Geometry of Integrable systems1
Appunti “sparsi” per il corso di Teoria dei Sistemi Dinamici.
Corso di Laurea Specialistica in matematica, A/A 2006/2007.
G. Falqui
Dipartimento di Matematica e applicazioni, Università di Milano–Bicocca
Contents
1 Hamiltonian systems
1.1 Prelimaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2 Non standard Poisson brackets
2.1 Magnetic Poisson brackets . . . . . . . . . . . . . . . . . . . . .
4
4
3 Lie algebras and Lie groups: a primer
3.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
8
10
4 Lie Poisson brackets
11
5 Liouville Integrable Systems
5.1 Action-Angle variables . . . . . . . . . . . . . . . . . . . . . . .
5.2 Hamilton-Jacobi equation and Separation of Variables . . . . . .
14
17
18
6 Lax Pairs
6.1 The Toda System . . . . . . . . . . . . . . . . . . . . . . . . . .
24
27
7 Lax equations with ‘spectral’ parameter
31
8 The Periodic Toda system revisited
34
8.1 Dual Lax representation for the Periodic Toda Lattice and r-matrix 36
9 R-matrices, Lax representation with spectral parameters, and
SoV
39
1
Note non rivedute e corrette: Commenti e correzioni saranno benvenuti.
1
10 Group actions and Moment maps: the Calogero System
10.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 The geometry . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 The Calogero model . . . . . . . . . . . . . . . . . . . . .
10.4 The Calogero Model as a Reduction . . . . . . . . . . . . .
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Hamiltonian systems
1.1
Prelimaries
We start by the local definition of the Hamilton equations of motion. Let Ω ⊂
R2n and let {q 1 , . . . , q n , p1 , . . . , pn } coordinates in Ω. Let H be a (sufficiently
regular, say C ∞ )) function on Ω, so that H = H(pi , q i). The Hamilton equations
are

∂H

 ṗi = − i
∂q i = 1, . . . , 2n.
(1.1)
∂H

i
 q̇ =
∂pi
Along with this structure we introduce the Poisson brackets between two functions F, G ∈ C ∞ (Ω) by
n
X
∂F ∂G ∂F ∂G
{F, G} =
−
.
∂q i ∂pi ∂pi ∂q i
i
(1.2)
In particular for the coordinate functions {q 1 , . . . , q n , p1 , . . . , pn } we have the
fundamental Poisson brackets:
{q i , pj } = −{pj , q i , =}δji ,
{q i , q j }{pi, pj } = 0.
(1.3)
The structural properties of the Poisson brackets (1.2) are:
1. Bilinearity and antisymmetry:
{F, G} = −{G, F };
{a1 F1 +a2 F2 , G} = a1 {F1 , G}+a2 {F2 , G}, ai constants.
2. Leibnitz rule:
{F1 F2 , G} = F1 {F2 , G} + F2 {F1 , G}.
3. Jacobi identity:
{{F, G, , }H} + {{G, H, , }F } + {{H, F, , }G} = 0
2
The Hamilton equations of motion (1.1) can be rewritten as

∂H

 ṗi = − i = {pi , H}
∂q
i = 1, . . . , 2n.
∂H

i
 q̇ =
= {q i , H}
∂pi
(1.4)
and, collectively, setting {x1 , . . . , x2n } = {q 1 , . . . , q n , p1 , . . . , pn } via
ẋa = {xa , H},
a = 1, . . . , 2n.
(1.5)
Historically, the canonical form (1.1) arises from the Lagrangian formulation of
the Newton equation of motion for systems with smooth constraints satisfying
the so-called D’Alembert principle, in which one has a Lagrangian function
L(q, q̇), and defines, on the space of paths γ : [0, 1] → Ω, γ(0) = q0 , γ(1) = q1
with endpoints in two fixed points q0 , q1 of Ω the action functional
Z
S = L(q, q̇)dt
γ
The differential of S is
X Z ∂L
d ∂L i
· h )dt
( i−
dt ∂ q̇ i
γ ∂q
i
where q i → q i + hi denotes the variation of the path γ. Hence the extrema of S
will be those paths satisfyng the Euler-lagrange equations
∂L
d ∂L
−
= 0, i = 1, . . . n.
i
∂q
dt ∂ q̇ i
(1.6)
To obtain the canonical equations (1.1) one argues as follows. One supposes
that the Legendre transformation
pi =
∂L
∂ q̇ i
(1.7)
is invertible w.r.t. the q̇ i , i.e. it is possible to get from the transformation (1.7)
q̇ i = q̇ i (q, p)
Then introduces the Hamiltonian (energy) function defined by the Lagrangian
as
X
H(p, q) =
pi q̇ i − L(q, q̇)|q̇i=q̇i (q,p) .
(1.8)
i
It is a straightforward computation to show that the system of n second order
differential equations (1.6) is equivalemt to the canonical Hamilton equations
(1.1).
3
Conversely (taking into account that the legendre transformation for convex
functions is involutive) starting from the definitions
q̇ i =
∂H
,
∂pi
,L =
X
q̇ i pi − H(p, q)|
q̇ i =
∂H
∂pi
one recovers, starting from (1.1) the E-L equations (1.6).
2
2.1
Non standard Poisson brackets
Magnetic Poisson brackets
Let M = T ∗ Q, let {q i , pi}i=1..n fibered coordinates on M and let
X
Ω=
Ωij (q)dq i ∧ dq j
(2.1)
i,j
a two form on Q.
Let us define a bracket on T ∗ Q by means of
{q i , pj }Ω = −{pj , q i}Ω = δji ,
{q i, q j }Ω = 0,
{pi , pj }Ω = Ωij (q).
(2.2)
Problem. Which are the conditions under which {·, ·}Ω is indeed a Poisson
bracket.
Let us first discuss what that means. Recall that a Poisson bracket is a
bilinear (over R) map C ∞ (M) × C ∞ (M) → C ∞ (M) satisfying antisymmetry,
the Leibnitz rule, and the Jacobi condition.
Let us first show that, indeed, the associated Poisson operator is a bivector.
Namely, let us consider a coordinate change q → q ′ in Q, that, as we know,
induces the coordinate change p → p′ in the fiber of T ∗ Q, where, matricially,
p′ = T J −1 p, J =
∂q ′
∂q
(2.3)
In terms of the coordinates x = (p, q) in T ∗ Q the jacobian of the transformation
will be a block matrix, given by


′
T −1 ∂p
J
J=
(2.4)
∂q ,
0
J
where we have indicated symbolically with
is
∂p′i
∂q j
4
∂p′
the matrix whose (i, j)-th entry
∂q
Since the matrix representing the Poisson tensor defined by (2.2) is given by
Ω −1
,
(2.5)
P =
1 0
we have to compare P ′ with J · P · T J. The latter explicitly equals


 −1

′
J
0
T −1 ∂p
Ω −1
 J
·  T ∂p′ T  =
∂q  ·
1 0
J
0
J
∂q

 

∂p′
J −1 0
T −1
T −1
 J Ω + ∂q − J  ·  T ∂p′ T  =
J
J
0
∂q


∂p′ −1 T −1T ∂p′
T −1
−1
( ) −1 
 J ΩJ + ( ∂q )J + J
∂q
1
0
(2.6)
Let us consider the upper block of this expression. It is given by
′
∂p −1 T −1 T ∂p′
T −1
−1
J ΩJ +
)J − J
(
∂q
∂q
′
∂p −1 T −1 T ∂p′
We notice that the second summand
identically van)J − J
(
∂q
∂q
ishes. If we multiply it on theleft by T J and on the right by J we get a new
′
′
∂p
∂p
n × n matrix B ′ = T J
J whose (i, j)-th element is given by
−T
∂q
∂q
!
n
X
∂q ′ k ∂p′k ∂p′k ∂q ′ k
.
(2.7)
− i
i ∂q j
j
∂q
∂q
∂q
k=1
Let us recall that the Liouville one-form θL is given by
θL =
X
i
pi dq i =
X
k
X X ∂q ′ k
p′k dq =
p′
i k
∂q
i
k
′k
!
dq i .
(2.8)
So we see that (2.7) is nothing but a complicated way to express the component
of ω = dθL along dq i ∧ dq j , which, by the first expression of θL , vanishes.
Finally, we see that P ′ = JP T J iff
Ω′ = T J −1 ΩJ −1 ,
(2.9)
that is if Ωij are indeed the components of a two-form on Q.
We thus see the (2.2) indeed defines a bilinear operation on C ∞ (M) that
satisfies the Leibnitz property. What we have to check the Jacobi identity.
5
This means that, if x1 , x2 , x3 are any three coordinates, we have to check
that
{x1 , {x2 , x3 }} + {x2 , {x3 , x1 }} + {x3 , {x1 , x2 }} = 0.
(2.10)
Recalling the definition of {·, ·}Ω , i.e.
{q i , pj }Ω = −{pj , q i }Ω = δji ,
{q i , q j }Ω = 0,
{pi , pj }Ω = Ωij (q),
we see that the only non trivially verified instance happens for triples {pi , pj , pk }.
We have:
{pi , {pj , pk }Ω }Ω + {pj , {pk , pi }Ω }Ω + {pk , {pi , pj }Ω }Ω =
{pi , Ωjk (q)}Ω + {pj , Ωki (q)}Ω + {pk , Ωij (q)}Ω =
∂Ωjk (q)
∂Ωki (q)
∂Ωij (q)
l
l
l
{pi , q }Ω +
{pj , q }Ω +
{pk , q }Ω =
l
l
l
∂q
∂q
∂q
l
X ∂Ωjk (q)
∂Ωki (q) l ∂Ωij (q) l
l
−
δi +
pj +
δk = −(∂i Ωjk + ∂j Ωki + ∂k Ωij )
∂q l
∂q l
∂q l
l
X
(2.11)
Summing up: (2.2) define Poisson brackets iff
X
Ω=
Ωij dq i ∧ dq j
ij
is a closed two form.
Before discussing the physical meaning of these brackets, let us check the
validity of Darboux lemma in this case, that is let us find a set of canonical
coordinates for {·, ·}Ω . Namely, we have to find a set of functions {yi, xi } on
(“every” chart of) T ∗ Q such that
1. the Jacobian
∂p, q
6= 0
∂y, x
2. {yi, yj }Ω = {xi , xj }Ω = 0,
{xi , yj }Ω = −{yj , xi }Ω = δji .
The form of the Poisson brackets {·, ·}Ω suggests the ansatz
xi = q i ; yi = pi + fi (q)
Indeed, with this definition we have
{xi , xj }Ω = 0, {xi , yj }Ω = −{yj , xi }Ω = δji .
We thus have only to check the vanishing of {yi , yj }Ω .
{yi, yj }Ω = {pi + fi (q), pj + fj (q)}Ω =
{pi, pj }Ω + {fi (q), pj }Ω + {pi , fj (q)}Ω = Ωij + (∂j fi − ∂i fj )
6
(2.12)
But from De Rham’s lemma, we know that this equation is locally solvable (say,
on a contractible – that is, homeomorphic to a ball in Rn – open set U ⊂ Q.
Let us now interpret the magnetic brackets (and justify the name). Let
H(p, q) be a function on T ∗ Q. The equation of motion (a/k/a/ the Hamiltonian
vector field) associated with H are

