finding critical points
of has a function of (x,y) with parameter z, and the KV-equivalence of Fcorresponds to
K-equivalence of 1-parameter families of gradients of functions. This is not the same as
unfolding equivalence (or bifurcation-equivalence), as the relation does not distinguish z
as a parameter. For example the differentials of the maps (h,j)=(x2−y2,z) and (x z ,z) are
KV-equivalent.
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