Conference PaperPDF Available

Relative equilibria and conserved quantities in symmetric Hamiltonian systems

September 2000

September 2000

DOI:10.1142/9789812792778_0008

Conference: INLN Summer School on Nonlinear Phenomena, 1998
At: Peyresq
Volume: Peyresq lectures on nonlinear phenomena

Authors:

James Montaldi

The University of Manchester

Level sets of a typical function on a sphere with symmetry D 3h , showing half of the 8 critical points

…

Figures - uploaded by James Montaldi

Content may be subject to copyright.

Content uploaded by James Montaldi

Content may be subject to copyright.

RELATIVE EQUILIBRIA AND CONSERVED QUANTITIES

IN SYMMETRIC HAMILTONIAN SYSTEMS

JAMES MONTALDI

Institut Non Lin´eaire de Nice

Published in:

Peyresq Lectures in Nonlinear Phenomena

R. Kaiser, J. Montaldi (eds)

World Scientiﬁc, 2000

1 Introduction

In this introduction, we ﬁrst recall the basic phase space structures involved

in Hamiltonian systems, the symplectic form, the Poisson brackets and the

Hamiltonian function and vector ﬁelds, and the relationship between them.

Afterwards we describe a few examples of Hamiltonian systems, both of

the classical ‘kinetic+potential’ type as well as others using the symplec-

tic/Poisson structure more explicitly.

There are many applications of the ideas in these notes that have been

investigated by diﬀerent people, but which I shall not cover. The major ex-

ample, the one for which classical mechanics was invented, is the gravitational

N-body problem. But there are many others too, such as rigid bodies, cou-

pled rigid bodies, coupled rods, underwater sea vehicles, . . . not to mention

the inﬁnite dimensional systems such as water waves, ﬂuid ﬂow, plasmas and

elasticity. The interested reader should consult the books in the list of refer-

ences at the end of these notes. Note that many of the items in the list of

references are not in fact referred to in the text!

1.1 Hamilton’s equations

The archetypal Hamiltonian system describes the motion of a particle in a

potential well. If the particle has mass m, and V(x) is the potential energy

at the point x(in whatever Euclidean space), then Newton’s laws state that

m¨x=−∇V(x).

In the 18th century, Lagrange introduced the phase space by deﬁning y= ˙x

and passing to a ﬁrst order diﬀerential equation, and Hamilton carried this

further by introducing his now-famous equations

˙q=∂H/∂p,

˙p=−∂H/∂q,

where qreplaces x, and p=m˙xis the momentum, and

H(q, p) = 1

2m|p|2+V(q)

J. Montaldi, Relative equilibria and conserved quantities . . . . 239

is the Hamiltonian, or total energy (kinetic + potential). This ﬁrst order

system is equivalent to Newton’s law above, as is easily checked.

The principal advantage of Lagrange/Hamilton’s approach is that it is

more readily generalized to systems where the conﬁguration space is not a

Euclidean space, but is a manifold. Such systems usually come about because

of constraints imposed (eg in the rigid body the constraints are that the dis-

tances between any two particles is ﬁxed, and the conﬁguration space is then

the set of rotations and translations in Euclidean 3-space).

The other advantage of the Hamilton’s approach is that it lends itself to

generalizations to systems that are not of the kinetic + potential type, such as

the model of a system of Npoint vortices in the plane or on a sphere, which

we will see below, or Euler’s equations modelling the “reduced” motion of a

rigid body.

These two generalizations lead to deﬁning the dynamics in terms of a

Hamiltonian on a phase space, where the phase space has the additional

structure of being a Poisson or a symplectic manifold. The ‘canonical’ Poisson

structure is given by

{f, g}=

j=1

∂f

∂pj

∂g

∂qj−∂g

∂pj

∂f

∂qj

where fand gare any two smooth functions on the phase space. The canonical

symplectic structure is given by,

ω=

j=1

dpj∧dqj=dα,

where α=Pn

j=1 pj∧dqjis the canonical Liouville 1-form.a

We shall see other examples of Poisson and symplectic structures in the

course of these lectures.

The Hamiltonian vector ﬁeld XHis determined by the Hamiltonian H, a

smooth function on the phase space, in either of the following ways:

F={H, F }

ω(v, XH) = dH(v),(1.1)

where ˙

F=XH(F) is the time-rate of change of the function Falong the

trajectories of the dynamical system. Combining these two expressions gives

aNot all authors agree on the choices of signs in the deﬁnition of the symplectic form or

the Poisson brackets, so when using formulae involving either from a text or paper, it is

necessary ﬁrst to check the deﬁnitions. Our choice ensures that X{f,g}= [Xf, Xg].

J. Montaldi, Relative equilibria and conserved quantities . . . . 240

the useful formula relating the symplectic and Poisson structures,

ω(Xf, Xg) = {f, g}(1.2)

for any smooth functions f, g.

The ﬁrst property of such Hamiltonian systems is their conservative na-

ture: the Hamiltonian function His conserved under the dynamics and so

too is the natural volume in phase space (Liouville’s theorem). This has an

important eﬀect, not only on the type of dynamics encountered in such sys-

tems, but also on the types of generic bifurcations that can occur. Indeed, the

ﬁrst consequence of these conservation laws (energy and volume) is that one

cannot have attractors in Hamiltonian systems, and in particular the notion

of asymptotic stability is not available.

A further feature of Hamiltonian systems is that symmetries lead to con-

served quantities. The two best-known examples of this are rotational sym-

metry leading to conservation of angular momentum, and translational sym-

metry to conservation of ordinary linear momentum. These further conserved

quantities could in principle complicate the types of dynamics and bifurca-

tions one sees. However a well deﬁned process called reduction (or symplectic

reduction) can be used to replace the symmetry and conservation laws by a

family of Hamiltonian systems, parametrized by these conserved quantities,

on which there are general Hamiltonian systems whose bifurcations are those

expected in generic systems.

That said, there is a further complication which is that the phase space(s)

for these reduced systems may be singular, or change or degenerate in a family,

and we are only just beginning to understand the eﬀects of these degenerations

on the dynamics and bifurcations.

1.2 Examples

Spherical pendulum A spherical pendulum is a particle constrained to move

on the surface of a sphere under the inﬂuence of gravity. As coordinates,

one can take spherical polar coordinates θ, φ (θmeasuring the angle with the

downward vertical, and φthe angle with a ﬁxed horizontal axis). Of course,

this system has the defect of being singular at θ= 0, π. The kinetic energy is

T(q,˙

q) = 1

2mℓ2˙

θ2+ sin2θ˙

φ2,

while the potential energy is V(q) = mgℓ(1 −cos θ). The momenta conjugate

to the spherical polars are

pθ=∂L/∂ ˙

θ=mℓ2˙

θ, pφ=∂L/∂ ˙

φ=mℓ2sin2θ˙

φ,

J. Montaldi, Relative equilibria and conserved quantities . . . . 241

where L=T−Vis the Lagrangian, and the Hamiltonian is then

H(q,p) = 1

2mℓ2p2

θ+1

sin2θp2

φ+V(q).

The equations of motion are given as usual by Hamilton’s equations. In

particular, one can see from these equations that the angular momentum pφ

about the vertical axis is conserved since His independent of φ(one has

˙pφ=∂H/∂φ = 0).

Point vortices Since the work of Helmholtz, Kirchhoﬀ and Poincar´e systems

of point vortices on the plane have been widely studied as ﬁnite dimensional

approximations to vorticity evolution in ﬂuid dynamics. Small numbers of

point vortices model the dynamics of concentrated regions of vorticity while

large numbers can be used to approximate less concentrated regions. The

equations of motion can be derived by substituting delta functions into Euler’s

equation for a two dimensional ideal ﬂuid. For general surveys of planar point

vortex systems see for example [17,3,15].

For this system of point vortices, each vortex has a vorticity—a real non-

zero number λ—and it is convenient to use complex numbers to describe

the positions of the vortices. The evolution is described by the diﬀerential

equation

¯zj=1

2πi X

k6=j

λk

zj−zk

,(1.3)

where zjis a complex number representing the position of the j-th vortex.

This system is Hamiltonian, with the Hamiltonian given by a pairwise inter-

action depending on the mutual distances:

H(z) = −1

4πX

j<k

λjλklog |zj−zk|2

This is clearly not of the form ‘kinetic+potential’. The Poisson structure is

{f, g}=X

λ−1

j(fzg¯z−gzf¯z)

and the symplectic form is ω(u,v) = Pjλj(uj¯vj−¯ujvj). Being of dimension

3, the Euclidean symmetry has 3 conserved quantities associated to it—see

equation (6.2).

A similar model can be obtained for point vortices on the sphere, providing

a simple model for cyclones and hurricanes in planetary atmospheres. See

Section 6.

J. Montaldi, Relative equilibria and conserved quantities . . . . 242

1.3 Symmetry

A transformation of the phase space T:P → P is a symmetry of the Hamil-

tonian system, if

(i) H(T x) = H(x) for all x∈ P,

(ii) Tpreserves the symplectic structure: T∗ω=ω, or

(ii’) Tpreserves the Poisson structure: {f◦T , g ◦T}(x) = {f, g}(T(x)).

There are three basic ways that symmetries aﬀect Hamiltonian systems:

(a) The image by Tof a solution is also a solution;

(b) A solution with initial point ﬁxed by Tlies entirely within the set

Fix(T, P), where

Fix(T, P) = {x∈ P | T x =x}.

quantities (Noether’s theorem).

The ﬁrst of these is clear, and in fact is also true of more generalized

symmetries for which H◦T−His constant, and T∗ω=cω (ca constant).

This occurs for example for homotheties of the plane in the planar point vortex

model described above. The second (b) is less obvious, but very well-known;

it follows from a very simple calculation as follows. If T x =xthen

σt(x) = σt(T x) = T σt(x),

where σtis the time tﬂow associated to the Hamiltonian system, and so σt(x)

is ﬁxed by T. If Tis part of a compact group, then not only is Fix(T , P) invari-

ant, but it is a symplectic submanifold, and the restrictions of the Hamiltonian

and the symplectic form (or Poisson structure) to Fix(T , P) is a Hamiltonian

system which coincides with the restriction of the given Hamiltonian system—

an exercise for the reader. This technique of restricting to ﬁxed point spaces

is sometimes called discrete reduction.

In these notes we will be concentrating on the eﬀect of (c). The central

force problem provides the basic motivating example of this.

1.4 Central force problem

Consider a particle of mass mmoving in the plane under a conservative

force, whose potential depends only on the distance to the origin (a sim-

ilar analysis is possible for the spherical pendulum). It is then natural

J. Montaldi, Relative equilibria and conserved quantities . . . . 243

to use polar coordinates (r, φ) which are adapted to the rotational sym-

metry of the problem, so that V=V(r). The velocity of the particle is

x= ( ˙rcos φ+ (rsin φ)˙

φ, ˙rsin φ−(rcos φ)˙

φ), so that the kinetic energy is

T=m

2|˙

x|2=m

2˙r2+r2˙

φ2.

The Lagrangian is given by L=T−V, and the Hamiltonian is H=T+V

with associated momentum variables given by pr=∂L/∂r =m˙rand pφ=

∂L/∂ ˙

φ=mr2˙

φ. Substituting for the velocities ˙rand ˙

φin terms of the

momenta determines the Hamiltonian to be:

H(r, φ, pr, pφ) = 1

2mp2

r+1

r2p2

φ+V(r).

Then Hamilton’s equations with respect to these variables are

(˙r=1

mpr,˙pr=−1

mr3p2

φ+V′(r)

φ=1

mr2pφ˙pφ= 0.(1.4)

The last equation says that pφis preserved under the dynamics. In fact, pφ

is the angular momentum about the origin r= 0.

Since pφis preserved, let us consider a motion with initial condition for

which pφ=µ. Then (r, pr) evolve as

(˙r=1

mpr,

˙pr=−1

mr3µ2+V′(r)(1.5)

This is in fact a Hamiltonian system, with Hamilton Hµ(r, pr) obtained by

substituting µfor pφ. So that

Hµ(r, pr) = 1

2mp2

r+µ2

mr2+V(r).

