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Available at: http://www.ictp.it/~pub- off IC/2007/030 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE DIFFERENTIAL-GEOMETRIC ASPECTS OF INTEGRABLE DYNAMICAL SYSTEMS Y.A. Prykarpatsky The AGH University of Science and Technology, Krakow 30059, Poland and Department of Differential Equations at the Institute of Mathematics at the NAS, Kyiv 00104, Ukraine, A.M. Samoilenko Department of Differential Equations at the Institute of Mathematics at the NAS, Kyiv 00104, Ukraine, A.K. Prykarpatsky The AGH University of Science and Technology, Krakow 30059, Poland, Department of Differential Equations at the Institute of Mathematics at the NAS, Kyiv 00104, Ukraine and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, N.N. Bogolubov (Jr.) V.A. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 117234, Russian Federation and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and D.L. Blackmore Department of Mathematical Sciences, NJIT, Newark, NJ 07102, USA and Institute of Mathematics, DTU, Lyngby, Denmark. MIRAMARE – TRIESTE May 2007

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Page 1: THE DIFFERENTIAL-GEOMETRIC ASPECTS OF ...users.ictp.it/~pub_off/preprints-sources/2007/IC2007030P.pdfstructure group G and base manifold N, on which the Lie group G acts [2, 1, 6]

Available at: http://www.ictp.it/~pub−off IC/2007/030

United Nations Educational, Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE DIFFERENTIAL-GEOMETRIC ASPECTS OF INTEGRABLE

DYNAMICAL SYSTEMS

Y.A. PrykarpatskyThe AGH University of Science and Technology, Krakow 30059, Poland

andDepartment of Differential Equations at the Institute of Mathematics at the NAS,

Kyiv 00104, Ukraine,

A.M. SamoilenkoDepartment of Differential Equations at the Institute of Mathematics at the NAS,

Kyiv 00104, Ukraine,

A.K. PrykarpatskyThe AGH University of Science and Technology, Krakow 30059, Poland,

Department of Differential Equations at the Institute of Mathematics at the NAS,Kyiv 00104, Ukraine

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,

N.N. Bogolubov (Jr.)V.A. Steklov Mathematical Institute of Russian Academy of Sciences,

Moscow 117234, Russian Federationand

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

D.L. BlackmoreDepartment of Mathematical Sciences, NJIT, Newark, NJ 07102, USA

andInstitute of Mathematics, DTU, Lyngby, Denmark.

MIRAMARE – TRIESTE

May 2007

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Abstract

The canonical reduction method on canonically symplectic manifolds is analyzed in detail,

and the relationships with the geometric properties of associated principal fiber bundles endowed

with connection structures are described. Some results devoted to studying geometrical prop-

erties of nonabelian Yang-Mills type gauge field equations are presented. A symplectic theory

approach is developed for partially solving the problem of algebraic-analytical construction of

integral submanifold embeddings for integrable (via the abelian and nonabelian Liouville-Arnold

theorems) Hamiltonian systems on canonically symplectic phase spaces. The fundamental role

of the so-called Picard-Fuchs type equations is revealed, and their differential-geometric and

algebraic properties are studied in detail. Some interesting examples of integrable Hamilton-

ian systems are are studied in detail in order to demonstrate the ease of implementation and

effectiveness of the procedure for investigating the integral submanifold embedding mapping.

1

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1. The geometric properties of reduced canonically-symplectic spaces with

symmetry, their relationship with structures on prinicipal fiber bundles, and

some applications

1.1. The Poisson structures and Lie group actions on manifolds: short introduction.

It has become increasingly clear in recent decades that many dynamical systems of classical

physics and mechanics are endowed with symplectic structures and associated Poisson brackets.

In many such cases, the structure of the Poisson bracket proves to be canonical and is given

on the dual space of the corresponding Lie algebra of symmetries, being augmented in some

cases with a 2-cocycle, and sometimes having a gauge nature. These observations give rise to

a deep group-theoretical interpretation of these Poisson structures for many dynamical systems

of mathematical physics, especially for the completely integrable ones.

The investigation of dynamical systems possessing a rich internal symmetry structure is usu-

ally carried out in three steps: 1) determining the symplectic structure (the Poisson bracket),

and recasting the initial dynamical system into Hamiltonian form; 2) determining conservation

laws (invariants or constants of the motion) in involution; 3) determining an additional set of

variables and computing their evolution under the action of Hamiltonian flows associated with

the invariants.

