Integrable Systems andConservation Laws

Tudor S. Ratiu

Section de Mathematiques, EPFL, Switzerland

Tudor.Ratiu@epfl.ch

Hanoi, April 2007

1

Bibliography

• F. Fasso: Notes on Finite Dimensional Integrable Hamiltonian

Systems, Universita di Padova, 1999, www.math.unipd.it/ fasso/

These notes are an excellent introduction and I will use some

of the examples he does there as well as his formulation of the

action-angle variables theorem. The “flower picture” for non-

Abelian integrability is due to Fasso and appears, as far as I

know, in these notes for the first time.

• J. Marsden and T.S. Ratiu: Introduction to Mechanics and

Symmetry, second edition, second printing Springer Verlag, 2003.

I have taken from here the presentation of the momentum map

and of the Lie-Poisson reduction theorem.

Hanoi, April 2007

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• R. Abraham and J. Marsden: Foundations of Mechanics, Addison-

Wesley, 1978.

• J.-P. Ortega and T.S. Ratiu: Momentum Maps and Hamiltonian

Reduction, Progress in Mathematics 222, Birkhauser, Boston,

2004.

Reduction theory is presented from these two sources. There

is more here: not just regular reduction, but also singular re-

duction. Also the first theorem on cotangent bundle reduction.

This latter topic is very well presented in

• J. Marsden, G, Misio lek, J.-P. Ortega, M. Perlmutter, and T.S.

Ratiu: Symplectic Reduction by Stages, Lecture Notes in Math-

ematics, 1913, Springer-Verlag, 2007

Hanoi, April 2007

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OVERVIEW OF THE COURSE

• Examples of integrable systems

• Free rigid body

• Abelian integrability

• Non-Abelian integrablity

• Symmetry reduction

EXAMPLES OF INTEGRABLE SYSTEMS

Harmonic Oscillator

Introduction of the main concepts. Point constrained

to a line subject to a linear attracting force −kx, k > 0,

whose potential is given by −k2x2. So the total energy is

H(x, v) :=m

2v2 +

k

2x2, m, k > 0, x, v ∈ R.

Changing coordinates (x, v) 7→ (q, p) :=(√

mx, m√kv)

and

letting ω := km > 0 yields the Hamiltonian

H(q, p) :=ω

2

(q2 + p2

), q, p ∈ R.

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Since the total energy is conserved, the trajectories of

this system lie on H(q, p) = h, which are concentric cir-

cles for h > 0 and the origin for h = 0. We look for

canonical coordinates in which the periodic motion ap-

pears as linear motion on the circle S1 = R/2πZ. This is

not possible for the equilibrium, so we restrict to R2\0.

Take as coordinates the energy h and the time τ on the

orbit. Since h is constant and τ runs with unit constant

speed, the equations are

τ = 1 and h = 0.

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To define τ need to fix an “origin” on each orbit that

should be smooth as we cut across the circles; so choose

a smooth section h 7→ (q0(h), p0(h)) of the foliation

given by H(q, p) = h. Then τ(q, p) is the time neces-

sary for the system to reach the point (q, p) if it started

at (q0(h), p0(h)). Example: q0(h) = 0, p0(h) =√

2h/ω.

The solution is

q(t; q0, p0) = q0 cosωt+ p0 sinωt

p(t; q0, p0) = −q0 sinωt+ p0 cosωt.

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If q0(h) = 0, p0(h) =√

2h/ω, then the coordinates (q, p)

are related to (τ, h) by

q(τ, h) =√

2h/ω cosωτ, p(τ, h) =√

2h/ω sinωτ.

Are (τ, h) really coordinates? Smoothness is clear. But

all points (τ + 2πn/ω, h), n ∈ Z, are mapped to the same

point of R2\0. So they are not coordinates. But they

are global coordinates of the covering ]0,∞[×R and we

have a map

C : (τ, h) ∈ R× ]0,∞[ 7→ (q(τ, h), p(τ, h)) ∈ R2 \ 0.

Its restriction to any open subset in which τ varies less

than a period is a diffeomorphism, so its inverse defines

coordinates.Hanoi, April 2007

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We also have dτ ∧ dh = dq ∧ dp, so (τ, h) are canonical

coordinates.

More geometric point of view: since C is invariant under

τ 7→ τ + 2πn/ω, it induces a map C : S1 × R → R2 \ 0,where the cylinder S1×R has the symplectic form dτ∧dh.

There is one more problem: the coordinate τ is not

really an angle. If (q, p) goes once around the orbit,

then τ(q, p) increases by 2π/ω instead of just 2π. So

change coordinates (α, a) 7→ (τω, h/ω) which gives

q(α, a) =√

2a sinα, p(α, a) =√

2a cosα.

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The Hamiltonian is H(α, a) = ωa and the equations of

motion are α = ω, a = 0.

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Two Uncoupled Harmonic Oscillators

This is a two-degree of freedom system. Hamiltonian is

H(q1, q2, p1, p2) : = H1(q1, p1) +H2(q2, p2)

=ω1

2

(q2

1 + p21

)+ω2

2

(q2

2 + p22

)

for ω1, ω2 > 0 constants. H1 and H2 are independent

integrals of motion, that is, their differentials on an open

dense set are linearly independent. In this case the set

is (R2 \ 0)× (R2 \ 0).

The common level setHanoi, April 2007

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(q1, q2, p1, p2) ∈ R4 | H1(q1, p1) = h1, H2(q2, p2) = h2

=

(q1, p1) ∈ R2 | q21 + p2

1 = 2h1/ω1

×(q2, p2) ∈ R2 | q2

2 + p22 = 2h2/ω2

is

• T2 if h1, h2 > 0

• S1 if h1 = or h2 = 0

• the origin, if h1 = h2 = 0.

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If we restrict to the generic case h1, h2 > 0 and proceed

as in the previous example, we get action-angle coordi-

nates (α1, α2, a1, a2) in which the Hamiltonian takes the

form H = ω1a1 + ω2a2 and the equations of motion are

αi = ωi, ai = 0, i = 1,2.

The motions are periodic iff ω1/ω2 ∈ Q. If not, they are

quasi-periodic, that is, each orbit fills densely the torus.

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Planar Kepler Problem

Movement of a point mass under the influence of the

gravitational potential. Set all constants equal to 1:

H(q, p) :=‖p‖2

2−

1

‖q‖, q, p ∈ R2, q 6= 0.

In polar coordinates (r, θ) in the punctured plane this

becomes

H(r, θ, pr, pθ) =p2r

2+

p2θ

2r2−

1

r, r > 0, θ ∈ S1, pr, pθ ∈ R.

Independent integrals of motion: H and J = pθ, the

angular momentum orthogonal to the plane.Hanoi, April 2007

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The common level set

(r, θ, pr, pθ) ∈ ]0,∞[×S1 × R2 | H = h, J = j

=

θ ∈ S1

×(r, pr) ∈ ]0,∞[×R | p

2r

2+ Vj(r) = h

,where Vj(r) = j2/2r2− 1/r is the amended potential for

the given value j. One verifies than that

• these level sets are compact iff h < 0 and j 6= 0

• if −1/2j2 < h < 0 then they are topologically two-tori

• if h = −1/2j2 then they are circlesHanoi, April 2007

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So, restrict to the open set where h < 0 and j 6= 0.

It is well known that all orbits are ellipses, so all motions

are periodic. Ultimately this is due to the existence

of another vector conserved quantity, the Laplace (or

Runge-Lenz) vector

L(q, p) := p× (q × p)−q

‖q‖.

So this set has a foliation by invariant circles, which is

dynamically far more restrictive than the coarser foliation

by invariant two-tori.

This is a non-Abelian integrable system.Hanoi, April 2007

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FREE RIGID BODY

so(3) and its dual. Special orthogonal group

SO(3) := A | A a 3×3 orthogonal matrix, det(A) = 1,

its Lie algebra

so(3) = 3× 3 skew symmetric matrices

(so(3), [·, ·]) is isomorphic to the Lie algebra (R3,×) by

u := (u1, u2, u3) ∈ R3 7→ u :=

0 −u3 u2

u3 0 −u1

−u2 u1 0

∈ so(3).

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Equivalently, this isomorphism is given by

uv = u× v for all u,v ∈ R3.

The following properties for u,v,w ∈ R3 are easily checked:

(u× v)ˆ = [u, v]

[u, v]w = (u× v)×w

u · v = −1

2trace(uv).

For A ∈ SO(3) and u ∈ so(3) denote AdA u := AuA−1

the adjoint action of SO(3) on its Lie algebra so(3).

Then

(Au)ˆ = AdA u := AuAT

since A−1 = AT , the transpose of A.Hanoi, April 2007

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Also

A(u× v) = Au×Av

for any u,v ∈ R3 and A ∈ SO(3). It should be noted that

this relation is not valid if A is just an orthogonal matrix;

if A is not in the component of the identity matrix, then

one gets a minus sign on the right hand side.

so(3)∗ is identified with R3 by the isomorphism Π ∈ R3 7→Π ∈ so(3)∗ given by Π(u) := Π · u for any u ∈ R3. Then

the coadjoint action of SO(3) on so(3)∗ is given by

Ad∗A−1 Π = (AΠ) .

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The coadjoint action of so(3) on so(3)∗ is given by

ad∗u Π = (Π× u)˜.

Euler angles. The Lie group SO(3) is diffeomorphic

to the real three dimensional projective space RP3. The

Euler angles provide charts for SO(3).