This is a 1-degree of freedom problem, called the reduced system, with

“eﬀective” potential energy

Vµ(r) = µ2

mr2+V(r),

and for a given potential energy function V(r), one can study how the be-

haviour of the system depends on µ.

For example, with the gravitational potential V(r) = −1/r, one obtains

an eﬀective potential of the form in Figure 1 below, where the ﬁst graph shows

the potential V(r) as a function of r, while the second and third show Vµ(r)

for increasing values of µ.

J. Montaldi, Relative equilibria and conserved quantities . . . . 244

It is clear that for µ= 0 there is no equilibrium for the reduced system,

while for µ > 0 there is an equilibrium, at rµsatisfying V′

µ(rµ) = 0—here

rµ= 3µ2/m. Indeed, in this example it is a stable equilibrium for the eﬀective

potential has a minimum at the rµ.

Figure 1. Eﬀective potential for increasing values of µ, for V(r) = −1/r

Since r=rµand pr= 0 is an equilibrium of the reduced system, it is

natural to substitute these values into the original equation (1.4). The two

remaining equations are then

(˙

φ=µ

mr2

˙pφ= 0.

This describes a simple periodic orbit in the original phase space:

(r, φ, pr, pφ)(t) = (rµ,µ

mr2

µt, 0, µ),

and there is one such periodic orbit for each value of µ6= 0 — or more in

the general case if V′

µ(r) = 0 has several solutions. In the case of a stable

equilibrium in the reduced space, the nearby solutions in the phase space will

be linear ﬂows on invariant tori.

There are three things one should learn from this example:

•the reduced dynamics can vary with the conserved quantity, giving rise

to bifurcation problems where the bifurcation parameter is an “internal”

variable;

•the reduction is possible due to the symmetry of the original problem,

•the relationship between the equilibrium in the reduced system and the

dynamic of the corresponding trajectory (relative equilibrium) in the full

phase space is given by the group action.

J. Montaldi, Relative equilibria and conserved quantities . . . . 245

1.5 Lie group actions

These notes assume the reader has a basic knowledge of actions of Lie groups

on manifolds. Here I recall a few basic formulae and properties that are used.

A useful reference is the new book by Chossat and Lauterbach [5].

Let Gbe a Lie group acting smoothly on a manifold P, and let gbe its

Lie algebra. We denote this action by (g, x)7→ g·x. The orbit through xis

G·x={g·x|g∈G}.

To each element ξ∈gthere is associated a vector ﬁeld on Pwhich we

denote ξP. It is deﬁned as follows

ξP(x) = d

dt t= 0 (exp(tξ)·x).

The tangent space to the group orbit at xis then g·x={ξP(x)|ξ∈g}.

A simple calculation relates the vector ﬁeld at xwith its image at g·x:

dgxξP(x) = (Adgξ)P(g·x),(1.6)

where Adgξis the adjoint action of gon ξ, which in the case of matrix groups

is just

Adgξ=gξg−1.

The adjoint representation of gon gis the inﬁnitesimal version obtained by

diﬀerentiating the adjoint action of G:

adξη=d

dt t= 0 Adexp(tξ)η= [ξ, η ].

Dual to the adjoint action on gis the coadjoint action on g∗:

hCoadgµ, ηi:= µ, Adg−1η,(1.7)

and similarly there is the inﬁnitesimal version,

hcoadξµ, ηi:= hµ, ad−ξηi=hµ, [η, ξ]i.(1.8)

Examples 2.5 describe the coadjoint actions for the groups SO(3),SE(2) and

SL(2).

Given x∈ P, the isotropy subgroup of xis

Gx={g∈G|g·x=x}.

The Lie algebra gxof Gxconsists of those ξ∈gfor which ξP(x) = 0, and the

ﬁxed point set of K

Fix(K, P) = {x∈ P | K·x=x},

J. Montaldi, Relative equilibria and conserved quantities . . . . 246

consists of those points whose isotropy subgroup contains K. It is not hard

to show that it’s a submanifold of P. Moreover, those points with isotropy

precisely Kform an open (possibly empty) subset of Fix(K, P).

Stratiﬁcation by orbit type If Gacts on a manifold P, then the orbit space

P/G is smooth at points where Gpis trivial, and more generally where the

orbit type in a neighbourhood of pis constant.

More generally, for each subgroup H < G one deﬁnes the orbit type stra-

tum P(H)to be the set of points pfor which Gpis conjugate to H. This is a

union of G-orbits, and its image in P/G is also called the orbit type stratum

(now in the orbit space). These orbit type strata are submanifolds of Pand

P/G, and they ﬁt together to form a locally trivial stratiﬁcation (i.e. locally

it has a product structure).

For dynamical systems, the importance of this partition in to orbit type

strata, is that for an equivariant vector ﬁeld, the strata are preserved by the

dynamics.

Slice to a group action Aslice to a group action at x∈ P is a submanifold of

Pwhich is transverse to the orbit through xand of complementary dimension.

If possible, the slice is chosen to be invariant under the isotropy subgroup Gx

(this is always possible if Gxis compact). A basic result of the theory of Lie

group actions is that under the orbit map P → P/G the slice pro jects to a

neighbourhood of the image of G·xin the orbit space.

Principle of symmetric criticality This principle is the variational version of

discrete reduction, and provides a useful method for ﬁnding critical points of

invariant functions. It states that, if Gacts on a manifold P, and if f:P → R

is a smooth invariant function, then x∈Fix(G, P) is a critical point of fif

and only if it is a critical point of the restriction f|Fix(G,P)of fto Fix(G, P).

One proof is to use an invariant Riemannian metric to deﬁne an equivariant

vector ﬁeld ∇f, which being equivariant, is tangent to Fix(G, P). For a full

proof, valid also in inﬁnite dimensions, see [58].

2 Noether’s Theorem and the Momentum Map

The purpose of this section is to bring together facts about symmetry and

conserved quantities that are useful for studying bifurcations. They are all

found in various places, more or less explicitly, but not together in a single

source. Furthermore, there appear to be misconceptions about whether “non-

equivariant” momentum maps cause extra problems. Essentially, anything

true for the equivariant ones remains true for non-equivariant ones (which are

J. Montaldi, Relative equilibria and conserved quantities . . . . 247

in fact equivariant, but for a modiﬁed action, as we shall see below). Many

of the details of this chapter can be found in the books [16,8,10,13].

Examples 2.1 We will be using 3 examples of symplectic group actions in

this section to illustrate various points. These arise in the models of systems

of point vortices.

(A) G=SO(3) acts by rotations on the sphere P=S2, with symplectic

form given by the usual area form with total area 4π(for example in spherical

polars, ω= sin(θ)dθ ∧dφ). The Lie algebra so(3) can be represented by skew-

symmetric matrices, and the vector ﬁeld corresponding to the skew-symmetric

matrix ξis simply x7→ ξx.

(B) G=SE(2) acts on the plane P=R2, with its usual symplectic form ω=

dx ∧dy. This group acts by translation and rotations; indeed, SE(2) ≃R2⋊

SO(2) (semidirect product), where R2is the normal subgroup of translations

of the plane, and SO(2) is the group of rotations about some point, e.g.

the origin. The Lie algebra se(2) is represented by constant vector ﬁelds

(corresponding to the translation subgroup) and by inﬁnitesimal rotations.

There are several ways to realize this action, of which perhaps the best-known

is to use M¨obius transformations on the upper-half plane. However, we will

use one that is more in keeping with the others, which is to represent the

hyperbolic plane as one sheet of a 2-sheeted hyperboloid in R3:

H={(x, y, z)∈R3|z2−x2−y2= 1, z > 0}.(2.1)

(The hyperbolic metric on His induced from the Minkowski metric (dx2+

dy2−dz2) on R3.) The symplectic form on His given by

ωx(u,v) = −1

2kxk2R(x)·u∧v,(2.2)

where R(x, y, z) = (x, y, −z) and u,v∈TxH. One way to realize the action

of SL(2) on His to embed Hinto the set of 2 ×2-trace zero matrices sl2(R):

x=



z

7→ ˆ

x=x y +z

y−z−x.(2.3)

Then A·ˆ

x=Aˆ

xA−1, for A∈SL(2). Note that the image of the embedding

consists of those matrices Xof trace zero, unit determinant and such that

X12 > X21. Under this identiﬁcation, the symplectic form (2.2) becomes the

Kostant-Kirillov-Souriau symplectic form on the coadjoint orbit (see Example

2.5(C)). 2

J. Montaldi, Relative equilibria and conserved quantities . . . . 248

2.1 Noether’s theorem

With such a set-up, the famous theorem of Emmy Noether states that any

1-parameter group of symmetries is associated to a conserved quantity for

the dynamics. In fact one needs some hypothesis such as the phase space

being simply connected, or the group being semisimple (see [8] for details).

For example, the circle group acting on the torus does not produce a globally

well-deﬁned conserved quantity.

How do these conserved quantities come about? We already have a pro-

cedure for passing from Hamiltonian function to Hamiltonian vector ﬁeld, and

here we apply the reverse procedure. For each ξ∈glet φξ:P → Rbe a

function that satisﬁes Hamilton’s equation

dφξ=ω(ξP,−),(2.4)

if such a function exists. Of course each of these φξis only deﬁned up to a

constant, since only dφξis determined.

Such functions are known as momentum functions, and a symplectic ac-

tion for which such momentum functions exist is said to be a Hamiltonian

action. See [8] for conditions under which symplectic actions are Hamilto-

nian.

Theorem 2.2 (Noether) Consider a Hamiltonian action of the Lie group G

on the symplectic (or Poisson) manifold P, and let Hbe an invariant Hamil-

tonian. Then the ﬂow of the Hamiltonian vector ﬁeld leaves the momentum

functions φξinvariant.

Proof: A simple algebraic computation:

XH(φξ) = {H, φξ}=−{φξ, H }=−ξP(H) = 0.

This last equation holds because His G-invariant. ❒

Momentum map We leave questions of dynamics now, and consider the

structure of the set of momentum functions. The ﬁrst observation is that the

momentum functions φξcan be chosen to depend linearly on ξ. For example,

let {ξ1,...,ξd}be a basis for g, and let φξ1,...,φξdbe Hamiltonian functions

for the associated vector ﬁelds. Then for ξ=a1ξ1+···+adξd, one can put,

φξ=a1φξ1+···+adφξd.

It is easy to check that such φξsatisfy the necessary equation (2.4).

Thus, for each point p∈ P we have a linear functional ξ7→ φξ(p), which

we call Φ(p), so that

Φ(p) = (φξ1(p),...,φξd(p))

J. Montaldi, Relative equilibria and conserved quantities . . . . 249

is a map Φ : P → g∗, where g∗is the vector space dual of the Lie algebra

g. Such a map is called a momentum map. The deﬁning equation for the

momentum map is,

hdΦp(v), ξi=ωp(ξP(p),v).(2.5)

for all p∈ P all v∈TpPand all ξ∈g. An immediate and important

consequence of (2.5) is:

ker(dΦp) = (g·p)ω(2.6)

im(dΦp) = g⊥

p⊂g∗(2.7)

Here, Uωis the linear space that is orthogonal to Uwith respect to the

symplectic form. In particular, the momentum map is a submersion in a

neighbourhood of any point where the action is free (or locally free: gp= 0).

The fact that Φ is only determined by the diﬀerential condition (2.5)

means that it is only deﬁned up to a constant. It follows that if Φ is a

momentum map, then Φ1is a momentum map if and only if there is some

C∈g∗for which, for all p∈ P,

Φ1(p) = Φ(p) + C.

We return to the possibility of choosing a diﬀerent Φ below.

Examples 2.3 Here we see momentum maps for three symplectic actions

that arise for the point-vortex models, ﬁrstly for point vortices on the sphere,

secondly for those in the plane and thirdly on the hyperbolic plane. We also

give the general formula for momentum maps for coadjoint actions.

