integrable systems, spectral curves and representation theory · integrable systems, spectral...

General Letters in Mathematics Vol. 3, No. 1, Aug 2017, pp.1-24

e-ISSN 2519-9277, p-ISSN 2519-9269

Available online at http:// www.refaad.com

Integrable Systems, Spectral Curves

and Representation Theory

A. Lesfari

Department of Mathematics, Faculty of Sciences, University of Chouaıb Doukkali,

B.P. 20, El Jadida, Morocco.

[email protected]

Abstract. The aim of this paper is to present an overview of the active area via the spectral linearization method for

solving integrable systems. New examples of integrable systems, which have been discovered, are based on the so called

Lax representation of the equations of motion. Through the Adler-Kostant-Symes construction, however, we can produce

Hamiltonian systems on coadjoint orbits in the dual space to a Lie algebra whose equations of motion take the Lax form.

We outline an algebraic-geometric interpretation of the flows of these systems, which are shown to describe linear motion

on a complex torus. These methods are exemplified by several problems of integrable systems of relevance in mathematical

physics.

Keywords: Integrable systems, Jacobian varieties, spectral curves.MSC2010 70H06, 14H40, 14H55.

1 Introduction

Last century, mechanics was dominated by the question whether a dynamical system can be solved by quadratures,i.e., by a finite number of algebraic operations including the inverting of functions.This was done, if at all possible, byfinding appropriate variables (α1, ..., α2) such that (α1, ..., αn, H1, ...,Hn) form a local system of canonical coordinatesin which the Hamiltonian vector fields XHi

take the simple form

XHj: αi = δij , Hi = 0.

Historically, the developments of mechanics and algebraic geometry (in particular the theory of Riemann surfaces)were closely intertwined. This comes from the fact that, in most examples, the quadratures αi were obtained interms of elliptic or hyperelliptic integrals:

αi =

g∑k=1

∫ sk(t) zn−idz√P (z, cj)

,

with P (z, cj) a polynomial of degree 2g+1 or 2g+2 in z and the sk(t) some appropriate variables, algebraically relatedto the originally given ones, for which the Hamilton-Jacobi equation could be solved by separation of variables. Thesolutions of these problems can be expressed in terms of theta functions related to Riemann surfaces.

The classical approach to proving that a system is integrable by quadratures (in terms of hyperelliptic integrals)was something very unsystematic and required a great deal of luck and ingenuity. Jacobi [24] himself was verymuch aware of this difficulty and in his famous ”Vorlesungen ber Dynamik”, in the contex of geodesic flow onthe ellipsoid (before introducing the elliptic coordinates), he wrote : ”Die Hauptschwierigkeit bei der Integrationgegebener Differentialgleichungen scheint in der Einfhrung des richtigen Variablen zu bestehen, zu deren Auffindunges kein allgemeine Regel giebt. Man daher das ungekehrte Verfahren einschlagen und nach erlangter Kenntniss

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2 A. Lesfari

einer merkwrdigen Substitution die Probleme aufsuchen, bei welchen dieselbe mit Glck zu brauchen ist” Finally, afterPoincare had recognized that most Hamiltonian systems are not completely integrable, the interest in this subjectdecreased for more than half a century.

The discovery some years ago by Gardner, Greene, Kruskal and Miura [15] that the Korteweg-de Vries (K-dV)equation [28] :

∂u

∂t− 6u

∂u

∂x+∂3u

∂x3= 0, u(x, 0) = u(x), x ∈ R

could be integrated via inverse spectral methods has generated an enormous number of ideas in the area of Hamil-tonian integrable systems. Lax [34] showed that(1) is equivalent to the so-called Lax equation

A = [A,B] ≡ AB −BA, (1)

where . =∂

∂tand A, B are the differential operators in x : A =

∂2

∂x2+ u, B =

∂3

∂x3+

3

2u∂

∂x+ u. The Lax equation

means that, under the time evolution of the system, the linear operator A(t) remain similar to A(0). So the spectrumof A is conserved, i.e. it undergoes an isospectral deformation. The eigenvalues of A, viewed as functionals, representthe integrals (constants of the motion) of the K-dV equation. Around 1974, Mc Kean and van Moerbeke [43],Dubrovin and Novikov [13] solved the periodic problem for the K-dV equation (for x ∈ S1) in terms of a linearmotion on a real torus. This torus is the real part of the Jacobi variety of a hyperelliptic curve with branch pointsdefined by the simple periodic and anti-periodic spectrum of A. A parallel theory related to these problems had itsorigin in the periodic Toda lattice (discretized version of the K-dV equation) and Krichever [21] generalized theseideas to differential operators of any order, inspired by specials examples of Zaharov-Shabat, among which is theimportant Kadomtsev-Petviashvili (KP) equation. Also this theory was generalized to difference operators of anyorder by van Moerbeke and Mumford [55]. They worked out a systematic method which provides an algebraic mapfrom the invariants manifolds defined by the intersection of the constants of the motion to the Jacobi variety of analgebraic curve associated to Lax equation.

These systems can be realized as straight line motions on a Jacobi variety of a spectral curve. Many problemson Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac-Moody Lie algebras)yield large classes of extended Lax pairs (1). A general statement leading to such situations is given by the Adler-Kostant-Symes theorem [1, 29, 53]. This theorem produces Hamiltonian systems having many commuting integrals;some precise results are known for interesting classes of orbits in both the case of finite and infinite dimensional Liealgebras. Paradoxically, the finite-dimensional Lie algebras usually lead to noncompact systems, and the infinite-dimensional ones to compact systems. Using the van Moerbeke-Mumford linearization method [55], Adler and vanMoerbeke [2, 3] showed that the linearized flow could be realized on the jacobian variety Jac(C) (or some sub-abelianvariety of it) of the algebraic curve (spectral curve)

C : P (h, z) = det(A− zI) = 0, (2)

associated to (1). We then construct an algebraic map from the complex invariant manifolds of these hamiltoniansystems to the jacobian variety Jac(C) of the curve C. Therefore all the complex flows generated by the constants ofthe motion are straight line motions on these jacobian varieties, i.e. the linearizing equations are given by

g∑j=1

∫ sj(t)

sj(0)

ωk = ckt , 0 ≤ k ≤ g,

where ω1, . . . , ωg span the g-dimensional space of holomorphic differentials on the curve C of genus g.In this paper, I will not discuss the notion of algebraic integrability of Hamiltonian systems. The concept

of algebraic integrability varies through the literature. A dynamical system is algebraic completely integrable (inthe sense of Adler-van Moerbeke [5, 6, 39, 56]) if it can be linearized on a complex algebraic torus Cn/lattice(=Abelian variety). The invariants (often called first integrals or constants) of the motion are polynomials and thephase space coordinates (or some algebraic functions of these) restricted to a complex invariant variety defined byputting these invariants equals to generic constants, are meromorphic functions on an Abelian variety. Moreover,in the coordinates of this Abelian variety, the flows (run with complex time) generated by the constants of themotion are straight lines. However, besides the fact that many hamiltonian completely integrable systems posses thisstructure, another motivation for its study which sounds more modern is : algebraic completely integrable systemscome up systematically whenever you study the isospectral deformation of some linear operator containing a rational

Integrable systems, spectral curves and representation theory 3

indeterminate. Indeed the Adler-Kostant-Symes theorem [1, 29, 53] applied to Kac-Moody algebras provides suchsystems which, by the van Moerbeke-Mumford theorem [55], are algebraic completely integrable. Therefore thereare hidden symmetries which have a group theoretical foundation. The concept of algebraic complete integrabilityis quite effective in small dimensions and has the advantage to lead to global results, unlike the existing criteria forreal analytic integrability, which, at this stage are perturbation results. I shall not discuss here the weaker notionof analytic integrability; the perturbation techniques developped in that context are of a totally different nature. Infact, the overwhelming majority of dynamical systems, hamiltonian or not, are non-integrable and possess regimesof chaotic behavior in phase space.

