Integrable Systems andConservation Laws
Tudor S. Ratiu
Section de Mathematiques, EPFL, Switzerland
Hanoi, April 2007
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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.
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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
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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.
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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.
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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.
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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.
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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 ·, ·−.
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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.
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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.
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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.
Hanoi, April 2007
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.
Hanoi, April 2007
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.
Hanoi, April 2007
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.
Hanoi, April 2007
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.
Hanoi, April 2007
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.
Hanoi, April 2007
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∗.
Hanoi, April 2007
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.
Hanoi, April 2007
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.
Hanoi, April 2007
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.
Hanoi, April 2007
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.
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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.
Hanoi, April 2007
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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