Let E1,E2,E3 be an orthonormal basis of R3 thought of

as the reference configuration. Points in the reference

configuration, called material or Lagrangian points,

are denoted by X and their components, called material

or Lagrangian coordinates by (X1, X2, X3).Hanoi, April 2007

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Another copy of R3 is thought of as the spatial or

Eulerian configuration; its points, called spatial or

Eulerian points are denoted by x whose components

(x1, x2, x3) relative to an orthonormal basis e1, e2, e3 are

called spatial or Eulerian coordinates.

A configuration is a map from the reference to the

spatial configuration that will be assumed to be an ori-

entation preserving diffeomorphism. If the configuration

is defined only on a subset of R3 with certain good prop-

erties such as being a submanifold, as will be the case for

the heavy top, then it is assumed that the configuration

is a diffeomorphism onto its image.Hanoi, April 2007

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A motion x(X, t) is a time dependent family of config-

urations. In what follows we shall only consider motions

that are given by rotations, that is, we shall assume that

x(X, t) = A(t)X with A(t) an orthogonal matrix. Since

the motion is assumed to be smooth and equal to the

identity at t = 0, it follows that A(t) ∈ SO(3).

Define the time dependent orthonormal basis ξ1, ξ2, ξ3

by ξi := A(t)Ei, for i = 1,2,3. This basis is anchored

in the body and moves together with it. The body or

convected coordinates are the coordinates of a point

relative to the basis ξ1, ξ2, ξ3.Hanoi, April 2007

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Note that the components of a vector V relative to the

basis E1,E2,E3 are the same as the components of the

vector A(t)V relative to the basis ξ1, ξ2, ξ3. In particular,

the body coordinates of x(X, t) = A(t)X are X1, X2, X3.

The Euler angles encode the passage from the spatial

basis e1, e2, e3 to the body basis ξ1, ξ2, ξ3 by means of

three consecutive counterclockwise rotations performed

in a specific order: first rotate around the axis e3 by the

angle ϕ and denote the resulting position of e1 by ON

(line of nodes), then rotate about ON by the angle θ

and denote the resulting position of e3 by ξ3, and finally

rotate about ξ3 by the angle ψ.Hanoi, April 2007

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Note that, by construction, 0 ≤ ϕ,ψ < 2π and 0 ≤ θ < π

and that the method just described provides a bijective

map between (ϕ,ψ, θ) variables and the group SO(3).

However, this bijective map is not a chart since its differ-

ential vanishes at ϕ = ψ = θ = 0. So for 0 < ϕ,ψ < 2π,

0 < θ < π the Euler angles (ϕ,ψ, θ) form a chart. Com-

pute explicitly the rotation just described. The resulting

linear map performing the motion x(X, t) = A(t)X has

the matrix relative to the bases ξ1, ξ2, ξ3 and e1, e2, e3

equal to

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A = cosψ cosϕ− cos θ sinϕ sinψ cosψ sinϕ+ cos θ cosϕ sinψ sin θ sinψ− sinψ cosϕ− cos θ sinϕ cosψ − sinψ sinϕ+ cos θ cosϕ cosψ sin θ cosψ

sin θ sinϕ − sin θ cosϕ cos θ

.

The total energy of the free rigid body. A heavy top

is by definition a rigid body moving about a fixed point

in R3. Let B be an open bounded set whose closure is

a reference configuration. Points on the reference con-

figuration are denoted, as before, by X = (X1, X2, X3),

with X1, X2, X3 the material coordinates relative to a

fixed orthonormal frame E1,E2,E3.Hanoi, April 2007

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η : B → R3, with enough smoothness properties so that

all computations below make sense, which is orientation

preserving and invertible on its image, is a configuration

of the free top. The spatial points x := η(X) ∈ η(B)

have coordinates x1, x2, x3 relative to an orthonormal

basis e1, e2, e3. Since the body is rigid and has a fixed

point, its motion ηt : B → R3 is necessarily of the form

ηt(X) := x(X, t) = A(t)X

with A(t) ∈ SO(3); this is a 1932 theorem of Mazur and

Ulam which states that any isometry of R3 that leaves

the origin fixed is necessarily a rotation.Hanoi, April 2007

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If ξ1, ξ2, ξ3 is the orthonormal basis of R3 defined by ξi :=

A(t)Ei, for i = 1,2,3, then the body coordinates of a

vector are its components relative to this basis anchored

in the body an moving together with it.

The material or Lagrangian velocity is defined by

V(X, t) :=∂x(X, t)

∂t= A(t)X.

The spatial or Eulerian velocity is defined by

v(x, t) := V(X, t) = A(t)X = A(t)A(t)−1x.

Hanoi, April 2007

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The body or convective velocity is defined by

V(X, t) : = −∂X(x, t)

∂t= A(t)−1A(t)A(t)−1x

= A(t)−1V(X, t) = A(t)−1v(x, t).

ρ0 density of the top in the reference configuration.

Then the kinetic energy at time t in material, spatial,

and convective representation is given by

K(t) =1

2

∫Bρ0(X)‖V(X, t)‖2d3X

=1

2

∫A(t)B

ρ0(A(t)−1x)‖v(x, t)‖2d3x

=1

2

∫Bρ0(X)‖V(X, t)‖2d3X

Hanoi, April 2007

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Denote

ωS(t) := A(t)A(t)−1

ωB(t) := A(t)−1A(t)

and take into account the definitions of spatial and body

velocity, we conclude that

v(x, t) = ωS(t)× x

V(X, t) = ωB(t)×X

which shows that ωS and ωB are the spatial and body

angular velocities respectively.

Note: ωS(t) = A(t)ωB(t).Hanoi, April 2007

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In Euler angles representation, ωS and ωB are

ωS =

θ cosϕ+ ψ sinϕ sin θθ sinϕ− ψ cosϕ sin θ

ϕ+ ψ cos θ

ωB =

θ cosψ + ϕ sinψ sin θ−θ sinψ + ϕ cosψ sin θ

ϕ cos θ + ψ

.

The kinetic energy in convective representation is

K(t) =1

2

∫Bρ0(X)‖ωB(t)×X‖2d3X =:

1

2〈〈ωB(t), ωB(t)〉〉.

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This is the quadratic form associated to the bilinear sym-

metric form on R3 defined by

〈〈a,b〉〉 :=∫Bρ0(X)(a×X) · (b×X)d3X = Ia · b,

where I : R3→ R3 is the symmetric isomorphism (relative

to the dot product) whose components are given by

Iij := IEj · Ei = 〈〈Ej,Ei〉〉, that is,

Iij = −∫Bρ0(X)XiXjd3X if i 6= j

and

Iii =∫Bρ0(X)

(‖X‖2 − (Xi)2

)d3X.

Hanoi, April 2007

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These are the expressions of the moment of inertia ten-

sor in classical mechanics, that is, I is the moment of

inertia tensor. Since I is symmetric, it can be diag-

onalized. The basis in which it is diagonal is called in

classical mechanics the principal axis body frame and

the diagonal elements I1, I2, I3 of I in this basis are called

the principal moments of inertia of the top. From now

on, we choose the basis E1,E2,E3 to be a principal axis

body frame.

Hanoi, April 2007

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Identify the linear functional 〈〈ωB, ·〉〉 on R3 with the vec-

tor Π := IωB ∈ R3. In Euler angles this equals

Π =

I1(ϕ sinψ sin θ + θ cosψ)I2(ϕ cosψ sin θ − θ sinψ)

I3(ϕ cos θ + ψ)

.

Let us show that Π is the angular momentum in the

body frame. To do this, use the identity (X×(ωB×X)) ·

a = (ωB × X) · (a × X) for any a ∈ R3 and the classical

expression ∫Bρ0(X)(X× V)d3X

of the angular momentum in the body frame to getHanoi, April 2007

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(∫Bρ0(X)(X× V)d3X

)· a =

∫Bρ0(X)((X× (ωB ×X)) · ad3X

=∫Bρ0(X)(ωB ×X) · (a×X)d3X

= 〈〈ωB, a〉〉 = IωB · a = Π · a

which proves the claim.

Using the formula for the kinetic energy in body repre-

sentation and ωB = I−1Π, the expression of the kinetic

energy on the dual of so(3)∗ identified with R3, is

K(Π) =1

2Π · I−1Π =

1

2

Π21

I1+

Π22

I2+

Π23

I3

.Hanoi, April 2007

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Chasing through the isomorphisms R3 ∼= so(3) ∼= so(3)∗,

the kinetic energy has an expression also on so(3), namely

K(ωB) =1

2ωB · IωB = −

1

4trace (ωB(IωB)ˆ)

= −1

4trace (ωB(ωBJ + JωB)) ,

where J is a diagonal matrix whose entries are given by

the relations I1 = J2+J3, I2 = J3+J1, and I3 = J1+J2,

that is, J1 = (−I1 +I2 +I3)/2, J2 = (I1−I2 +I3)/2, and

J3 = (I1 + I2 − I3)/2. The last equality above follows

from the identity (IωB)ˆ = ωBJ+JωB, proved by a direct

verification.

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From here we get the kinetic energy on the tangent

bundle TSO(3):

K(A, A) = −1

4trace((JA−1A+A−1AJ)A−1A).

Since left translation of SO(3) on itself lifts to the left

action B · (A, A) := (BA,BA) on TSO(3), this formula

shows that K is invariant relative to this action. Thus,

the kinetic energy of the free top is left invariant.

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Left translating the inner product 〈〈·, ·〉〉 from the tangent

space to the identity to the tangent space at an arbitrary

point of SO(3), defines a left invariant Riemannian met-

ric on SO(3) whose kinetic energy is the formula above.

Thus the vector field of the free rigid body motion on

TSO(3) is the geodesic spray of the left invariant metric

on SO(3) given by I.

Relative to this metric, the Legendre transformation

gives the canonically conjugate variables

pϕ :=∂K

∂ϕ, pψ :=

∂K

∂ψ, pθ :=

∂K

∂θ.