2 Lax equations and spectral curves

A Lax equation (with parameter h) is given by a differential equation of the form

A(h, t) = [A(h, t), B(h, t)] or [B(h, t), A(h, t)],

(. ≡ d

dt

)(3)

where

A(h, t) =

m∑k=l

Ak(t)hk, B(h, t) =

m∑k=l

Bk(t)hk,

are functions depending on a parameter h (spectral parameter) whose coefficients Ak and Bk are matrices in Liealgebras. The pair (A,B) is called Lax pair. Thereafter, the notation A(h, t) or A(h) or A(t) or simply A will beused indifferently. The bracket [, ] is the usual Lie bracket of matrices.

The equation (3) establishes a link between the Lie group theoretical and the algebraic geometric approaches tocomplete integrability. The solution to (3) has the form A(t) = g(t)A(0)g(t)−1, where g(t) is a matrix defined asg(t) = −A(t)g(t). We form the polynomial

P (h, z) = det(A− zI) or det(zI −A)

where z is another variable and I the n × n identity matrix. We define the curve (spectral curve) C, to be thenormalization of the complete algebraic curve whose affine equation is

P (h, z) = 0. (4)

Theorem 2.1. The polynomial P (h, z) is independent of t. Moreover, the functions tr (An) are first integrals for(3).

Proof. Let us call C ≡ A− zI. Observe that

P = detC.tr(C−1C

)= detC.tr

(C−1BC −B

)= 0,

since trC−1BC = trB. On the other hand

An = AAn−1 +AAAn−2 + · · ·+An−1A,

= [A,B]An−1 +A [A,B]An−2 + · · ·+An−1 [A,B] ,

= (AB −BA)An−1 + · · ·+An−1 (AB −BA) ,

= ABAn−1 −BAn + · · ·+AnB −An−1BA,= A

(BAn−1

)−(BAn−1

)A+ · · ·+A

(An−1B

)−(An−1B

)A.

Since tr (X + Y ) = trX + trY, trXY = trY X, X, Y ∈Mn(C), we obtain

d

dttr (Anh) = tr

d

dt(Anh) = 0,

and consequently tr (An) are first integrals of motion.

4 A. Lesfari

3 Completely integrable systems and Kac-Moody Lie algebras

We have shown that a hamiltonian flow of the type (3) preserves the spectrum of A and therefore its characteristicpolynomial. The curve

C : P (z, h) = det (A(h)− zI) = 0,

is time independent, i.e., its coefficients tr (An) are integrals of the motion (equivalently, A(t) undergoes an isospectraldeformation). Some hamiltonian flows on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Liealgebras (Kac-Moody Lie algebras) yield large classes of extended Lax pairs (3). A general statement leading tosuch situations is given by the Adler-Kostant-Symes theorem [1, 29, 53] and that we will study below (for moreinformation, one can consult for example [2, 6, 37]).

Theorem 3.1. Let L be a Lie algebra paired with itself via a nondegenerate, ad-invariant bilinear form 〈, 〉, Lhaving a vector space decomposition L = K +N with K and N Lie subalgebras. Then, with respect to 〈, 〉, we havethe splitting L = L∗ = K⊥ +N⊥ and K⊥ ≈ N ∗(≡ the dual of N ) paired with N via an induced form 〈〈, 〉〉 inheritsthe coadjoint symplectic structure of Kostant and Kirillov; its Poisson bracket between functions H1 and H2 on N ∗reads

H1, H2 (a) = 〈〈a, [∇N∗H1,∇N∗H2]〉〉 , a ∈ N ∗.

Let V ⊂ N ∗ be an invariant manifold under the above co-adjoint action of N on N ∗ and let A(V ) be the algebraof functions defined on a neighborhood of V , invariant under the coadjoint action of L (which is distinct from theN −N ∗ action). Then the functions H in A(V ) lead to commuting Hamiltonian vector fields of the Lax isospectralform

a = [a, prK(∇H)] , prK projection onto K

Proof. Recall that the gradient ∇H of a function H on a vector space E is defined by

dH = (∇H, dv)V , v ∈ E, ∇H ∈ E∗,

where E∗ is the dual of E and (., .)V the pairing between E, E∗. In the above, ∇H ∈ (L∗)∗ = L is the gradient ofH when viewed as a function of L∗(≈ L), while we have in general that prK, prN , prK⊥ , prN⊥ are respectively theprojections onto K, N , K⊥, N⊥ along K⊥, K⊥, N⊥, K⊥, respectively. Note that if H ∈ L∗ ≈ L, then

∇K⊥H = prN (∇H), ∇N⊥H = prK(∇H).

Let V ⊂ K⊥ be an invariant manifold under the coadjoint action of N on K⊥ ≈ N ∗. From the identity

d

dtH(Adg(t)(a)

)∣∣∣∣t=0

= 0,

whereg(t) = 1 + bt+ o(t), b ∈ L, a ∈ V,

we deduce the relation[∇H(a), a] = 0, a ∈ V,

or what amounts to the same[a,∇K⊥H] = − [a, prK(∇H)] (5)

The Poisson bracket between two functions H1 et H2 on N ∗ is written

H1, H2 (a) = 〈〈a, [∇N∗H1,∇N∗H2]〉〉 , a ∈ N ∗,

where 〈〈., .〉〉 is the natural pairing between N and N ∗, and where ∇N∗H1 ∈ N is the natural gradient of H definedby

dH1(X) = 〈〈dX,∇N∗H1, 〉〉 .

Since K⊥ ≈ N ∗ et 〈〈., .〉〉 = 〈., .〉|K⊥×N , hence

H1, H2(a) = 〈a, [∇K⊥H1,∇K⊥H2]〉 . (6)


Suppose now that H1, H2 ∈ A(V ) and satisfying the relation (5). Then, by virtue of the relations (5) and (6) andthe fact that 〈., .〉 is ad-invariant, we obtain

H1, H2 = 〈[a,∇K⊥H1],∇K⊥H2〉,= −〈[a, prKH1],∇K⊥H2〉,= −〈a, [prKH1,∇K⊥H2]〉.

Using a similar reasoning for H2, we get

H1, H2 = 〈a, [prKH1,∇K⊥H2]〉.

Since K is a Lie algebra and a ∈ K⊥, we obtain H1, H2 = 0. The Hamiltonian vector field is written

XH1(H2) = H1, H2 = 〈[∇K⊥H1, a],∇K⊥H2〉,

andXH1(a) = prK⊥ [∇K⊥H1, a].

Consequently, the corresponding hamiltonian flow is

a = prK⊥ [∇K⊥H1, a], H1 ∈ A(V ),

and from (5), we havea = prK⊥ [a,∇K⊥H1].

Or [K⊥,K] ⊂ K⊥, and thusa = [a,∇K⊥H1],

which completes the proof.This theorem produces Hamiltonian systems having many commuting integrals; some precise results are known

for interesting classes of orbits in both the case of finite and infinite dimensional Lie algebras. Paradoxically, thefinite-dimensional Lie algebras usually lead to noncompact systems, and the infinite-dimensional ones to compactsystems.

Any finite dimensional Lie algebra L with bracket [, ] and Killing form 〈, 〉 leads to an infinite dimensional formalLaurent series extension

L =

m∑−∞

Aihi : Ai ∈ L, m ∈ Z free,

with bracket [∑Aih

i,∑

Bjhj]

=∑i,j

[Ai, Bj ]hi+j ,

and ad-invariant, symmetric forms ⟨∑Aih

i,∑

Bjhj⟩k

=∑

i+j=−k

〈Ai, Bj〉 ,

depending on k ∈ Z. The forms 〈, 〉k are non degenerate if 〈, 〉 is so. Let Lp,q (p ≤ q) be the vector subspace of L,corresponding to powers of h between p and q.

A first interesting class of problems is obtained by taking L = Gl(n,R) and by putting the form 〈, 〉1 on theKac-Moody extension. Then we have the decomposition into Lie subalgebras

L = L0,∞ + L−∞,−1 = K +N ,

with K = K⊥, N = N⊥ and K = N ∗. Consider the invariant manifold Vm, m ≥ 1 in K = N ∗, defined as

Vm =

A =

m−1∑i=1

Aihi + αhm , α = diag(α1, · · · , αn) fixed

,

with diag (Am−1) = 0. From [2, 3], we have

6 A. Lesfari

Theorem 3.2. The manifold Vm has a natural symplectic structure, the functions

H =⟨f(Ah−j), hk

⟩1,

on Vm for good functions f lead to complete integrable commuting hamiltonian systems of the form

A =[A, prK(f ′(Ah−j)hk−j)

], A =

m−1∑i=0

Aihi + αhm,

and their trajectories are straight line motions on the jacobian of the curve C of genus (n− 1) (nm− 2) /2 defined by

P (z, h) = det (A− zI) = 0.