In many cases the above programme is too difficult to realize because of the lack of regular

methods for seeking both symplectic structures and a system of the related invariants. Conse-

quently, one seeks additional structures that make the programme more tractable; for example, of

particular interest are those dynamical systems with a deep intrinsic group nature [20, 21, 22, 30],

for which there exists a possibility of investigating their symmetry structure in exact form. The

corresponding symplectic manifolds on which these systems lie, in general, are pull-backs of the

corresponding group actions, related to the coadjont action of a Lie group on the dual space G∗

to its Lie algebra G, together with the natural Poisson structure upon them. In many cases these

spaces carry a principle fiber bundle structure and can be endowed with a variety of connections,

which play a very important role in describing the symmetry structure of the related dynamical

systems.

1.2. The Lie group actions on Poisson manifolds and the orbit structure. Let us recall

some definitions. The Poisson structure on a smooth manifold M is given by the pair (M, ., .),where

(1.1) ., . : D(M)×D(M)→ D(M),

is a Poisson bracket mapping onto the space of real-valued smooth functions on M , satisfying

the conditions: 1) it is bilinear and skew-symmetric; 2) it is a differentiation with respect to

each of the arguments; 3) it obeys the Jacobi identity. Any function H ∈ D(M) determines the2

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vector field sgrad : H (symplectic gradient) for all f ∈ D(M) via the formula:

(1.2) sgrad : H(f) := H, f.

The vector field sgrad : H : M → T (M) is called Hamiltonian, with the Hamiltonian function

H ∈ D(M).

A symplectic structure ω(2) on M supplies the manifold M with a Poisson bracket in a natural

manner. For any function H ∈ D(M), the vector field sgrad : H : M → T (M) is defined via

the rule:

(1.3) isgrad:Hω(2) := −dH,

whence

(1.4) H, f := −ω(2)( sgrad : H, sgrad : f )

for all f ∈ D(M), where isgrad:fω(2) := −df by definition.

The Poisson, or its associated Hamiltonian, structure comprises a wider class than do sym-

plectic ones. It is not hard to convince ourselves that any Poisson structure ., . on a manifold

M is stratified by symplectic structures.

The next class of Poisson structures appeared to be very important for applications [1, 2, 4,

13, 14]. Let G be a connected real Lie group, G its Lie algebra over the field R, and G∗ its linear

space dual to G. To each element x ∈ G∗ there is the associated endomorphism ad : x : G → G,ad x(y) := [x, y], y ∈ G, where [., .] is the Lie structure of the Lie algebra G. To each element

X ∈ G there is the associated automorphism Ad X : G → G via the rule:

(1.5) Ad X : y → dlX∗ drX(y),

where y ∈ G, dlX and drX are the tangent mappings for left and right translations on the Lie

group G, respectively.

Denote by ad∗ and Ad∗ adjoint mappings to ad and Ad, respectively, on G. Then for all

α ∈ G∗, x, y ∈ G the following identity obtains:

(1.6) 〈 ad∗ x(α), y 〉 := 〈 α, [x, y] 〉,

where 〈., .〉 is the convolution of G∗ with G.The representation ad∗ of the Lie algebra G and Ad∗ of the Lie group G in the space End G∗

are called co-adjoint.

Let f ∈ D(G∗); then one can determine the gradient ∇f : G∗ → G∗ via the rule:

(1.7) 〈 m,∇f(α) 〉 := df(α;m) :=d

dεf(α + εm)|ε=0,

where (α;m) ∈ G∗×G ∼= T (G∗). The structure of the Poisson bracket on G∗ is defined as follows:

(1.8) f, g(α) := 〈 α, [∇f(α),∇g(α)] 〉

for any f, g ∈ D(G∗). A proof that the bracket (1.8) is a Poisson bracket, is given below. The

corresponding Hamiltonian vector field for a function H ∈ D(G∗) takes the form:3

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(1.9) sgrad : H(α) = ( α; ad∗ : ∇H(α)(α) ) ∈ T (G∗),

where α ∈ G∗ is arbitrary. The vector field sgrad : H(α) is tangent to the orbit Oα(G) of the Lie

group G through an element α ∈ G under the Ad∗-action. These orbits are symplectic strata of

the manifold M . To each element uν := ( α; ad∗ xν(α)) ∈ Tα(Oα) ≡ Vα, ν = 1, 2, define

(1.10) ωα(u1, u2) := 〈 α, [x1, x2] 〉.