Hanoi, April 2007

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The kinetic energy in the variables (ϕ,ψ, θ, pϕ, pψ, pθ) is

a left invariant function on T ∗SO(3) given by

K =1

2

[(pϕ − pψ cos θ) sinψ + pθ sin θ cosψ]2

I1 sin2 θ

+[(pϕ − pψ cos θ) cosψ − pθ sin θ sinψ]2

I2 sin2 θ+p2ψ

I3

.

For completeness we summarize in the table below the

relationship between the variables introduced till now.

Hanoi, April 2007

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Π1 = [(pϕ − pψ cos θ) sinψ + pθ sin θ cosψ]/ sin θ

= I1(ϕ sin θ sinψ + θ cosψ)

Π2 = [(pϕ − pψ cos θ) cosψ − pθ sin θ sinψ]/ sin θ

= I2(ϕ sin θ cosψ − θ sinψ)

Π3 = pψ = I3(ϕ cos θ + ψ)

pϕ = I1(ϕ sin θ sinψ + θ cosψ) sin θ sinψ

+ I2(ϕ sin θ cosψ − θ sinψ) sin θ cosψ

+ I3(ϕ cos θ + ψ) cos θ

pψ = I3(ϕ cos θ + ψ)

pθ = I1(ϕ sin θ sinψ + θ cosψ) cosψ

− I2(ϕ sin θ cosψ − θ sinψ) sinψ

38

The equations of motion. In a chart on T ∗SO(3) given

by the Euler angles and their conjugate momenta, the

equations of motion are

ϕ = ∂K∂pϕ

, ψ = ∂K∂pψ

, θ = ∂K∂pθ

pϕ = −∂H∂ϕ , pψ = −∂H∂ψ , pθ = −∂K∂θ

Consider now the map

J : (ϕ,ψ, θ, pϕ, pψ, pθ) 7→ Π

given by the formulas above. This is not a change of

variables!

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A lengthy direct computation, using the formulas above,

shows that these equations imply the Euler’s equations

Π = Π×Ω

where Ω := ωB = I−1Π.

These equations can be obtained in two ways.

(i) The canonical Poisson bracket of two functions f, h :

T ∗SO(3) → R in a chart given by the Euler angles and

their conjugate momenta is

Hanoi, April 2007

40

f, h =∂f

∂ϕ

∂h

∂pϕ−∂f

∂pϕ

∂h

∂ϕ+∂f

∂ψ

∂h

∂pψ−∂f

∂pψ

∂h

∂ψ+∂f

∂θ

∂h

∂pθ−∂f

∂pθ

∂h

∂θ.

A direct long computation shows that if F,H : R3×R3→

R, then

F J, H J = F,H− J,

where

F,H−(Π) = −Π · (∇F ×∇H)

An additional long computation shows that this defines

a Poisson bracket, that is, it is bilinear, skew symmetric,

and satisfies both the Jacobi and the Leibniz identities.Hanoi, April 2007

41

F = F,K− for any F : R3 × R3 → R is equivalent to

Euler’s equations Π = Π×Ω. Indeed, since ∇K(Π) = Ω,

F,K−(Π) = −Π · (∇F (Π)×Ω) = ∇F (Π) · (Π×Ω).

On the other hand, by the chain rule

d

dtF (Π) = ∇F (Π) · Π

which proves the statement.

The bracket of any function with an arbitrary function

of ‖Π‖2 is zero. Functions of ‖Π‖2 are the Casimir

functions of the bracket Lie-Poisson bracket ·, ·−.

Hanoi, April 2007

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(ii) Euler’s equations can also be obtained from a vari-

ational principle. Given is the Lagrangian

L(Ω,Γ) :=1

2IΩ ·Ω.

Consider the variational principle for L

δ∫ baL(Ω)dt = 0

but only subject to the restricted variations of the form

δΩ := Σ + Ω×Σ

where Σ(t) is an arbitrary curve vanishing at the end-

points a and b, i.e Σ(a) = Σ(b) = 0.

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Integration by parts, the vanishing conditions at the end-

points, and ∇L(Ω) = IΩ = Π, yield

0 = δ∫ baL(Ω)dt =

∫ ba∇L(Ω) · δΩdt =

∫ ba

Π · δΩdt

=∫ ba

Π · (Σ + Ω×Σ)dt

= −∫ ba

Π ·Σdt+∫ ba

Π · (Ω×Σ)dt

=∫ ba

(−Π + Π×Ω

)·Σdt.

The arbitrariness of Σ yields Euler’s equations.

Hanoi, April 2007

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The motion of the rigid body takes place on Π-spheres

of constant radius. The solutions of the Euler equation

Π = Π×Ω are therefore obtained by intersecting concen-

tric spheres Π | ‖Π‖ = R with the family of ellipsoids

Π | Π · I−1Π = C for any constants R,C ≥ 0. In this

way one immediately sees that there are six equilibria,

four of them stable and two of them unstable. The sta-

ble ones correspond to rotations about the short and

long axes of the moment of inertia and the unstable one

corresponds to rotations about the middle axis.

Hanoi, April 2007

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Solutions of the rigid body equations; I1 < I2 < I3Hanoi, April 2007

46

The angular momentum in the spatial frame is

π :=∫A(B)

ρ(x)(x× v)d3x,

where ρ(x) := ρ0(X) is the spatial mass density and

v = ωS × x is the spatial velocity. For any a ∈ R3:

π · a =(∫A(B)

ρ(x)(x× v)d3x

)· a

=∫A(B)

ρ(x)(x× (ωS × x)) · ad3x

=∫A(B)

ρ(x)(ωS × x) · (a× x)d3x

=∫Bρ0(X)(ωS ×AX) · (a×AX)d3X

=∫Bρ0(X)(ATωS ×X) · (ATa×X)d3X

=⟨⟨ATωS, A

Ta⟩⟩

=⟨⟨ωB, A

Ta⟩⟩

= IωB ·ATa = AΠ · a

Hanoi, April 2007

47

Thus π = AΠ and we have

π = AΠ +AΠ = AA−1AΠ +A(Π×Ω)

= ωSπ +AΠ×AΩ = ωS × π + π × ωS = 0.

Thus, the spatial angular momentum is conserved during

the motion.

Let π0 ∈ R3 and k ∈ R be given. If k 6= ‖π0‖2/2Ii for

i = 1,2,3, then the common level set given by K = k

and π = π0 is diffeomorphic to a two-dimensional torus.

Show bijectivity, rest routine technical verification.Hanoi, April 2007

48

Note that ‖Π‖ = ‖AΠ‖ = ‖π‖ and hence Π belongs to

the intersection of the energy ellipsoid I−1Π · Π = 2k

with the momentum sphere ‖Π‖ = ‖π0‖. k 6= ‖π0‖2/2Ii

for i = 1,2,3, iff Π is not equal to one of the semi-axes

of the ellipsoid. In this case, this intersection is a closed

curve. Fix now a Π on such a closed curve and look at

the set A ∈ SO(3) | π = AΠ = π0. We claim that this

is also a circle. Indeed, let A0 be the matrix giving the

anti-clocksie rotation with axis Π×π0 that moves Π to

π0. If Π = π0, take A0 = identity. Then every matrix

A ∈ SO(3) satisfying AΠ = π0 can be uniquely written

as A = AψA0 where Aψ is the rotation about the fixed

axis π0 by the angle ψ.Hanoi, April 2007

49

Note that the invariant tori are two-dimensional and that

this is a direct consequence of the existence of the ad-

ditional vector conserved quantity π.

The same phenomenon as in the case of the planar Ke-

pler problem.

Hanoi, April 2007

50

ABELIAN INTEGRABILITY

Review the classical geometry of integrable systems.

Some preliminary geometry.

Ehresmann Fibration Theorem: A proper surjective sub-

mersion f : M → B is a locally trivial fibration.

If B is connected, all fibers f−1(b) are diffeomorphic.

Hanoi, April 2007

51

Given a locally trivial fibration f : M → B, B con-

nected, n = dimM , k = dim f−1(b), a system of co-

ordinates (y1, . . . , yn−k, z1, . . . , zk) is said to be adapted

to the fibration if yi(x) = yi(x′) for all i = 1, . . . , n−k, iff

f(x) = f(x′). (z1, . . . , zk) are not required to be global

coordinates of the fiber. (y1, . . . , yn−k) can be consid-

ered to be coordinates on the base. One can always

construct locally such a system of adapted coordinates.

Hanoi, April 2007

52

The classical setup.

Let (M,Ω) be a 2n-dimensional symplectic manifold and

H a Hamiltonian. A set of smooth functions f1, . . . , fk :

M → R is said to be

• in involution, if fi, fj = 0 for all i, j = 1, . . . , k;

• independent if the set

σ(F ) := x ∈M | df1(x), . . . ,dfk(x) are linearly dependent

of critical points of F := f1× . . .× fk is of measure zero

in M (relative to the Liouville volume Ωn);Hanoi, April 2007

53

• a system of integrals of the system determined by H

if fi, H = 0, for all i = 1, . . . , k;

• completely integrable if the fi, i = 1, . . . , k, are inde-

pendent integrals in involution and k = n.

Let Σ(F ) ⊂ Rk be the bifurcation set of F : the set over

which F : M → Rk fails to be a locally trivial fibration.

Σ(F ) includes the critical values F (σ(F )) of F . By Sard,

F (σ(F )), and hence Σ(F ) has measure zero in Rk.

Liouville-Mineur-Arnold: Let f1, . . . , fn be a completely

integrable system U ⊂ Rn open such that U ∩ σ(F ) = ∅.Hanoi, April 2007

54

• If F |F−1(U) : F−1(U) → U is a proper map, then each

Xfi is complete, U ⊂ Rn \ Σ(F ), and the fibers of the

locally trivial fibration F |F−1(U) are disjoint unions of

submanifolds diffeomorphic to the torus Tn.