∂H

i
i


 q̇ = {q , H}Ω = ∂pi
(2.13)
∂H X j
∂H X ∂H


+
Ω
=
−
+
q̇
Ω
ṗ
=
{p
,
H}
=
−
ij
ij
i
i
Ω


∂q i
∂pj
∂q i
j
j
So we see that the second of these equation contains, in addition to the usual
∂H
force term −
, the additional term
∂q
X
Fi =
q̇ j Ωij
j
which is linear in the velocity q̇ and has the property of being “orthogonal” to
the velocity, in the sense that, owing to the antisimmetry of Ωij ,
X
X
(q̇, F) =
q̇ i Fi =
q̇ i q̇ j Ωij = 0
i
ij
The best known instance of such a forse is the Lorentz force experienced by a
point charge e in a magnetic field B,
FL =
e
q̇ × B
c
To enhance the analogy, notice that under the identification R3 ≃ ∧2 R3 given
by
X
Bi =
ǫijk Ωjk
jk
the closure equation ∂i Ωjk + ∂j Ωki + ∂k Ωij = 0 transforms into one of the
Maxwell’s equations, namely, the Gauss law
divB = 0.
3
Lie algebras and Lie groups: a primer
In this Section we will collect a few of the basic notions of the theory of Lie
groups and algebras.
7
3.1
Lie Algebras
We already introduced the notion of Lie algebra by considering vector fileds
on manifolds, and the space C ∞ (M) endowed with the Poisson brackets. The
formal definition is the following:
Definition 3.1 A Lie algebra g is a vector space over R(or C endowed with a
product [·, ·], that is a bilinear operation g × g → g satisfying
1. Antisymmetry: [X, Y ] = −[Y, X], ∀ X, Y ∈ g
2. Jacobi identity: [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 ∀X.Y.Z. ∈ g
We notice the following: any associative algebra endowed with the associative
product A, B → A · B can be turned into a Lie algebra setting
[A, B] := A · B − B · A.
(3.1)
In this way the sapce of endomorphisms of a vector space can be seen as a Lie
algebra, and the space of n × n matrices with entries in R is a Lie algebra.
If g and h are Lie algebras, a homomrphism of g to h is a linear map L : g → h
such that
[L(X1 ), L(X2 )]h = L([X1 , X2 ]g), ∀X1 , X2 ∈ g.
(3.2)
We have already encountered an example of Lie algebra homomorphism. Indeed,
if M and M ′ are manifolds, and F : M → M ′ is an injective C ∞ map, then we
have seen that, for the tangent map F∗ it holds
[F∗ X, F∗ Y ] = LF ∗X F∗ Y = F∗ (LX Y ) = F∗ ([X, Y ]).
That is, given any injective map (in particular, a diffeomorphism) between
two manifolds, the tangent map gives rise to a homomorphism Vect(M) →
Vect(M ′ ).
Let us now turn our attention to finite dimensional Lie algebras. A notion
which will be used in the sequel is that of structure constants. Namely, let g
be an n-dimensional Lie algebra, and let {X 1 , . . . , X n } be a basis (w.r.t. the
vector space structure) of g. Then the (Lie) product of two elements Y, Z of g
is determined by the (Lie) products of the basis elements. Since [X i , X j ] ∈ g
it can be uniquely developped along the basis {X 1 , . . . , X n } that is, there exist
n3 numbers cij
k such that
X ij
ck X k .
(3.3)
[X i , X k ] =
k
8
These numbers are the structure constants of the Lie algebra g, corresponding to
the basis {X 1 , . . . , X n }. We notice that, fo all i, j, k and all bases we necessarily
have, in correspondence with the axioms 1 and 2 of Definition 3.1,
ji
cij
k = −ck .
X
mk
mi
ki mj
=0
cij
+ cjk
m cl
m cl + cm cl
(3.4)
m
Remark. (Trivial) Any vector space V can be turned into a (somewhat uninteresting) Lie algebra. Namely, one can define
[v1 , v2 ] = 0, ∀v1 , v2 ∈ V.
This is an instance of a commutative Lie algebras. Much more interesting is
the determination of (maximal) commutative subalgebras of a given Lie algebra g (this leads to the Cartan-Dynkin classification scheme for simpe finite
dimensional Lie algebras.
Example. Let us consider the space of (complex) 2 × 2 matrices M2 (C).
The space sl(2, C) can be (for instance) endowed with the following two bases:
1 0
0 1
0 0
h=
= X3 , e =
= X1 , f =
= X2 (Cartan base)
0 −1
0 0
1 0
(3.5)
and
1 0 I
1 I 0
1 0 −1
, σ2 =
, σ3 =
(Pauli base) (3.6)
σ1 =
2 1 0
2 I 0
2 0 −I
The commutation relations in the two bases are:
[X1 , X2 ] = [e, f ] = h = X3 , [X1 , X3 ] = [e, h] = −2e = −2X1 ,
[X2 , X3 ] = [f, h] = 2f = 2X2 ,
⇒
13
23
{c12
3 = 1, c1 = −2, c2 = 2}
(3.7)
and
[σ1 , σ2 ] = σ3 , [σ2 , σ3 ] = σ1 , [σ3 , σ1 ] = σ2
23
31
⇒ {c12
3 = c1 = c2 = 1}
(3.8)
So we see that the structure constants in the two bases differ. Clearly, they are
related by the consequnce of the relations
σ3 =
1
−I
−I
X3 , σ1 = (f − e) σ2 =
(e + f )
2
2
2
We finally remark that σ1 , σ2 , σ3 are a basis in the real Lie algebra su(2) of the
traceless antiselfadjoint 2 × 2 matrices, that is those 2 × 2 matrices A satisfying
A† ≡ T Ā = −A, Tr(A) = 0.
9
(3.9)
Proposition 3.2 The Lie algebra su(2) is isomorphic to the Lie algebra so(3)
of real 3 × 3 antisymmetric matrices.
Proof. A base in so(3) is given by






0 0 0
0 0 1
0 −1 0
Ω1 =  0 0 −1  , Ω2 =  0 0 0  , Ω3 =  1 0 0  .
0 1 0
−1 0 0
0 0 0
(3.10)
A straightforward computation shows that the commutation rules between the
elements of such a basis are:
[Ω1 , Ω2 ] = Ω3 , [Ω2 , Ω3 ] = Ω1 , [Ω3 , Ω1 ] = Ω2 ,
(3.11)
that is, they coincide with those given by (3.8)
3.2
Lie groups
Definition 3.3 A Lie group is a group G endowed with the structure of a differentiable manifold, such that the maps
G×G → G
(g1 , g2 ) 7→ g1 g2
and
G → G
g 7→ g −1
(3.12)
are smooth (sometimes required to be analytic). Clearly, the differential structure on G × G s the product one.
An equivalent definition is that the map
G×G →
G
(g, h) 7→ g h−1
(3.13)
be smooth (analytic).
Remark. By definition a Lie Group G is a manifold equipped with a distinguished set (indeed it is the manifold G itself) of diffeomorphisms. Indeed,
if we consider the first “half” of the definition of a Lie group, we have a map
defined on the Cartesian product G × G → G. If we “freeze” the first entry to
g1 = h, we obtain a map
φh : G →
G
g 7→ φh (g) = h g.
(3.14)
This map is smooth by definition, and is invertible (the inverse is φh−1 .
Definition 3.4 A vector field X on G is called left-invariant if, for all h, g ∈ G,
(φh )∗ X(g) = X(φh (g)) = X(hg).
10
(3.15)
4
Lie Poisson brackets
Let g be a (finite dimensional) Lie algebra, endowed with a basis {E1 , . . . , En }
and let cij k be its structure constant. Let us consider Rn with euclidean coordinates {x1 , . . . , xn }. Let us define in Rn the following brackets
X
{xi , xj } :=
cij k xk .
(4.1)
k
Proposition 4.1 The brackets (4.1) are indeed Poisson brackets.
Proof. Antisymmetry is obvious. The characteristic condition
{xi , {xj , xk }} + cyclic = 0
(4.2)
is equivalent to the characteristic property of the structure constants of g.
Let now g be reductive, that is let us suppose that an ad-invariant nondegenerate inner product
g : g × g → R(C),
(4.3)
that is a symmetric non-degenerate bilinear form g(X, Y ) s.t.
g([X, Y ], Z) = g(X, [Y, Z]), ∀ X, Y, Z, ∈ g.
(4.4)
In terms of the structure constants we have, calling g ij = g(E i, E j ),
P
cij l E l
P
cjk l E l
z l }| {
z l }| {
i
j
k
i
g( [E , E ], E ) = g(E , [E j , E k ])
P ij || l k
P jk || i l
l c l g(E , E ) =
l c l g(E , E )
||
P ij lk
P jk ||li
ijk
jki
γ := l c l g
=
l c l g =: γ
(4.5)
Thus, taking into account the obvious antisymmetry relation γ ijk = −γ jik we
can state:
Proposition 4.2 The completely contravariant structure constants γ ijk of a reductive Lie algebra are completely antisymmetric w.r.t. permutation of indices.
This property will allow us to give a remarkable characterization of the Hamiltonian vector fields generated by the brackets (4.1).
Before examining this, let us first give an example of Hamiltonian vector
field with respect to these structures. Let us consider so(3), endowed with the
basis (3.10), that is with the structre constants cij k = ǫij k . The Poisson tensor,
in the cartesian coordinates {x, y, z}, is represented by the matrix