The coefficients of this polynomial provide the orbit invariants of Vm and an independent set of integrals of the motion(of particular interest are the flows where j = m, k = m+ 1 which have the following form

A =[A , adβ ad

−1α Am−1 + βh

], βi = f ′ (αi) , (7)

the flow depends on f through the relation βi = f ′ (αi) only).

Another class is obtained by choosing any semi-simple Lie algebra L . Then the Kac-Moody extension L equippedwith the form 〈, 〉 = 〈, 〉0 has the natural level decomposition

L =∑i∈Z

Li, [Li,Lj ] ⊂ Li+j , [L0, L0] = 0, L∗i = L−i.

LetB+ =

∑i≥0

Li, B− =∑i<0

Li.

Then the product Lie algebra L × L has the following bracket and pairing[(l1, l2) , (l

′

1, l′

2)]

=(

[l1, l′

1],−[l2, l′

2]),⟨

(l1, l2) , (l′

1, l′

2)⟩

= 〈l1, l′

1〉 − 〈l2, l′

2〉.

It admits the decomposition into K +N with

K = (l,−l) : l ∈ L , K⊥ = (l, l) : l ∈ L ,

N =

(l−, l+) : l− ∈ B−, l+ ∈ B+, pr0(l−) = pr0(l+),

N⊥ =

(l−, l+) : l− ∈ B−, l+ ∈ B+, pr0(l+ + l−) = 0,

where pr0 denotes projection onto L0. Then from the last theorem , the orbits in N ∗=K⊥ possesses a lot ofcommuting hamiltonian vector fields of Lax form.

In fact the following result [55] could be thought of as an example of theorem 3.2.

Theorem 3.3. The N-invariant manifolds

V−j,k =∑−j≤i≤k

Li ⊆ L ' K⊥,

has a natural symplectic structure and the functions H(l1, l2) = f(l1) on V−j,k lead to commuting vector fields of theLax form

l =

[l, (pr+ − 1

2pr0)∇H(l)

], pr+ projection onto B+,

their trajectories are straight line motions on the Jacobian of a curve defined by the characteristic polynomial ofelements in V−j,k thought of as functions of h. (where ∇H(l) ∈ N is the gradient of H thought of as a function onL).


Using the van Moerbeke-Mumford linearization method, Adler and van Moerbeke [2, 3] showed that the linearizedflow could be realized on the jacobian variety Jac(C) (or some sub-abelian variety of it) of the algebraic curve(spectral curve) C associated to (3). We then construct an algebraic map from the complex invariant manifolds ofthese hamiltonian systems to the jacobian variety Jac(C) of the curve C. Therefore all the complex flows generatedby the constants of the motion are straight line motions on these jacobian varieties, i.e. the linearizing equations aregiven by ∫ s1(t)

s1(0)

ωk +

∫ s2(t)

s2(0)

ωk + · · ·+∫ sg(t)

sg(0)

ωk = ckt , 0 ≤ k ≤ g,

where ω1, . . . , ωg span the g-dimensional space of holomorphic differentials on the curve C of genus g. In an unifyingapproach, Griffiths [18] has found necessary and sufficient conditions on B for the Lax flow (3) to be linearizable onthe Jacobi variety of its spectral curve, without reference to Kac-Moody Lie algebras.

4 Applications

4.1 Geodesic flow on SO(n)

In the particular case m = 1, i.e., for V1, we choose

A = X + αh, X ∈ so(n).

In this case, the Hamiltonian flow described by equation (7) (where αi and βi can be taken arbitrarily) is reducedto the study of the Euler-Arnold equations for the geodesic flow on SO(n),

X = [X,λX], (λX)ij = λijXij , λij =βi − βjαi − αj

,

for a left-invariant diagonal metric ΣλijXij . The natural phase space for this motion is an orbit defined in SO(n) by[n/2] orbit invariants. By theorem 3.3, the problem is completely integrable and the trajectories are straight lines onJac(C) of dimension (n− 2)(n− 1)/2 and more specifically, on the Prym variety Prym(C/C0) ⊂ Jac(C) of dimension(n(n− 1)/2− [w/2])/2 induced by the natural involution

C −→ C, (z, h) 7−→ (−z,−h),

on C as a result of X ∈ so(n); C0 is the curve obtained by identifying (z, h) with (−z,−h). This problem is studiedin detail for n = 3 and n = 4 in several sections through various methods.

4.2 The problem of the rotation of a solid body about a fixed point

One of the most fundamental problems of mechanics is the study of the motion of rotation of a solid body around afixed point. The differential equations of this problem are written in the form

M = M ∧ Ω + µgΓ ∧ L, (8)

Γ = Γ ∧ Ω,

where ∧ is the vector product in R3, M = (m1,m2,m3) the angular momentum of the solid, Ω = (m1

I1, m2

I2, m3

I3) the

angular velocity, I1, I2 and I3, moments of inertia, Γ = (γ1, γ2, γ3) the unitary vertical vector, µ the mass of the solid,g the acceleration of gravity, and finally, L = (l1, l2, l3) the unit vector originating from the fixed point and directedtowards the center of gravity; all these vectors are considered in a mobile system whose coordinates are fixed to themain axes of inertia. The configuration space of a solid with a fixed point is the group of rotations SO(3). This isgenerated by the one-parameter subgroup of rotations

A1 =

1 0 00 cos t − sin t0 sin t cos t

, A2 =

cos t 0 sin t0 1 0

− sin t 0 cos t

,

A3 =

cos t − sin t 0sin t cos t 0

0 0 1

.

8 A. Lesfari

Recall that this is the group of n× n orthogonal matrices A and the motion of this solid is described by a curve on

this group. The angular velocity space of all rotations (the set of derivatives A(t)∣∣∣t=0

of the differentiable curves in

SO(3) passing through the identity in t = 0 : A(0) = I) is the Lie algebra of the group SO(3); it is the algebra so(3)of the 3× 3 antisymmetric matrices. This algebra is generated as a vector space by the matrices

e1 = A1(t)∣∣∣t=0

=

0 0 00 0 −10 1 0

, e2 = A2(t)∣∣∣t=0

=

0 0 10 0 0−1 0 0

,

e3 = A3(t)∣∣∣t=0

=

0 −1 01 0 00 0 0

,

which verify the commutation relations :

[e1, e2] = e3 [e2, e3] = e1, [e3, e1] = e2.

We will use the fact that if we identify so(3) to R3 by sending (e1, e2, e3) on the canonical basis of R3, the bracketof so(3) corresponds to the vector product. In other words, consider the application

R3 −→ so(3), a = (a1, a2, a3) 7−→ A =

0 −a3 a2a3 0 −a1−a2 a1 0

,

which defines an isomorphism between Lie algebras(R3,∧

)et (so(3), [, ]) where

a ∧ b 7−→ [A,B] = AB −BA.

By using this isomorphism, the system (8) can be rewritten in the form

M = [M,Ω] + µg [Γ, L] , (9)

Γ = [Γ,Ω] ,

where

M = (Mij)1≤i,j≤3 ≡3∑i=1

miei ≡

0 −m3 m2

m3 0 −m1

−m2 m1 0

∈ so (3) ,

Ω = (Ωij)1≤i,j≤3 ≡3∑i=1

ωiei ≡

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

∈ so (3) ,

Γ = (γij)1≤i,j≤3 ≡3∑i=1

γiei ≡

0 −γ3 γ2γ3 0 −γ1−γ2 γ1 0

∈ so (3) ,

and

L =

0 −l3 l2l3 0 −l1−l2 l1 0

∈ so (3) .