• If F |F−1(U) : F−1(U)→ U is not proper but we assume

that each Xfi|F−1(U) is complete and that U ⊂ Rn\Σ(F ),

then the fibers of the submersion F |F−1(U) are disjoint

unions of submanifolds diffeomorphic to the cylinder Rk×

Tn−k for some k = 0, . . . , n.

We think of Tp := Rp/Zp.Hanoi, April 2007

55

For ν ∈ Rn define the translation flow Ψt : Rn → Rn by

Ψt(v) := v+tν. Let π : Rn→ Rk×Tn−k be the canonical

projection. The flow Ψt induces the translation type

flow ψt : Rk × Tn−k → Rk × Tn−k by π Ψt = ψt π, i.e.

ψ(v1, . . . , vk, θk+1, . . . θn

)=

(v1 + tν1, . . . , vk + tνk,

θk+1 + tνk+1(mod 1), . . . , θn + tνn(mod 1))

If k = 0, the flow ψt is called quasi-periodic and ν1, . . . , νn

are the frequencies of the flow.

Dirichlet: Each orbit of ψt is dense in Tn iff∑ni=1 `iν

i = 0,

for `i ∈ Z implies `i = 0 for all i = 1, . . . , n.Hanoi, April 2007

56

• Let Ic be a connected component of Ic := F−1(c)

and let ϕt be the flow of any of XH or Xfi. Then ϕt is

differentiably conjugate to a translation type flow, that

is, there exists a diffeomorphism χ : Rk × Tn−k → Ic and

a translation type flow ψt on Rk × Tn−k such that

ϕt|Ic χ = χ ψt.

Put these results together.

Hanoi, April 2007

57

Assume that F |F−1(Rn\Σ(F ) is a locally trivial fibration.

So if c0 /∈ Σ(F ) there is an open neighborhood U0 of c0

in Rn \ Σ(F ) and a smooth map λ0 : F−1(U0) → Ic0 =

F−1(c0) such that

λ := F |F−1(U0) × λ0 : F−1(U0)→ U0 × Ic0

is a diffeomorphism (and in particular for each c ∈ U0,

λ0|F−1(c) : Ic→ Ic0 is a diffeomorphism). Ic0 is a disjoint

union of cylinders. Is λ∗(XH|F−1(U0)

)Hamiltonian? In

general NO.

Ic0 is compact⇒ ∃λ for which this push forward is Hamil-

tonian. Its components are the action-angle variables.Hanoi, April 2007

58

Action-angle variables.

In R2n define the equivalence relation which identifies

(q,p) with (q′,p′) iff q = q′ and p−p′ ∈ Z. The quotient

space is Rn × Tn which inherits a symplectic structure

from R2n. Let Bn ⊂ Rn be the open unit ball and de-

note the coordinates on Bn×Rn by (I1, . . . In, ϕ1, . . . , ϕn):

Ii = qi and ϕi = pi(mod 1). A Hamiltonian H(I1, . . . , In)

yields the equations of motion

Ii = 0, ϕi = −∂H

∂Ii=: νi(I

1, . . . , In)

Hanoi, April 2007

59

and the maps I1, . . . , In are thus n (everywhere) inde-

pendent integrals in involution. Given initial conditions

Ii(0) = Ii0 and ϕ(0) = ϕ0i , the solution of the system is

ϕi(t) = νi(I10 , . . . , I

n0)t+ ϕ0

i (mod 1), Ii(t) = Ii0.

So Ii = Ii0 describe the invariant tori and the motion

on them is periodic or quasi-periodic with frequencies

νi(I10 , . . . , I

n0).

This is the standard model of action-angle coordinates.

In general one proceeds as follows.

Hanoi, April 2007

60

A Hamiltonian H ∈ C∞(M) admits action-angle coor-

dinates in an open set U ⊂M if

• there is a symplectic diffeomorphism ψ : U → Bn × Tn;

• H ψ−1 ∈ C∞(Bn × Tn) admits standard action-angle

coordinates in the sense above, which is equivalent to

ψ∗ (XH|U) = −∂(H ψ−1)

∂Ii∂

∂ϕi

Classical example: the Delaunay variables in the two-

body problem.

Hanoi, April 2007

61

A submanifold N ⊂M is said to be isotropic if Ω(y)(u, v) =

0 for all y ∈ N and u, v ∈ TyN . A submanifold L ⊂ M is

Lagrangian if it is isotropic and dimL = 12 dimM .

In the hypotheses of the Liouville-Mineur-Arnold Theo-

rem, the manifolds F−1(c) are Lagrangian for c /∈ F (σ(F )).

Indeed, by hypothesis, df1(x), . . . ,dfn(x) are linearly in-

dependent if x ∈ F−1(c). So Xf1(x), . . . , Xfn(x) are lin-

early independent if x ∈ F−1(c).

Hanoi, April 2007

62

Since Tx(F−1(c)

)= ker TxF , TxF = df1(x)× . . .×dfn(x),

and 0 = fi, fj =⟨dfi, Xfj

⟩, we see that Xfj(x) annihi-

lates each component dfi(x) of TxF , that is, Xfj(x) ∈

ker TxF . So the vector fields are tangent to the fiber

F−1(c) whose dimension is n. Since Ω(Xfi, Xfj) = fi, fj =

0 on F−1(c), it follows that F−1(c) is isotropic. Since

its dimension is n, it is Lagrangian.

The converse is also true.

Hanoi, April 2007

63

Let (M,Ω) be a symplectic manifold of dimension 2n

and π : M → B a locally trivial fibration with connected

Lagrangian fibers. Then, locally, there exist independent

functions f1, . . . , fn in involution whose common level

sets are the fibers of π.

To see this, let y1, . . . , yn be a system of coordinates on

some open set V ⊂ B. Define

fi := yi π : π−1(V )→ R, i = 1, . . . , n.

Then clearly F := f1 × . . . × fn : π−1(V ) → Rn is a sub-

mersion whose levels sets are those of π.Hanoi, April 2007

64

We show that these functions are in involution. If X is

any vector field tangent to the fibers of π, then Txπ(X(x))

= 0 and hence 〈dfi(x), X(x)〉 =⟨dyi(π(x)), TxπX(x)

⟩=

0. Therefore Ω(x)(Xfi(x), X(x)

)= 0 for all i = 1, . . . , n.

Since the vector fields Xfj are tangent to the fibers (be-

cause the common level sets of the fi are the fibers of

π) it follows that 0 = Ω(Xfi, Xfj) = fi, fj.

So the Liouville-Mineur-Arnold Theorem is really a state-

ment about compact Lagrangian fibration of a symplec-

tic manifold. So we can define action-angle coordinates

in this context.Hanoi, April 2007

65

(M,Ω) a 2n-dimensional symplectic manifold and π :

M → B a locally trivial fibration with compact connected

Lagrangian fibers. A local system of action-angle

coordinates for the fibration π is a diffeomorphism

I × ϕ : U := π−1(V )→W × Tn

for W ⊂ Rn such that

• Ω|U = dIi ∧ ϕi;

• I(x) = I(x′) iff x, x′ belong to the same fiber of π.

Hanoi, April 2007

66

Non-uniqueness of action-angle coordinates.

Let SL(n,Z)± denote the group of n × n matrices with

integer entries whose determinant is ±1.

To simplify notations, denote by A−T := (A−1)T .

Let I × ϕ : U → W × Tn and I ′ × ϕ′ : U ′ → W ′ × Tn

be two local systems of action-angle coordinates for the

Lagrangian fibration π : M → B. Assume that U∩U ′ 6= ∅

is connected (otherwise the statement applies to each

connected component).

Hanoi, April 2007

67

Then there exist a matrix Z ∈ SL(n,Z)±, a vector z ∈ Rn,and a map Γ : I(U ∩ U ′)→ Rn which satisfy

∂

∂Ii

(Z−TΓ

)j

=∂

∂Ij

(Z−TΓ

)i, i, j = 1, . . . , n (1)

and on U ∩ U ′ we have

I ′ = ZI + z, ϕ′ = Z−Tϕ+ Γ(I) (mod 1) (2)

Conversely, given a system of local action-angle coordi-

nates I × ϕ : U → W × Tn, a matrix Z ∈ SL(n,Z)±, a

vector z ∈ Rn, and a map Γ : I(U ∩ U ′) → Rn satisfying

(1), the map I ′ × ϕ′ : U ′ → W ′ × Tn defined by (2) is

also a local system of action-angle coordinates of the

fibration π on U .Hanoi, April 2007

68

So the action-angle variables are never unique but all of

them are obtained from a given one by (2).

In addition, (2) shows that B has an affine structure (the

atlas has affine transition functions). This is a general

property of bases of Lagrangian fibrations (Weinstein

[1971]).

Hanoi, April 2007

69

Construction of action-angle variables.

This is a difficult task. There is procedure that works,

in principle, due to Arnold. Since this is local, we work

from the beginning on an open set U ⊂ R2n the range

of a symplectic chart of M . The Darboux coordinates

are (q1, . . . , qn, p1, . . . , pn). f1, . . . , fn are n everywhere in-

dependent integrals in involution on U ⊂ Rn \Σ(F ) and

we are already in the situation that F−1(U) is diffeo-

morphic to U × Tn. We need to construct a symplectic

diffeomorphism ψ : F−1(U)→ Bn × Tn.

Locally, Ω = dqi ∧ dpi is exact, Ω− dΘ, for Θ = pidqi.