0
z −y
x 
P =  −z 0
(4.6)
y −x 0
11
............................
Let us now turn to the general properties of these linesar Poisson brackets.
Let H be a function on Rn ; then XH = P dH is expressed by
X
X
ẋi = {xi , H} =
{xi , xl }∂l H =
cil k xk ∂l H.
(4.7)
l
lk
Let us introduce two g-valued functions on Rn as follow:
X
X
X=
gin xi E n , ∇H =
∂l HE l , gil g lk = δik .
in
(4.8)
l
From (4.7) we have:
Ẋ =
X
gin ẋi E n =
in
X
gin cil k xk ∂l HE n .
On the other hand, let us compute the commutator [∇H, X], as
X
X
[∇H, X] = [
∂l HE l ,
gin xi E n ]
=
X
(4.9)
inkl
in
l
∂l H xi gin [E l , E n ] =
X
∂l H xi gin cln k E k .
(4.10)
inlk
inl
Let us change a few indices in the last equation, namely first i ↔ k to get
X
∂l H xk gkn cln i E i .
inlk
and then i ↔ n. This yields
[∇H, X] =
X
gki cli n (∂l H xk E n ),
(4.11)
inlk
to be compared with (4.9), i.e.,
X
Ẋ =
(xk ∂l HE n )gin cil k .
(4.12)
inkl
We want to prove that
X
i
gki cli n =
X
gin cil k , ∀l, k, n.
i
We multiply both sides by g mk g ns to get, implicitly unserstanding summation
over repated indices
g mk gki cli n g ns = g ns gin cil k g mk
12
Taking into account the symmetry g ab = g ba , gab = gba we have:
δm
γ lis
γ ilm
δs
i
z }|
{ z }| {
z }|i { z }| {
mk
li
ns
g gki c n g
versus g ns gin cil k g mk
||
||
γ lms
versus
γ slm ,
(4.13)
which is the equality (4.5). We have thus proven:
Proposition 4.3 The Hamiltonian vector fields on Rn , equipped with the Poisson brackets (4.1) admit the formulation, called Lax formulation
Ẋ = [∇H, X]
P
Definition 4.4 Let H = ij Bij xi xj . The Hamilton equation associate with H
are quadratic equations in the xi ’s that are called Euler equations on g, meaning
that
X
Bij xi E j .
(4.14)
Ẋ = [∇H, X], ∇H =
ij
Let us turn to discuss the intrinsic meaning of these Poisson brackets. Let F
and G be functions on the dula space g. On C ∞ (g∗ ) we can naturally define
a bracket as follows, called Lie-Poisson(-Kostant-Kirillov) bracket. We start
noticing that if
F, G : g∗ → R(C)
then dF |α is, for every α an element of Hom g∗ , R(C) ), that is an element of
the double dual g∗∗ , i.e., can be naturally identified with an element of g. Se
we define
{F, G}(α) =< α, [dF |α, dG|α] > .
(4.15)
In particular, let us put on g∗ coordinates x1 , . . . , xn obtained considering the
basis E i in g and developing α on teh dual basis ǫi , i.e.:
X
α=
xi (α)ǫi , < ǫi , E j >= δij .
(4.16)
i
So we see that, setting as it follows from the definition of xi ,
dF |α =
X ∂F
l
E l , dG|α =
l
∂x
13
X ∂G
l
∂xl
El,
we get
{F, G} =<
=
X
ilk
=
X
ilkm
=
X
lkm
X
i
X ∂F
X ∂G
l
xi ǫi , [
E
,
Ek >
l
k
∂x
∂x
l
k
∂F ∂G
xi l k < ǫi , [E l , E k ] >
∂x ∂x
xi
(4.17)
∂F ∂G lk
c m < ǫi , E m >
| {z }
∂xl ∂xk
δim
xm
∂F ∂G lk
c m
∂xl ∂xk
which is indeed what we get from the definition (4.1).
5
Liouville Integrable Systems
Let us consider a system XH endowed with n = 21 dim(M) mutually commuting
functionally independent integrals of the motion {f1 (p, q), . . . , fn (p, q)}. We
shall show that in this situation it is possible (let us say, in principle), to define a
generating function of a canonical transformation that sends {p, q} into another
set of canonical coordinates, whose first set is exactly the set {f1 , . . . , fn }.
The key to understand this property is the geometry of the situation, as
depicted in Figure 2.
In the open set U where df1 ∧ df2 . . . dfn 6= 0 define a foliation of M, whose
leaves Sc are the common level surface of the {f1 (p, q), . . . , fn (p, q)}, that is the
submanifolds defined by the equations
ci ∈ V ⊂ R n .
f1 (p, q) = c1 , . . . , fn (p, q) = cn ,
(5.1)
That is, we have defined a map
πF : U → V, π(p, q) = (f1 (p, q), . . . , fn (p, q).
(5.2)
Let us momentarily fix a (smooth) section of this map, that is, a function
σ0 : V → U s.t. fi (σ0 (p), σ0 (q)) = ci , i = 1, . . . , n, and, for all m = (p, q) ∈ U
define an “object” according to the following recipe:
1. Determine to which leaf Sc m belongs, i.e., compute the values ci = fi (m).
2. Consider the point m0 = σ0 (c), and a path γ in Sc joining m0 and m.
3. Compute
S(m; σ0 ; γ) :=
14
Z
θL .
γ
(5.3)
Figure 1: The Geometry
p
f(p,q)=f
i
f
111111111111111111111111
000000000000000000000000
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
m(q,f)
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
γ
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
m(q,f)
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
0
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
000000000000000000000000
111111111111111111111111
q
q(m)
0
q
f
What we want to show is that S is, whenever one fixes σ0 a “(possibly multivalued) function” of m. This means that, if γ ′ is another path joining m0 to m,
that can be continuously deformed to γ in Sc , then
S(m; σ0 ; γ) = S(m; σ0 ; γ ′ ).
(5.4)
Remark that the above condition means that, if γ and γ ′ are parametrized
as γ(t), γ ′ (t), t ∈ [0, 1] then there exists a continuous map
F : [0, 1] × [0, 1] → Sc , s.t.F (t, 0) = γ(t), F (t, 1) = γ ′ (t), ∀ t.
(5.5)
Actually this holds true thanks to the involutivity of the fi′ s (and to Stokes’
lemma). Indeed, (see Figure 3) we have that
Z
Z
Z
Z
′
S(m; σ0 ; γ) − S(m; σ0 ; γ ) = θL − θL =
dθL =
ω.
(5.6)
γ
γ
Ω
Ω
where Ω is the 2-dimensional manifold contained in Sc and bounded by γ and
γ ′ . Now, in coordinates,
Z
ω
Ω
15
is given by the following. The map F of (5.5) gives, in coordinates xα where
P
ω = α,β ωα,β dxα ∧ dxβ ,
xα = xα (t, s).
Thus, dxα =
ω=
X
∂xα
∂xα
dt +
ds. Hence, on Ω,
∂t
∂s
ωα,β ((x1 , . . . , x2n )dxα ∧ dxβ =
α,β
X
ωα,β ((x1 (t, s), . . . , x2n (t, s))(
α,β
∂xα
∂xα
∂xβ
∂xβ
dt +
ds) ∧ (
dt +
ds = (5.7)
∂t
∂s
∂t
∂s
< ωα,β ((x1 (t, s), . . . , x2n (t, s)), X1 (t, s) ∧ X2 (t, s) > dt ∧ ds,
that is the (oriented) ordinary surface integral of the function
ω
e (t, s) =< ωα,β ((x1 (t, s), . . . , x2n (t, s)), ξ1(t, s) ∧ ξ2 (t, s) >,
∂
∂
), ξ2 (t, s) = F∗ ( ) are the images of the two basis tangent
∂t
∂s
vectors to the square (t, s) ∈ [0, 1] × [0, 1] under F .
R
Remark. The above formula shows that the value Ω ω is actually independent of the choice made in its definition. Indeed, if one changes coordinates in
U, the function ω
e stays unchanged, while if one changes the parametrization
of Ω, that is, the map F , then one recovers the familiar formula for change of
variables in two-dimensional integrals.
For our purposes, the important point is the fact that ξ1 , ξ2 are vectors in
the tangent space to Sc . As such, they will be written as linear combinations of
the vectors
Xfi = P dfi.
(5.8)
where ξ1 (t, s) = F∗ (
Indeed, the condition {fj , fi } =< dfj , P dfi >= 0 shows that Xfi are tangent
to Sc while the nondegeneracy condition df1 ∧ df2 . . . dfn 6= 0, together with the
fact that ω is symplectic, implies that they are at every point of Sc a basis in
T Sc .
P
Writing now ξa = i gia Xfi , we see that
X
X
< ω, ξ1 ∧ ξ2 >=
gi1 gj2 < ω, Xfi ∧ Xfj >=
gi1 gj2 {fi , fj } = 0.
(5.9)
i,j
i,j
At this point our analysis bifurcates: we will at first define the so-calles Action–
Angle variables; then, by looking more carefully at the definition of S, we will
discuss the Hamilton-Jacobi equation and the problem of Separation of Variables.
16
5.1
Action-Angle variables
Action Angle variables can be defined for those regions in which the leaves Sc
are compact. It relies on the following theorem:
Proposition 5.1 A compact connected n-dimensional manifold admitting a
globally defined basis Xi in its tangent bundle, for which [Xi , Xj ] = 0, is diffeomeophic to the n dimensional torus Tn , that is the quotient Rn /Λ of Rn
modulo the discrete group Zn generated by a non degenerate lattice Λ ⊂ Rn over
the integers.
Let us now consider the set W in phase space for which Sc is compact, which
we can regard as a manifold fibered in tori over F (W ). Possibly restricting the
base, this will be a product Tn × F (W ′ ).
Let us fix a set of n generators γi of the homology of Tn , and let us change
generators in the ring of the constants of the motion as:
I
1
Ii =
pdq.
(5.10)
2π γi
Notice that the Ii ’s depend only on the fj .
These new constants of the motion will be called Action variables. We will
∂I
be non zero, so that, actually, the I ′ s are indeed
suppose that the Jacobian
∂f
a complete set of generators for the ring of integrals of XH . So, the generating
function S(q, f ) can be seen as a function of the 2n variables (q, I). Let us call
ψ i the variables canonically conjugated to the Ii , namely,
ψi =
∂
S(q, I).
∂Ii
(5.11)
We want to show that these variables are indeed angles on Tn . To this end it
suffices to show that their variation along the k-th cycle equals 2π if i = k and
vanishes otherwise. So:
I
X I ∂ψ i
∂ψ i
j
i
dq
+
dIj =
dψ = ((q, I)are coordinates) =
j
∂Ij
γk
γk ∂q
j
I
I
∂ψ i j
∂ 2 S(q, I) j
(γi is tangent to Sc ) =
dq =
dq
=
(5.11)
=
j
q j Ii
γk ∂q
γk
(5.12)
∂S
(the transformation is canonical so j = pj )
∂q
I
I
∂
∂
∂
=
pj dq j = (γk is fixed) =
2πIk = 2πδik ,
pj dq j =
∂Ii γk
∂Ii
γk Ii
H
that is γk dψ i = 2πδik which was what we had to prove.
17
Remark. The notion of action-angle variables is of great importance for
what the structural aspects of integrable systems are concerned. Indeed our
analysis can be restated as follows. By the introduction of the action variables,
and of their conjugated variables (which are angles on tori) we still have the
property the the Hamitonian equation of motion read