(A) Let G=SO(3) act diagonally on the product P=S2×... ×S2(N

copies). On the ith factor we put the symplectic form ωi=λiω0, where ω0is

the canonical area form on the unit sphere (RS2ω0= 4π). Then a momentum

map is given by

Φ(x1,...,xN) = X

λjxj,(2.8)

where the Lie algebra so(3) (consisting of skew symmetric 3 ×3 matrices)

is identiﬁed with R3in the “usual way”: B∈so(3) corresponds to b∈R3

satisfying Bu =b×ufor all vectors u∈R3. It is clear that this momentum

map is equivariant with respect to the coadjoint action, which under this

identiﬁcation becomes the usual action of SO(3) on R3.

(B) An analogous example is G=SE(2) acting on a product of Nplanes,

with symplectic form ω=⊕jλjωj, where ωjis the standard symplectic form

J. Montaldi, Relative equilibria and conserved quantities . . . . 250

on the jth plane. If we write SE(2) as R2⋊SO(2) (semidirect product), then

the natural momentum map is

Φ(x1,...,xN) = 

X

λjJxj,1

2Pjλj|xj|2

.(2.9)

where J=0−1

1 0 is the matrix for rotation through π/2.

copies of the hyperbolic plane, P=HN, with symplectic form ω=⊕jλjωj,

where ωjis the standard symplectic form on the jth plane, see (2.2). The

natural momentum map is given by,

Φ(x1,...,xN) = X

λjxj.(2.10)

Remark 2.4 Consider any group acting on a manifold X, the conﬁguration

space. Classical mechanics of the “kinetic + potential” type takes place on the

cotangent bundle of a conﬁguration space, and in this setting the given action

on Xinduces a symplectic action on the cotangent bundle, by the formula

g·(x, p) = (g·x, (dgx)−Tp),

where A−Tis the inverse transpose of the operator A. Such actions are called

cotangent actions or cotangent lifts and they always preserve the canonical

symplectic form ωon the cotangent bundle.

The momentum map for cotangent actions always exists, and is given by

hΦ(x, p), ξi=hp, ξP(x)i,

where the pairing on the right is between T∗

xXand TxX. For example, if

P=T∗R3is the phase space for a central force problem, which has SO(3)

symmetry, then after identifying so(3)∗with R3as above, the momentum

map Φ : P → so(3)∗is just the angular momentum.

2.2 Equivariance of the momentum map

A natural question arises: since a momentum map Φ : P → g∗is deﬁned on a

space Pwith an action of the group G, is there an action of Gon g∗for which

Φ commutes with (intertwines) the two actions? The answer was given in the

aﬃrmative by Souriau [16]. Usually, but not always, this turns out to be the

coadjoint action of Gon g∗, and before stating Souriau’s result we give some

examples of coadjoint action.

J. Montaldi, Relative equilibria and conserved quantities . . . . 251

Examples 2.5 The coadjoint actions for the groups described in Examples

2.1 are as follows.

(A) For G=SO(3), the Lie algebra g=so(3) consists of the 3 ×3 skew-

symmetric matrices, and via the inner product hA, Bi= tr(ABT) we can

identify the dual g∗with g. An easy computation shows that the coadjoint

action is given by

CoadAµ=AµAT.

With the usual identiﬁcation of so(3) with R3(see Example 2.3) the coadjoint

action is just the usual action of SO(3) by rotations, so that the orbits for the

coadjoint actions are spheres centered at the origin, and the origin itself. For

each of the 2 types of orbit, the isotropy subgroup is either all of SO(3), or

it is a circle subgroup SO(2) of SO(3). This variation of the symmetry type

of the orbits has interesting repercussions for the dynamics, and in particular

for the families of relative equilibria.

(B) Consider now the 3-dimensional non-compact group G=SE(2) of

Euclidean motions of the plane. As a group this is a semidirect product

R2⋊SO(2), where R2acts by translations and SO(2) by rotations about

some given point (the “origin”). Elements (u, R)∈R2⋊SO(2) can be iden-

tiﬁed with elements

Ru

0 1 ∈GL(R2×R),

by introducing homogeneous coordinates. A calculation shows that the adjoint

action is

Ad(u,R)(v, B) = (Rv−Bu, B),

since Band Rcommute. The coadjoint action is then

Coad(u,R)(ν, ψ) = (Rν, ψ +RνuT).(2.11)

Note that since elements of so(2) are skew-symmetric, it follows that only

the skew-symmetric part of ψ+RνuTis relevant. One can therefore replace

ψ+RνuTby ψ+1

2(RνuT−uνTRT). A nice representation of this using

complex numbers is given in Section 6.

The coadjoint orbits are again of two types: ﬁrst the cylinders with axis

along the 1-dimensional subspace of g∗which annihilates the translation sub-

group (or its subalgebra), and secondly the individual points on that axis;

that is, the points of the form (0, ψ). In this case the two types of isotropy

subgroup for the coadjoint action are ﬁrstly the translations in a given direc-

tion (orthogonal to ν), so isomorphic to R, or in the case of a single point

J. Montaldi, Relative equilibria and conserved quantities . . . . 252

Figure 2. Coadjoint orbits for SO(3) and for SE(2)

on the axis, it is the whole group. In both cases the isotropy subgroup is

non-compact, a fact to be contrasted with the modiﬁed coadjoint action to be

deﬁned below.

CoadAµ=A−TµAT.

Notice that the determinant of the matrix in sl(2)∗is constant on each coad-

joint orbit. In fact the coadjoint orbits are of 4 types: (i) the 1-sheeted

hyperboloids (where Gµ≃R), (ii) one sheet of the 2-sheeted hyperboloids

(where Gµ≃SO(2)), (iii) each sheet of the cone with the origin removed

(where Gµ≃R), and (iv) the origin itself (where Gµ=SL(2)). 2

It is easy to see that the momentum maps given in Examples 2.3(A,C)

are equivariant with respect to the coadjoint actions described above, which

is not surprising in the light of the theorem below. However this is not always

true in the case of G=SE(2).

To describe the action that makes Φ equivariant we follow Souriau and

deﬁne the cocycle

θ:G→g∗

g7→ Φ(g·x)−CoadgΦ(x),(2.12)

It is of course necessary to show that this expression is independent of x, which

it is provided Pis connected. We leave the details to the reader: it suﬃces to

diﬀerentiate with respect to xand use the invariance of the symplectic form.

The map θdeﬁned above allows one to deﬁne a modiﬁed coadjoint action,

Coadθ

gµ:= Coadgµ+θ(g).(2.13)

A short calculation shows that this is indeed an action. Moreover, this action

is by aﬃne transformations whose underlying linear transformations are the

coadjoint action.

J. Montaldi, Relative equilibria and conserved quantities . . . . 253

Theorem 2.6 (Souriau) Let the Lie group Gact on the connected symplec-

tic manifold Pin such a way that there is a momentum map Φ : P → g∗. Let

θ:G→g∗be deﬁned by (2.12). Then Φis equivariant with respect to the

modiﬁed action on g∗:

Φ(g·x) = Coadθ

gΦ(x).

Furthermore, if Gis either semisimple or compact then the momentum map

can be chosen so that θ= 0.

For proofs see [16] or [8]; the ﬁrst proof of equivariance in the compact

case appears to be in [48].

Examples 2.7 In Examples 2.3 we gave the momentum maps for the three

point-vortex models, and pointed out that for SO(3) and SL(2) the mo-

mentum map is equivariant with respect to the usual coadjoint action (not

surprisingly in view of the theorem above since SO(3) is compact and SL(2)

is semisimple). However, this is not always true in the planar case:

(B) Consider the momentum map given in Example 2.3(B) for the point

vortex model in the plane:

Φ(x1,...,xN) = 

X

λjJxj,1

2Pjλj|xj|2

.

To ﬁnd the action that makes this momentum map equivariant, we compute

Φ(Ax1+u,...,AxN+u) = AX

λjJxj,1

2Pjλj|xj|2+PjλjAxj.u+

+ ΛJu,1

2|u|2

= Coad(u,A)Φ(x1,...,xN) + Λ(Ju,1

2|u|2),(2.14)

where Λ = Pjλj∈R, and Coad is given in (2.11). The cocycle associated to

this momentum map is thus given by θ(u, A) = Λ(Ju,1

2|u|2). If Λ = 0 then

Φ is equivariant with respect to the usual coadjoint action, while if Λ 6= 0

it is equivariant with respect to a modiﬁed coadjoint action. Furthermore,

one can show that in this latter case there is no constant vector C∈g∗for

which Φ + Cis coadjoint-equivariant. Indeed, it is enough to see that the

orbits for this modiﬁed coadjoint action are in fact paraboloids, with axis the

annihilator in g∗of R2⊂gand these are not translations of the coadjoint

orbits, which are either cylinders or points. See Figure 3. Furthermore, a

short calculation shows that the isotropy subgroups for this action are all

compact: Gµ≃SO(2), for all µ∈se(2)∗, which is quite diﬀerent from the

coadjoint action. 2

J. Montaldi, Relative equilibria and conserved quantities . . . . 254

Figure 3. Modiﬁed coadjoint orbits for SE(2)

2.3 Reduction

Since by Noether’s theorem the dynamics preserve the level sets of the mo-

mentum map Φ, it makes sense to treat the dynamical problem one level set at

a time. However, these level sets are not in general symplectic submanifolds,

and the system on them is therefore not a Hamiltonian system. However, it

turns out that if one passes to the orbit space of one of these level sets of

Φ, then the resulting reduced space is symplectic, and the induced dynamics

are Hamiltonian. Historically, this process of reduction was ﬁrst used by Ja-

cobi, in what is called “elimination of the nodes”. However, its systematic

treatment is much more recent, and is due to Meyer [47] and independently

to Marsden and Weinstein [43].

As usual suppose the Lie group Gacts in a Hamiltonian fashion on the

symplectic manifold P, and let Φ : P → g∗be a momentum map which is

equivariant with respect to a (possibly modiﬁed) coadjoint action as discussed

above. Consider a value µ∈g∗of the momentum map. Then since Φ is

equivariant, the isotropy subgroup Gµof this modiﬁed coadjoint action acts

on the level set Φ−1(µ). Deﬁne the reduced space to be

Pµ:= Φ−1(µ)/Gµ.(2.15)

That is, two points of Φ−1(µ) are identiﬁed if and only if they lie in the same

group orbit. This deﬁnes Pµas a set, but to do dynamics one needs to give

it more structure.

If the group Gacts freely near p(i.e. Gpis trivial), then Pµis a smooth

manifold near the image of pin Pµ. This is for two reasons: ﬁrstly Φ is a

submersion near pby (2.7), and secondly the orbit space by Gµwill have

no singularities. This is the case of regular reduction. We discuss singular

reduction brieﬂy below.

It is important to know whether the induced dynamics on the reduced

J. Montaldi, Relative equilibria and conserved quantities . . . . 255

spaces are also Hamiltonian. The answer of course is “yes”. To see this it

is necessary to deﬁne the symplectic form ωµon Pµ. Let ¯u, ¯v∈T¯pPµ, be

projections of u, v ∈TpP, and deﬁne

ωµ(¯u, ¯v) := ω(u, v).

Of course, one must show this to be well deﬁned, non-degenerate and closed:

exercises left to the reader!

The Hamiltonian is an invariant function on Pso its restriction to Φ−1(µ)

is Gµ-invariant and so induces a well-deﬁned function on Pµ, denoted Hµ. The

dynamics induced on the reduced space is then determined by a vector ﬁeld

Xµwhich satisﬁes Hamilton’s equation: dHµ=ωµ(Xµ,−).

The orbit momentum map It is clear that Pµ= Φ−1(µ)/G can also be

deﬁned by Pµ= Φ−1(Oµ)/G, where Oµis the coadjoint orbit through µ.

Since Φ−1(Oµ)/G ⊂ P/G it is natural to use the orbit momentum map:

ϕ:P/G →g∗/G deﬁned by

PΦ

−→ g∗

↓ ↓

P/G ϕ

−→ g∗/G

(2.16)

where the vertical arrows are the quotient maps. Then Pµ=ϕ−1(Oµ). This is

very useful for studying bifurcations as the momentum value varies. However,

it may not be so useful if Gµis not compact, for there the orbit space g∗/G

is not in general a reasonable space (it is not even Hausdorﬀ for example for

the coadjoint action of SE(2) near the ψ-axis).