Taking into account that M = IΩ, then the above equations become

M = [M,ΛM ] + µg [Γ, L] , (10)

Γ = [Γ,ΛM ] ,

where

ΛM =≡3∑i=1

λimiei ≡

0 −λ3m3 λ2m2

λ3m3 0 −λ1m1

−λ2m2 λ1m1 0

∈ so (3) , λi ≡1

Ii


The solution of this problem was analyzed first by Euler [14] and in 1758 he published the equations (case µ = 0)which carry his name. Euler’s equations were integrated by Jacobi [24] in terms of elliptic functions and around 1851,Poinsot [51] gave them a remarkable geometric interpretation. Before, around 1815 Lagrange [33] found another case(I1 = I2, l1 = l2 = 0) of integrability, that subsequently Poisson lengthily examined. The problem continued toattract mathematicians but for a long time no new results could be obtained. It was then around 1888-1989 that amemoir [31], of the highest interest, appears containing a new case (I1 = I2 = 2I3, l3 = 0) of integrability discoveredby Kowalewski. For this remarkable work, Kowalewski was awarded the Bordin Prize of the Paris Academy ofSciences. In fact, although Kowalewski’s work is quite important, it is not at all clear why there would be no othernew cases of integrability. This was to be the starting point of a series of fierce research on the question of theexistence of new cases of integrability. Moreover, among the remarkable results obtained by Poincare [50] with theaid of the periodic solutions of the equations of dynamics, we find the following (around 1891): in order to exist inthe motion of a solid body around of a fixed point, an algebraic first integral not being reduced to a combination ofthe classical integrals, it is necessary that the ellipsoid of inertia relative to the point of suspension is of revolution.In 1896, R. Liouville (not to be confused with Joseph Liouville, well known in complex analysis) also competed forthe Bordin prize, presented a paper [41] indicating necessary and sufficient conditions (I3 = 0, 2I3/I1 = integer) ofexistence of a fourth algebraic integral. These conditions have been reproduced in most conventional treatises (egWhittaker [57]) and in scientific journals. And it was not until the year 1906, when Husson [23], working under thedirection of Appell and Painleve, discovered an erroneous demonstration in the work of Liouville. Indeed, paragraphsI and III of Liouville’s dissertation devoted to the search for the necessary conditions seem at first satisfactory, buta more careful study shows that the demonstrations are at least insufficient and that it is impossible to acceptconclusions. In fact, although the conditions found by Liouville are necessary, they can not be deduced from thecalculations indicated and these conditions are not sufficient. And it was Husson who first solved completely thequestion of looking for new cases of integrability. Inspired by Poincare’s research on the problem of the three bodiesand Painleve [48] on the generalization of Bruns’s theorem, Husson demonstrated that any algebraic integral is acombination of classical integrals except in the cases of Euler, Lagrange and Kowalewski. Moreover, the questionof the existence of analytic integrals has been studied rigorously by Ziglin [58, 59] and Holmes-Marsden [22]. somespecial cases could be found : cases of Hesse-Appel’rot [21, 7], Goryachev-Chaplygin [16, 11] and Bobylev-Steklov.

The Euler rigid body motion : In this case we have l1 = l2 = l3 = 0, that is, the fixed point is its centerof gravity. The Euler rigid body motion (also called Euler-Poinsot motion of the solid) express the free motion of arigid body around a fixed point. Then the motion of the body is governed by

M = [M,ΛM ] , or M = [M,Ω] (11)

Equation (11) admits two first integrals

H1 =1

2

(λ1m

21 + λ2m

22 + λ3m

23

), (Hamiltonian)

and

H2 =1

2

(m2

1 +m22 +m2

3

).

We shall use the Lax representation of the equations of motion to show that the linearized Euler flow can be realizedon an elliptic curve. The solution to the equation (11), has the form

M(t) = O(t)M(t)M>(t),

where O(t) is one parameter sub-group of SO(3). So the hamiltonian flow (11) preserves the spectrum of M andtherefore its characteristic polynomial

det (M − zI) = −z(z2 +m2

1 +m22 +m2

3

).

Unfortunately, the spectrum of a 3×3 skew-symmetric matrix provides only one piece of information; the conservationof energy does not appear as part of the spectral information. Therefore one is let to considering another formulation.The basic observation, due to Manakov [42], is that equation (11) is equivalent to the Lax equation

A = [A,B],

whereA = M + αh, B = ΛM + βh,

10 A. Lesfari

with a formal indeterminate h and

α =

α1 0 00 α2 00 0 α3

, β =

α1 0 00 β2 00 0 β3

,

λ1 =β3 − β2α3 − α2

, λ2 =β1 − β3α1 − α3

, λ3 =β2 − β1α2 − α1

,

and all αi distinct. The characteristic polynomial of A is

P (h, z) = det (A− zI) ,

= det (M + αh− zI) ,

=

3∏j=1

(αjh− z) +

3∑j=1

αjm2j

h−

3∑j=1

m2j

z.

The spectrum of the matrix A = M + αh as a function of h ∈ C is time independent and is given by the zeroes ofthe polynomial P (h, z) , thus defining an algebraic curve (spectral curve). Letting w = h/z, we obtain the followingelliptic curve

z23∏j=1

(αjw − 1) + 2H1w − 2H2 = 0.

Finally, we have the

Theorem 4.1. The Euler rigid body motion is a completely integrable system and the linearized flow can be realizedon an elliptic curve.

The Lagrange top : In this case, we have I1 = I2, l1 = l2 = 0, i.e., the Lagrange top is a rigid body, inwhich two moments of inertia are the same and the center of gravity lies on the symmetry axis. In other words, theLagrange top is a symmetric top with a constant vertical gravitational force acting on its centre of mass and leavingthe base point of its body symmetry axis fixed. As in the case of Euler, we show that in this case also the problemis solved by elliptic integrals. Or what amounts to the same, the integration is done using elliptic functions. TheLagrange top is another example of V2 in theorem 3.3, we consider the Lagrange top which evolves on an orbit oftype V2; n = 3,

A = Γ +Mh+ ch2.

Recall that Γ ∈ so(3) ' R3 is the unit vector in the direction of gravity and M ∈ so(3) ' R3 is the angularmomentum in body coordinates with regard to the fixed point; moreover c = (λ + µ)Υ where Υ ∈ so(3) ' R3

expresses the coordinates of the center of mass and where (λ+µ, λ+µ, 2λ) is the inertia tensor in diagonalized form.The situation then leads to a linear flow on an elliptic curve.

The Kowalewski top : This top is special symmetric top with a unique ratio of the moments of inertia satisfythe relation : I1 = I2 = 2I3, l3 = 0; in which two moments of inertia are equal, the third is half as large, and thecenter of gravity is located in the plane perpendicular to the symmetry axis (parallel to the plane of the two equalpoints). Moreover, we may choose l2 = 0, µgl1 = l and I3 = 1. The system in question admits four first integrals :

H1 ≡ H,

H2 = m1γ1 +m2γ2 +m3γ3, (12)

H3 = γ21 + γ22 + γ23 ,

H4 =

((m1 + im2

2

)2

− (γ1 + iγ2)

)((m1 − im2

2

)2

− (γ1 − iγ2)

).

The fourth first integral was obtained by Kowalewski [31]. During several years and despite many attempts concerningthere was no known Lax form for the Kowalewski spinning top for its resolution using the isospectral deformationmethod. Subsequently, various forms of Lax have been proposed. A detailed geometrical study of Kowalewski’s casehas revealed rational relationships between various systems. For example, for a long time no one had suspected a


rational relationship between the Kowalewski spinning top, the geodesic flow over the SO(4) group for the Manakovmetric and the Hnon-Heiles differential system. Such a relation was obtained by Adler-van Moerbeke [4] and the useof our results obtained in [26] allowed them to provide a pair of Lax for the case of Kowalewski , Which had beenopen for several years. Similarly, Haine-Horozov [20] show that on each level surface of the invariants, the equationsof the Kowalewski top are equivalent to a Neumann system describing the motion of a mass point on the sphere. Thisallows us to write a global Lax pair for the Kowalewski system and to show that Kowalewski’s original reduction ofthe integration of the equations of motion to Jacobi’s inversion theorem is identical to the introduction of ellipticalspherical coordinates for integrating C. Neumann’s system. In what follows, we will use the form of Lax (see [6] andreferences cited in bibliography) in order to linearize the Kowalewski problem via the spectral curve method.