 I˙i = 0
∂H
(5.13)
 ψ̇ i =
= ωi (I1 , . . . , In )
∂Ii
with the additional property that the second half of the equation describe indeed
a regular motion on a torus, dexscribed in angular coordinates. In particular,
we see that, whenever the frequencies are linearly dependent on the integers,
i.e., there is a relation of the form
X
ni ωi = 2πN,
i
then the trajectory is periodic. For this reason we call such motions on tori
quasi - or conditionally periodic motion.
Thus the upshot is that, in those regions of phase space in which the fibers
(leaves) Sc are compact, the trajectory of an integrable system are quasi-periodic
orbits that, in the generic case, fill the torus Sc .
Unfortunately, the Action-Angle approach is less effective for what the “applicative” problem of finding the solutions of an integrable system is concerned,
also in the regions where the invariant tori exist. Indeed, the main problems
are two:
1. identify tori in M and the generators of their homology.
2. (This is a general problem.) Explicitly express the momenta pi in terms
of the variables q, I (or, at least, q, f ).
The following – indeed very classical – examples will show how to use these
techniques, and pave the way to the solution to the second problem mentioned
above.
5.2
Hamilton-Jacobi equation and Separation of Variables
Let us reconsider the defining equation of the canonical transformation that
“trivializes” the dynamics, that is (5.3), which we rewrite:
Z
S(m; m0 ) = θL
(5.14)
γ
18
Under the hypothesis - that we imnplicitly made - that the relation fi (p, q) = ai
(ai are parameters that parametrize the leaves Sc ) can be inverted w.r.t. the pi ,
that is yield pi = pi (q, a) we can choose as section m0 of the IM map (see the
Figure)
m → (f1 (m), . . . , fn (m))
the one given by q = q0 . In this way the integral (5.14) becomes an integral in
the q space, i.e.:
Z qX
(5.15)
pk (q; f )|f=a dq
S(q, f ) =
q0
k
where the constants of the motion fi play the role of parameters (just think of
the examples on the previous lecture), and label the specific leaf of the foliation
we are in. We remark that the same “passive” role is played by these constants
of the motion in the second half of the ‘characteristic equations’ of a second
kind canonical transformation, that is
pi =
∂S(q, f )
.
∂qi
(5.16)
We notice, furthermore, that once fixed the values of these parameters, S becomes a function of q, depending on these additional parameters fi , and the
Hamiltonian H is a quantity h depending (as we discussed before) on these parameters, or, to cut a long story short, becomes one of these parameters. Thus
we see that
S = S(q1 , · · · , qn ; f1 , . . . , fn ) ≡ S{f
1 ,...,f
()
n}
(q1 , · · · , q − n)
satisfies, the (nonlinear) PDE
H(qi ,
∂S
) = h.
∂qi
(5.17)
This equation is called the (stationary) Hamilton-Jacobi equation.
It is outside the size of this lecture course to discuss in details the features of
the HJ equation. We will limit our analysis to some specific topics, taylored on
the needs of the present lecture course, and aimed at unravelling the applicative
aspects of these equations for what integrable systems are concerned.
Definition 5.2 A solution S(q; f ) of the HJ equation (5.17), depending on
n parameters f1 , . . . , fn is called a complete integral of the Hamilton-Jacobi
equation if the matrix of second derivatives
∂ 2 S(q; f )
∂q i ∂fj
is non-degenerate.
19
Remark. In many texts the Hamilton-Jacobi equation is thought of as an
equation for a function S(q, t), depending explicitly on time, and reads
∂S
∂S(q, t)
+ H(q,
) = 0.
∂t
∂q
(5.18)
Given a solution of (5.17), we can obtain a solution of this equation setting
S(q, t) = S(q) − h t.
Proposition 5.3 (Jacobi) Let us suppose that we know a complete integral
Sf (q) of the HJ equation (5.17). Then the Hamiltonian vector field XH is completely integrable.
Proof. The crucial observation is that the relation
∂ 2 S(q; f ) 6= 0
Det
∂q i ∂fj
implies that the relation
pi =
(5.19)
∂S
∂q i
can be locally solved to yield
fi = fi (p, q).
(5.20)
Since
∂fi
∂ 2 S(q; f ) −1
∂pj −1
=
=
∂pj
∂fi
∂q i ∂fj
that the quantities {q1 , . . . , qn , f1 . . . , fn } are a set of coordinates on the phase
space. Then we can use the complete integral Sf (q) as a generator of a second
kind canonical transformation. This shows that the parameters fi , which, according to (5.20) are to be thought of as functions on the phase space M, are
mutually commuting, and, since H = H(fi), they are indeed mutually commuting constants of the motion.
This general proposition displays its effectiveness when a further peculiarity,
which unfortunately depends on the choice of canonical coordinates in M, takes
place. The following proposition well exemplifes this situation.
Proposition 5.4 Suppose that we are given canonical coordinates (p, q) in M
and an Hamiltonian H(p, q) that has the functional form:
P
φi (pi , qi )
,
(5.21)
H(p, q) = Pi
i ψi (pi , qi )
that is is the ration of two sums of functions φi , ψi that depend on the pair of
canonical coordinates (pi , qi ). Then the HJ equation associated with (5.21) can
be solved by quadratures, and, under mild hypotheses on the φi , ψi , a complete
integral can be found.
20
Proof. The Hamilton-Jacobi equation for S = S(q 1 , . . . , q n ) reads
P
∂S
, qi )
∂qi
= h.
P
∂S
, qi )
i ψi (
∂qi
i
φi (
Multiplying by the denominator we get
X ∂S
∂S
φi ( , qi ) − h ψi ( , qi ) = 0.
∂qi
∂qi
i
(5.22)
At this point we make the following Ansatz, (the “Separation Ansatz”):
We seek for a function S of n variables, depending on n parameters α =
{α1 , α2 , . . . , αn }, written as a sum of n functions Si which depend only on the
i-th variable q i (and on the parameters), that is:
X
S(q 1 , . . . , q n ; α) =
S(i) (q i; α).
(5.23)
i
Plugging the Ansatz (5.23) into (5.22) we get
X ∂S(i)
∂S(i)
φi (
, qi ) − h ψi (
, qi ) = 0,
∂q
∂q
i
i
i
(5.24)
and notice that this equalities relate n expressions
Φi = φi (
∂S(i)
∂S(i)
, qi ) − h ψi (
, qi )
∂qi
∂qi
in which each Φi depends only on the variable qi . Hence, this functional relation
might hold if and only if each of these expression is separately equal to a constant
αi , with the relation
X
αi = 0
i
So, as a set of independent constants we can take h and the first n − 1 α’s.
The procedure for solving the HJ equation is now clear: we suppose that (it is
the “mild assumption” we mentioned in the statement of the Proposition) that
each of the relations
φi (
can be inverted to yield
∂S(i)
∂S(i)
, qi ) − h ψi (
, qi ) = αi
∂qi
∂qi
∂S(i)
= Gi (qi ; αi , h),
∂qi
21
and so, remarking that S(i) is a function of a single variable q i , each of the
summands S(i) in (5.23) is given by the integral
Z q
Gi (si ; αi , h)ds.
(5.25)
q0i
Let us now consider
Φei = φi (pi , q i) − H(p, q)ψi (pi , q i)
(5.26)
For what we have said before, these expressions coincide the parameters αi
appearing in the definition of the HJ complete integral, and thus, provide a set
of mutually commuting integrals of the motion (remark that the matrix
∂ 2 S(q; f )
∂q i ∂αj
is easily computed in terms of known quantities.)
Exercise. Explicitly check the involutivity of the n functions:
f1 , . . . , Φ
fn
H, Φ
Let us now reconsider the equations (5.26) written as follows:
e 1, . . . , Φ
e n )ψ(q i , pi ) = 0,
Φei − φi (q i , pi ) − H(Φ
i = 1, . . . , n.
(5.27)
In this form they appear as n relations (to be called Jacobi separation relations)
e i (also through the Hamiltotying the mutually Poisson-commuting quantities Φ
nian H) and, for each value of the index i, a single pair of canonical coordinates,
namely q i and pi . They are the key ingredient (see equations (5.25)) to get the
expression of the generating function of the canonical II kind transformation we
are looking for.
Definition 5.5 A complete integral S(q; α) of the HJ equation (5.17) is called
a separated complete integral if it is
a A complete integral
b It has the additive form
S(q 1 , . . . , q n ; α) =
X
S(i) (q i ; α).
(5.28)
i
A set of canonical coordinates (pi , q i ) in which the HJ equation admits a separated solution is called a set of separated (or separation) coordinates for H.
22
Proposition 5.6 A set of coordinates is a separated set iff there are n relations
of the form

1

 Φ1 (p1 , q ; f1 , . . . , fn ) = 0

 Φ2 (p2 , q 2 ; f1 , . . . , fn ) = 0
(5.29)
..

.


 Φ (p , q n ; f , . . . , f ) = 0
n n
1
n
with
∂Φi
6= 0
∂pi
and Det
∂Φi
6= 0,
∂fk
(5.30)
tying n functionally independent first integrals f1 , . . . , fn of a Hamiltonian vector
field XH and pairs of canonically conjugated variables.
Proof. Let us show sufficiency. Under the hypotheses of (5.30), we can solve
the separation relation with respect to the pi ’s, as
pi = pi (q i ; f1 , . . . , fn ).
(5.31)
So we can substitute in the definition (5.15) of the (a) generating function
associated with the foliation defined by fi = constanti these expressions to get
S(q; f ) =
n Z
X
i=1
qi
pi (si ; f1 , . . . , fn )dsi =
q0i
X
S(i) (q i ; f1 , . . . , fn ).
(5.32)
i
∂S i
, q ) = h(f1 , . . . , fn ), and, from the second of (5.30) we see that
∂q i
this is a complete integral.
The converse statement is trivial. Indeed, the definitions
Clearly, H(
∂ X
∂
S(j) (q j ; f1 , . . . , fn ) = i S(i) (q i ; f1 , . . . , fn )
i
∂q j
∂q
(5.33)
provide the separation relations we seek for.
Example Let us reconsider the motion in a central force field (now in R3 ).
In Cartesian coordinates, the Hamiltonian reads
1
H = (p2x + p2y + p2z ) + V ((x2 + y 2 + z 2 )).
2
(5.34)
Fromm the fact that each of the compnents of the angular momentum of the
particle computed with respect to the “source” of the potential vanishes, and
that {H, Lz , L2 } are mutually commuting, we know that the system fill the
hypotheses of the Liouville theorem. To separate variables in the HJ equation
(i.e., to solve the problem up to quadratures), we pass to polar coordinates
23
(r, ϑ, ϕ), with equatorial plane orthogonal to the angular momentum opf the
particle. In these coordinates, the Hamiltonian reads
1
1
1
H = (p2r + 2 p2ϑ + 2 2 p2ϕ ) + V (r 2 )
2
r
r sin (ϑ)
(5.35)
and the defining equations for the other two integrals of the motion are
Lz = pϕ ;
2
L =
p2ϑ
p2ϕ
.
+
sin2 (ϑ)
So we see that the two relations (5.35,5.36) can be written as