2.4 Singular reduction

If the action of Gon Pis not free, then the reduced space is no longer a

manifold. However, it was shown by Sjamaar and Lerman [67] that if Gis

compact then the reduced space can be stratiﬁed—that is, decomposed into

ﬁnitely many submanifolds which ﬁt together in a nice way—and each stratum

has a symplectic form which determines the dynamics on that stratum. In

fact these strata are simply the sets of constant orbit type in Φ−1(µ), or

rather their images in Pµ. For the case of a proper action of a non-compact

group see [20]. The theorem follows from the local normal form of Marle and

Guillemin-Sternberg [8,42].

J. Montaldi, Relative equilibria and conserved quantities . . . . 256

2.5 Symplectic slice and the reduced space

Recall that a slice to a group action at a point p∈ P is a submanifold S

through psatisfying TpS⊕g·p=TpP. If Gpis compact, it can be chosen

to be invariant under Gp. The slice, or more precisely S/Gp, provides a local

model for the orbit space P/G.

In the symplectic world, one needs to take into account the symplectic

structure, and one wants the “symplectic slice” to provide a local model for

the reduced space Pµ. In the case of free (or locally free) actions this is fairly

straightforward, but in the general case, this is more delicate because the

momentum map is singular. For this reason, the symplectic slice is usually

taken to be a subspace of TpPrather than a submanifold of P.

Deﬁnition 2.8 Suppose Gpis compact. Deﬁne N⊂TpPto be a Gpinvariant

subspace satisfying TpP=N⊕g·p. The symplectic slice is then

N1:= N∩ker(dΦp).

It follows from the implicit function theorem that local coordinates can

be chosen that identify a transversal to Gµ·pwithin the possibly singular set

Φ−1(µ) with a subset of the symplectic slice N1.

3 Relative Equilibria

An equilibrium point is a point in the phase space that is invariant under the

dynamics: p∈ P for which XH(p) = 0, or equivalently dHp= 0, and one way

to deﬁne a relative equilibrium is as a group orbit that is invariant under the

dynamics. Although geometrically appealing, this is not the most physically

transparent deﬁnition.

Deﬁnition 3.1 Arelative equilibrium is a trajectory γ(t) in Psuch that for

each t∈Rthere is a symmetry transformation gt∈Gfor which γ(t) =

gt·γ(0).

In other words, the trajectory is contained in a single group orbit. It

is clear that if a group orbit is invariant under the dynamics, then all the

trajectories in it are relative equilibria; and conversely, if γ(t) is the trajectory

through p, then g·γ(t) is the trajectory through g·pand consequently the

entire group orbit is invariant as claimed above.

For N-body problems in space, relative equilibria are simply motions

where the shape of the body does not change, and such motions are always

rigid rotations about some axis.

Proposition 3.2 Let Φbe a momentum map for the G-action on Pand let

Hbe a G-invariant Hamiltonian on P. Let p∈ P and let µ= Φ(p). Then

J. Montaldi, Relative equilibria and conserved quantities . . . . 257

the following are equivalent:

1The trajectory γ(t)through pis a relative equilibrium,

2The group orbit G·pis invariant under the dynamics,

3∃ξ∈gsuch that γ(t) = exp(tξ)·p, ∀t∈R,

4∃ξ∈gsuch that pis a critical point of Hξ=H−φξ,

5pis a critical point of the restriction of Hto the level set Φ−1(µ).

Remarks 3.3 (i) The vector ξappearing in (3) is the angular velocity of the

relative equilibrium. It is the same as the vector ξappearing in (4). The

angular velocity is only unique if the action is locally free at p; in general it

is well-deﬁned modulo gp.

(ii) If Φ−1(µ) is singular then it has a natural stratiﬁcation (see §2.4), and

condition (4) of the proposition should be interpreted as being a stratiﬁed

critical point; that is all derivatives of Halong the stratum containing p

vanish at p.

(iii) Notice that (3) implies that relative equilibria cannot meandre around

a group orbit, but must move in a rather rigid fashion. It follows from this

equation that the trajectory is in fact a dense linear winding on a torus, at

least if Gis compact. The dimension of the torus is at most equal to the rank

of the group G. For G=SO(3), the rank is 1, and this means that any re

that is not an equilibrium is in fact a periodic orbit.

Proof: The equivalence of (1) and (2) is outlined above.

Equivalence of (1) and (3): (3) ⇒(1) is clear. For the converse, since

γ(t) lies in G·pso its derivative XH(p) = ˙γ(0) lies in the tangent space

g·p=Tp(G·p). Let ξ∈gbe such that XH(p) = ξP(p). Then by equivariance

XH(g·p) = (Adgξ)P(g·p), see (1.6). Let gt= exp(tξ). Then since Adgtξ=ξ,

dt(gt·p) = ξP(gt·p) = XH(gt·p).

That is, t7→ gt·pis the unique solution through p.

Equivalence of (3) and (4): (3) is equivalent to XH(p) = ξP(p). Using

the symplectic form, this is in turn equivalent to dH(p) = dφξ(p).

Equivalence of (4) and (5): If Φ is submersive at pthen Φ−1(µ) is a

submanifold of Pand this is just Lagrange multipliers, since dφξ=hdΦ(p), ξi.

In the case that Φ is singular, the result follows from the principle of symmetric

criticality (see the end of the Introduction for a statement). If Φ is singular at

p, then by (2.7) gp6= 0. Consider the restriction of Hto Fix(Gp,P). By the

J. Montaldi, Relative equilibria and conserved quantities . . . . 258

theorem of Sjamaar and Lerman, the set Φ−1(µ) is stratiﬁed by the subsets

of constant orbit type—see §2.4. Consider then the set PGpof points with

isotropy precisely Gp(this an open subset of Fix(Gp,P) containing p), and

restrict both Φ and Hto this submanifold. Now Φ restricted to PGpis of

constant rank, so that Φ−1(µ)∩PGpis a submanifold. The result then follows

as before, since by the principle of symmetric criticality Hrestricted to PGp

has a critical point at pif and only if Hhas a critical point at p.❒

One of the earliest systematic investigations of relative equilibria was in

a paper of Riemann where he classiﬁed all possible relative equilibria in a

model of aﬃne ﬂuid ﬂow that is now called the Riemann ellipsoid problem

(or aﬃne rigid body or pseudo-rigid body)—see [25] and [66] for details. The

conﬁguration space is the set of all 3×3 invertible matrices, and the symmetry

group is SO(3)×SO(3). In that paper he identiﬁed the 6 conserved quantities

and found geometric restrictions on the possible forms of relative equilibria

using the fact that the momentum is conserved, and that the solutions are

of the form given in (3) of the proposition above. In terms of general group

actions, the geometric condition Riemann used is the following, which follows

immediately from Proposition 3.2 above together with the conservation of

momentum.

Corollary 3.4 Let p∈ P be a point of a relative equilibrium, of angular

velocity ξ, and let θbe the cocycle associated to the momentum map, then

coadθ

ξΦ(p) = 0.

If Gis compact, so the adjoint and coadjoint actions can be identiﬁed and

θ= 0 for a suitable choice of momentum map, this means that the angular

velocity and momentum of a relative equilibrium commute. For example, for

any system with symmetry SO(3) this means that at any relative equilibrium,

the angular velocity and the value of the momentum are parallel.

Deﬁnition 3.5 A relative equilibrium through pis said to be non-degenerate

if the restriction of the Hessian d2Hξ(p) to the symplectic slice N1is a non-

degenerate quadratic form.

Deﬁnition 3.6 A point µ∈g∗is a regular point of the (modiﬁed) coadjoint

action if in a neighbourhood of µall the isotropy subgroups are conjugate.

Examples of regular points are: all points except the origin for the coad-

joint action of SO(3), all points except the special axis for the coadjoint action

of SE(2), and all points for the modiﬁed coadjoint action of SE(2) described

in Example 2.7(B).

The following result, the ﬁrst on the structure of the families of relative

equilibria, was observed by V.I. Arnold in [2]. The proof is an application of

J. Montaldi, Relative equilibria and conserved quantities . . . . 259

the implicit function theorem.

Theorem 3.7 (Arnold) Suppose that plies on a non-degenerate relative

equilibrium, with Gp= 0 and µ= Φ(p)a regular point of the (modiﬁed)

coadjoint action. Then in a neighbourhood of pthere exists a smooth family

of relative equilibria parametrized by µ∈g∗.

Lyapounov stability A compact invariant subset Sof phase space is said to be

Lyapounov stable if “any motion that starts nearby remains nearby”, or more

precisely, for every neighbourhood Vof Sthere is another neighbourhood

V′⊂Vsuch that every trajectory intersecting V′is entirely contained in V.

A compact subset is said to be Lyapounov stable relative to or modulo G

if the Vand V′above are only required to be G-invariant subsets.

The principal tool for showing an equilibrium to be Lyapounov stable is

Dirichlet’s criterion, which is that if the equilibrium point is a non-degenerate

local minimum of the Hamiltonian, then it is Lyapounov stable. The proof

consists of noting that in this case the level sets of the Hamiltonian are topolog-

ically spheres surrounding the equilibrium, and so by conservation of energy,

if a trajectory lies within one of these spheres, it remains within it.

It is reasonably clear ﬁrstly that it is suﬃcient if any conserved quantity

has a local minimum at the equilibrium point, not necessarily the Hamiltonian

itself, and secondly that the local minimum may in fact be degenerate. These

observations lead to the notion of . . .

Extremal relative equilibria A special role is played by extremal relative

equilibria. This is partly due to Dirichlet’s criterion for Lyapounov stability,

and partly because of their robustness.

A relative equilibrium is said to be extremal if the reduced Hamiltonian

Hµon Pµhas a local extremum (max or min) at that point. This is usually

established by showing the restriction to the symplectic slice of the Hessian

of Hξ=H−φξto be positive (or negative) deﬁnite, for some ξfor which Hξ

has a critical point at p, see Proposition 3.2.

It is of course conceivable that a relative equilibrium is extremal while the

Hessian matrix is degenerate. The simplest case of this is for H(x, y) = x2+y4

in the plane. In fact this arises in the case of 7 identical point vortices in the

plane. The conﬁguration where they lie at the vertices of a regular heptagon

is a relative equilibrium, and the Hessian of Hon the symplectic slice is only

positive semi-deﬁnite. However, a lengthy calculation shows that the relevant

fourth order terms do not vanish, and the reduced Hamiltonian does indeed

have a local minimum there.

For the following statement, recall that any momentum map is equivariant

with respect to an appropriate action of Gon g∗(Section 2), and for µ∈g∗

J. Montaldi, Relative equilibria and conserved quantities . . . . 260

we write Gµfor the isotropy subgroup of µfor that action.

Theorem 3.8 ([48,34]) Let Gact properly on Pwith momentum map Φ,

and suppose p∈Φ−1(µ)is an extremal relative equilibrium, with Gµcompact.

Then,

(i) The relative equilibrium is Lyapounov stable, relative to G;

(ii) There is a G-invariant neighbourhood Uof psuch that, for all µ′∈Φ(U)

there is a relative equilibrium in U∩Φ−1(µ′).

The proof is mostly point-set topology on the orbit space and using the

orbit momentum map (2.16), though part (ii) uses the deeper property of

(local) openness of the momentum map. In fact the proof of (i) also holds if µis

a regular point for the (modiﬁed) coadjoint action, even if Gµis not compact.

By Proposition 2.4 of [35] this result can be reﬁned to conclude that the

relative equilibrium is stable relative to Gµ, as described by Patrick [59]. Note

that the compactness of Gµensures that gµhas a Gµ-invariant complement in

g, as required in [35]. A consequence of using point-set topology is that there

is very little information on the structure of the family of relative equilibria;

for such information see Section 5.