Then obviously, ωα is a symplectic structure on Oα for all α ∈ G∗. Thereby it is clear that

the above symplectic stratification of the Poisson structure (1.1) is realized by means of Ad∗ G

- the expansion in orbits in the space G∗. Notice here that each orbit Oα(G), α ∈ G∗, is a

uniform symplectic submanifold in G∗; i.e., the action Ad∗ of the Lie group is symplectic and

transitive. The restriction of the vector field sgrad : H upon an orbit Oα is defined uniquely via

the restriction of the Hamiltonian function H ∈ D(G∗) upon the orbit Oα(G), α ∈ G∗.

1.3. The canonical reduction method on canonically symplectic spaces and related

geometric structures on principal fiber bundles: introductory background. The canon-

ical reduction method in application to many geometric objects on symplectic manifolds with

symmetry turns out to be a very powerful tool; especially for finding the effective phase space

variables [2, 1, 12, 7, 13, 9] on integral submanifolds of Hamiltonian dynamical systems on which

they are integrable [2, 1, 11] via the Liouville-Arnold theorem, and for investigating related sta-

bility problems [2, 5, 15] of Hamiltonian dynamical systems under small perturbations. Let

G denote a given Lie group with the unity element e ∈ G and the corresponding Lie algebra

G ≃ Te(G). Consider a principal fiber bundle M(N ;G) with the projection p : M → N, the

structure group G and base manifold N, on which the Lie group G acts [2, 1, 6] by means of a

smooth mapping ϕ : M ×G→M. Namely, for each g ∈ G there is a group of diffeomorphisms

ϕg : M → M, generating for any fixed u ∈ M the following induced mapping: u : G → M,

where

(1.11) u(g) = ϕg(u).

On the principal fiber bundle p : (M,ϕ) → N there is a natural [12, 10, 6] connection Γ(A)

associated to such a morphism A: : (T (M), ϕg,∗) → (G, Adg−1), that for each u ∈ M the

mapping A(u) : Tu(M) → G is a left inverse of the mapping u∗(e) : G → Tu(M), and the

mapping A∗(u) : G∗ → T ∗u (M) is a right inverse of the mapping u∗(e) : T ∗

u (M)→ G∗; that is,

(1.12) A(u)u∗(e) = 1, u∗(e)A∗(u) = 1.

As usual, denote by ϕ∗g : T ∗(M)→ T ∗(M) the corresponding lift of the mapping ϕg : M →M

at any g ∈ G. If α(1) ∈ Λ1(M) is the canonical G - invariant 1-form on M, the canonical

symplectic structure ω(2) ∈ Λ2(T ∗(M)) given by

(1.13) ω(2) := d pr∗Mα(1)

4

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generates the corresponding momentum mapping l : T ∗(M)→ G∗, where

(1.14) l(α(1))(u) = u∗(e)α(1)(u)

for all u ∈M. Observe here that the principal fiber bundle structure p : M → N subsumes, in

part, the exactness of the following two adjoint sequences of mappings:

0← G u∗(e)← T ∗u (M)

p∗(u)← T ∗p(u)(N)← 0,

(1.15) 0→ G u∗(e)→ Tu(M)p∗(u)→ Tp(u)(N)→ 0,

that is

(1.16) p∗(u)u∗(e) = 0, u∗(e)p∗(u) = 0

for all u ∈M. Combining (1.16) with (1.12) and (1.14), one obtains the embedding:

(1.17) [1−A∗(u)u∗(e)]α(1)(u) ∈ range p∗(u)

for the canonical 1-form α(1) ∈ Λ1(M) at u ∈M. The expression (1.17) means, of course, that

(1.18) u∗(e)[1 −A∗(u)u∗(e)]α(1)(u) = 0

for all u ∈M. Now taking into account that the mapping p∗(u) : T ∗p(u)(N)→ T ∗

u (M) is injective

for each u ∈M , it has the unique inverse mapping (p∗(u))−1 upon its image p∗(u)T ∗p(u)(N) ⊂

T ∗u (M). Thus, for each u ∈M one can define a morphism pA : (T ∗(M), ϕ∗

g)→ (T ∗(N), id) as

(1.19) pA(u) : α(1)(u)→ (p∗(u))−1[1−A∗(u)u∗(e)]α(1)(u).

Based on definition (1.19) one can easily check that the diagram

(1.20)T ∗(M)

pA→ T ∗(N)prM↓ ↓prN

Mp→ N

is commutative.

Let now an element ξ ∈ G∗ be G-invariant, that is Ad∗g−1ξ = ξ for all g ∈ G. Denote also

by pξA the restriction of the mapping (1.19) upon the subset Mξ := l−1(ξ) ∈ T ∗(M), that is

the mapping pξA :Mξ → T ∗(N), where for all u ∈M

(1.21) pξA(u) :Mξ → (p∗(u))−1[1−A∗(u)u∗(e)]Mξ .