 Lz − pϕ 2= 0
L2 − sinL2z(ϑ) = 0

H − 21 (p2r + r12 L2 ) − V (r) = 0
(5.36)
(5.37)
that are indeed separation relations of the kind (5.29).
6
Lax Pairs
Lax pairs were introduced in the theory of integrable PDEs, and not of ODEs.
In particular they were introduced in the study of the KdV equation
ut = uxxx + 6uux .
However, as it often happens (a similar case is the theory of R-matrices, such
a tool revealed its relevance also in the theory of finite dimensional systems.
We have already encountered Lax pairs in the theory of Hamiltonian equations
defined on duals of Lie-Poisson reductive algebras. We are presently going to
analyze such formulations of the equations of the motion.
Definition 6.1 Let X be a vector field on a manifold M. We say that the pair
(L, P ) is a Lax pair for X or, equivalently, that X admits a Lax representation
if there are two smooth maps
L : M → g,
P :M →g
(6.1)
from M to a Lie algebra g which is not restrictive to assume to be (a subspace
of ) the Lie algebra of n × n matrices such that
1. The map L is non-trivial (possibly injective).
2. the evolution along X entails the equations
d
L = [P, L],
dt
24
(6.2)
Remark. The first requirement of this definition is purposedly left mathematically somewhat vague. The reason for this choice will be clear from the study
of the Toda system and of the Calogero system. For the moment let us just say
that the meaning of the word “non-trivial” is to be discussed case by case.
The method of Lax representation is, on general grounds, a very efficient although non-algorithmic - way to find constants of the motion for X.
Let us consider, along with the Lax matrix L (who is the principal character
in this play, P being playing a side role) its characteristic polynomial. For
simplicity, let us suppose that L be diagonalizable.
n
p(λ) = Det(λ − L) = λ −
n
X
pi (L)λn−1 .
(6.3)
i=1
The coefficients pi (L) are polynomials in the entries of L (namely, pi is (up to
a sign) the sum of the determinant of the principal minors of L of order n − i,
P
e.g., p1 = Tr(L) = i Lii , and pn = Det(L)). As L is to be thought of as a
function of m ∈ M, so pi are functions of m ∈ M as well.
In the sequel we will prove the following
Proposition 6.2 The coefficients pi of the characteristic polynomial of the Lax
matrix L are constants of the motion for X.
Remark: the eigenvalues of L are functions of these p′i s, being the roots of the
characteristic polynomial. Thus they are constant of the motion as well. This
is why one usually refers to a Lax formulation of an evolution equation (a/k/a/
Lax representation) as an isospectral flow.
We will divide the proof of this proposition (which is very easy) in two parts
for simplicity. The first part is essentially an algebraic lemma. Let us consider,
along with the pi ’s another set of polynomial functions of the matrix elements
of L, namely the polynomials
1
Ii = Tr(Li ).
i
(6.4)
Proposition 6.3 (Newton’s formulæ) There is a polynomial relation (with polynomial inverse) between the pi ’s and the Ii ’s. Namely, one has:
pi = Ii + Pi (I1 , . . . , Ii−1 );
Ii = pi + Pi′ (p1 , . . . , pn ), i = 1, . . . , n; P0 = P0′ = 0.
The polynomials Pi , Pi′ are universal, that is their form is independent of the
number n.
25
In view of Proposition 6.3 to prove the statement, that is, to prove isospectrality of the Lax flow, it suffices to prove that the Ii ’s are constant of the motion
as well. But this is true, thanks to the following chain of equalities:
l
dL l−k−1
1d
1X
d
Tr(Lk ·
Il =
Tr(Ll ) =
·L
)=
dt
l dt
l k=1
dt
(6.5)
dL k−1
Tr(
· L ) = Tr([P, L] · Lk−1 ) = Tr(P, [L, Lk−1 ]) = 0,
dt
where we have used the ad-invariance (a/k/a cyclicity) of the trace.
At the formal level, we finally remark that a Lax representation allows one
to “factorize” the problem of finding solutions of the equations of motion (that
is, of the integration of X) as follows. Let us consider the Cauchy problem
associated with (a coordinate representation of) X:
dxi
= X i (x)
dt
.
(6.6)
i
x (0) = xi0
It will be trasformed, on g = Matn (C) into the problem
d
L = [M, L]
dt
L(0) = L(xi0 )
(6.7)
Let us assume that L(0) be diagonalizable with distinct eigenvalues λ1 , . . . , λn ,
i.e. let us write
L(0) = G(0)ΛG(0)−1, λ = Diag(λ1 , . . . , λn )
(6.8)
where G(0)−1 is the matrix sending “the j-th eigenvector of L(0)” into the j-th
element of the standard basis of Cn (that is, the j-th column of G(0) is a non
zero element of the j-th eigenspace of L(0)).
Since the eigenvalues of the solution L(t) of the Cauchy problem (6.8) are
constant along the flow of X, we seek for a solution of the form:
L(t) = G(t)ΛG(t)−1 .
(6.9)
Taking the derivative w.r.t. t (that is, applying the vector field X, we have,
taking into account that the matrix product (6.9) is multilinear, we get (with
˙
= ∗:
the notation d∗
dt
−1
˙
L̇ = (G)ΛG
− GΛG−1 ġG−1
(6.10)
we get, inserting 1 = G−1 G before the term Λ in the first summand,
−1
−1
−1
−1
−1
˙
L̇ = (G)G
|GΛG
| {z } ġG = [ĠG , L],
{z } − GΛG
=L
=L
26
(6.11)
i.e., a Lax equation with second member of the Lax pair M = ĠG−1 .
This yields the following scheme (the so-called “Lax scheme” - a/k/a Inverse
scattering method of the KdV theory):
Let X admit a Lax representation L̇ = [M, L] and suppose that for initial
data x(0) the matrix L0 = L(x0 ) be diagonalizable. The solution of the Cauchy
problem (6.6) can be obtained via the following steps:
1. Construct the Lax pair, i.e. the map x ∈ M → (L, M) of the system.
2. Diagonalize L0 as L0 = G0 ΛG−1
0 ;
3. Solve the Cauchy problem
(
d
G = MG
dt
G(0) = G0
(6.12)
4. With the solution of (6.12), and with the constant diagonal matrix Λ
construct L(t) = G(t)ΛG(t)−1
5. Inverte the first half of the Lax map to get the solution of the equation of
motion xi = xi (t) from L = L(t).
Remark. As we shall see in the sequel, this scheme is less algorithmic as it
might look. In particuar, points 1,3, and 5 in general need a lot of skill for
being effectively implemented. However, the Lax scheme, and its generalizations have emerged, in the last 30 years, as the most effective way for studying
(and solving!) integrable systems, especially systems of evolutionary PDEs,
and, moreover, a method that shows and enhances connections of the theory of
integrable systems with algebra (e.g. the R-matrix theory) and geometry.
We will discuss these issues later (and in the proper Ph.D. course). For the
moment let us show how the Lax method proves integrability of a remarkable
N particle system, the Toda system.
6.1
The Toda System
We will now discuss a system who has a certain interests in the applications,
and will be used to discuss features, as well as “problems” of the Lax scheme.
The system is a N-body system on the line; that is, its phase space is M =
∗ n
T R , endowed with canonical coordinates {p1 , . . . , q n }, whose Hamiltonian is
HN =
X1
i
2
p2i +
X
exp (q i − q i+1 ), mod N,
i
27
(6.13)
where mod N means that pi+N ≡ pi , qi+N ≡ q i (e.g., q N +1 = q 1 , q 0 = q N .
In the two particle case, we see that
1
H2 = (p21 + p22 ) + 2 cosh (q1 − q2 )
2
is the Hamiltonian of two particles on the line, interactint through a anharmonic
potential of special kind.
In general one can see that, actually, the potential VN has equilibrium points
(qi = c, ∀ i) with nonnegative Hessian.
The Hamilton equation of motion associated with H can be compactly written as follow:
dpi
= exp (qi−1 − qi ) − exp (qi − qi+1 )
dt
(6.14)
dq i
= pi
dt
We will show that this Hamiltonian flow is, irrespectively on the number of
particles N, integrable in the Liouville sense, that is that (6.14) admits N
mutually communting integrals of the motion. We will divide the discussion in
two steps:
1. We will find N constants of the motion.
2. We will show that they are in involution.
For the solution of both problems we will extensively use a Lax representation
for the Hamiltonian flow (6.14).
This is constructed as follows. To simplify notations, define the following
functions (“Flaschka coordinates”):
1
bi = pi , ai = exp ( qi − qi+1 ) mod N, i = 1, . . . , N.
2
(6.15)
We notice that the Hamiltonian can be expressed through these functions as
H=
X1
i
2
b2i +
X
a2i .
i
The 2N functions {bi , ai } are not a coordinate system in M, but they are not
“far” from being coordinates; indeed, they are not functionally independent,
Q
since i ai = 1, but if we replace, say, aN by the center of mass coordinate
Q=
1 X
qi
N i
we get a honest coordinate system on M.
28
Let Ei,j be the N × N matrix whose elements are all zeroes, except for the
elemnt on the i-th row and j-th column, which equals 1. (In other words,
Eij = ei ⊗ ǫj , where ei is the standard basis in Rn and ǫi is its dual basis).
Let us define the two matrices L and M by:
X
L=
pi Eii + ai (Ei,i+1 + Ei+1,i )
i
M=
X
(6.16)
ai (Ei,i+1 − Ei+1,i ),
i
that is,

p1 a1 0
···
 a1 p2 a2
0

.. ..
..

.
.
.
 0
L=
.
.

..
..

 0
aN −2
aN
···

0
a1 0
···
 −a1 0 a2
0

.. ..
..

.
.
.
 0
M =
..
..

.
.

 0
−aN −2
aN
···
···
aN
0
..
.
..
0
.

pN −1 aN −1
aN −1 pN








···
−aN
0
..
.
..
0
.
0
aN −1
−aN −1 pN
(6.17)









(6.18)
Proposition 6.4 Along the Hamiltonian vector field XH (6.14), the matrix L
evolves as
1
d
L = [ M, L].
(6.19)
dt
2
Proof. The proof of this well known fact (Flaschka, 1973) is somewhat elementary, and requires only computations. For what the left hand side of the
equation is concerned, we can compute ṗi and ȧi through (6.14) and the definitions (6.15) as:
1
(6.20)
ṗi = a2i−1 − a2i , ȧi = (pi − pi+1 )ai .
2
For what the left hand side of this equation is concerned, we can algebraically
compute the commutator [M, L] recalling that
[Eij , Ekl ] = δjk Eil − δil Ekj ,
and thus verify the validity of the assertion.
From the general theory of Lax evolution equations we thus get that the N
quantities
1
Ii = TrLi
(6.21)
i
29
are constants of the motion.
We remark that the Ii are functionally independent. Indeed, it is clear that
P
I1 = i pi , I2 = H, and, generically,
Ik =
X1
pki + o(pk−2
).
i
k
i
(6.22)
To show that they are mutaully commuting is somewhat more complicated.
At first we remark that, since the Ii′ s are functionally independent, the coefficients pi of the characteristc polynomial of L are functionally independent,
and hence, generically (that is, on an open dense set of M), the eigenvalues
λ1 , . . . , λN of L are independent. We will show that such eigenvalues are mutually commuting.
Let us take two of them, say λ 6= µ and coinsider two normalized eigenvectors
u, v of L associated with λ and µ, that is two vectors satisfying
L u = λu, L v = µv,
(u, u) = (v, v) = 1,
(6.23)
whose existence is guaranteed by the symmetry of L.
Since, e.g., λ = (u, Lu), we have
∂λ
=
∂pi
∂u
∂L
∂u ∂
(u, Lu) = (
, Lu) + (u,
u) + (u, L
) =
∂pi
∂pi
∂pi
∂pi
∂L
∂u
∂u ∂L
∂
(u,
u) + ((
, λu) + (L† u,
) = (u,
u) + λ
(u, u) =
∂pi
∂pi
∂pi
∂pi
∂pi
∂L
u) = u2i .
(u,
∂pi
(6.24)
In complete analogy, since
∂L
= ai (Ei,i+1 + Ei + 1, i) − ai−1 (Ei−1,i + Ei,i−1 )
∂qi
we get
∂λ
= ai uiui+1 − ai−1 ui−1ui .
∂qi
(6.25)
Taking into account the analogous formulæ for µ, we get that the Hamiltonian
vector field associated with µ is
Xµ =
X
− (ai vi vi+1 − ai−1 vi−1 vi )
i
30
∂
∂ + vi2
∂pi
∂qj
(6.26)
so that
{λ, µ} = − u2i (ai vi vi+1 − ai−1 vi−1 vi ) − vi2 (ai uiui+1 − ai−1 ui−1ui ) ,
(6.27)
that is,
{λ, µ} = −
X
(ui vi (Ri + Ri−1 )), with Ri = ai (ui vi+1 − vi ui+1).
(6.28)
i
We now consider the eigenvector equation L u = λ u, which reads for the
components ui (still understanding ui+N ≡ ui , vi+N ≡ vi ),
ai−1 ui−1 + pi ui + ai ui+1 = λui,
(6.29)
and the corresponding equation for v:
ai−1 vi−1 + pi vi + ai vi+1 = µvi ,
(6.30)
Now, multiplying (6.29) by vi and (6.30) by ui, and subtracting we get
(λ − µ)ui vi = ai−1 ui−1 vi − ai ui+1 vi − ai−1 vi−1 ui + ai vi+1 ui ,
that is,
ui vi =
1
(Ri − Ri−1 ).
λ−µ
(6.31)
(6.32)
Substituting this result into (6.28) we get
{λ, µ} =
1 X 2
(Ri−1 − Ri2 )
λ−µ i
which vanishes for the periodicity conditions Rj+N = Rj
7
Lax equations with ‘spectral’ parameter
Let us reconsider §, and the Lie-Poisson brackets. The simplest example of this
instance is well known basic courses in Mechanics: the Euler equations for the
free rigid body in the Euclidean space.
We recall that, in vectorial language, these equations are written as
d
M = Ω × M,
dt
(7.1)
where M and Ω are the angular momentum and the angular velocity in the socalled body reference frame and × denotes the vector product. Recalling that in
31
a suitable body reference frame we have M i = Ii Ωi , (the Ii ’s are called the principal momenta of inertia) we can explicitly write the equations of motion (7.1)
as
1
1
ddtM1 = ( − )M2 M3 = Ω2 M3 − Ω3 M2 and cyclic.
(7.2)
I2 I3
Writing these equations as
d i
M = ǫi lk Ωl M k ,
dt
(7.3)
we see that these equations are Hamiltonian equations on so(3)∗ , with Hamiltonian function
!
1
1 X i 2
(M ) /Ii = (Ω · M) = T
HL =
2
2
i
According to the general recipe of equation (4.8) we can find the Lax representation for (7.1). Let us consider the basis in g = so(3) given by the antisymmetrc
3 × 3 matrices (3.10), which now we rebaptise as