The crucial remaining point is how to determine whether a given relative

equilibrium is extremal. In the case of free actions this was done by the

so-called energy-momentum and/or energy-Casimir methods of Arnold and

Marsden and others. Recently this was extended to the general case of proper

actions:

Proposition 3.9 (Lerman, Singer [35]) Let p∈Φ−1(µ)be a relative equi-

librium satisfying the same hypotheses as the theorem above, and let ξ∈gbe

any angular velocity of the re as in Proposition 3.2. If the restriction to the

symplectic slice of the quadratic form d2Hξis deﬁnite, then the re is extremal,

and so Lyapounov stable relative to Gµ.

4 Bifurcations of (relative) equilibria

In this section we take an extremely brief look at the typical bifurcations of

equilibria in families of Hamiltonian systems as a single parameter is varied.

These results also apply to relative equilibria, provided the reduced space is

smooth in a neighbourhood of the relative equilibrium, and failing that, it

applies to the stratum containing the relative equilibrium. Bifurcations of

relative equilibria near singular points of the reduced space have not been

investigated systematically.

J. Montaldi, Relative equilibria and conserved quantities . . . . 261

4.1 One degree of freedom

Saddle-centre bifurcation

This is the generic bifurcation of equilibria in 1 d.o.f. See Fig. 4. For

λ < 0 there are two equilibrium points: at (x, y) = (±√−λ, 0), one local

extremum and one saddle. As λ→0 these coalesce and then disappear

for λ > 0. One notices also a homoclinic orbit for λ < 0, connecting the

saddle point with itself, which can also be seen as a limit of the family of

periodic orbits surrounding the stable equilibrium. In terms of eigenvalues of

the associated linear system, this bifurcation can be seen as a pair of simple

imaginary eigenvalues (a centre) decreases along the imaginary axis, collide at

0 and “then” emerge along the real axis (a saddle). This description is slightly

misleading as the centre and saddle coexist. At the point of bifurcation, the

linear system is 0 1

0 0 ; that is, it has non-zero nilpotent part.

Note that this bifurcation is compatible with an antisymplectic (i.e. time-

reversing) symmetry (x, y)→(x, −y).

λ < 0λ= 0 λ > 0

H(x, y) = 1

2y2+1

3x3+λx + h.o.t.

Figure 4. Saddle-centre bifurcation

Symmetric pitchfork This is usually caused by symmetry, and is 2 saddle

centre bifurcations occurring simultaneously. On one side of the critical pa-

rameter value, there are 3 coexisting equilibria, while on the other side there

is only one. In fact there are two types of pitchfork: a supercritical pitchfork

involves two centres collapsing into a central saddle, leaving a single centre,

and a subcritical pitchfork involves two saddles collapsing into a central centre,

leaving a single saddle. See Figures 5 and 6.

These results and normal forms are derived from Singularity/Catastrophe

Theory.

J. Montaldi, Relative equilibria and conserved quantities . . . . 262

λ < 0λ= 0 λ > 0

H(x, y) = 1

2y2+x4+λx2+ h.o.t.

Figure 5. Supercritical Pitchfork

λ < 0λ= 0 λ > 0

H(x, y) = 1

2y2−x4−λx2+ h.o.t.

Figure 6. Subcritical Pitchfork

4.2 Higher degrees of freedom

From the point of view of bifurcations of equilibria, or of critical points of

the Hamiltonian, the results for 1 degree of freedom carry over to higher

dimensions, by adding a sum of quadratic terms in the other variables. So for

example the saddle-centre bifurcations has as “normal form”,

Hλ(x,y) = 1

2y2

1+1

3x3

1+λx1+1

2Pj>1(±x2

j±y2

j) + h.o.t.

Whether any of the equilibria are stable depends of course on the signs of the

quadratic terms.

On the other hand, it is a much more subtle question as to whether any

of the associated dynamics, such as the heteroclinic connections, survive this

J. Montaldi, Relative equilibria and conserved quantities . . . . 263

passage into higher dimensions. For the saddle-centre bifurcation, see [22] and

for the time-reversible case, see the lectures of Eric Lombardi in this volume,

and for more detail [39].

5 Geometric Bifurcations

In this section we discuss bifurcations of the families of relative equilibria

(res) due to degenerations in the geometry of the momentum map. One

understands fairly well now the geometry of the family of relative equilibria

in the neighbourhood of a point where the reduced phase spaces change in

dimension, which occurs at special values of the momentum map, provided

however that the group action on the phase space is (locally) free, so that

the momentum map is submersive. However, the general structure of relative

equilibria near points with continuous isotropy is not so well understood,

although some recent progress has been made.

Notation Throughout the theorems below, we will suppose that p∈ P is a

point on a non-degenerate re, that the angular velocity of this re is ξand the

momentum value is µ. Recall that an re is said to be non-degenerate if the

Hessian d2

Hξrestricted to the symplectic slice is a non-degenerate quadratic

form.

In an important paper [60], George Patrick investigates the structure

of the set of relative equilibria as quoted in the following theorem. He also

studied the nearby dynamics and introduced the notion of drift around relative

equilibria in terms of the linearized vector ﬁeld there, but we will not be

describing that aspect here.

Theorem 5.1 (Patrick [60]) Assume Gis compact, Gpis ﬁnite and Gξ∩

Gµis a maximal torus. Then in a neighbourhood of pthe set of relative

equilibria forms a smooth symplectic submanifold of Pof dimension dim(G)+

rank(G).

Note that, as matrices, since ξand µcommute they are simultaneously

diagonalizable. Consequently, they are both contained in a common maximal

torus, so that Gξ∩Gµalways contains a maximal torus. The condition of the

theorem is therefore a generic condition. For example, for SO(3) the condition

is satisﬁed if and only if ξand µare not both zero. This theorem has been

reﬁned by Patrick and Mark Roberts [62], where they show that assuming

a generic transversality hypothesis and that as before Gpis ﬁnite, the set of

relative equilibria near pis a stratiﬁed set, with the strata corresponding to

the conjugacy class of the group Gξ∩Gµ.

J. Montaldi, Relative equilibria and conserved quantities . . . . 264

The following result is more of a bifurcation theorem, as it is aimed at

counting the number of relative equilibria on each reduced phase space, near a

given non-degenerate re. Recall that, if µis a regular point for the coadjoint

action and Gpis trivial, then by Arnold’s theorem (Theorem 3.7) there is a

unique re on each nearby reduced space.

Theorem 5.2 (Montaldi [48]) Assume Gµis compact and Gptrivial.

Then for regular µ′near µthere are at least w(Gµ)res on the reduced space

Pµ′, where w(Gµ)is the order of the Weyl group of Gµ. Furthermore, if ξis

regular then there are precisely this number of relative equilibria on Pµ′. (For

non-regular µ′see [48].)

Note that if Gµis a torus, then w(Gµ) = 1, and this result reduces

to Arnold’s in the case that Gis compact. The lower bound of w(G) for

general ξfollows from the Morse inequalities on the coadjoint orbits, and

so presupposes that the res on Pµare all non-degenerate. Without that

assumption one can use the Lyusternik-Schnirelman category giving a lower

bound of 1

2dim(Oµ) + 1.

The proof of this result relies on the local normal form of Marle [42] and

Guillemin-Sternberg [28]. Here I will outline the idea of the proof in the case

that µ= 0. The idea is to use the reduced Hamiltonian rather than the

augmented Hamiltonian Hξ. So, the relative equilibria in question are critical

points of the Hamiltonian restricted to Pµ′, and near pone has

Pµ′≃ P0× Oµ′⊂ P0×g∗.

The reduced space P0can be identiﬁed with the symplectic slice N1(see §2.5),

so that by hypothesis p∈ P0=P0× {0}is a non-degenerate critical point of

the restriction of Hto P0. Write coordinates (y, ν )∈ P × g∗. Then for each

ν, the function H(·, ν) has an isolated non-degenerate critical point y=y(ν).

Deﬁne h:g∗→Rby

h(ν) = H(y(ν), ν).

Then one can show that the restriction hµ′of hto Oµ′has a critical point

at νiﬀ H|Pµ′has a critical point at (y(ν), ν). In this manner, the problem

is reduced to ﬁnding critical points of the restrictions of a smooth function

hto coadjoint orbits Oµ′, and then one can use Morse theory or Lyusternik-

Schnirelman techniques.

The above proof assumes that Gpis trivial. However, if Gpis ﬁnite,

then the proof can be modiﬁed to show that the nearby relative equilibria

correspond to critical points of a smooth Gp-invariant function h, constructed

in the same manner as above, but on P0×g∗rather than on the full orbit

space P0×Gpg∗. This time though, diﬀerent critical points correspond to

J. Montaldi, Relative equilibria and conserved quantities . . . . 265

the same relative equilibrium if they lie in the same Gp-orbit on Oµ′. This

argument gives the following ‘equivariant’ version of Theorem 5.2.

Theorem 5.3 (Montaldi, Roberts [50]) If Gpis ﬁnite, then it acts on

nearby coadjoint orbits Oµ′. If a subgroup Σof Gpis such that Fix(Σ,Oµ′)

consists of isolated points, then they are all relative equilibria.

This result uses the fact that His Gp-invariant, and not that Gpacts

symplectically. In [50] this bifurcation result is applied to ﬁnding relative

equilibria of molecules, and in [37] it is applied to ﬁnding relative equilibria

of systems of point vortices on the sphere, and in both we use antisymplectic

symmetries of Has well as symplectic ones. An action of Gwhere the elements

act either symplectically or antisymplectically, i.e. g∗ω=±ω, is said to be

semisymplectic [51].

In [50], the stability of the bifurcating relative equilibria is also calculated

using these methods. The reader should also see [65,36,62,57] for further

developments.

6 Examples

Let us look at three examples of symmetric Hamiltonian systems.

•Point vortices on the sphere.

•Point vortices in the plane.

•Molecules (as classical mechanical systems).

The motivation for choosing these models is that the ﬁrst is relatively

simple: it has no points where the action fails to be free (unless there are only

2 point vortices), and the group is compact. The bifurcations that arise are

therefore of the types described in Section 5.

The second is similar, except that the group of symmetries is no longer

compact.

The study of the classical mechanics of molecules is also of interest in

molecular spectroscopy, where it is common practice to label particular quan-

tum states in terms of the corresponding classical behaviour. We treat this

example extremely brieﬂy!

6.1 Point vortices on the sphere

The model is a ﬁnite set of point vortices on the unit sphere in 3-space. A

point vortex is an inﬁnitesimal region of vorticity in a 2-dimensional ﬂuid

ﬂow, though we ignore the ﬂuid, and just concentrate on the vortices. The

equations of motion for this system were obtained by V.A. Bogomolov [21].

A study of the dynamics of 3 point vortices has been carried out by Kidambi

J. Montaldi, Relative equilibria and conserved quantities . . . . 266

and Newton [30] and by Pekarsky and Marsden [63]. The case of Nidentical

point vortices has been treated in [37].

If x1,...,xNare the distinct locations of these point-vortices (unit vectors

in R3), then the diﬀerential equation describing their motion is

˙xj=X

k6=j

λk

xk×xj

1−xk·xj

Here λ1,...,λNare the strengths of the vortices: each λjis a non-zero real

number. It turns out that this vector ﬁeld is Hamiltonian, with

H(x) = −1

4πX

j<k

λjλklog (1 −xj·xk),

and Poisson structure

{f, g }(x) = −X

λ−1

jdjf×djg·xj,

where djf=djf(x)∈R3is the diﬀerential of fwith respect to the point

xj∈S2, and x= (x1,...,xN).

The phase space is P=S2×S2×... ×S2\∆, where ∆ is the ‘big

diagonal’ where at least one pair of points coincides, which is removed to

avoid collisions. The symplectic form on Pis given by

ω=λ1ω1⊕ · ·· ⊕ λNωN,

where ωjis the standard area form on the jth copy of S2.

This system has full rotational symmetry G=SO(3), and hence a 3-

component conserved quantity. After identifying so(3) with R3as usual, this

momentum map is the so-called centre of vorticity:

Φ(x) = X

λjxj.

There are a number of immediate general consequences for this system

that can be drawn from the Hamiltonian structure. For example, if all the

vorticities are of the same sign, then as x→∆ in P, so H(x)→+∞and

it follows that Hattains its minimum at some point; this point is necessar-

ily an equilibrium point, and the set on which this minimum is attained is

Lyapounov stable (one would expect this set to be a ﬁnite union of SO(3)-

orbits). Moreover, for any µ∈so(3)∗≃R3the same argument can be applied

on Φ−1(µ), and the minimum on that set is necessarily a relative equilibrium,

by Proposition 3.2.