Hanoi, April 2007

70

For a given c, the common level set Ic is diffeomorphic

to Tn. Let γ1(c), . . . γn(c) be the fundamental n cycles

of Ic corresponding to the n factors S1 (the basis of the

firs homology group). Define λ = (λ1, . . . λn) : U → Rn

by

λi(c) :=∫γi(c)

i∗cΘ, i = 1, . . . n,

where ic : Ic → U .

ASSUMPTION: λ is a diffeomorphism onto its image.

So λ F : F−1(U) → λ(U). Shrinking U we can arrange

so that λ(U) = Bn.

Hanoi, April 2007

71

Roughly, this is half of the desired diffeomorphism ψ :

F−1(U)→ Bn × Tn.

We want a map Γ : F−1(U)→ Tn such that (λ F )×Γ :

F−1(U)→ Bn×Tn is a diffeomorphism. Γ gives the angle

coordinates.

STEP 1 : i∗cΘ ∈ Ω1(Ic) is closed.

The vector fields Xf1, . . . , Xfn are independent on U . In-

deed, U ⊂ Rn \ Σ(F ) =⇒ U ∩ σ(F ) = ∅ since f1, . . . , fn

are independent by hypothesis.

Hanoi, April 2007

72

Thus Xf1(q,p), . . . , Xfn(q,p) is a basis of T(q,p)Ic. So to

show that di∗cΘ = 0, it suffices to prove that

(di∗cΘ

) (Xfi|Ic, Xfj|Ic

)= 0 i, j = 1, . . . n.

This is easy:

(di∗cΘ

) (Xfi|Ic, Xfj|Ic

)=

(i∗cdΘ

) (Xfi|Ic, Xfj|Ic

)= −

(i∗cΩ

) (Xfi|Ic, Xfj|Ic

)= fi, fj ic = 0

by involutivity.

Hanoi, April 2007

73

STEP 2 : An auxiliary function.

Fix I ∈ Rn. Want to solve for p in the equation F (q,p)−

λ−1(I) = 0. Since the matrix with entries ∂fi/∂pj is

nonsingular by the independence hypothesis, fixing q0 ∈

Rn, the implicit function theorem gives a solution p =

p(q, I) for q in a neighborhood of q0. Define

S(q, I) :=∫ (q,p)

(q0,p0)i∗λ−1(I)Θ

where the integral is taken over any path joining (q0,p0)

to (q,p). Since i∗λ−1(I)

Θ is closed, the integral does not

depend on the path if (q,p) is close to (q0,p0).Hanoi, April 2007

74

But globally, it does depend on the path since Tn is not

simply connected. This is a multi-valued function.

STEP 3 : The map Γ = (Γ1, . . . ,Γn) : F−1(U)→ Tn.

Define the multi-valued functions

Γi(q,p) :=∂S(q, I)

∂Ii

∣∣∣∣∣∣I=(λF )(q,p)

.

The variation of Γi on the cycle γk(λ−1(I) is δki .

Indeed,

Hanoi, April 2007

75

∫γk(λ−1(I))

d(Γi iλ−1(I)

)=

∫γk(λ−1(I))

d

(∂S

∂Ii iλ−1(I)

)

=∂

∂Ii

∫γk(λ−1(I))

dS =∂

∂Ii

∫γk(λ−1(I))

i∗λ−1(I)Θ

=∂

∂Iiλk(λ−1(I)) =

∂Ik

∂Ii= δki .

So, mod 1, the Γi are well defined and we get a well

defined map Γ : F−1(U)→ Tn.

STEP 4 : The action-angle map.

Define ψ = (λ F ) × Γ : F−1(U) → Bn × Tn and assume

that it is bijective. Locally true by what we do below.Hanoi, April 2007

76

We show that

∂S(q, I)

∂qi= pi(q, I)

To see this, fix I and note that on the torus Iλ−1(I), the

map S(q, I) can be written as

S(q, I) =∫ (q,p)

(q0,p0)pi(q, I)dq

i = constant+∫ q

q0pi(q, I)dq

i

by taking the path of integration the union of the fol-

lowing two segments:

(q0,p0), (q0,p(q, I)) and (q0,p(q, I)), (q,p(q, I)).

Hanoi, April 2007

77

Conclusion: we have the two relations

Γi(q, I) =∂S(q, I)

∂Iiand pi(q, I) =

∂S(q, I)

∂qi

which means that S is the generating function of the

symplectic map ψ : (q,p) 7→ (I, ϕ), where Γ = ϕ. Being

symplectic between manifolds of the same dimension, ψ

is a local diffeomorphism and since it is assumed to be

a bijective it is a diffeomorphism.

Note that, by construction, Ii = λi F ψ−1.

Hanoi, April 2007

78

STEP 5 : The Hamiltonian is independent of the angle

variables.

By Hamilton’s equations

∂(H ψ−1)

∂ϕi=dIi

dt=

⟨dIi, XHψ−1

⟩= 〈d(λi F ), XH〉 ψ−1 = (dλi TF ) (XH) ψ−1.

But

TF (XH) = (df1(XH), . . . ,dfn(XH))

= (f1, H, . . . , fn, H) = 0.

Hanoi, April 2007

79

REMARKS.

• The construction involves many choices and various

assumptions that are heavily local. The obstruction to

global action-angle variables was studied by Duistermaat

[1978].

• Many of the constructions above cannot be carried

out explicitly in many cases. Yet, one still would like

to linearize the flows. This is possible by methods of

algebraic geometry. But remember, that linearizing the

flow is less than finding action-angle variables.Hanoi, April 2007

80

• What happens when one passes from one region to

another through singular values of the map F? This has

to do with the bifurcation of tori. The level set of a sin-

gular value is no longer a cylinder; even in the compact

case it can give surfaces of higher genus (Flaschka has

a very simple example of this sort). The general bifur-

cation of the tori is extremely difficult. For two degrees

of freedom these bifurcations are classified by Fomenko

and his collaborators. Even in this simple case compli-

cated surfaces can appear, for example the Klein bottle.

This is a very difficult current area of research.

Hanoi, April 2007

81

• Action-angle variables are crucially needed in quantiza-

tion. There are very few integrable systems where this

is carried out explicitly beyond the few classical ones.

For example the finite Toda lattice has been completely

solved by Kostant [1978] and the periodic one by Good-

man and Wallach in a long series of paper in the 80s.

• What are the analogues of these theorems in the non-

Abelian integrability case? Some things are known, oth-

ers are current research.

Hanoi, April 2007

82

NON-ABELIAN INTEGRABILITY

So far, “Abelian” referred to the fact that the integrals

were in involution, that is, the Lie algebra they generate

under Poisson bracket is Abelian. What happens if this

is not the case? We have seen this in the Kepler problem

and in the free rigid body case.

There is a generalization of the Liouville-Mineur-Arnold

theorem to this non-Abelian, or degenerate, case. Only

the compact level surfaces case will be treated. The link

with reduction theory will be done later.Hanoi, April 2007

83

Mischenko-Fomenko, Fasso: Let (M,Ω) be a symplec-

tic manifold, dimM = 2n, and U ⊂ M and open set.

Assume that there is a submersion F = (f1, . . . , f2n−k) :

U → R2n−k, k ≤ n, which has compact connected level

sets and has the property that there exist functions

Pij : F (U)→ R such that

fi, fj = Pij F, i.j = 1, . . .2n− k,

rank(P (F (x)) = 2n− 2k, for all x ∈ U

where P := [Pij]. Then on U every level set of the map F

is diffeomorphic to the torus Tk and has a neighborhood

U and a diffeomorphismHanoi, April 2007

84

b× ϕ : U → B × Tk,

where B = b(U) ⊂ R2n−k is open, with the following

properties:

• On U , the level sets of F coincide with the sets b =

constant;

• Writing b = (q1, . . . , qn−k, p1, . . . , pn−k, I1, . . . Ik), the re-

striction to U of the symplectic form is

2n−k∑i=1

dqi ∧ dpi +k∑

j=1dIj ∧ dϕj.

Hanoi, April 2007

85

Ij and ϕj, j = 1, . . . , k, are action-angle variables. But

there are additional canonical variables that are neither

actions nor angles, namely qi and pi, i = 1, . . . ,2n − k.

The diffeomorphism b×ϕ is called a generalized system

of action-angle coordinates.

A system having 2n − k integrals in involution as in

the theorem, is called noncommutatively integrable.

Sometimes, if k < n it is said to be degenerate, because

of the following.

Hanoi, April 2007

86

Since the n-tori F = constant are invariant under the

flow, the local representative of the Hamiltonian H de-

pends only on the actions Ij. Hamilton’s equations are

qi = 0, pi = 0, Ij = 0, ϕj = −∂H

∂Ij=: νi(I

1, . . . , ik),

i = 1, . . . ,2n − k, j = 1, . . . , k. So the invariant tori are

lower dimensional than in the Abelian case.

The second condition will become clear later when we

study the geometry of these systems.

Hanoi, April 2007

87

The first condition in the theorem states that the func-

tions f1, . . . , f2n−k generate a Lie algebra, in general in-

finite dimensional. Since the Poisson bracket of any two

integrals is again an integral, this condition is automat-

ically verified if f1, . . . , f2n−k form a maximal set of inte-

grals in the sense that any other integral is functionally

dependent on them.

Hanoi, April 2007

88

The planar Kepler problem

We have seen that the angular momentum J in the di-

rection orthogonal to the plane and the two components

L1, L2 of the Laplace vector

L(q, p) := p× (q × p)−q

‖q‖.

are integrals of motion. For negative energies, it is con-

venient to consider the rescaled Laplace vector L :=

L/√−2H which remains an integral. Take f1 = L1, f2 =

L2, f3 = J. Then the matrix P in the theorem has rank

2 whenever F 6= 0 (equivalent to H < 0 and J 6= 0) and

is given byHanoi, April 2007

89

P =

0 −f3 f2f3 0 −f1−f2 f1 0

.The computation uses ‖F‖2 = − 1

2H . Here n = 2 and

k = 1.