Theorem 4.2. The Kowalewski system admits the following Lax pair

A = i[A,B],

with

A =

0 −Γ2 − 1

2x2h −γ3Γ1 0 γ3

12x1h

− 12x1h −γ3 −m3h Γ1 − h2γ3

12x2h −Γ2 + h2 m3h

,

B =

−m3 0 1

2x2 00 m3 0 − 1

2x112x1 0 m3 h0 − 1

2x2 −h m3

where x1 −m1 + im2, x2 = m1 − im2, Γ1 = γ1 + iγ2, Γ2 = γ1 − iγ2.

Proof. We sketch a proof. The projective complex algebraic curve (or spectral curve), defined by the affineequation

C : P (h, z) = det(A− zI) = 0,

is written explicitlyz4 + (h4 − c1h2 + 2c3)z2 + c4h

4 + (c22 − c− 1c− 3)h2 + c23 = 0,

where the constants (supposed generic) c1, c2, c3, c4 denote respectively the constants H1, H2, H3, H4 (12) and thisequation describes isospectral deformation. The curve C is smooth and is a double cover

ϕ : C −→ H, (z, h) 7−→ (w, h), w =2z2 + h4 − c1h2 + 2c3

h,

of a hyperelliptic curve H defined by

H : w2 =((h2 − c1)2 + 4(c3 − c4)

)h2 − 4c22.

The projection π : H −→ C, (w, h) 7−→ h, realizes H as a double cover of C branched in six points. Consequently,the compactified H of H is a hyperelliptic curve of genus 2. The covering ϕ has four branch points (w, h) on H forwhich z = 0, i.e., wh = h4− c1h2 + 2c3. It is shown that the genus of C is 5. Moreover, the curve H is an unramifieddouble cover H −→ E , (w, h) 7−→ (ζ, ξ) = (wh, h2), of an elliptic curve E ,

E : ζ2 =((ξ − c1)2 + 4(c3 − c4)

)ξ2 − 4c2ξ.

Moreover, the curve C can be seen as an unramified double cover C −→ S, (z, h) 7−→ (z, ξ) = (z, h2), of a new curveS, defined by

S : z4 + (h4 − c1h2 + 2c3)z2 + c4h4 + (c22 − c− 1c− 3)h2 + c23 = 0.

The curve S itself is a double cover of the elliptic curve E ,

S −→ E , (z, ξ) 7−→ (ζ, ξ) = (2z2 + ξ2 − c1ξ + 2c3, ξ),

nd the genus of S is equal to 3. Finally, it is easily verified that the linearization of the flow is done on the Jacobianvariety of the curve C.

12 A. Lesfari

The Goryachev-Chaplygin top : The differential equations of the Goryachev-Chaplygin top correspond tothe case I1 = I2 = 4I3, l2 = l3 = 0 and are explicitly written in this case in the form (without restricting generality,values have been given to the constants in order not to weigh down the notations),

m1 = 3m2m3, γ1 = 4m3γ2 −m2γ3,

m2 = −3m1m3 − 4γ3, γ2 = m1γ3 − 4m3γ1 (13)

m3 = 4γ2, γ3 = m2γ1 −m1γ2.

This system admits the following four invariants :

H1 = m21 +m2

2 + 4m23 − 8γ1 = 6b1,

H2 = (m21 +m2

2)m3 + 4m1γ3 = 2b2,

H3 = γ21 + γ22 + γ23 = b3,

H4 = m1γ1 +m2γ2 +m3γ3 = 0,

where b1, b2, b3 are generic constants. The system (13) is integrable in the sense of Liouville, H1 (the Hamiltonian)and H4 are in involution while H2, H3 are Casimir invariants. The differential system (13) has a pair of Lax of theform

A = [A,B], (14)

with

A =

0 γ312x1h

−γ3 −m3h Γ1 − h212x2h −Γ2 + h2 m3h

, B =

3im3 0 −ix10 2im3 2ih−ix2 −2ih −2im3

,

where x1 = m1 + im2, x2 = m1 − im2, Γ1 = γ1 + iγ2, Γ1 = γ1 − iγ2. The transformation

(t,m1,m2,m3, γ1, γ2, γ3) 7−→ (−t,−m1,−m2,−m3,−γ1,−γ2,−γ3),

shows that the systems (13) and (14) are equivalent. The spectral curve, C : P (h, z) = det(A − zI) = 0, is writtenexplicitly

C : w3 +

(h4 − 3

2b1h

2 + b3

)w +

b22h3 = 0,

and one easily checks that the flow linearizes on the Jacobian of this curve.

4.3 The Manakov geodesic flow on the group SO(4)

Consider the group SO(4) and its Lie algebra so(4) paired with itself, via the customary inner product

〈X,Y 〉 = −1

2tr (X.Y ) ,

where

X = (Xij)1≤i,j≤4 =

6∑i=1

xiei =

0 −x3 x2 −x4x3 0 −x1 −x5−x2 x1 0 −x6x4 x5 x6 0

∈ so(4).

A left invariant metric on SO(4) is defined by a non-singular symmetric linear map Λ : so(4) −→ so(4), X 7−→ Λ.X,and by the following inner product; given two vectors gX and gY in the tangent space SO(4) at the point g ∈ SO(4),

〈gX, gY 〉 =⟨X,Λ−1.Y

⟩,

regardless of g. Then the geodesic flow for this metric takes the following commutator form (Euler-Arnold equations):

X = [X,Λ.X] , . ≡ d

dt(15)


where

Λ.X = (λijXij)1≤i,j≤4 =

6∑i=1

λixiei =

0 −λ3x3 λ2x2 −λ4x4

λ3x3 0 −λ1x1 −λ5x5−λ2x2 λ1x1 0 −λ6x6λ4x4 λ5x5 λ6x6 0

∈ so(4).

In view of the isomorphism between(R6,∧

), and (so(4), [, ]) we write the system (15) as

x1 = (λ3 − λ2)x2x3 + (λ6 − λ5)x5x6,

x2 = (λ1 − λ3)x1x3 + (λ4 − λ4)x4x6,

x3 = (λ2 − λ1)x1x2 + (λ5 − λ4)x4x5, (16)

x4 = (λ3 − λ5)x3x5 + (λ6 − λ2)x2x6,

x5 = (λ4 − λ3)x3x4 + (λ1 − λ6)x1x6,

x6 = (λ2 − λ4)x2x4 + (λ5 − λ1)x1x5.

This flow is Hamiltonian with regard to the usual Kostant-Kirillov symplectic structure induced on the orbit

O =Ad∗g(X) = g−1Xg : g ∈ SO(4)

,

formed by the coadjoint action Ad∗g(X) of the group SO(4) on the dual Lie algebra so(4)∗ ≈ so(4). Let z1, z2 ∈ so(4)and consider ξ1 = [X, z1] , ξ2 = [X, z2] two tangent vectors to the orbit at the point X ∈ so(4). Then the symplecticstructure is defined by

ω(X) (ξ1, ξ2) = 〈X, [z1, z2]〉 .

This orbit is 4-dimensional and is defined by setting two trivial quadratic invariants H1 and H2 equal to genericconstants c1 and c2 :

H1 =√

detX = x1x4 + x2x5 + x3x6 = c1 (17)

H2 = −1

2tr(X2)

= x21 + x22 + · · ·+ x26 = c2 (18)

Fonctions H defined on the orbit lead to Hamiltonian vector fields

X = [X,∇H] .

In particular

H =1

2〈X,ΛX〉 =

1

2

(λ1x

21 + λ2x

22 + · · ·+ λ6x

26

), (19)

induces geodesic motion (15) The constants of the motion are given by the two quadratic invariants H1 (17), H2 (18)and the Hamiltonian H (17). Since the system is Hamiltonian on a 4-dimensional symplectic manifold

H1 = c1 ∩ H2 = c2 ,

to make it completely integrable, one needs one independent invariant.Note that the equations (16) can be written as a Hamiltonian vector field in another form

x(t) = J∂H

∂x, x ∈ R6,

with

J =

0 −x3 x2 0 −x6 x5x3 0 −x1 x6 0 −x4−x2 x1 0 −x5 x4 0

0 −x6 x5 0 −x3 x2x6 0 −x4 x3 0 −x1−x5 x4 0 −x2 x1 0

∈ so(6).