0 0 0
0 0 1
0 −1 0
E 1 =  0 0 −1  , E 2 =  0 0 0  , E 3 =  1 0 0  . (7.4)
0 1 0
−1 0 0
0 0 0
As ad-invariant inner product on g we can take
1
g(X, Y ) = − Trace(XY )
2
so that g ij = δ ij . Then we set:
X
L=
M iEi
,Ω =
i
which gives
since
X
X ∂H
i
E
=
Ωi E i ,
i
∂M
i
i
(7.5)
d
L = [Ω, L]
dt
(7.6)
X
X
X
ǫi lk Ωl M k E i .
[Ω, L] = [
Ωl E l ,
M kEk] =
(7.7)
l
k
ilk
However, this Lax representation is somewhat deceiving: indeed, although
it is true that the traces of the Lax matrix are constants of the motion we see
that
1
(7.8)
T r(L) = 0; Tr(L2 ) = −2(M · M),
2
that is, it yields the trivial constant of the motion (that is, the Casimir of the
Lie Poisson brackets on g). Indeed this somewhat unpleasant feature is not a
specific feature of this example. Indeed it can be proven (see below) that
32
Proposition 7.1 The Casimir functions of the Lie-Poisson brackets are the
ad∗ -invariant functions on g∗ .
The simplest loop off the hole for this situation is the following trick due to
Manakov, to be further discussed and justified in the next lectures stems form
the following idea. Let us suppose that we have a Lax representation of a
dynamical system,
d
L = [M, L]
dt
in which L (and, possibly, M) depend (say, polynomially) on a parameter z;
In this case, we say that this is a Lax representation with a parameter (that
is usually called spectral parameter for a sort of historical reasons) if the above
equation holds identically in the parameter z.
In such a case, the characteristic polynomial of L(z) will be, identically in
z, constant along the Lax flow. The novelty is that, now, in the expansion
X
Det(λ − L(z)) = λn −
pi (z)λn−i
(7.9)
i
the coefficients pi are themselves polynomials in z, say of degree mi ,
pi (z) =
mi
X
pki z k .
(7.10)
k=0
dpi (z)
= 0
dt
(k)
identically in z, all the coeffcients pi are constants of the motion. In this way
we may hope to recover the interesting constants of the motion.
Let us illustrate such a procedure in the case of the Euler equations for the
rigid body.
Let us deform the matrices




0
−Ω3 Ω2
0
−M 3 M 2
0
−Ω1 
0
−M 1  , Ω =  Ω3
(7.11)
M =  M3
2
1
2
1
−Ω
Ω
0
−M
M
0
where the pki are functions on the phase space M. Now, since
into:
M(z) = zA + M; Ω(z) = zB + Ω
(7.12)
with constant diagonal matrices A = diag(ai ), B = diag(bi ). The Lax equations
will read
d
(zA + M) =
[zB + Ω, zA + M]
dt
||
||
(7.13)
dM
2
= z [B, A] +z([B, M] + [Ω, A]) + [Ω, M]
| {z }
dt
=0
33
So, these equations will be equivalent to the Euler equations iff
[B, M] + [Ω, A] = 0.
(7.14)
One can notice that the general solution to this equation, taking into account
that A and B are diagonal, is
Mij =
ai − aj
Ωij (i 6= j).
bi − bj
(7.15)
Since Ωij = ǫijk Mij /Ik one can see that the only non trivial solution to our
problem is
1
ai = b2i ;
a1 = (I2 + I3 − I1 )/ > and cyclic.
(7.16)
2
In this way we get


 
I2 + I3 − I1
0
0
0
−M 3 M 2
1
 + M 3
0
−M 1 
0
I1 + I3 − I2
0
L(z) = z 
2
−M 2 M 1
0
0
I1 + I2 − I3
(7.17)
A straightforward computation shows that, calling C == m − 12 + m22 + m23 ,
the characteristic integrals of the Lax representations are given by:
Tr(L(z)) = F1 (I1 , I2 , I3 )
1
Tr(L(z)2 ) = z 2 F2 (I1 , I2 , I3 ) − C
2
1
Tr(L(z)3 ) = z 3 F3 (I1 , I2 , I3 )+
3
(7.18)
1
2
2
2
z M3 I2 I1 + M2 I3 I1 + M1 I2 I3 − C(I1 + I2 + I3 ) ,
2
2
2
2
that is, the coefficient of z in the last constant of the motion is
(I1 I2 I3 )H + F4 (I1 , I2 , I3 ))C,
(7.19)
which is what we were seeking for.
8
The Periodic Toda system revisited
In this Section we will re-examine the Toda system. First of all we notice that
the Lax pair (6.16) can be deformed as follows:
First of all we can insert a “spectral” parameter u, considering the elementary matrices E′ik where
E′1N = E1N /u, E′1N = uEN 1 ,
34
all others unchanged,
(8.1)
that is, by considering the matrices

p1
a1
 a1
p2

..

.
 0
L(u) = 


 0
u aN
0
a2
..
.
..
.
···
aN /u
0
···
0
..
..
.
.
..
..
.
.
0
aN −2 pN −1 aN −1
···
aN −1 pN






,



0
a1 0
···
−aN /u
 −a1
0
a
0
·
·
·
0
2

..
.. ..
..

.
.
.
0
.

M(u) = 
.
.
.

..
..
..
0


0
−aN −2
0
aN −1
u/, aN
···
−aN −1
pN
(8.2)









(8.3)
Indeed, it is easy to check that, along the Toda flow (6.19) we have
1
L̇(u) = [ M(u), L(u)], identically in u,
2
(8.4)
and since the coefficients of the expansion of Det(λ − L(u)) of u and u−1 are
Q
(−1)N +1 N
i=1 ai ≡ 1, the constants of the motion defined by L(u) and by L =
L(1) are the same.
Futher on, we remark the following fact, that essentially states that the Lax
representation of a vector field in by no means unique.
Proposition 8.1 Let (L, M) be a rank n Lax representation (possibly with spectral parameter) of a vector field X on M, and let G be any function G : M →
GL(n) (possibly depending on the spectral parameter as well) . Then
L′ = GLG−1 , M ′ = GMG−1 + ĠG−1 .
(8.5)
provided another Lax representation for X.
Proof. Let us blindly proceed with the computation:
L̇′ = (GLG−1 )˙ = ĠLG−1 + GL̇G−1 − GLG−1 ĠG−1 =
−1
−1
−1
−1
ĠG−1 |GLG
{z } − GLG
| {z } ĠG + G[L, M]G
L′
L′
−1
= [ĠG−1 , L′ ] + [GMG−1 , |GLG
{z }] =
L′
[ĠG−1 + GMG−1 , L′ ] = [M ′ , L′ ].
35
(8.6)
Remark A) Notice that the characteristic polynomials (and hence the constants
of the motion) of L and L′ coincide.
B) Geometrically the second member of a Lax pair transforms, under the
gauge transformation L′ = GLG−1 transforms as a connection 1-form.
Using this gauge freedom, taking
G = diag(1, a1 , a1 a2 , . . . , a1 a2 · · · aN −1 ),
(8.7)
we arrive at the following Lax matrix for Toda:
′
L (u) =

N
X
(pi Eii + a2i Ei+1,i + Ei+1,i ) =
i=1
p1 1
0
···
a2N /u
 a21 p2 1
0
···
0

..
.. ..
..

.
.
.
.
 0

.
.
.

..
..
..
0

2
 0
aN −2 pN −1
1
u
···
a2N −1 pN
8.1





,



(8.8)
Dual Lax representation for the Periodic Toda Lattice and r-matrix
In the previous Section we have presented a Lax representation for the periodic
Toda lattice of rank N, equal to the number of particles, and of degree 2 in the
paramenter u. It turns of that it admits as well a Lax representation of rank
2 and degree N in another parameter v (whose relation with u will be clarified
later on).
The idea is the look at the label of the particle as that of a labeling of a
lattice, and (cleverly) associate a 2 × 2 matrix Li to each site.
This “site” matrix is defined as follows:
pi − v − exp −qi
.
(8.9)
Li(v) =
exp qi
0
By means of this matrix one defines the Lax matrix as:
L(v) = LN (v) · LN −1 · · · L2 · L1 .
We notice that Det(L) = 1, while for instance, for N = 2 we get
2
v − v(p1 + p2 ) + p1 p2 − exp q1 − q2 (v − p2 ) exp −q1
L(v) =
;
(p2 − v) exp q2
− exp q2 − q1
36
(8.10)
(8.11)
so
Tr(L2 )(v) = v 2 −v(p1 +p2 )+(p1 p2 −exp (q1 − q2 )−exp (q2 − q1 )) ≡ v 2 +I11 v +I01 .
and we see that the physical 2-particle Hamiltonian H2 is a function of these
coefficients,
1
H1 = (I11 )2 − I01 .
2
The usefulness of this iterative definition is elucidated by the following arguments. Let us consider, along with the site matrix L the two 4 × 4 matrices
1
2
Li = Li ⊗ 1, Li = 1 ⊗ Li
(8.12)
and denote by
1
2
{Li (v), Lj (v ′ )}
(8.13)
the 4 × 4 matrix whose elements are the Poisson brackets of the matrix elements
of Li (u) and Lj (u′ ). A straightforward computation shows that
1
2
1
2
{Li (v), Lj (v ′ )} = δij [r12 (v − v ′ ), Li (v)· Lj (v ′ )]
where

1

1
 0
r12 (v − v ′ ) = ′
v −v  0
0
0
0
1
0
0
1
0
0

0
0 
.
0 
1
(8.14)
(8.15)
Thanks to the fact that matrix elements located at different sites Poisson commute, and to the Leibnitz property both of the Poisson and the Lie bracket, we
see that (8.14) implies the relation
1
2
1
2
{L (v), L (v ′ )} = [r12 (v − v ′ ), L (v)· L (v ′ )].
(8.16)
Let us now quit for the moment the concrete case of the Toda system, and
give general definitions and a theorem.
˙
Definition 8.2 Let (L)
= [P, L] be a Lax representation for a Hamiltonian
vector field XH (possibly with spectral parameter). We say that the lax representation admits a multiplicative (or quadratic) r-matrix structure if (8.16)
holds. We say that it admits an additive (or linear) r-matrix structure if it holds
1
2
1
2
{L (v), L (v ′)} = [r12 (v − v ′ ), L (v)+ L (v ′ )].
(8.17)
A matrix r12 satisfying (8.16) or (8.17) is called a classical r-matrix. An rmatrix that – unlike the one of (8.15) – depends non trivially on m ∈ M is
called a dynamical r-matrix.
37
The importance of r-matrices in the (modern) theory of integrable systems
relies in the fact that they are the missing ring in the chain connecting Lax
representations and the Liouville theorem. Indeed it holds:
˙ = [P, L] be a Lax representation for a Hamiltonian
Proposition 8.3 Let (L)
vector field XH (possibly with spectral parameter), which admits an r-matrix
structure (multiplicative or additive). Then the spectral invariants of L are
mutually commuting quantities.
Proof. By using the Leinbitz property of both sides of the equations we see
that it holds:

1
2


 [Anm , L (v)+ L (v ′ )] additive
1
2
{Ln (v), Lm (v ′ )} =
(8.18)

1
2

 nm
[A , L (v)· L (v ′ )] multiplicative
where
n−1,m−1
nm
A
=
X
1
n−p−1
L
2
m−q−1
(v)· L
1
2
(v ′ ) · r12 (v − v ′ )· Lp (v)· Lq (v ′ )
(8.19)
p=0,q=0
The proof then follows noticing that the trace of the tensor product is multiplicative, that is,
1
2
TrV1 ⊗V2 (T , S) = TrV1 ⊗V2 (T ⊗ S) = TrV1 (T )TrV2 (S)
(8.20)
Reading this equality in reverse, and taking the trace of (8.18) we get the thesis.
After stressing once again this is a general property of Lax representations
with r-matrix structure, let us come back to our Toda system. We will now show
that the spectral invariants of the 2×2 matrix L(v) of (8.10) coincide with those
of the N × N matrix L′ (8.8) and hence of the Flaschka Lax matrix (6.17). This
argument is due to Sklyanin.
Let θ1 be an eigenvector of the Lax matrix L(v) (8.10), corresponding to an
eigenvalue u, i.e. let θ1 solve
L(u)θ1 = uθ1 .
(8.21)
θj+1 = Lj θj , j = 1, . . . , n.
(8.22)
Let us define iteratively
38
Thus (8.21) reads
θn+1 = uθ1
Writing
θj =
φj
ψj
(8.23)
we can explicitate equations (8.228.23) as the following recurrence relations:
φj+1 = vφj − pj φj − exp(−qj )ψj , j = 1, . . . , n − 1
uφ1 = vφn − pn φn − exp(−qn )ψn
(8.24)
ψj+1 = exp(qj )φj , j = 1, . . . , n − 1
uψ1 = exp(qn )φn
Eliminating the ψj ’s from the first set of equations thanks to the second set we
get the following set of equations:

exp(qn − q1 )


φ2 = vφ1 − p1 φ1 −
φn



u


 ...
(8.25)
φj+1 = vφj − pj φj − exp(qj−1 − qj )φj−1



.