The Euler top

Here n = 3 and k = 2. The four independent non-

commuting integrals are the kinetic energy and the three

components of the spatial angular momentum.

Hanoi, April 2007

90

Bifoliations or dual pairs

This theory was started by Weinstein [1983] and contin-

ued by Fasso [1994], [1999] in the context of integrable

systems. The original idea goes back to Lie and appears

under the name of “function groups”.

Definitions.

(M,Ω), N ⊂ M submanifold. Define for x ∈ N the

symplectic orthogonal space to TxN by

(TxN)Ω := v ∈ TxM | Ω(x)(u, v) = 0, ∀u ∈ TxNHanoi, April 2007

91

Essential property:

dimTxN + dim (TxN)Ω = dimTxM.

A (injectively immersed) submanifold N is said to be

isotropic (resp. coisotropic) if all of its tangent spaces

are contained (resp. contain) their own symplectic com-

plements. Isotropic submanifolds (injectively immersed)

have dimensions ≤ (dimM)/2 and coisotropic ones have

dimensions ≥ (dimM)/2.

Lagrangian submanifolds are those that are simulta-

neously isotropic and coisotropic which implies that their

dimension is (dimM)/2.Hanoi, April 2007

92

Symplectic submanifolds are those for which the tan-

gent spaces to M equal the direct sum of their tangent

spaces and their symplectic orthogonal complements.

So, if i : N →M is a submanifold, it is symplectic iff i∗Ω

is a symplectic form on N .

A foliation F on M is called isotropic (coisotropic, La-

grangian, symplectic) if its leaves are isotropic (coisotropic,

Lagrangian, symplectic).

Hanoi, April 2007

93

If F is a foliation on M , its polar foliation, if it exists, is

the unique foliation FΩ on M whose leaves have tangent

spaces the symplectic orthogonal tangent spaces to F.

A foliation F that admits a polar is called symplectically

complete or a bifoliation. Note (FΩ)Ω = F.

Examples:

- Lagrangian foliations coincide with their polars.

- The orbits of a Hamiltonian vector field give locally

and generically a foliation (straightening out theorem).

The polar is formed by the level sets of the Hamiltonian.Hanoi, April 2007

94

- Every coisotropic foliation admits a polar which is nec-

essarily an isotropic foliation (will be seen later)

- Not every isotropic foliation admits a polar. This turns

out to be a problem. Consider a non-integrable distri-

bution with leaves of codimension one. The symplecti-

cally orthogonal distribution is integrable and isotropic,

being one dimensional, but its polar is the original non-

integrable distribution; so there is no polar distribution.

- The orbits of a Hamiltonian action and the level sets of

the momentum map are polar to each other at generic

points (this is the reduction lemma, to be proved later)Hanoi, April 2007

95

If both F and FΩ are given by surjective submersions

B1π1←−M π2−→ B2

then the pair (F ,FΩ) is also called a dual pair, as the

classical analogue of the same term in Lie group or Lie

algebra theory. If both π1 and π2 are locally trivial fibra-

tions, then the diagram is also called a bifibration.

Notation: If F is a foliation, F denotes its integrable

distribution defining it. Similarly, if FΩ is a foliation,

its distribution is denoted by FΩ; note that FΩ always

exists.Hanoi, April 2007

96

Properties.

A first integral of F is a function (defined possibly only

on an open subset) that is constant on the leaves of F.

If F is given by a surjective submersion, a first integral is

necessarily the lift of an arbitrary function on the base.

Let F be a foliation on the symplectic manifold (M,Ω)

and F the associated distribution.

1. f is a first integral of F iff Xf is a section of ∈ FΩ.

Hanoi, April 2007

97

Indeed, f is a first integral iff 〈df,X〉 = 0 for all sections

X of F . Hence 0 = Ω(Xf , X) for all sections X of F iff

Xf is a section of FΩ.

2. F has a polar foliation iff the Poisson bracket of any

two local integrals of F is again a local integral of F.

If FΩ exists, then FΩ is a completely integrable distri-

bution. If f, g are first integrals of F, then Xf , Xg are

sections of FΩ by 1. By Frobenius, Xf,g = −[Xf , Xg]

is also a section of FΩ, so f, g is a section of FΩ by 1.

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98

To prove the converse, we show first the following gen-

eral fact: the space of sections of FΩ has local bases

consisting of Hamiltonian vector fields of first integrals

of F. Indeed, taking a foliated chart at x ∈ M , the co-

ordinates transverse to the leaf L through x are codimL

many functions whose differentials are independent in

this neighborhood and are clearly first integrals of F in

this neighborhood. By 1., the Hamiltonian vector fields

of these coordinate functions are sections of FΩ; they

are linearly independent and have the dimension of the

fiber of FΩ, hence are a local basis.

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99

Choose f1, . . . , fm local integrals of F such that Xf1, . . . , Xfm

is a local basis of FΩ. Then fi, fj are also local

integrals of F for any i, j (by hypothesis) and hence

−[Xfi, Xfj

]= Xfi,fj is a local section of FΩ by 1. Thus,

if X = aiXfi, Y = bjXfj are arbitrary local sections of FΩ,

ai, bj smooth functions, then

[X,Y ] =[aiXfi, b

jXfj

]= aiXfi[b

j]Xfj − bjXfj[a

i]Xfi + aibj[Xfi, Xfj

]

is also a local section of FΩ so, by Frobenius, FΩ is

integrable and hence FΩ exists.

Hanoi, April 2007

100

Corollary: If F is coisotropic then FΩ exists.

Let f, g be local integrals of F which, by 1, is equivalent

to Xf , Xg local sections of FΩ. Then for every local

section Z of F we have

−Ω([Xf , Xg

], Z

)= Ω

(Xf,g, Z

)= 〈df, g, Z〉

= Z [f, g] = Z[Ω(Xf , Xg

)]= 0

since Xf is a local section of FΩ ⊂ F (F is coisotropic)

and Xg is a local section of FΩ. Thus −[Xf , Xg

]= Xf,g

is a local section of FΩ so, by 1, f, g is a local integral

of F. Then 2 guarantees the existence of FΩ.Hanoi, April 2007

101

3. Assume FΩ exists. Then:

3(i) the leaves of F are generated by the flows of Hamil-

tonian vector fields of the first integrals of FΩ;

3(ii) f is a local first integral of F iff it is in involution

with every local first integral of FΩ.

The leaves of any foliation are generated by the flows

of local bases of sections of the associated distribution.

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102

But we just saw that the space of sections of F has

local bases consisting of Hamiltonian vector fields of first

integrals of FΩ (apply the previous statement in the

proof with F replaced by FΩ). This proves 3(i).

By 3(i), f is a first integral of F iff 0 = 〈df,Xg〉 = f, g

for any local first integral g of FΩ. This proves 3(ii).

To state the next properties we need some elementary

facts from the theory of Poisson manifolds.

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103

(P, ·, ·), Xf := ·, f Hamiltonian vector field of f ∈

C∞(P ). The collection of subspaces Sx := Xf(x) |

f ∈ C∞(U), U open containing x, forms a smooth gen-

eralized distribution so by Stefan-Sussmann there is a

generalized foliation of P such that the tangent space

to the leaf through x is Sx. Each leaf is symplectic and

the associated Poisson bracket on it coincides with the

given one on P . The leaves are the equivalence classes

of the relation that identifies two points if they can be

joined by a broken curve each of whose segments is a

part of an integral curve of a Hamiltonian vector field.

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104

The dimension of the symplectic leaf through x is called

the rank of the Poisson bracket at x. If (x1, . . . , xn) are

coordinates around x ∈ P then the rank equals the rank

of the matrix [xi, xj(x)].

Weinstein Coordinates: Around each x0 ∈ P there

are coordinates (q1, . . . , qr, p1, . . . , pr, z1, . . . , zn) such that

qi, qj = pi, pj = zk, qi = zk, pi = 0, qi, pj = δij,

zk, z` is a function of only z1, . . . , zn, and zk, z`(x0) =

0. The coordinates qi, pj are Darboux coordinates on the

symplectic leaf through x0.

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105

The Poisson structure defined by z1, . . . , zn is intrinsic, in

the sense that for any other Darboux-Weinstein coordi-

nates (q1, . . . , qr, p1, . . . , pr, z1, . . . , zn) at x0, the Poisson

structures defined by z1, . . . , zn and z1, . . . , zn are isomor-

phic. This is the transverse Poisson structure at x0.

If the rank of the Poisson bracket at x0 is maximal,

then the transverse Poisson structure is trivial and the

coordinates zk are the local Casimir functions in a neigh-

borhood of x0.

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106

Let (M,Ω) be symplectic, π : M → B a surjective sub-

mersion with connected fibers, and F the foliation whose

leaves are the fibers of π. Denote by ·, ·M the Poisson

bracket on M .

4. F has a polar distribution FΩ iff there is a Poisson

structure on B that turns π into a Poisson map. Such a

Poisson structure on B, if it exists, is necessarily unique.

If there is a Poisson structure on B such that π is a

Poisson map, then it is unique by injectivity of π∗.

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107

Assume FΩ exists. Let U ⊂ B open, f, g ∈ C∞(U).

Then f π, g π are first integrals of F on π−1(U). By

2, so is f π, g πM , that is, this function is constant

on the fibers of π. Since they are connected, there is

a unique smooth function f, gB ∈ C∞(B) such that

f, gB π = f π, g πM . It is obvious that ·, ·B so

defined is a Poisson structure (since π∗ is injective) and

that π is a Poisson map by construction.