Now one can characterize the structure of the reduced phase space Mξ :=Mξ/G by means of

the following simple lemma.

Lemma 1.1. The mapping pξA(u) :Mξ → T ∗(N) is a principal fiber G-bundle with the reduced

space Mξ =Mξ)/G

Denote by < ., . >G the standard Ad-invariant non-degenerate scalar product on G∗ × G. It

follows directly from Lemma 1.1 that has the following characteristic theorem.5

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Theorem 1.2. Given a principal fiber G-bundle with a connection Γ(A) and a G-invariant

element ξ ∈ G∗, then every such connectionΓ(A) is a symplectomorphism νξ : Mξ → T ∗(N)

between the reduced phase space Mξ and cotangent bundle T ∗(N). Moreover, the following

equality

(1.22) (pξA)(d pr∗Nβ(1) + pr∗N Ω

(2)ξ ) = d pr∗Mξ

α(1)

holds for the canonical 1-forms β(1) ∈ Λ(1)(N) and α(1) ∈ Λ(1)(M), where Mξ := prMMξ ⊂M,

and the 2-form Ω(2)ξ ∈ Λ(2)(N) is the ξ-component of the corresponding curvature 2-form Ω(2) ∈

Λ(2)(M)⊗ G.

Proof. One has that on Mξ ⊂ T ∗(M) the following equation is owing to (1.19):

p∗(u)pξA(α(1)(u)) := p∗(u)β(1)(prN (u)) = α(1)(u)−A∗(u)u∗(e)α(1)(u)

for any β(1) ∈ T ∗(N), α(1) ∈Mξ and u ∈Mξ. Therefore, for such α(1) ∈Mξ one obtains

α(1)(u) = (pξA)−1β(1)(pN (u)) = p∗(u)β(1))(prN (u))+ < A(u), ξ >G

for all u ∈ Mξ. Recall now that in virtue of (1.20) one gets for Mξ and Mξ the following

relationships:

p · prMξ= prN · pξ

A, pr∗Mξ· p∗ = (pξ

A)∗ · pr∗N .

Consequently, for any u ∈M

pr∗Mξα(1)(u) = pr∗Mξ

p∗(u)β(1)(p(u)) + pr∗Mξ< A(u), ξ >

= (pξA)∗pr∗Nβ(1)(u) + pr∗Mξ

< A(u), ξ >,

so upon taking the external differential, one arrives at the following equality:

d pr∗Mξα(1)(u) = (pξ

A)∗d(pr∗Nβ(1))(u) + pr∗Mξ< d A(u), ξ >

= (pξA)∗d(pr∗Nβ(1))(u) + pr∗Mξ

< Ω(2)(u)), ξ >

= (pξA)∗d(pr∗Nβ(1))(u) + pr∗Mξ

p∗ < Ω(2), ξ > (u)

= (pξA)∗d(pr∗Nβ(1))(u) + (pξ

A)∗pr∗N < Ω(2), ξ > (u)

= (pξA)∗[d(pr∗Nβ(1))(u) + pr∗NΩ

(2)ξ (u)].

When deriving the above expression we made use of the following property satisfied by the

curvature 2-form Ω(2) := dA+A ∧A ∈ Λ2(M)⊗ G :

< dA(u), ξ >G=<dA(u) +A(u) ∧A(u), ξ >G

− < A(u) ∧A(u), ξ >G=< Ω(2)(u), ξ >G=

= < Ω(2)(u), Ad∗gξ >G=< AdgΩ(2)(u), ξ >G=

= < Ω(2), ξ > (p(u))G := p∗Ω(2)ξ (u)

at any u ∈M, since for any A,B ∈ G , < [A,B], ξ >G=< B, ad∗Aξ >G= 0 holds in virtue of the

invariance condition Ad∗gξ = ξ for any g ∈ G, which finishes the proof.⊲6

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Remark 1.3. As the canonical 2-form d pr∗Mα(1) ∈ Λ(2)(T ∗(M)) is G-invariant on T ∗(M)

due to its construction, it is evident that its restriction upon the G-invariant submanifold

Mξ ⊂ T ∗(M) is effectively defined only on the reduced space Mξ, which ensures the validity

of the equality sign in (1.22).

As a simple but useful consequence of Theorem 1.2, one can formulate the following result,

which is useful for applications.