 ..


uφ1 = vφn − pn φn − exp(qn−1 − qn )φn−1
that is, rearranging terms,

exp(qn − q1 )


vφ1 = φ2 + p1 φ1 +
φn



u


.
 ..
 vφj = φj+1 + pj φj + exp(qj−1 − qj )φj−1


..


.



vφn = uφ1 + pn φn + exp(qn−1 − qn )φn−1
(8.26)
This means that the N component vector T (φ1 , · · · , φn ) is an eigenvector of the
matrix L′ (u) relative to the eigenvalue v iff the 2 component vector θ1 is an
eigenvector of the matrix L(v) relative to the eigenvalue u. This means that
Det(v − L′ (u)) = C Det(u − L(v)),
and our proof if finished noticing that C = CN =
9
(8.27)
(−1)N+1
.
x
R-matrices, Lax representation with spectral
parameters, and SoV
Our next step is to show that Lax representations with spectral parameter,
admitting an r-matrix structure may solve the SoV problem, that is provide
39
an effective way not only to ascertain integrability of a system, but to actually
integrate it. Furthermore, the method introduces the notion of Algebraically
Completely Integrable Hamiltonian System.
We will confine our discussion to the case of rank 2 Lax matrices of the form
A(x) B(x)
L(x) =
,
(9.1)
C(x) D(x)
where A, B, C, D are polynomials in x of degree, respectively,
degA = NdegB = degC = N − 1, degD = N − 2,
such as the Lax matrix (8.10) of the Toda system, with the constraint
A(x)D(x) − B(x)C(x) = 1.
(9.2)
Actually, as it will be shortly seen, this constraint simply amount to fixing the
value of a Casimir of the r-Poisson brackets. Indeed we will assume that the
multiplicative r-matrix brackets hold:
1
2
1
2
{Ln (x), Lm (y)} = [r12 (x − y), L (x)· L (y)], i. e.
with

1
 x−y

 0

r12 (x − y) = 

 0


0
0
0
0
0
1
x−y
0
0
0
1
x−y
(9.3)





.