14 A. Lesfari

Since det J = 0, then m = 2n + k and m − k = rk J . Here m = 6 and rgJ = 4, then n = k = 2. In summary, thesystem (16) has beside the energy H (19), two trivial constants of motion H1 (17) and H2 (18) because :

J∂H2

∂x= J

∂H3

∂x= 0.

and in order that the hamiltonian system (16) be completely integrable, it is suffices to have one more integral, whichwe take of the form

H4 =1

2

(µ1x

21 + µ2x

22 + · · ·+ µ6x

26

).

The four invariants must be functionally independent and in involution, so in particular

H4, H3 =

⟨∂H4

∂x, J∂H3

∂x

⟩= 0,

i.e.,

((λ3 − λ2)µ1 + (λ1 − λ3)µ2 + (λ2 − λ1)µ3)x1x2x3

+ ((λ6 − λ5)µ1 + (λ1 − λ6)µ5 + (λ5 − λ1)µ6)x1x5x6

+ ((λ4 − λ6)µ2 + (λ6 − λ2)µ4 + (λ2 − λ4)µ6)x2x4x6

+ ((λ5 − λ4)µ3 + (λ3 − λ5)µ4 + (λ4 − λ3)µ5)x3x4x5 = 0.

Then(λ3 − λ2)µ1 + (λ1 − λ3)µ2 + (λ2 − λ1)µ3 = 0,

(λ6 − λ5)µ1 + (λ1 − λ6)µ5 + (λ5 − λ1)µ6 = 0,

(λ4 − λ6)µ2 + (λ6 − λ2)µ4 + (λ2 − λ4)µ6 = 0,

(λ5 − λ4)µ3 + (λ3 − λ5)µ4 + (λ4 − λ3)µ5 = 0.

Put

A =

λ3 − λ2 λ1 − λ3 λ2 − λ1 0 0 0λ6 − λ5 0 0 0 λ1 − λ6 λ5 − λ1

0 λ4 − λ6 0 λ6 − λ2 0 λ2 − λ40 0 λ5 − λ4 λ3 − λ5 λ4 − λ3 0

.

The number of solutions of this system is equal to the number of columns of the matrix A minus the rank of A.- If rkA = 4, we have two solutions : µi = 1 lead to the invariant H2 and µi = λi lead to the invariant H3. This

is unacceptable.- If rkA = 3, each four-order minor of A is singular. Now

λ3 − λ2 λ1 − λ3 λ2 − λ1 0λ6 − λ5 0 0 0

0 λ4 − λ6 0 λ6 − λ20 0 λ5 − λ4 λ3 − λ5

= − (λ6 − λ5)C,

λ1 − λ3 λ2 − λ1 0 0

0 0 0 λ1 − λ6λ4 − λ6 0 λ6 − λ2 0

0 λ5 − λ4 λ3 − λ5 λ4 − λ3

= (λ1 − λ6)C,

λ2 − λ1 0 0 0

0 0 λ1 − λ6 λ5 − λ10 λ6 − λ2 0 λ2 − λ4

λ5 − λ4 λ3 − λ5 λ4 − λ3 0

= − (λ2 − λ1)C,

where

C ≡ λ1λ6λ4 + λ1λ2λ5 − λ1λ2λ4 + λ3λ6λ5 − λ3λ6λ4 − λ3λ2λ5+λ4λ2λ5 + λ4λ1λ3 − λ4λ1λ5 + λ6λ2λ3 − λ6λ2λ5 − λ1λ6λ3,


and it follows that the condition for which these minors are zero is C = 0. Notice that this relation holds by cyclingthe indices : 1→ 4, 2→ 5, 3→ 6. Under Manakov [42] conditions,

λ1 =β2 − β3α2 − α3

, λ2 =β1 − β3α1 − α3

, λ3 =β1 − β2α1 − α2

, (20)

λ4 =β1 − β4α1 − α4

, λ5 =β2 − β4α2 − α4

, λ6 =β3 − β4α3 − α4

,

where αi, βi ∈ C,∏i<j

(αi − βj) 6= 0, equations (16) admits a Lax equation with an indeterminate h :

˙(︷︸︸︷X + αh) = [X + αh,ΛX + βh] , (21)

α =

α1 0 0 00 α2 0 00 0 α3 00 0 0 α4

, β =

β1 0 0 00 β2 0 00 0 β3 00 0 0 β4

.

m

X = [X,Λ.X]⇔ (15),

[X,β] + [α,Λ.X] = 0⇔ (20),

[α, β] = 0 trivially satisfied for diagonal matrices.

The parameters µ1, . . . , µ6 can be parameterized (like λ1, . . . , λ6) by :

µ1 =γ2 − γ3α2 − α3

, µ2 =γ1 − γ3α1 − α3

, µ3 =γ1 − γ2α1 − α2

µ4 =γ1 − γ4α1 − α4

, µ5 =γ2 − γ4α2 − α4

, µ6 =γ3 − γ4α3 − α4

.

To use the method of isospectral deformations, consider the Kac-Moody extension (n = 4) :

L = ˜gl(n,R) =

N∑−∞

Aihi : N arbitrary ∈ Z, Ai ∈ gl(n,R)

,

of gl(n,R) with the bracket :

[A(h), B(h)] =[∑

Aihi,∑

Bjhj]

=∑k

∑i+j=k

[Ai, Bj ]

hk,

and the nondegenerate, invariant inner product

〈A(h), B(h)〉 =⟨∑

Aihi,∑

Bjhj⟩

=∑

i+j=−1〈Ai, Bj〉,

where 〈, 〉 is the usual form defined on gl(n,R).This Lie algebra has a natural decomposition

L = L−∞,−1 + L0,∞, Lij =

∑i≥0

Akhk

.

Observe that L⊥−∞,−1 = L−∞,−1 and L⊥0,∞ = L0,∞ where ⊥ is taken with respect the form above. The infinite-

dimensional Lie group underlying L−∞,−1 acts coadjointly on the dual Kac-Moody Lie algebra L∗−∞,−1 ≈ L⊥0,∞ =

16 A. Lesfari

L0,∞, according to the rule of customary conjugation followed by registering the non-negative powers of h only. Theorbits described in this way come equipped with a symplectic structure with Poisson bracket

H1, H2 (α) =⟨α,[∇L∗−∞,−1

H1,∇L∗−∞,−1H2

]⟩where α ∈ L∗−∞,−1 and ∇L∗−∞,−1

H ∈ L−∞,−1. The functions defined on this orbit are all in involution and the flow

(21) evolves on the coadjoint orbit through the point X + ah ∈ L0,∞, X ∈ so(4).According to the Adler-Kostant-Symes theorem (where K = L0,∞ and N = L−∞,−1 are respectively the ≥ 0

and < 0 powers of h in L), the flow (21) is hamiltonian on an orbit through the point X + ah, X ∈ so(4)) formedby the coadjoint action of the subgroup GN ⊂ SL(n) of lower triangular matrices on the dual Kac-Moody algebraL∗−∞,−1 ≈ L⊥0,∞ = L0,∞. As a consequence, the coefficients of zihi appearing in curve :

C :

(z, h) ∈ C2 : det (X + ah− zI) = 0, (22)

associated to the equation (21) are invariant of the system in involution for the symplectic structure of this orbit.Notice that

det(gXg−1

)= detX = (x1x4 + x2x5 + x3x6)

2,

tr(gXg−1

)2= tr

(gX2g−1

)= tr

(X2)

= −2(x21 + x22 + · · ·+ x26

).

Also the complex flows generated by these invariants can be realized as straight lines on the Abelian variety definedby the periods of the curve C. Explicitly, equation (22) looks as follows

C :

4∏i=1

(αih− z) + 2H4h2 − 2H1zh+ 2H2z

2 +H23 = 0, (23)

whereH1(X) = c1, H2(X) = c2, H3(X) = 2H = c3, H4(X) = c4,

with c1, c2, c3, c4 generic constants. C is a curve of genus 3 and it has a natural involution σ : C −→ C, (z, h) 7−→(−z,−h). Therefore the jacobian variety Jac(C) of C splits up into an even and old part : the even part is an ellipticcurve C0 = C/σ and the odd part is a 2−dimensional Abelian surface Prym(C/C0) (which is also noted, Prymσ(C))called the Prym variety :

Jac(C) = C0 + Prymσ(C).