J. Montaldi, Relative equilibria and conserved quantities . . . . 267

However, if the vorticities are of mixed signs, then there is no general

statement about the existence of equilibria or relative equilibria, except for

N= 3 where all res are known (see below).

If there are some identical vortices, then there is an extra ﬁnite symmetry

group—a subgroup of the permutation group SN. This extra ﬁnite symmetry

group is used to considerable eﬀect in [37], from which Figures 7 and 8 are

taken. Moreover, the time-reversing symmetries obtained from the reﬂexions

in O(3) are also used together with the principle of symmetric criticality to

prove the existence of many types of relative equilibria.

For example, let Chdenote the group of order 2 generated by reﬂexion in

the horizontal plane. Then x∈Fix(Ch) if all the vortices are on the equator.

An application of the arguments above shows that if all the vorticities are of

the same sign then there is a point on Fix(Ch,P) where the restriction of the

Hamiltonian attains its minimum, and by the principle of symmetric critical-

ity (see §1.5) it follows that this is an equilibrium point for the full system

(though no longer a local minimum in general). And the same extension to this

argument as before provides relative equilibria in Fix(Ch,Φ−1(µ)). In general

there will be several relative equilibria in each component of Fix(Ch,Φ−1(µ)).

However, it is shown in [37] that if all the vorticities are of the same sign then

in each component of Fix(Ch,P) there is a unique equilibrium point.

2 vortices If there are only 2 point vortices, then every solution is a relative

equilibrium. Indeed, if Φ(x) = µ6= 0 then the two points rotate at the same

angular velocity about the axis containing µ. The only possible case where

µ= 0 is if λ1=λ2and x1=−x2which is an equilibrium point.

3 vortices There have been two recent studies of the system of 3 point

vortices, by Kidambi and Newton [30] (who also treat the 2 vortex case in

an appendix) and by Pekarsky and Marsden [63]. The former describes not

only the relative equilibria, but also self-similar collapse, where triple collision

occurs in ﬁnite time with the three vortices retaining their same shape up to

similarity. The latter describes the res and their stability using the techniques

described in these lectures (many of which were in fact developed by Marsden

and co-workers).

One of the results of Kidambi and Newton is that there are two classes

of re: those lying on a great-circle and those lying at the vertices of an

equilateral triangle, and any re belongs to one of these classes. Moreover,

every equilateral triangle on the sphere is an re, as we shall prove below.

The following results are mostly taken from [30] and [63], and some are

discussed below. Denote λ1λ2+λ2λ3+λ3λ1by σ2(λ).

Proposition 6.1 For the system of N= 3 point vortices on the sphere,

J. Montaldi, Relative equilibria and conserved quantities . . . . 268

C (R)

3v C (R,p)

Figure 7. Relative equilibria for 3 identical vortices on the sphere

(i) There exist equilibria iﬀ σ2(λ)>0, and they always lie on a great circle.

(ii) All equilateral conﬁgurations are res, and they are Lyapounov stable

modulo SO(2) if σ2(λ)>0, and are unstable if σ2(λ)<0.

(iii) Self-similar collapse occurs iﬀ σ2(λ) = 0.

(iv) The conﬁgurations where the vortices lie on a great circle, at the vertices

of a right-angled isosceles triangle is always an re. Furthermore, the triangle

with x1at the right-angle is Lyapounov stable provided

λ2

2+λ2

3>2σ2(λ).

The phase space is of dimension 6 in this case, and so the orbit space

P/SO(3) is of dimension 3, and points in the orbit space correspond to the

shapes of the triangle formed by the 3 vortices. The obvious set of coordinates

consisting of the three pairwise distances has a problem for great circle con-

ﬁgurations since nearby such a conﬁguration these distances do not determine

the conﬁguration uniquely. Indeed, these three distances are O(3) invariants,

as they do not distinguish the orientation, and conﬁgurations on a great circle

have non-trivial isotropy for the O(3)-action so it is not surprising that these

coordinates have a problem there. For a good set of SO(3)-invariants, one

must use the oriented volume as well, which is what is done in the two works

cited above.

However, the three distances do form a good set of coordinates away from

the great circle conﬁgurations, and we will restrict our attention to those. So,

let r1be the chord distance kx2−x3ketc. Then the Hamiltonian and the

orbit momentum map (2.16) are given by

H(r1, r2, r3) = −λ1λ2

2πlog(r3)−λ2λ3

2πlog(r1)−λ3λ1

2πlog(r2)

ϕ(r1, r2, r3) = |Φ|2= (λ1+λ2+λ3)2−λ1λ2r2

3−λ2λ3r2

1−λ3λ1r2

(6.1)

J. Montaldi, Relative equilibria and conserved quantities . . . . 269

The reduced spaces are then Pµ:= ϕ−1(µ2). The relative equilibria

are determined by the critical points of the restriction of Hto the reduced

spaces, and are therefore critical points of H−ηϕ, for some η(the Lagrange

multiplier). A short calculation shows that the only solutions to this equation

are

r1=r2=r3=r, say, and η= 1/(4πr2).

Thus all relative equilibria away from great circle conﬁgurations are equilateral

triangles. Note that the relation between the angular velocity ξ(Proposition

3.2) and ηis simply ξ= 2ηΦ(x), since

0 = d(H−ηϕ) = d(H−η|Φ|2) = dH −2ηΦ.dΦ.

Of course, Φ 6= 0 since the vortices do not all lie on a great circle.

For the stability of these equilateral res, we ﬁrst calculate the Hessian of

H−ηϕ (this is in fact Arnold’s “energy-Casimir” method):

d2(H−ηϕ)(r) = 4

r2



λ2λ30 0

0λ3λ10

0 0 λ1λ2

.

Now, the tangent space to ϕ−1(µ2) is spanned by (λ1,−λ2,0),(λ1,0,−λ3),

and a computation then shows that the Hessian of the reduced Hamiltonian

is deﬁnite if and only if σ2(λ)>0.

The computations for the conﬁgurations of vortices lying on great circles

are longer, and I will not go into them further here; details can be found in

the original papers [30,63].

Remarks 6.2 (i) The case σ2(λ) = 0 remains to be understood.

(ii) It is not known whether any other form of collapse (i.e. not self-similar)

can occur.

(iii) A further bifurcation occurs at equilateral res which lie on a great

circle which to my knowledge has not been investigated.

4 vortices Much less is known in general about the case of 4 vortices. It is

shown in [63] that the conﬁguration where the four vortices lie at the vertices

of a regular tetrahedron is always a relative equilibrium. It is also easy to show

(from the equations of motion) that a square lying in a great circle is always

an re, independently of the values of the vorticities. However, in contrast to

the 3-vortex case, squares not lying in great circles are not res unless all the

vortices are identical.

In the case that all the vorticities coincide, a classiﬁcation of symmetric

res is given in [37], from which Figure 8 is taken. Moreover, using the implicit

J. Montaldi, Relative equilibria and conserved quantities . . . . 270

C (R,R’)

2v C (2R)

C (R,p)

C (R,2p)

C (R)

Figure 8. Relative equilibria for 4 identical vortices on the sphere

function theorem one can show that many of the res shown to exist in [37]

persist under small perturbations of the Hamiltonian, and so in particular

under small changes of the values of the vorticities. There is one interesting

occurrence of a symmetric pitchfork bifurcation: consider the family of square

conﬁgurations, on the co-latitude θ—type C4v(R) in the ﬁgure. When the

ring is close to the North pole (say), the re is stable. As θincreases, one

pair of eigenvalues approaches 0, and at θ= arccos(1/√3) there is a pitchfork

bifurcation (of type I in the terminology of §4.1). The bifurcating pair of

relative equilibria are of type C2v(R, R′).

Stability of a ring of vortices It was shown by Dritschel and Polvani [64]

that the stability of a single ring of identical vortices depends on the latitude.

They show that if θis the angle subtended by any of the vortices with the axis

of symmetry of the ring (the colatitude), then the conﬁguration of a regular

J. Montaldi, Relative equilibria and conserved quantities . . . . 271

ring of Nidentical vortices is linearly stable as follows

Nrange of stability Nrange of stability

3 all θ4 cos2θ > 1/3

5 cos2θ > 1/2 6 cos2θ > 4/5

while for N > 6 the ring is never stable. See also [38], where it is shown that

the rings are not only linearly stable but Lyapounov stable.

Bifurcations The changes in stability that occur in the table above involve

bifurcations of the relative equilibria, and indeed the supercritical pitch-

fork bifurcation described in Section 4, since they all involve a loss of C2

symmetry. Consider for example the case of N= 4 identical vortices. A

single ring (square) near the pole is a Lyapounov stable relative equilib-

rium. As θ→cos−1(1/√3), so one of the eigenvalues tends to 0, and for

θ > cos−1(1/√3), there appears a new family of relative equilibria consisting

of vortices alternately above and below the vortices in the now-unstable square

conﬁguration. These bifurcating res—denoted C2v(R, R′) in [37]—are then

Lyapounov stable. Other stability transitions have been observed in [37,38],

but the corresponding bifurcations have not been studied.

The other type of bifurcation that occurs in this problem is the geometric

bifurcation due to the diﬀerent geometry of the reduced spaces for µ= 0 and

µ6= 0—see Section 5. Consider a relative equilibrium peon µ= 0, for example

the ring of Nidentical vortices on the equator. This corresponds to a point

with symmetry DN h in the phase space (the dihedral group in the equatorial

plane together with inversion in that plane). Nearby reduced spaces are then

locally of the form Pµ≃ P0× Oµ, where Oµis the coadjoint orbit through

µ, which here is a sphere, as described very brieﬂy in Section 5. The relative

equilibria on Pµnear peare the critical points of some function h:Oµ→R,

and moreover this function is invariant under some action of Dnh ≃DN×C2

on Oµ. An analysis of this action shows that there must be critical points with

symmetry of types CN v and C2v; see Figure 9. The corresponding relative

equilibria have conﬁgurations of types CN v (R) (a regular ring) and if N= 2m

then C2v(mR) and C2v((m−1)R, 2p), while if N= 2m+ 1 then C2v(mR, p).

Here the conﬁguration C2v(mR, ℓp) consists of mpairs and ℓpoles all lying

on a common great circle, while the great circle is rotating rigidly about an

axis containing the poles. See Figure 8, and see [37] for details.

6.2 Point vortices in the plane

This system has a much older history than the model of vortices on the sphere,

going back to Helmoltz and Kirchhoﬀ, and has been studied by many people

J. Montaldi, Relative equilibria and conserved quantities . . . . 272

Figure 9. Level sets of a typical function on a sphere with symmetry D3h, showing half of

the 8 critical points

since; for reviews see [17,3,15]. Consider an ideal ﬂuid in the plane whose

vorticity is concentrated in Npoint vortices, of strengths λ1,...,λN. These

points move according to the diﬀerential system

zj=1

2πi X

k6=j

λk

zj−zk

where zjis a complex number representing the position of the j-th vortex,

after identifying the plane with C, see (1.3). The Hamiltonian for this system

H(z) = −1

4πX

j<k

λjλklog |zj−zk|2.

The symmetry here is the group of 2-D Euclidean motions SE(2)—which is

not compact. The corresponding conserved quantity (momentum map) is

Φ(z1,...,zN) = iX

λjzj,1

2Pλj|zj|2.(6.2)

From the geometric point of view, this is interesting because if the total

vorticity Λ ≡Pjλjis non-zero, the coadjoint action on se(2)∗must be mod-

iﬁed in order that the momentum map be equivariant (see Section 2). If we

identify SE(2) with C⋊U(1) and u(1) with R, then the modiﬁed coadjoint

action (2.7) becomes

CoadΛ

(u,θ)(ν, ψ) = eiθ ν, ψ +ℑ(eiθν¯u)+ Λ iu, 1

2|u|2(6.3)

where ℑ(z) is the imaginary part of z.