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108

Conversely, suppose that (B, ·, ·B) is a Poisson man-

ifold and that f, gB π = f π, g πM for all f, g ∈

C∞(U), U ⊂ B open. Every local first integral of F is of

the form f π for some f ∈ C∞(U) with U ⊂ B open. So

the identity in the working hypothesis guarantees that

the Poisson bracket of any two local integrals of F is

again a local integral of F so, by 2, the polar foliation

FΩ exists.

5. Same hypotheses and notations and suppose that FΩ

exists. Then the following statements are equivalent:

Hanoi, April 2007

109

(i) The leaves of F are isotropic.

(ii) The rank of the Poisson structure on B is everywhere

equal to 2 dimB − dimM .

(iii) The leaves of F are generated by Hamiltonian vector

fields of lifts to M of local Casimirs of B.

(iv) The (local) integrals of FΩ are exactly the lifts to

M of the (local) Casimirs of B.

Hanoi, April 2007

110

First we rephrase condition (ii). The rank of the Poisson

structure on B is everywhere equal to 2 dimB−dimM iff

the number of local independent Casimirs around each

point is dimB−(2 dimB−dimM) = dimM−dimB which

is the dimension of the fibers of π. Thus (ii) states that

the number of local independent Casimirs of B coincides

in a neighborhood of each point with the fiber dimension

of π.

Since FΩ exists, there is a unique Poisson structure on

B that makes π into a Poisson map. Let nb be the

number of independent Casimirs in an open subset of B

containing b ∈ B.Hanoi, April 2007

111

(i) =⇒ (iv). Assume that F ⊂ FΩ and let f be a local

integral of FΩ on U ⊂M . By 1 (with F replaced by FΩ)

this means that Xf is a local section of F . Therefore,

vx ∈ ker Txπ = Fx =⇒ 〈df(x), vx〉 = Ω(Xf(x), vx

)= 0,

i.e., f is constant on the fibers of F and hence ∃f ∈

C∞(π(U)) such that f = f π.

If g ∈ C∞(π(U)), then f , gB π = f π, g πM =

f, g πM = −⟨d(g π), Xf

⟩= −

⟨dg, Tπ Xf

⟩= 0 since

Xf is a local section of F = ker Tπ. Since π is surjective,

we get f , gB = 0, ∀g ∈ C∞(π(U)), i.e., f is a local

Casimir of B.Hanoi, April 2007

112

(iv) =⇒ (iii) by 3(ii).

(iii) =⇒ (i). If g1, . . . , gnb are local independent Casimirs

on the open set V ⊂ B, b ∈ B then, by (iii), for any x ∈

π−1(V ), the linear span of Xg1π(x), . . . , Xgnbπ(x) equals

Fx. Therefore, Ω(x)(Xgiπ(x), Xgjπ(x)

)= gi π, gj

πM(x) = gi, gjB(π(x)) = 0 which shows that if vx ∈ Fxis given and wx ∈ Fx is arbitrary, then Ω(x) (vx, wx) = 0,

that is, vx ∈ FΩx and hence Fx ⊂ FΩ

x , i.e., the leaves of

F are isotropic.

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113

(iii) =⇒ (ii). Since Xg1π(x), . . . , Xgnbπ(x) are linearly

independent in TxM iff d(g1π)(x) = dg1(π(x)), . . . ,d(gnb

π)(x) = gnb(π(x)) are linearly independent which is true

by choice, it follows that Xg1π(x), . . . , Xgnbπ(x) is a ba-

sis of Fx. Hence nb = dimFx = dimM − dimB =⇒

rankbB = dimB − nb = dimB − (dimM − dimB) =

2 dimB − dimM .

Hanoi, April 2007

114

(ii) =⇒ (iii). rankbB = 2 dimB − dimM =⇒ nb =

dimM − dimB = dimFx, ∀x ∈ π−1(b). As above, if

g1, . . . , gnb are local independent Casimirs on the open

set V ⊂ B, b ∈ B, it follows that Xg1π(x), . . . , Xgnbπ(x)

are linearly independent in TxM . Since Txπ(Xgiπ(x)) =

Xgi(π(x)) = 0 =⇒ Xgiπ(x) ∈ Fx, it follows that Xg1π(x), . . . , Xgnbπ(x)

is a basis of Fx and hence the leaves of F are gener-

ated by Hamiltonian vector fields of lifts to M of local

Casimirs of B.

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115

π : M → B surjective submersion with connected isotropic

fibers, (M,Ω) symplectic. Assume FΩ exists and its

leaves are the fibers of a surjective submersion ρ : M →

A. Then the symplectic leaves of B are the connected

components of the fibers of the unique induced map

ρ : B → A which is also a surjective submersion.

(M,Ω)

B A

@@

@@

@@

@@

@@

@@R

-

π ρ

ρ

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116

Define ρ : B → A by ρ(b) = ρ(x) for any x ∈ π−1(b).

The map ρ is well defined since by isotropy of F, the

leaves of FΩ = fibers of ρ are disjoint unions of leaves

of F = fibers of π. Since ρ π = ρ and both π and ρ are

surjective submersions, so is ρ.

(iv) shows that ker Tbρ coincides with the tangent space

to the symplectic leaf of B through b.

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117

The following is a geometric reformulation of the state-

ment about generalized action-angle coordinates.

π : (M,Ω) → B locally trivial fibration with compact

connected isotropic fibers of dimension k. Let Fbe the

associated foliation. Assume that FΩ exists. Then:

• The fibers of π are diffeomorphic to Tn.

• Every fiber of π has an open neighborhood U that ad-

mits generalized action-angle coordinates b×ϕ : U→b(U)×Tk (i.e., b×ϕ satisfies the two conclusions in the gener-

alized action-angle coordinates theorem).Hanoi, April 2007

118

The generalized action-angle coordinates theorem is a

local version of this one.

• The fibers in the submersion F : M → F (M) in the gen-

eralized action-angle coordinates theorem are isotropic

and the polar foliatiation exists.

• Every locally trivial fibration that satisfies the hypothe-

ses of the theorem above can be described locally by

dimM−dim(fiber) functions as in the generalized action-

angle coordinates theorem.

Hanoi, April 2007

119

Non-uniqueness of the generalized action-angle co-

ordinates.

Any two different generalized action-angle coordinates

(qi, pi, Ir, ϕr) and (qi, pi, I

r, ϕr) of A←M → B are related

in each connected component of the intersection of their

domain by

I = ZI + z

(qi, pi) = ∆(qi, pi, Ir)

ϕ = Z−Tϕ+ Γ(qi, pi, Ir),

where Z ∈ SL(k,Z)±, z ∈ Rk, and ∆,Γ smooth functions.

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120

Noncommutative integrability.

A Hamiltonian system on (M,Ω) is noncommutatively

integrable on an open subset if there is an isotropic bifo-

liation π : M → B whose fibers are compact, connected,

and invariant under the flow of the given system.

The flow is linear on the tori and on all tori over the same

symplectic leaf of B the motion has the sam frequencies.

In addition, the tori are generated by the Hamiltonian

vector fields of the lifts of the Casimirs of B. Ik are

coordinates on A which are also local Casimirs on B and

qi, pi are coordinates on the symplectic leaves of B.Hanoi, April 2007

121

Fasso Flowers: ρ−1(a) is a flower on the meadow A

whose center is the symplectic leaf ρ−1(a) and the petals

are the tori π−1(b), where ρ(b) = a.

Hanoi, April 2007

122

MOMENTUM MAPS AND REDUCTION

(M,ω) symplectic manifold, G connected Lie group with

Lie algebra g, G×M →M free proper symplectic action

J : M → g∗ equivariant momentum map: XJξ = ξM ,

where Jξ := 〈J, ξ〉 and ξM infinitesimal generator of ξ ∈ g

Noether’s Theorem: The fibers of J are preserved by

the Hamiltonian flows associated to G-invariant Hamil-

tonians. Equivalently, J is conserved along the flow of

any G-invariant Hamiltonian.

Hanoi, April 2007

123

Proof Let h ∈ C∞(M) be G-invariant, so h Φg = h for

any g ∈ G. Take the derivative of this relation at g = e

and get £ξMh = 0. But ξM = XJξ so we get Jξ, h =⟨

dh,XJξ⟩

= £ξMh = 0, which shows that Jξ ∈ C∞(M)

is constant on the flow of Xh for any ξ ∈ g, that is J is

conserved.

Example: lifted actions on cotangent bundles. Let

G be a Lie group acting on the manifold Q and then by

lift on its cotangent bundle T ∗Q.

〈J(αq), ξ〉 = 〈αq, ξQ(q)〉,

for any αq ∈ T ∗Q and any ξ ∈ g.Hanoi, April 2007

124

Example: Cayley-Klein parameters and the Hopf

fibration. SU(2) acts on C2 by isometries of the Her-

mitian metric, so it is symplectic and therefore has a

momentum map J : C2→ su(2)∗ given by

〈J(z, w), ξ〉 =1

2ω(ξ(z, w)T , (z, w)), z, w ∈ C, ξ ∈ su(2).

su(2) consists of 2×2 skew Hermitian matrices of trace

zero. This Lie algebra is isomorphic to so(3) and there-

fore to (R3,×) by the isomorphism given by

x = (x1, x2, x3) ∈ R3 7−→

x :=1

2

−ix3 −ix1 − x2

−ix1 + x2 ix3

∈ su(2).

Hanoi, April 2007

125

Thus we have

[x, y] = (x× y)˜, ∀x,y ∈ R3.

Other useful formulas are

det(2x) = ‖x‖2 and trace(xy) = −1

2x · y.