(9.4)
1
x−y
A straightforward computation shows that these r-brackets are a shorthand
notation for the following brackets between the polynomial valued functions
A(x), B(x), C(x), D(x):
0
0
{A(x), A(y)} = 0, {B(x), B(y)} = 0, {C(x), C(y)} = 0 {D(x), D(y)} = 0
C(x)A(y) − A(x)C(y)
A (x) B (y) − B (x) A (y)
, {A(x), C(y)} =
{A(x), B(y)} =
x−y
x−y
B(x)C(y) − C(x)B(y)
D(x)A(y) − A(x)D(y)
{A(x), D(y)} =
{B(x), C(y)} =
x−y
x−y
C(x)D(y) − D(x)C(y)
D(x)B(y) − B(x)D(y)
{C(x), D(y)} =
{B(x), D(y)} =
x−y
x−y
(9.5)
We notice that the characteristic polynomial of L
Γ(z, x) := Det(z − L(x)) = z 2 − zTr(L(x)) + 1
40
(9.6)
defines, via Γ(z, x) = 0 a Riemann surface (actually, the affine part of it), the
spectral curve whose coefficients are, thanks to the Lax evolution equations,
constants of the motion. In other words, integrability translates in the fact that
the curve is left invariant by the Hamiltonian vector fields associated with any
of the spectral invariants of L(x) – that is, with any of the coefficients of the
polynomial
N
X
Tr(L(x)) = xN +
ti xN −i .
(9.7)
i=1
The idea, actually quite natural, is to use the equation of the spectral curve (9.6)
to get Separation Relations.
The only non trivial point in this respect in thus to find N − 1 pairs (xi , yi)
that are canonical coordinates and satisfy the spectral curve equation, or, said
the other way around, to find N −1 points Pi lying on Γ, depending on the phase
space M, whose x, y coordinates (x(Pi ), y(Pi)) (or suitable functions thereof, see
below) satisfy canonical commutation relations.
Remark that, for the Toda case, we actually need only N − 1 non trivial
separation relations, since, thanks to the translational invariance of the problem,
the missing one is the definition of the total momentum.
Looking at the Lax matrix (8.10), and taking into account the first line of
(9.5), we see that the zeroes xi of L12 (x) = B(x) provide N − 1 functionally
independent quantities that are in involution,
{xi , xj } = 0, where B(xi ) = 0, i = 1, . . . , N − 1.
(9.8)
The reason for this property to hold is simply that, as a simple inductive arPN
N −i
gument shows, the coefficients bi of the polynomial B(x) =
are
i=1 bi x
functionally independent, while the R-matrix relation
{B(x), B(y)} = 0
(9.9)
is a compact way of stating that {bi , bj } = 0, i, j = 1, . . . , N, which implies the
involutivity of the zeroes of B(x).
Next we consider the N − 1 points yi defined by the spectral curve relation
Γ(xi , yi) = 0. Indeed, for x = xi , with B(xi ) = 0 we have that, explicitly, yi are
given either by yi = A(xi ) or by yi = 1/A(xi ). Indeed, yi is a solution of
y 2 − (A(xi ) + D(xi ))y + 1 = 0,
(9.10)
that is, thanks to the constraint (9.2) – which also guarantees that A(xi ) 6= 0 –
and the fact that B(xi ) = 0, solution of
y 2 − A(xi ) +
1 y + 1 = 0.
A(xi )
41
(9.11)
So we are left with the computation of the Poisson brackets
{A(xi ), A(xj )}, {A(xi ), xj }.
(9.12)
The crucial point is to compute these brackets knowing the R-brackets (9.5)
between the polynomials A(x), B(x) that are, we recall,
{A(x), A(y)} = 0, {A(x), B(y)} =
A (x) B (y) − B (x) A (y)
.
x−y
(9.13)
The key idea to perform such computations is the following. The polynomial
A(x) should be regarded as a function
A(x) : M → C[[x]],
(9.14)
and its differential as a section of T ∗ (M)[[x]]. Explicitly, if m ∈ M,
A(x) =
N
+1
X
N +1−i
ai (m)x
,
dA(x) =
N
+1
X
dai (m)xN +1−i
If g is a smooth function on M we can consider A(g) =
as an ordinary function on M. Clearly,
d(A(g)) =
N
+1
X
i=1
dai (m)(g(m))N +1−i +
(9.15)
i=1
i=1
N
X
PN +1
i=1
ai (m)(g(m))N +1−i
ai (m)(N + 1 − i)g(m))N −i dg, (9.16)
i=1
that is,
∂A(x) dg
d(A(g)) = dA(x)x=g +
∂x x=g
So, if A1 (x), A2 (x) are two such polynomials2 we have:
{A1 (g1 ), A2 (g2 )} ≡ hd(A1 (g1 )), P dA2 (g2 ))i =
∂A1 (x) x=g1 +
{A1 (x), A2 (y)}y=g
{g
,
A
(y)}
1
2
y=g2
2
∂x x=g1
∂A1 (x) ∂A2 (y) ∂A2 (y) {A
(x),
g
}
+
{g1 , g2 }.
+
1
2
x=g1
∂y y=g2
∂x x=g1 ∂y y=g2
(9.17)
(9.18)
We will now use this formula to compute the brackets (9.12). Plugging, in
(9.18), A1 = A2 = A = L12 , g1 = xi , g2 = xj with B(xi ) = 0, we get
2
∂A(x) x=xi +
{x
,
A(y)}
{A(xi ), A(xj )} = {A(x), A(y)}y=x
i
y=xj
j
∂x x=xi
∂A(y) ∂A(x) ∂A(y) +
{A(x),
x
{xi , xj }.
j } x=x +
y=x
x=x
j
i
i
∂y
∂x
∂y y=xj
(9.19)
Actually, the restriction to polynomials is only a matter of convenience, the result hold
for any meromorphic function of x.
42
Remarking that the first and last terms are vanishing, we notice that to show
that {A(xi ), A(xj )} = 0 it suffices to show that
{A(x), xj }x=xi = 0 for i 6= j.
(9.20)
Next, we proceed by noticing that, since B(xi ) ≡ 0, we have
∂B(x) 0 = d B(xj ) = dB(x)x=xj +
dxj ,
∂x x=xj
implying that, for any F (possibly a polynomial function),
{F, B(x)}x=xj
.
{F, xj } = −
∂B(x) ∂x x=xj
At this point, we notice that using the explicit form (9.13) we have:
{A(x), B(y)} x=xi = 0if i 6= j,
y=xj
(9.21)
(9.22)
(9.23)
since the left hand side is a polynomial in x, y (and this proves, with (9.22) the
validity of (9.20). Finally, for x = y we remark that the second of (9.13) reads
{A(x), B(x)} = A′ (x)B(x) − B ′ (x)A(x).
Thus, for x = xi where B(xi ) = 0, we have:
{A(x), B(x)}x=xi = −B ′ (x)x=xi A(xi )
(9.24)
(9.25)
Summing up we have that the zeroes xi of B(x) and the solutions yi = A(xi ) of
Det(yi − L(xi )) = 0 satisfy the following Poisson brackets:
{xi , xj } = {yi , yj } = 0,
{yi, xi } = yi ,
(9.26)
so that, finally, the canonically conjugated variables (xi , zi = log yi) satisfy the
separation relations:
Det(exp (zi ) − L(xi )) = 0, i = 1, . . . , N − 1.
(9.27)
We finally remark that, mutatis mutandis, the same arguments can be used
to discuss SoV (a/k/a/ algebraic complete integration) in the case of linear
r-brackets.
In this case (as we shall see later) one is lead to consider a traceless matrix
A(x) B(x)
′
L (x) =
,
(9.28)
C(x) −A(x)
43
whose characteristic polynomial is
1
Γ′ (y, x) = y 2 − 2Tr(L′ (x)2 ) = y 2 − (A(x)2 + B(x)C(x)).
/
(9.29)
In this case the R-brackets between the polynomials are given by
{A(x), A(y)} = 0, {B(x), B(y)} = 0, {C(x), C(y)} = 0
B(x) − B(y)
{A(x), B(y)} =
,
x−y
C(y) − C(x)
{A(x), C(y)} =
x−y
2(A(x) − A(y))
{B(x), C(y)} =
x−y
(9.30)
Supposing that the degree of the polynomial B(x) gives the correct number of
roots xi , we see that once again, the complementary set of coordinates can be
taken as the values yi = A(xi ), and, from the second line of (9.30), we see that
the brackets between these separated coordinates are
{xi , xj } = {yi , yj } = 0,
10
{yi, xi } = −1.
(9.31)
Group actions and Moment maps: the Calogero
System
In the next lectures we will examine the complete integrability of another manybody problem, known as (rational) Calogero-Moser system.
We will use it as a tool to introduce some fundamental concepts in the
geometry of integrable systems:
1. Hamiltonian Group Actions (and Lie algebra actions)
2. Moment maps
3. Hamiltonian reductions
10.1
The model
It consists in N particles on the real line, interacting one another with an inverse
square potential, i.e.:
N
N
1
1X 2 X
pi +
H=
2 i=1
(qk − qi )2
j<i
44
(10.1)
The system admits the following Lax representation:
X
X 1
L=
pj Ejj − i
Ejk ,
q
−
q
j
k
j
j6=k
X
X
X
1
1
M =i
Ejj (
)
+
i
E .
2
2 jk
(q
−
q
)
(q
−
q
)
k
j
k
j
j
k6=j
j6=k
(10.2)
As, along the evolution induced by the Hamiltonian (10.1) we have
dL
= [L, M],
dt
(10.3)
we have that Ik = k1 Tr(Lk ) is a constant of the motion (and, since its leading
P
term is k1 j pkj they are functionally independent). Moreover (Perelomov-book)
it holds the following. Let Q the matrix
X
Q=
qj Ejj
j
then it holds
d
Q = [Q, M] + L.
(10.4)
dt
The equations (10.2) and (10.4), sometimes referred to as Extended Lax representation allow one to integrate at once the equations of motion. Indeed, let us
consider, along with the Ik the functions Jl defined as
Jk = Tr(Q Lk−1 ), k = 1, . . . , N.
(10.5)
It is not difficult to realize that the Jacobian of the 2N functions {I1 , . . . , IN , J1 . . . , JN }
w.r.t. {p, q} is generically non zero, so that {I1 , . . . , IN , J1 . . . , JN } can be regarded as a set of coordinates on the phase space of the Calogero model.
We already know that Ik ’s are constants of the motion; let us look at the
time evolution of the Jk ’s.
d
d
Jk = Tr(QLk−1 )
dt
dt
k−1
X
k−1
= Tr(Q̇L
+
QLk−1−i L̇ Li−1
i=1
k−1
= Tr([Q, M]L
k
+L +Q
z
k−1
X
[QLk−1 ,M ]
= kIk .
45
}|
k−1−i
L
i=1
k−1
= Tr(Lk + [Q, M]Lk−1 + Q[L
{z
|
[Lk−1 ,M ]
, M]
}
{
i−1
[L, M] L
(10.6)
Actually, the solution of the equations of motion
d
I =0
dt l
d
J = lIl
dt l
is, clearly,
10.2
Il (t) = Il (0)
Jl (t) = lIl (0) t + Jl (0).
(10.7)
(10.8)
The geometry
Definition 10.1 Let G be a Lie group, and M a manifold. An action of G on
M is a homomorphism from G to the group D(M) of diffeomorphisms of M.
That is, a map
g 7→ ψg : M → M
(10.9)
∈G
x
7→ ψg (x)
such that, ∀ x ∈ M,
ψg1 g2 (x) = ψg1 (ψg2 (x)).
(10.10)
An action of G induces an “action” of its Lie Algebra g on M, which is defined
as follows. We notice, in particular, that the very definition of action of a Lie
group means that, if eis the identity element of G, then ψe (x) = x, ∀, x ∈ M. If
gt = exp (tξ) is the one parameter group associated with the element ξ ∈ g, the
tangent to the curve
γ(t) = ψgt (x)
is, for any x ∈ M an element Xξ (x) ∈ Tx M. Letting x vary in M, we obtain
a vector field Xξ ∈ X (M). We notice that the map associating, in this way, a
vector field Xξ with an element ξ ∈ g is linear and satisfies
[Xξ , Xη ]X (M ) = X[ξ,η]g .
(10.11)
In words, with a small abuse of language:
Definition 10.2 An action of a Lie algebra (sometimes called infinitesimal action) g on a manifold M is a homomorphism from (g, [·, ·]g) to the Lie algebra
X (M) of vector fields on M.
It is thus quite natural to give the subsequent
Definition 10.3 Let (M, P ) a Poisson manifold, G a Lie group, and g its Lie
algebra. An action of g on M is called a Poisson action if the map ξ 7→ Xξ
linearly factors through P , that is, there exists a linear map ξ 7→ Fξ from g to
C ∞ (M), such that, for all ξ ∈ g,
Xξ = P dFξ ,
46
(10.12)
that is, the vector field Xξ associated with ξ ∈ g is, actually, the Hamiltonian
vector field associated with Fξ .
A Lie group action (in the sense of Definition 10.1) is called a Poisson action
if the induced Lie algebra action fulfills the above requirement.
We remark that, given a Poisson action of g, the map
F : g → C ∞ (M)
is, by definition, linear. Hence, evaluating this at any x ∈ M, we obtain a linear
map F (x) : g → R(or C), that is, we can associate, in a smooth way, an element
of g∗ to each point x ∈ M. This map is called the moment map. Summarizing:
Definition 10.4 Let Ψ be a Poisson action of G on a Poisson manifold M, and
let ξ 7→ X (M) the infinitesimal action of g on M. A moment map associated
with Ψ is any map µ : M → g∗ satisfying
Xξ = P dhµ(x), xii, ∀ξ ∈ g, x ∈ M.
(10.13)
Remarks.
1. A necessary condition for an action to be Poissonian is that every vector
field Xξ preserves the Poisson tensor, LXξ P = 0.
2. Furthermore, from [Xξ , Xη ] = X[ξ,η] we have:
[P dFξ , P dFη ] = P dF[ξ,η] = P d({Fξ , Fη })
(10.14)
where the first equality follows from the fact that the action is Poissonian,
while the second is the usual equality due to the fact that P is Poisson.
In particular, if the manifold is symplectic, we get that the moment map
is uniquely defined, and furthermore,
F[ξ,η] = {Fξ , Fη }.
10.3
(10.15)
The Calogero model
Let u(n) be the space of n × n self-adjoint matrices, and G = U(n) the group
of unitary operators in Cn . Explicitly,
x ∈ u(n) iff u† = u, X ∈ U(n) iff U † = U −1 , Lie(U(n)) = iu(n)
Let the manifold M = u(n) × u(n) be the space of pairs (y, x) of self-adjoint
matrices. Let us define, in M the form
ω = Tr(dy ∧ dx).
47
(10.16)
This means the following: being u(n) a vector space, a tangent vector to M at
m = (a, b) is a pair of matrices X = (y, x). Thus, if X1 = (y1 , x1 ), X2 = (y2 , x2 ),
the value of the two form omega on (X1 , X2 ) is
ω(X1, X2 ) = Tr(y1 x2 − y2 x1 ).
(10.17)
Since ω is antisymmetric, non-degenerate and constant, it is a symplectic twoform. Thus M is a symplectic manifold. We remark that Hamiltonian vector
fields associated with (10.17) have the form:
ẏF
ẋF
∂F
= −
∂x
∂F ,
=
∂y
(10.18)
where in the coordinates given by the matrix elements (Yij , Xij ), the matrices
∂F ∂F
,
are given by:
∂y ∂x
(
X ∂F
X ∂F
∂F ∂F
,
)=(
Eji,
Eji ).
∂y ∂x
∂Y
∂X
ij
ij
i,j
i,j
Let the group G = U(n) act on M = u(n) × u(n) by
ψg (y, x) = (gyg †, gxg †),
so that the infinitesimal action of g = iu(n) is
ẏ = [ξ, y]
Xξ =
ẋ = [ξ, x]
(10.19)
(10.20)
Proposition 10.5 The action (10.19) is a symplectic action. Under the identification g ≃ g∗ given by the trace form, its moment map is
Proof. Let us consider
µ (y, x) = [x, y]
(10.21)
Fξ ((y, x)) = Tr(ξ [x, y]).
We have, for any tangent vector (ẏ, ẋ) that the first order term in t of Fξ (y +
tẏ, x + tẋ) − Fξ (y, x) is
t (Tr(ξ[x, ẏ]) + Tr(ξ[ẋ, y])) = t (Tr(ẏ[ξ, x]) − Tr(ẋ[ξ, y])) .
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(10.22)
This essentially proves the statement. Indeed, form this relation, we see that
the differential of F is represented by
dF = ([ξ, x], −[ξ, y]);
(10.23)
Also, the meaning of (10.22) is
hdF, Xi = Tr(dF1 X2 − dF2 X2 ) = −ω(XF , X)
Let us now consider the “free” Hamiltonian
1
H = Tr(y 2).
2
(10.24)
(10.25)
Since g † = g −1, the Hamiltonian is invariant under G, since
1
1
1
Tr(ψg (y)ψg (y)) = Tr(gyg † gyg † = Tr(y 2).
2
2
2
(10.26)
Infinitesimally, for all ξ ∈ u(n) we have LXξ H = 0. Since Xξ is the Hamiltonian
vector field associated with Fξ we have that
{H, Fξ } = 0, ∀ ξ ∈ g.
(10.27)
This means that the g∗ -valued momentum map µ(x) is a constant of the motion.
In the present case, the hyper-surfaces defined by
µ((x, y)) = [x, y] = Constant
(10.28)
are left invariant by the Hamiltonian flow associated with H.
10.4
The Calogero Model as a Reduction
Let |ei be the vector with all elements equal to 1, and let us consider the matrix
X
C = 1 − |ei ⊗ he| = −
Eij .
(10.29)
i6=j
The little group GC of C is isomorphic to the product
GC = U(n − 1) × U(1);
(10.30)
more explicitly, it is the group of matrices that admit the vector |ei as an
eigenvector.
Proposition 10.6 The reduced space µ−1 (i C)/GC is 2n-dimensional, and is
isomorphic to the space of pair of matrices (Q, L) having the “Calogero” form
X
X 1
Q = diag(q1 , . . . , qn ), L =
pj Ejj − i
Ejk .
(10.31)
q
−
q
j
k
j
j6=k
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Proof. (Perelomov) The manifold of pairs of matrices satisfying
i[x, y] = C
(10.32)
is given by
x = gQg †,
y = gLg †.
(10.33)
with Q, L given by (10.31), g ∈ U(n). Indeed, x is diagonalizable, and so
there is g ∈ U(n) such that the first equation of (10.33) is satisfied; then, a
straightforward computation shows that [Q, L] = −iC implies that L has the
form given by the second of eq. (10.31).
What we need to show is that the matrix g can be taken in Gc . Let us
consider the equation i[x, y] = C and conjugate it by the adjoint of any matrix
g that diagonalizes x:
i[Q, g † yg] = g †Cg = g † (1−|ei⊗he|)g = 1−|f i⊗ hf¯ |, where f = g †|ei. (10.34)
Setting g̃ = gΦ, Φ = diag(f1 , . . . , fn ) we get another matrix g̃ diagonalizing x,
which leaves |ei invariant, that is, an element of Gc .
Let us now show that from the representation (10.31) we recover the extended
Lax representation for the Calogero system. The keys to get this result are two.
The first is that, since H = 21 Tr(y 2), the Hamilton equation of motion are
ẏ = 0
(10.35)
ẋ = y
Furthermore, as we have seen in the proof of the above proposition, there is an
element g ∈ GC such that
y = gLg †;
x = gQg †.
(10.36)
So, deriving w.r.t. time, and using (10.35) we get
0 = ġLg † + g L̇g † + gLġ † =
since g † = g −1 ⇒ 0 = g −1 ġL + L̇ − Lg −1 ġ
⇔ L̇ = [L, M], with M = −g −1 ġ.
(10.37)
and
g −1 yg = g −1ġQ − Qg −1ġ + Q̇
| {z }
L
⇔ Q̇ = [Q, M] + L.
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