J. Montaldi, Relative equilibria and conserved quantities . . . . 273

If Λ = 0 then the orbits are points on the ψ-axis, and cylinders around

that axis, while if Λ 6= 0 the orbits are all paraboloids—see Examples 2.5 and

2.7 respectively and Figures 2 and 3. Indeed, one can show that the modiﬁed

coadjoint orbits are given by the level sets of f:C×R→Rdeﬁned by

f(ν, ψ) = |ν|2−2Λψ, (6.4)

at least if Λ 6= 0. If Λ = 0 then the non-zero level sets are the cylindrical

orbits, but the zero level set is the whole ψ-axis.

2 vortices in the plane As in the case of 2 vortices on the sphere, here they

are also always relative equilibria. It is simple to prove from the diﬀerential

equation that if Λ ≡λ1+λ26= 0 then the two point vortices rotate about the

ﬁxed point (λ1z1+λ2z2)/Λ. On the other hand, if Λ = 0 then they translate

together towards inﬁnity, in the direction orthogonal to the segment joining

them. These two types of motion are both relative equilibria.

From a geometrical point of view, the reduced spaces are all just single

points, so the corresponding motions are indeed all relative equilibria, and

furthermore the relative equilibria are trivially extremal, and so provided Gµ

is compact (ie. Λ 6= 0, so Gµ≃SO(2)) they are stable modulo Gµ.

3 vortices in the plane The classical work on three planar point vortices is

a beautiful paper by J.L. Synge [69]. We approach this problem as we did

for three point vortices on the sphere. That is, points in the quotient of the

phase space (R2)3≃C3by SE(2) correspond to shapes of oriented triangles.

Again we ignore the orientation, which only causes problems near collinear

conﬁgurations. Then a point in the quotient space is determined by the three

lengths (r1, r2, r3), and on that space

4πH(r1, r2, r3) = −λ1λ2log(r3)−λ2λ3log(r1)−λ3λ1log(r2)

ϕ(r1, r2, r3) = −λ1λ2r2

3−λ2λ3r2

1−λ3λ1r2

2,(6.5)

the second is just f◦Φ. It is remarkable that apart from constant term in ϕ,

these are identical to equations (6.1) for 3 point vortices on the sphere.

The non-collinear relative equilibria are given by the critical points of H

restricted to the level-sets of ϕ, and the computation has already been done.

Thus the relative equilibria are again equilateral triangles, of side rsay, with

Lagrange multiplier η= 1/(4πr2) again. Note however, that this time the

relation between ηand the “angular velocity” ξis not so simple:

0 = d(H−ηϕ)(p) = d(H−ηf ◦Φ)(p) = dH (p)−η df(µ)dΦ(p),

so that ξ=η df (µ). Thus if Φ(p) = (ν, ψ) then

ξ= 2η(ν, −Λ).

J. Montaldi, Relative equilibria and conserved quantities . . . . 274

One consequence of this expression for ξis that if Λ = 0 then the “angu-

lar velocity” is in fact rectilinear motion, with constant velocity ξ= 2ην =

(1/2πr2)ν, where ν=iPjλjzj, which in modulus is independent of the rep-

resentative triangle. Note that if the 3 vortices are not collinear, then ν6= 0

so that the function fseparates the relevant coadjoint orbits even in the case

Λ = 0. On the other hand, if Λ 6= 0, and η6= 0, then the relative equilibrium

is a periodic orbit.

Furthermore, the Lyapounov stabilities modulo Gof these equilateral

relative equilibria are the same as for the spherical case.

The symplectic reduction for a particular class of planar 4-vortex prob-

lems has recently been considered by Patrick [61]. In particular he treats the

case Λ = 0 so that the momentum map is coadjoint-equivariant; it would be

interesting to see how the results are aﬀected by changing to Λ 6= 0.

6.3 Molecules

Consider a molecule consisting of Natoms. The Born-Oppenheimer approxi-

mation consists of ignoring the movement of the electrons (which is reasonable

as they are so light). The system then has 3Ndegrees of freedom, the 3 di-

rections of motion for each nucleus, or 3N−3 after ﬁxing the centre of mass.

The rotational symmetry of the system gives rise to the conservation of

angular momentum J. If we put J= 0, then there are 3N−6 degrees of

freedom which describes the shape of the molecule, and correspond to the

vibrational motions. In other words, the J= 0 reduced space is of dimension

6N−12. The simplest motions beyond the equilibria are the periodic orbits,

which near the stable equilibrium are given by Lyapounov’s theorem and

its generalizations. For J6= 0, the motion has a rotational aspect, and the

simplest type of motion is the relative equilibrium. Beyond that are motions

that are a combination of rotations and vibrations, so-called rovibrational

states. The reduced spaces for J6= 0 are of dimension 6N−10.

Consider now the simplest interesting case of a triatomic molecule. The

reduced spaces are of dimension 6 (for J= 0) or 8 (for J6= 0). There is

one complication that we will not discuss here, namely the reduced space for

J= 0 is singular at points corresponding to collinear equilibria. Consider then

an equilibrium of a triatomic molecule which is not collinear. The geometric

bifurcation methods discussed in Section 5, and at the end of §6.1, show that

there are relative equilibria which bifurcate from the equilibria, and in fact

there at at least 6 such families of re parametrized by kJkand corresponding

to critical points of functions on a sphere. The stabilities of these bifurcating

families are discussed in [50]. For a complete investigation into the relative

J. Montaldi, Relative equilibria and conserved quantities . . . . 275

equilibria of a speciﬁc molecule with 3 identical atoms—namely H+

3—see [33].

Another interesting example is the tetra-atomic molecule ammonia NH3.

This has an equilibrium where the three hydrogen atoms form an equilateral

triangle, and the nitrogen atom is slighly above (or below) the centre of the

triangle—so that the molecule is nearly planar. The analysis is similar to the

triatomic case, with 6 families of res that bifurcate from each equilibrium.

The stability analyses should also be similar to the triatomic case. However,

the presence of two very close stable equilibria (with the nitrogen atom on

one side and on the other of the hydrogen-plane) and an unstable planar

equilibrium betewen them suggests that there will be further bifurcations. An

interesting “semilocal” analysis could be obtained by adding a new parameter

λso that the equilibria undergo a pitchfork bifurcation (of type I) at λ= 0,

with the genuine system corresponding to say λ=−1 and an artiﬁcial one

for λ > 0 with the symmetric planar equilibrium being stable. One needs

to investigate how the pitchfork bifurcation within the reduced space P0,

obtained by varying λ, couples with the geometric bifurcation, obtained by

varying kJk.

References

Books

1. R. Abraham and J. Marsden. Foundations of Mechanics. Benjamin-

Cummings, 1978.

2. V.I. Arnold. Mathematical Methods of Classical Mechanics. Springer-

Verlag, 1978.

3. V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics,

Springer-Verlag, New York, 1998.

4. V.I. Arnold, V.V. Kozlov and A.I. Neishtadt. Mathematical Aspects of

Classical and Celestial Mechanics, 2nd edition. Springer-Verlag, 1997.

5. P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcation The-

ory and Dynamical Systems. World Scientiﬁc, to appear 2000.

6. R.H. Cushman and L.M. Bates, Global Aspects of Classical Integrable

Systems. Birkhuser Verlag. 1997.

7. M. Golubitsky, I. Stewart and D. Schaeﬀer. Singularities and Groups in

Bifurcation Theory, Vol. II. Springer-Verlag, New York. 1988.

8. V. Guillemin and S. Sternberg Symplectic Techniques in Physics. Cam-

bridge University Press. 1984.

9. V. Guillemin, E. Lerman and S. Sternberg Symplectic Fibrations and

Multiplicity Diagrams. Cambridge University Press. 1996.

J. Montaldi, Relative equilibria and conserved quantities . . . . 276

10. P. Libermann and C.-M. Marle. Symplectic Geometry and Analytical

Mechanics. Reidel, 1987.

11. R. MacKay and J. Meiss. Hamiltonian Dynamical Systems, a reprint

collection. Adam Hilger, Bristol. 1988.

12. J. Marsden. Lectures on Mechanics. L.M.S. Lecture Note Series 174,

Cambridge University Press, 1992.

13. J. Marsden and T. Ratiu. Introduction to Mechanics and Symmetry.

Springer-Verlag, New-York, 1994. [Second ed. 1999]

14. K. Meyer and G. Hall. Hamiltonian Systems and the N-Body Problem.

Springer-Verlag, New-York, 1992.

15. P.G. Saﬀman, Vortex Dynamics. Cambridge University Press, Cam-

bridge, 1992.

16. J.-M. Souriau Structure des Syst`emes Dynamiques. Dunod, Paris, 1970.

[English translation: Structure of Dynamical Systems: A Symplectic View

of Physics, Birkhauser, Boston, 1997.]

Research papers

17. H. Aref, Integrable, chaotic and turbulent vortex motion in two-

dimensional ﬂows, Ann. Rev. Fluid Mech. 15 (1983), 345–389.

18. J.M. Arms, A. Fischer and J.E. Marsden, Une aproche symplectique pour

des th´eor`emes de d´ecomposition en g´eom´etrie ou relativit´e g´en´erale. C.

R. Acad. Sci. Paris 281 (1975), 517 – 520.

19. J.M. Arms, J.E. Marsden and V. Moncrief, Bifurcations of momentum

mappings. Comm. Math. Phys. 78 (1981), 455–478.

20. L.M. Bates and E. Lerman, Proper group actions and symplectic strati-

ﬁed spaces. Paciﬁc J. Math. 181 (1997), 201–229.

21. V.A. Bogomolov, Dynamics of vorticity at a sphere, Fluid Dynamics 6

(1977), 863–870.

22. H. Broer, S.-N. Chow, Y. Kim and G. Vegter, A normally elliptic Hamil-

tonian bifurcation. Z. Angew. Math. Phys. 44 (1993), 389–432.

23. M. Dellnitz, I. Melbourne and J. Marsden, Generic bifurcation of Hamil-

tonian vector ﬁelds with symmetry. Nonlinearity 5(1992), 979–996.

24. J.J. Duistermaat, Bifurcations of perodic solutions near equilibrium

points of Hamiltonian systems. In Bifurcation Theory and Applications,

Montecatini, 1983 (ed. L. Salvadori), LNM 1057, Springer, 1984.

25. F. Fass`o and D. Lewis, Stability properties of the Riemann ellipsoids.

Preprint, 2000.

26. I.M. Gelfand and L.D. Lidskii, On the structure of stability of linear

Hamiltonian systems of diﬀerential equations with periodic coeﬃcients.

J. Montaldi, Relative equilibria and conserved quantities . . . . 277

Usp. Math. Nauk. 10 (1955), 3–40. (English translation: Amer. Math.

Soc. Translations (2) 8(1958), 143–181.)

27. M. Golubitsky and I. Stewart, Generic bifurcations of Hamiltonian sys-

tems with symmetry. Physica D 24 (1987), 391–405.

28. V. Guillemin and S. Sternberg, A normal form for the moment map. In

Diﬀerential Geometric Methods in Mathematical Physics (S. Sternberg

ed.) Mathematical Physics Studies, 6. D. Reidel Publishing Company

(1984).

29. G. Iooss and M.-C. P´erou`eme, Perturbed homoclinic solutions in re-

versible 1:1 resonance vector ﬁelds. J. Diﬀerential Equations 102 (1993),

62–88.

30. R. Kidambi and P. Newton, Motion of three point vortices on a sphere,

Physica D 116 (1998), 143–175.

31. Y. Kimura, Vortex motion on surfaces with constant curvature. Proc. R.

Soc. Lond. A 455 (1999), 245–259.

32. F.C. Kirwan, The topology of reduced phase spaces of the motion of

vortices on a sphere, Physica D 30 (1988), 99–123.

33. I. Kozin, R.M. Roberts and J. Tennyson, Symmetry and structure of

rotating H+

3.Preprint, University of Warwick, 1999.