The van Moerbeke-Mumford linearization method provides then an algebraic map from the complex affine variety⋂4i=1 Hi(X) = ci ⊂ C6 to the Jacobi variety Jac(C). By the antisymmetry of C, this map sends this variety to the

Prym variety Prymσ(C) :4⋂i=1

Hi(X) = ci → Prymσ(C), p 7→3∑k=1

sk,

and the complex flows generated by the constants of the motion are straight lines on Prymσ(C). Finally, we havethe

Theorem 4.3. The geodesic flow (15) is a hamiltonian system with

H ≡ H1 =1

2

(λ1x

21 + λ2x

22 + · · ·+ λ6x

26

),

the hamiltonian. It has two trivial invariants

H2 =1

2

(x21 + x22 + · · ·+ x26

),

H3 = x1x4 + x2x5 + x3x6.

Moreover, if

λ1λ6λ4 + λ1λ2λ5 − λ1λ2λ4 + λ3λ6λ5 − λ3λ6λ4 − λ3λ2λ5+λ4λ2λ5 + λ4λ1λ3 − λ4λ1λ5 + λ6λ2λ3 − λ6λ2λ5 − λ1λ6λ3 = 0,


the system (16) has a fourth independent constant of the motion of the form

H4 =1

2

(µ1x

21 + µ2x

22 + · · ·+ µ6x

26

).

Then the system (16) is completely integrable and can be linearized on the Prym variety Prymα(C).

The well-known Kirchhoff’s equations of motion of a solid in an ideal fluid can be regarded as the equations ofthe geodesics of the right-invariant metric on the group E(3) = SO(3) ×R3 of motions of 3-dimensional euclideanspace R3, generated by rotations and translations. Hence the motion has the trivial coadjoint orbit invariants 〈p, p〉and 〈p, l〉. As it turns out, this is a special case of a more general system of equations written as

x = x ∧ ∂H∂x

+ y ∧ ∂H∂y

, y = y ∧ ∂H∂x

+ x ∧ ∂H∂y

,

where x = (x1, x2, x3) ∈ R3 and y = (y1, y2, y3) ∈ R3. The first set can be obtained from the second by putting(x, y) = (l, p/ε) and letting ε → 0. The latter set of equations is the geodesic flow on SO(4) for a left invariantmetric defined by the quadratic form H.

4.4 Jacobi’s geodesic flow on an ellipsoid and Neumann’s problem

For m = 2, i.e., V2, if one choosesA = αh2 − hx ∧ y − y ⊗ y x, y ∈ Rn,

(which can also be considered as a rank 2 perturbation of the diagonal matrix α, see [2, 3, 44, 45]) then equation (7)reduces to

A = [A, adβad−1α (y ∧ x) + βh], βi = f ′(αi).

This equation can be reduced to the following Hamiltonian system :

x = −(adβad−1α (y ∧ x))x− βy = −∂Hβ

∂y,

y = −(adβad−1α (y ∧ x))y =

∂Hβ

∂x,

where

Hβ =1

2

∑i

βi

y2i +∑j 6=i

(xiyj − xjyi)2

αi − αj

,

which

- for f(z) = ln z, i.e., βi =1

αi, we obtain the problem of Jacobi’s geodesic flow on the ellipsoid

x21α21

+ · · ·+ x2nα2n

= 1,

expressing the motion of the tangent line x+ sy : s ∈ R to the ellipsoid in the direction y of the geodesic.

- for f(z) =1

2z2, i.e., βi = αi, we get the Neumann’s motion [47] of a point on the sphere Sn−1, |x| = 1, under

the influence of the force −αx.From Theorem 3.3, both motions are straight lines on Jac(C), where C turns out to be hyperelliptic of genus

n− 1 (much lower than the generic one) ramified at the following 2n points : some point at ∞, the n points αi andn− 1 other points λi of geometrical significance, based on the observation that generically a line in Rn touches n− 1confocal quadrics. To be precise, the set of all common tangent lines to n− 1 confocal quadrics

Qλi(x, x) + l = 0, i = 1, ..., n− 1,

where Qu(x, y) = 〈(u− α)−1x, y〉, can be parametrized by the quotient of the Jacobian of the hyperelliptic curve Cabove by an abelian group G. The group is generated by the discrete action obtained by flipping the signs of xk and

18 A. Lesfari

yk and some trivial one-dimensional action. Letting h→ 0 in the matrx A and excising the largest eigenvalue fromthis matrix leads to a new isospectral symmetric matrix

L = (I − Py)(α− x⊗ x)(I − Py),

and a flowL = [adβad

−1α x ∧ y, L],

where the spectrum of L is given by the n− 1 branch points λi above and zero. From these considerations, it followsthat the tangent line x + sy : s ∈ R to the ellipsoid remains tangent to n − 2 other confocal quadrics and thecorresponding n− 1 eigenfunctions of L provide the orthogonal set of normals to the n− 1 quadrics at the points oftangency, hence recovering a theorem of Chasles. The close relationship between Jacobi’s and Neumann’s problems,which in fact live on the same orbits, was implemented by Knorrer [26, 27], who showed that the normal vector tothe ellipsoid moves according to the Neumann problem, when the point moves according to the geodesic. Also theset of all n− 1 dimensional linear subspaces in the intersection of two quadrics

X21 + · · ·+X2

n − Y 21 − · · · − Y 2

n−1 = 0,

α1X21 + · · ·+ αnX

2n − λ1Y 2

1 − · · · − λn−1Y 2n−1 = X2

0 ,

in P2n−1 is the Jacobian of the curve C defined above. This is done by observing that the set of linear subspaces inthe above quadrics is the same as the set of (n− 2)-dimensional linear subspaces tangent to n− 1 quadrics

(α1 − λj)X21 + · · ·+ (αn − λj)X2

n = X20 , j = 1, 2, ..., n− 1

which is dual to the set of tangents to the confocal quadrics. The Neumann problem is also strikingly related to theKorteweg-de Vries (KdV) equation and various other nonlinear partial differential equations.

4.5 A family of integrable systems

We consider the hamiltonian

H =1

2

(x21 + x22 + a1y

21 + a2y

22

)+

1

4y41 +

1

4a3y

42 +

1

2a4y

21y

22 , (24)

where a1, a2, a3, a4 are arbitrary constants. The corresponding system is given by

y1 +(a1 + y21 + a4y

22

)y1 = 0, (25)

y2 +(a2 + a3y

22 + a4y

21

)y2 = 0.

The integrability of this hamiltonian system has been studied by several authors. It arises in connection withsome problems in scalar field theory and in the semi-classical method in quantum field theory. It corresponds to theGarnier system and to the anisotropic harmonic oscillator in a radial quartic potential. Also, this hamiltonian canbe connected to the coupled nonlinear Schrodinger equations. This section deals with the problem of integrability ofthe system (25) corresponding to the choice : a1, a2 arbitrary constants and a3 = a4 = 1, i.e.,

H =1

2

(x21 + x22 + a1y

21 + a2y

22

)+

1

4

(y21 + y22

)2. (26)

The corresponding system is given by

y1 +(a1 + y21 + y22

)y1 = 0, (27)

y2 +(a2 + y22 + y21

)y2 = 0.

For a1 = a2, it is easy to show that the problem can be integrated in terms of elliptic functions. We suppose thata1 6= a2.

Using the results given in [12], we consider the Lax representation in the form

Ah = [Bh, Ah] = BhAh −AhBh, ˙≡ ∂

∂t(28)


with the following ansatz for the Lax operator

Ah =

(u(h) v(h)w(h) −u(h)

), Bh =

(0 1

r(h) 0

),

where

v(h) = −(a1 + h)(a2 + h)

(1 +

1

2

(y21

a1 + h+

y22a2 + h

)),

u(h) =1

2(a1 + h)(a2 + h)

(x1y1a1 + h

+x2y2a2 + h

), (29)

w(h) = (a1 + h)(a2 + h)

(1

2

(x21

a1 + h+

x22a2 + h

)− h+

1

2

(y21 + y22

)),

r(h) = h− y21 − y22 .