Identify su(2)∗ with R3 by the map µ ∈ su(2)∗ 7→ µ ∈ R3

defined by

µ · x := −2〈µ, x〉

for any x ∈ R3.

The symplectic form on C2 is given by minus the imag-

inary part of the Hermitian inner product.Hanoi, April 2007

126

With these notations, the momentum map J : C2 → R3

can be explicitly computed in coordinates: if x ∈ R3

J(z, w) · x = −2〈J(z, w), x〉

=1

2Im

−ix3 −ix1 − x2

−ix1 + x2 ix3

zw

· zw

= −

1

2(2 Re(wz),2 Im(wz), |z|2 − |w|2) · x.

Therefore

J(z, w) = −1

2(2wz, |z|2 − |w|2) ∈ R3.

Hanoi, April 2007

127

J is a Poisson map from C2, endowed with the canoni-

cal symplectic structure, to R3, endowed with the + Lie

Poisson structure. Therefore, −J : C2 → R3 is a canon-

ical map, if R3 has the − Lie-Poisson bracket relative

to which the free rigid body equations are Hamiltonian.

Pulling back the Hamiltonian

H(Π) =1

2Π · I−1Π, I−1Π :=

Π1

I1,Π2

I2,Π3

I3

to C2 gives a Hamiltonian function (called collective) on

C2. I = diag(I1, I2, I3) is the moment of inertia tensor

written in a principal axis body frame of the free rigid

body.Hanoi, April 2007

128

The classical Hamilton equations for this function are

therefore projected by −J to the rigid body equations

Π = Π× I−1Π.

In this context, the variables (z, w) are called the Cayley-

Klein parameters. They represent a first attempt to

understand the rigid body equations as a Hamiltonian

system, before the introduction of Poisson manifolds.

In quantum mechanics, the same variables are called

the Kustaanheimo-Stiefel coordinates. A similar con-

struction was carried out in fluid dynamics making the

Euler equations a Hamiltonian system relative to the so-

called Clebsch variables.Hanoi, April 2007

129

Now notice that if

(z, w) ∈ S3 :=(z, w) ∈ C2 | |z|2 + |w|2 = 1

,

then ‖−J(z, w)‖ = 1/2, so that −J|S3 : S3→ S21/2, where

S21/2 is the sphere in R3 of radius 1/2.

One checks that −J|S3 is surjective and that its fibers

are circles. Indeed, given (x1, x2, x3) = (x1 + ix2, x3) =

(reiψ, x3) ∈ S21/2, the inverse image of this point is

− J−1(reiψ, x3) =eiθ

√√√√1

2+ x3, eiϕ

√√√√1

2− x3

∈ S3∣∣∣∣ ei(θ−ϕ+ψ) = 1

.Hanoi, April 2007

130

One recognizes now that −J|S3 : S3 → S21/2 is the Hopf

fibration. In other words:

the momentum map of the SU(2)-action on C2, the

Cayley-Klein parameters, the Kustaanheimo-Stiefel co-

ordinates, and the family of Hopf fibrations on concentric

three-spheres in C2 are the same map.

Hanoi, April 2007

131

Properties of the momentum map.

Freeness of the action is equivalent to the regularity of

the momentum map: rangeTmJ = (gm). This is also

called the Bifurcation Lemma.

We have TmM = Xf(m) | f ∈ C∞(U), U open neigh-

borhood of m. For any ξ ∈ g we have

⟨TmJ

(Xf(m)

), ξ⟩

= dJξ(m)(Xf(m)

)= Jξ, f(m)

= −df(m)(XJξ(m)

)= −df(m) (ξM(m)) .

Hanoi, April 2007

132

So

ξ ∈ gm⇐⇒ ξM(m) = 0⇐⇒

df(m) (ξM(m)) = 0, ∀f ∈ C∞(U)⇐⇒⟨TmJ

(Xf(m)

), ξ⟩

= 0, ∀f ∈ C∞(U)⇐⇒

ξ ∈ (rangeTmJ)

ker TmJ = (g ·m)ω.

vm ∈ ker TmJ if and only if for all ξ ∈ g

0 = 〈TmJ(vm), ξ〉 = dJξ(m)(vm) = ω(m)(XJξ(m), vm

)= ω(m) (ξM(m), vm)

⇐⇒ vm ∈ (g ·m)ω

Hanoi, April 2007

133

Reduction Lemma: J : M → g∗ equivariant (not nec-

essary). Then

gJ(m) ·m = g ·m ∩ ker TmJ = g ·m ∩ (g ·m)ω.

ξM(m) ∈ g ·m ∩ ker TmJ

⇐⇒ 0 = TmJ (ξM(m)) = − ad∗ξ J(m)

⇐⇒ ξ ∈ gJ(m).

So we get a bifoliation, or dual pair (up to technical

conditions)

M/Gπ←−M J−→ g∗

Hanoi, April 2007

134

Gµ • z

J–1(µ)

G • z

• z

symplectically

orthogonal spaces

The geometry of the reduction lemma.

Hanoi, April 2007

135

Marsden-Weinstein Reduction Theorem

• J : M → g∗ equivariant (not essential)

• µ ∈ J(M) ⊂ g∗ regular value of J

• Gµ-action on J−1(µ) is free and proper, where Gµ :=

g ∈ G | Ad∗g µ = µ

then (Mµ := J−1(µ)/Gµ, ωµ) is symplectic: π∗µωµ = i∗µω,

where iµ : J−1(µ) → M inclusion and πµ : J−1(µ) →

J−1(µ)/Gµ projection.

Hanoi, April 2007

136

The flow Ft of Xh, h ∈ C∞(M)G, leaves the connected

components of J−1(µ) invariant and commutes with the

G-action, so it induces a flow Fµt on Mµ by

πµ Ft iµ = Fµt πµ.

Fµt is Hamiltonian on (Mµ, ωµ) for the reduced Hamil-

tonian hµ ∈ C∞(Mµ) given by

hµ πµ = h iµ.

Moreover, if h, k ∈ C∞(M)G, then h, kµ = hµ, kµMµ.

Hanoi, April 2007

137

Proof: Since πµ is a surjective submersion, if ωµ exists,

it is uniquely determined by the condition π∗µωµ = i∗µω.

This relation also defines ωµ by:

ωµ(πµ(z)) (Tzπµ(v), Tzπµ(w)) := ω(z)(v, w),

for z ∈ J−1(µ) and v, w ∈ TzJ−1(µ).

To see that this is a good definition of ωµ, let

y = Φg(z), v′ = TzΦg(v) and w′ = TzΦg(w),

where g ∈ Gµ. If, in addition Tg·zπµ(v′′) = Tg·zπµ(v′) =

Tzπµ(v) and Tg·zπµ(w′′) = Tg·zπµ(w′) = Tzπµ(w), then

v′′ = v′ + ξM(g · z) and w′′ = w′ + ηM(g · z) for some

ξ, η ∈ gµ and hence

138

ω(y)(v′′, w′′) = ω(y)(v′, w′) (by the reduction lemma)

= ω(Φg(z))(TzΦg(v), TzΦg(w))

= (Φ∗gω)(z)(v, w)

= ω(z)(v, w) (action is symplectic).

Thus ωµ is well-defined. It is smooth since π∗µωµ is

smooth. Since dω = 0, we get

π∗µdωµ = dπ∗µωµ = di∗µω = i∗µdω = 0.

Since πµ is a surjective submersion, we conclude that

dωµ = 0.

To prove nondegeneracy of ωµ, suppose that

ωµ(πµ(z))(Tzπµ(v), Tzπµ(w)) = 0

for all w ∈ Tz(J−1(µ)). This means that

ω(z)(v, w) = 0 for all w ∈ Tz(J−1(µ)),

i.e., that v ∈ (Tz(J−1(µ)))ω = Tz(G · z) by the Reduction

Lemma. Hence

v ∈ Tz(J−1(µ)) ∩ Tz(G · z) = Tz(Gµ · z)

so that Tzπµ(v) = 0, thus proving nondegeneracy of ωµ.

Let Y ∈ X(Mµ) be the vector field whose flow is Fµt .

Therefore, from πµ Ft iµ = Fµt πµ it follows

Tπµ Xh = Y Tπµ on J−1(µ).

Also, hµ πµ = h iµ implies that dhµ Tπµ = dh on

J−1(µ). Therefore, on J−1(µ) we get

π∗µ (iY ωµ) = iXhπ∗µωµ = iXhi

∗µω = i∗µ

(iXhω

)= i∗µdh

= d(h iµ) = d(hµ πµ) = π∗µdhµ

= π∗µ(iXhµωµ

),

so iY ωµ = iXhµωµ since πµ is a surjective submersion.

Hence Y = Xhµ because ωµ is nondegenerate.

Finally, for m ∈ J−1(µ) we have

hµ, kµMµ(πµ(m)) = ωµ(πµ(m))(Xhµ(πµ(m)), Xkµ(πµ(m))

)= ωµ(πµ(m)) (Tmπµ(Xh(m)), Tmπµ(Xk(m)))

= (π∗µωµ)(m) (Xh(m), Xk(m))

= (i∗µω)(m) (Xh(m), Xk(m))

= ω(m) (Xh(m), Xk(m))

= h, k(m)

= h, kµ(πµ(m)),

which shows that hµ, kµMµ = h, kµ.

Problems with the reduction procedure

• Momentum map inexistent

• How does one recover the conservation of isotropy?

• Mµ is not a smooth manifold

• G is discrete so momentum map is zero

• M is not a symplectic but a Poisson manifold

Hanoi, April 2007

The flow of the original system can be completely re-

constructed from the reduced flows.

A completely integrable system is, by definition, one that

has generically the reduced spaces zero dimensional.

Hanoi, April 2007

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