34. E. Lerman, J. Montaldi and T. Tokieda, Persistence of extremal relative

equilibria. In preparation

35. E. Lerman and S.F. Singer, Relative equilibria at singular points of the

momentum map. Nonlinearity 11 (1998), 1637-1649

36. E. Lerman and T.F. Tokieda, On relative normal modes. C. R. Acad.

Sci. Paris Sr. I 328 (1999), 413-418.

37. C.C. Lim, J. Montaldi and R.M. Roberts, Systems of point vortices on

the sphere. Preprint, INLN, 2000.

38. C.C. Lim, J. Montaldi and R.M. Roberts, Stability of relative equilibria

for point vortices on the sphere. In preparation.

39. E. Lombardi, Oscillatory integrals and phenomena beyond any alge-

braic order; with applications to homoclinic orbits in reversible systems.

Springer Verlag Lecture Notes in Mathematics. To appear.

40. A.M. Lyapounov, Probl`eme g´en´erale de la stabilit´e du mouvement. Ann.

Fac. Sci. Toulouse 9, (1907). (Russian original: 1895.)

41. R. MacKay, Stability of equilibria of Hamitonian systems. Nonlinear

Phenomena and Chaos (1986), 254–70. Also reprinted in [11].

42. C.-M. Marle, Mod`ele d’action hamiltonienne d’un groupe the Lie sur une

vari´et´e symplectique. Rend. Sem. Mat. Univers. Politecn. Torino 43

(1985), 227–251.

43. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with

J. Montaldi, Relative equilibria and conserved quantities . . . . 278

symmetry. Pep. Math. Phys 5(1974), 121–130.

44. J.-C. van der Meer, The Hamiltonian Hopf bifurcation, Lecture Notes in

Math. 1160, Springer, 1985.

45. I. Melbourne, Versal unfoldings of equivariant linear Hamiltonian vector

ﬁelds. Math. Proc. Camb. Phil. Soc. 114 (1993), 559–573.

46. I. Melbourne and M. Dellnitz, Normal forms for linear Hamiltonian vector

ﬁelds commuting with the action of a compact Lie group. Math. Proc.

Camb. Phil. Soc. 114 (1993), 235–268.

47. K. Meyer, Symmetries and integrals in mechanics. Dynamical Systems

(M. Peixoto, ed.), 259–273. Academic Press, New York, 1973.

48. J. Montaldi, Persistence and stability of relative equilibria. Nonlinearity

10 (1997), 449–466.

49. J. Montaldi, Perturbing a symmetric resonance: the magnetic spheri-

cal pendulum. In SPT98 – Symmetry and Perturbation Theory II, (A.

Degasperis and G. Gaeta eds.), World Scientiﬁc 1999.

50. J. Montaldi and R.M. Roberts, Relative equilibria of molecules. J. Non-

lin. Sci. 9(1999), 53–88.

51. J. Montaldi and R.M. Roberts, A note on semisymplectic actions of Lie

groups. In preparation.

52. J. Montaldi, R.M. Roberts and I. Stewart, Periodic Solutions near Equi-

libria of Symmetric Hamiltonian Systems. Proc. Roy. Soc. London 325

(1988), 237–293.

53. J. Montaldi, R.M. Roberts and I. Stewart, Existence of nonlinear normal

modes of symmetric Hamiltonian systems. Nonlinearity 3(1990), 695–

730.

54. J. Moser, Lectures on Hamiltonian Systems. Memoirs of the A.M.S. 81

(1981). (Also reprinted in [11]).

55. J. Moser, Periodic orbits near equilibrium and a theorem by Alan Wein-

stein. Communs. Pure Appl. Math. 29 (1976), 727–747.

56. J.P. Ortega, Symmetry, Reduction ans Stability in Hamilonian Systems.

Thesis, University of California, Santa Cruz, 1988.

57. J.P. Ortega and T.S. Ratiu, Stability of Hamiltonian relative equilibria.

Nonlinearity 12 (1999), 693–720.

58. R. Palais. The principle of symmetric criticality, Commun. Math. Phys.

69 (1979), 19–30.

59. G.W. Patrick, Relative equilibria in Hamiltonian systems: The dynamic

interpretation of nonlinear stability on the reduced phase space. J. Geom.

Phys. 9(1992), 111-119.

60. G.W. Patrick, Relative equilibria of Hamiltonian systems with symmetry:

linearization, smoothness and drift. J. Nonlin. Sci. 5(1995), 373–418.

J. Montaldi, Relative equilibria and conserved quantities . . . . 279

61. G.W. Patrick, Reduction of the planar 4-vortex system at zero momen-

tum. Preprint math-ph/9910012

62. G.W. Patrick and R.M. Roberts, The transversal relative equilibria of

Hamiltonian systems with symmetry. Preprint, University of Warwick

(1999).

63. S. Pekarsky and J.E. Marsden, Point vortices on a sphere: Stability of

relative equilibria, J. Mathematical Physics 39 (1998), 5894–5906.

64. L.M. Polvani and D.G. Dritschel, Wave and vortex dynamics on the sur-

face of a sphere, J. Fluid Mech. 255 (1993), 35–64.

65. R.M. Roberts and M.E.R. Sousa Dias, Bifurcations from relative equilib-

ria of Hamiltonian systems. Nonlinearity 10 (1997), 1719–1738.

66. R.M. Roberts and M.E.R. Sousa Dias, Symmetries of Riemann ellipsoids.

To appear in Resenhas do IME - USP, 2000.

67. R. Sjamaar and E. Lerman, Stratiﬁed symplectic spaces and reduction.

Ann. Math 134 (1991), 375–422.

68. S. Smale, Topology and mechanics I, II. Invent. Math. 10 (1970), 305–

331, and 11 (1970), 45–64.

69. J.L. Synge, On the motion of three vortices. Can. J. Math. 1(1949),

257–270.

70. A. Vanderbauwhede and J.C. van der Meer, A general reduction method

for periodic solutions near equilibria in Hamiltonian systems. Fields In-

stitute Comm. 4(1995), 273–294.

71. A. Weinstein, Normal modes for nonlinear Hamiltonian systems. Invent.

Math. 20 (1973), 47–57.

72. A. Weinstein, Bifurcations and Hamilton’s principle. Math. Z. 159

(1978), 235–248.

J. Montaldi, Relative equilibria and conserved quantities . . . . 280

Generalized point vortex dynamics on \begin{document}$ \mathbb{CP} ^2 $\end{document}

Article

Jan 2019

Reduction and relative equilibria for the 2-body problem in spaces of constant curvature

Article

Full-text available

Jun 2018
CELEST MECH DYN ASTR

We perform the reduction of the two-body problem in the two dimensional spaces of constant non-zero curvature and we use the reduced equations of motion to classify all relative equilibria (RE) of the problem and to study their stability. In the negative curvature case, the nonlinear stability of the stable RE is established by using the reduced Hamiltonian as a Lyapunov function. Surprisingly, in the positive curvature case this approach is not sufficient and the proof of nonlinear stability requires the calculation of Birkhoff normal forms and the application of stability methods coming from KAM theory. In both cases we summarize our results in the Energy-Momentum bifurcation diagram of the system.

Geography of the rotational resonances and their stability in the ellipsoidal full two body problem

Article

Full-text available

Feb 2016
ICARUS

A fourth-order Hamiltonian describing the planar full two body problem is obtained, allowing for a mapping out of the geography of spin–spin–orbit resonances. The expansion of the mutual potential function up to the fourth-order results in the angles to come through one single harmonic and consequently the rotation of both bodies and mutual orbit are coupled. Having derived relative equilibria, stability analysis showed that the stability conditions are independent of physical and orbital characteristics. Simultaneously chaotic motion of bodies is investigated through the Chirikov diffusion utilizing geographic information of the complete resonances. The results show that simultaneous chaos among the binary asteroids is not expected to be prevalent due to the mass distribution of primary in compare with secondary. If mass distribution of bodies is of the same order, simultaneous chaos and global instability are achievable.

Point vortices on the hyperbolic plane

Article

Full-text available

Mar 2014

We investigate the dynamical system of point vortices on the hyperboloid. This system has non-compact symmetry $SL(2, R)$ and a coadjoint equivariant momentum map. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on 2 and 3 vortices. Unlike the system on the sphere, this system has relative equilibria with non-compact momentum isotropy subgroup, and these are used to illustrate the different stability types of relative equilibria.

Rotopulsators of the curved N-body problem

Article

Oct 2012

We consider the N-body problem in spaces of constant curvature and study its rotopulsators, i.e.\ solutions for which the configuration of the bodies rotates and changes size during the motion. Rotopulsators fall naturally into five groups: positive elliptic, positive elliptic-elliptic, negative elliptic, negative hyperbolic, and negative elliptic-hyperbolic, depending on the nature and number of their rotations and on whether they occur in spaces of positive or negative curvature. After obtaining existence criteria for each type of rotopulsator, we derive their conservation laws. We further deal with the existence and uniqueness of some classes of rotopulsators in the 2- and 3-body case and prove two general results about the qualitative behaviour of rotopulsators. More precisely, for positive curvature we show that there is no foliation of the 3-sphere with Clifford tori such that the motion of each body is confined to some Clifford torus. For negative curvature, a similar result is proved relative to foliations of the hyperbolic 3-sphere with hyperbolic cylinders.

Vortex Crystals

Article

Dec 2003
ADV APPL MECH

Vortex crystals is one name in use for the subject of vortex patterns that move without change of shape or size. Most of what is known pertains to the case of arrays of parallel line vortices moving so as to produce an essentially two-dimensional (2D) flow. The possible patterns of points indicating the intersections of these vortices with a plane perpendicular to them have been studied for almost 150 years. Analog experiments have been devised, and experiments with vortices in a variety of fluids have been performed. Some of the states observed are understood analytically. Others have been found computationally to high precision. Our degree of understanding of these patterns varies considerably. Surprising connections to the zeros of ‘special functions’ arising in classical mathematical physics have been revealed. Vortex motion on 2D manifolds such as the sphere, the cylinder (periodic strip) and the torus (periodic parallelogram) has also been studied because of the potential applications, and some results are available regarding the problem of vortex crystals in such geometries. Although a large amount of material is available for review, some results are reported here for the first time. The subject seems pregnant with possibilities for further development.

1975: Une approche symplectique pour des théorèmes de décomposition en géométrie ou relativité générale (A symplectic approach for the decomposition theorems in geometry or general relativity)

Article

Full-text available

Jul 1975

Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A

A Normal Form for the Moment Map

Article

Jan 1984

A general reduction method for periodic solutions near equilibria in Hamiltonian systems

Chapter

Aug 1995

On the motion of three vortices

Article

J.L. Synge

Introduction to Mechanics and Symmetry

Book

Jan 1998

Symmetries and integrals in mechanics. In: Peixoto, M. (ed.) Dynamical Systems

Article

Jan 1973

K. Meyer

Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique

Article

Jan 1985

C.-M. Marle

Lectures on Hamiltonian systems

Article

Jan 1968

Jürgen K. Moser

Symplectic Fibrations and Multiplicity Diagrams

Article

Sep 1996

Multiplicity diagrams can be viewed as schemes for describing the phenomenon of "symmetry breaking" in quantum physics. The subject of this book is the multiplicity diagrams associated with the classical groups U(n), O(n), etc. It presents such topics as asymptotic distributions of multiplicities, hierarchical patterns in multiplicity diagrams, lacunae, and the multiplicity diagrams of the rank 2 and rank 3 groups. The authors take a novel approach, using the techniques of symplectic geometry. The book develops in detail some themes which were touched on in the highly successful Symplectic Techniques in Physics by V. Guillemin and S. Sternberg (CUP, 1984) , including the geometry of the moment map, the Duistermaat-Heckman theorem, the interplay between coadjoint orbits and representation theory, and quantization. Students and researchers in geometry and mathematical physics will find this book fascinating.

Foundations of Mechanics

Book

Jan 1978

Relative equilibria and conserved quantities in symmetric Hamiltonian systems

Figures

Recommended publications

Restoring number conservation in quadratic bosonic Hamiltonians with dualities

Stability Analysis of Large-Scale Time-Delay Systems Via Evolutionary Programming Algorithm

Almost periodic solutions of the KdV equation

Point vortices on the sphere: A case with opposite vorticities