The system (25) admits a Lax representation given by (28). The proof is straightforward and based on directcomputation : we have

[Bh, Ah] =

(w(h)− v(h)r(h) −2u(h)

2u(h)r(h) v(h)r(h)− w(h)

),

and it follows from (26), (27) and (29) that

u(h) = w(h)− v(h)r(h),

v(h) = −2u(h),

w(h) = 2u(h)r(h).

Equation (28) means that for h ∈ C and under the time evolution of the system, Ah(t) remain similar to Ah(0).So the spectrum of Ah is conserved, i.e., it undergoes an isospectral deformation. The eigenvalues of Ah, viewedas functionals, represent the integrals (constants of the motion) of the system. To be precise, a hamiltonian flow ofthe type (28) preserves the spectrum of Ah and therefore its characteristic polynomial det(Ah − zI). We form theRiemann surface in (z, h) space :

Γ : det(Ah − zI) = 0,

whose coefficients are functions of the phase space. Explicitly, this equation looks as follows

Γ : z2 = u2(h) + v(h)w(h) ≡ P5(h), (30)

= (a1 + h)(a2 + h)(h3 + (a1 + a2)h2 + (a1a2 −H1)h−H2),

where H1 = H is defined by (26) with a1, a2 arbitrary, a3 = a4 = 1 and a second quartic integral H2 of the form :

H2 =1

4(a2y

41 + a1y

42 + (a1 + a2)y21y

22 + (x1y2 − x2y1)2) (31)

+1

2(a2x

21 + a1x

22 + a1a2(y21 + y22)).

The curve Γ determined by the fifth-order equation (30) is smooth, hyperelliptic and its genus is two. Obviously, Γis invariant under the involution (h, z) 7−→ (h,−z). The second hamiltonian vector field is written as

y1 =1

2(x1y2 − x2y1)y2 + a2x1,

y2 = −1

2(x1y2 − x2y1)y1 + a1x2,

x1 = −a2y31 −1

2(a1 + a2)y1y

22 +

1

2(x1y2 − x2y1)x2 − a1a2y1,

x2 = −a1y32 −1

2(a1 + a2)y21y2 −

1

2(x1y2 − x2y1)x1 − a1a2y2.

These vector fields are in involution with respect to the associated Poisson bracket. For generic c = (c1, c2) ∈ C2,

Vc = H1 = c1, H2 = c2, (32)

20 A. Lesfari

is a smooth affine surface. The linearized flow could be realized on the jacobian variety Jac(Γ) of the curve (30).Indeed, as the structure of the matrices Ah and Bh (degree of the polynomials in h) is the same as for the well-knownNeumann system, so both systems are linearized in the same way. We can construct an algebraic map from Vc tothe Jacobi variety Jac(Γ) :

Vc −→ Jac(Γ), p ∈ Vc 7−→ (s1 + s2) ∈ Jac(Γ),

and the flows generated by the constants of the motion are straight lines on Jac(Γ), i.e., the linearizing equationsare given by

2∑i=1

∫ si(t)

si(0)

ωk = ckt, 0 ≤ k ≤ 2,

where ω1, ω2 span the two-dimensional space of holomorphic differentials on the curve Γ and s1, s2 two appropriatevariables, algebraically related to the originally given ones, for which the Hamilton-Jacobi equation could be solvedby separation of variables. Consequently, we have

Theorem 4.4. The system (23) is completely integrable for all a1, a2, a3 = a4 = 1 (i.e., system (27)) and admitsa Lax representation given by (28). The invariants of Ah are integrals of motion in involution. The first integralis given by the hamiltonian H1 = H (26) whereas the second integral H2 is also quartic and has the form (31).The flows generated by H1 and H2 are straight line motions on the jacobian variety Jac(Γ) of a smooth genus twohyperelliptic curve Γ (30) associated to Lax equation (28).

We introduce coordinates s1 and s2 on the surface Vc (32), such that v(s1) = v(s2) = 0, a1 6= a2,

y21 = 2(a1 + s1)(a1 + s2)

a1 − a2, y22 = 2

(a2 + s1)(a2 + s2)

a2 − a1,

i.e.,

s1 + s2 =1

2(y21 + y22)− a1 − a2, s1s2 = −1

2(a2y

21 + a1y

22) + a1a2.

After some manipulations, we obtain the following equations :

s1 = 2

√P5(s1)

s1 − s2, s2 = 2

√P5(s2)

s2 − s1,

where P5(s) is defined by (30). These equations can be integrated by the Abelian mapping

Γ −→ Jac(Γ) = C2/L, p 7−→(∫ p

p0

ω1,

∫ p

p0

ω2

),

where the genus two hyperelliptic curve Γ is given by the equation (30), L is the lattice generated by the vectorsn1 + Ωn2, (n1, n2) ∈ Z2,Ω is the matrix of period of Γ, (ω1, ω2) is a basis of holomorphic differentials on the curveΓ, i.e.,

ω1 =ds√P5(s)

, ω2 =sds√P5(s)

,

and p0 is a fixed point. Consequently, we have

Theorem 4.5. The system of differential equations (27), can be integrated in terms of genus two hyperellipticfunctions of time.

4.6 The coupled nonlinear Schrodinger equations

The system of two coupled nonlinear Schrodinger equations is given by

i∂a

∂z+∂2a

∂t2+ Ω0a+

2

3

(|a|2 + |b|2

)a+

1

3

(a2 + b2

)a = 0, (33)

i∂b

∂z+∂2b

∂t2− Ω0b+

2

3

(|a|2 + |b|2

)b+

1

3

(a2 + b2

)b = 0,


where a(z, t) and b(z, t) are functions of z and t, the bar “−” denotes the complex conjugation, “||” denotes themodulus and Ω0 is a constant. These equations play a significant role in mathematics, with a important number ofphysical applications. We seek solutions of (33) in the following form

a(z, t) = y1(t) exp(iΩz), b(z, t) = y2(t) exp(iΩz),

where y1(t) et y2(t) are two functions and Ω is an arbitrary constant. Then we obtain the system

y1 +(y21 + y22

)y1 = (Ω− Ω0) y1,

y2 +(y21 + y22

)y2 = (Ω + Ω0) y2.

The latter coincides obviously with (27) for a1 = Ω0 − Ω and a2 = −Ω0 − Ω.

4.7 The Yang-Mills equations

We consider the Yang-Mills system for a field with gauge group SU(2) :

DjFjk =∂Fjk∂τj

+ [Aj , Fjk] = 0,

where Fjk, Aj ∈ TeSU(2), 1 ≤ j, k ≤ 4 and

Fjk =∂Ak∂τj− ∂Aj∂τk

+ [Aj , Ak] .

The self-dual Yang-Mills (SDYM) equations is an universal system for which some reductions include all classical topsfrom Euler to Kowalewski (0+1-dimensions), K-dV, Nonlinear Schrdinger, Sine-Gordon, Toda lattice and N-wavesequations (1+1-dimensions), KP and D-S equations (2+1-dimensions). In the case of homogeneous double-componentfield, we have ∂jAk = 0, j 6= 1, A1 = A2 = 0, A3 = n1U1 ∈ su(2), A4 = n2U2 ∈ su(2) where ni are su(2)-generators(i.e., they satisfy commutation relations : n1 = [n2, [n1, n2]], n2 = [n1, [n2, n1]]). The system becomes

∂2U1

∂t2+ U1U

22 = 0,

∂2U2

∂t2+ U2U

21 = 0,

with t = τ1. By setting Uj = yj ,∂Uj∂t

= xj , j = 1, 2, Yang-Mills equations are reduced to Hamiltonian system

x = J∂H

∂x, x = (y1, y2, x1, x2)>, J =

(O −II O

),

with H = 12 (x21 + x22 + y21y

22), the Hamiltonian. The symplectic transformation

x1 ←−√

2

2(x1 + x2), x2 ←−

√2

2(x1 − x2),

y1 ←−1

2(

4√

2)(y1 + iy2), y2 ←−1

2(

4√

2)(y1 − iy2),

takes this Hamiltonian into

H =1

2(x21 + x22) +

1

4y41 +

1

4y42 +

1

2y21y

22 ,

which coincides with (26) for a1 = a2 = 0.

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