Theorem 1.4. Let a momentum mapping value l(α(1))(u) = u∗(e)α(1)(u) = ξ ∈ G∗ have the

isotropy group Gξ acting naturally on the subset Mξ ⊂ T ∗(M) invariantly, freely and properly,

so that the reduced phase space (Mξ , ω(2)ξ ) is symplectic. By definition [2, 13], of the natural

embedding mapping πξ :Mξ → T ∗(M) and the reduction mapping rξ :Mξ → Mξ the defining

equality

(1.23) r∗ξ ω(2)ξ := π∗

ξ (d pr∗Mα(1))

holds onMξ. If an associated principal fiber bundle p : M → N has a structure group coinciding

with Gξ, then the reduced symplectic space (Mξ, ω(2)ξ ) is symplectomorphic to the cotangent

symplectic space (T ∗(N), σ(2)ξ ), where

(1.24) σ(2)ξ = d pr∗Nβ(1) + pr∗NΩ

(2)ξ ,

and the corresponding symplectomorphism is given by a relation of the form. (1.22)

Concerning some applications, the following criterion can be useful when constructing asso-

ciated fibre bundles with connections related to the symplectic structure reduced on the space

Mξ.

Theorem 1.5. In order that two symplectic spaces (Mξ , ω(2)ξ ) and (T ∗(N), d pr∗Nβ(1))

be symplectomorphic, it is necessary and sufficient that the element ξ ∈ ker h, where for a

G-invariant element ξ ∈ G∗ the mapping h : ξ → [Ω(2)ξ ] ∈ H2(N ; Z) with H2(N ; Z) is the

cohomology group of 2-forms on the manifold N.

1.4. The structure of reduced symplectic structures on cotangent spaces to Lie group

manifolds and associated canonical connections. When one has a Lie group G, the tangent

space T (G) can be interpreted as a Lie group G isomorphic to the semidirect product G⊗Ad G≃ G of the Lie group G and its Lie algebra G under the adjoint Ad-action of G on G. The

Lie algebra G of G is the semidirect product of G with itself, regarded as a trivial abelian

Lie algebra, under the adjoint ad-action and therefore has the induced bracket defined as

[(a1,m1), (a2,m2)] := ([a1, a2], [a1,m2] + [a2,m1]) for all (aj ,mj) ∈ G ⊗ad G, j = 1, 2. Take now

any element ξ ∈ G∗ and compute its isotropy group Gξ under the co-adjoint action Ad∗ of G on

G∗, and denote by Gξ its Lie algebra. The cotangent bundle T ∗(G) is obviously diffeomorphic to

M := G× G∗ ,on which the Lie group Gξ acts freely and properly (due to its construction) by7

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left translation on the first factor and Ad∗-action on the second. The corresponding momentum

mapping l : G× G∗ → G∗ξ can be obtained as

(1.25) l(h, α) = Ad∗h−1α∣∣G∗

ξ

and has no critical points. Let η ∈ G∗ and η(ξ) := η|G∗ξ. Then the reduced space (l−1(η(ξ))/G

η(ξ)ξ , ω

(2)ξ )

has to be symplectic due to the well-known Marsden-Weinstein reduction theorem [2, 1, 13],

where Gη(ξ)ξ is the isotropy subgroup of the Gξ -coadjoint action on η(ξ) ∈ G∗ξ , and the con-

dition r∗ξ ω(2)ξ := π∗

ξd pr∗α(1) naturally induces the symplectic form ω(2)ξ on the reduced space

Mξ := l−1(η(ξ))/Gη(ξ)ξ from the canonical symplectic structure on T ∗(G). Define for η(ξ) ∈ G∗ξ

the one-form α(1)η(ξ) ∈ Λ1(G) as

(1.26) α(1)η(ξ)(h) := R∗

hη(ξ),

where Rh : G→ G is right translation by an element h ∈ G. It is easy to check that the element

(1.26) is right G-invariant and left Gη(ξ)ξ -invariant, thus inducing a one-form on the quotient

Nξ := G/Gη(ξ)ξ . Denote its pull-back to T ∗(Nξ) by pr∗Nξ

α(1)η(ξ), and form the symplectic manifold

(T ∗(Nξ), σ(2)ξ ), where σ

(2)ξ := dpr∗Nξ

β(1)+ dpr∗Nξα

(1)η(ξ)) and d pr∗Nξ

β(1) ∈ Λ(2)(T ∗(Nξ)) is the

canonical symplectic form in the next theorem.

Theorem 1.6. Let ξ, η ∈ G∗ and η(ξ) := η|G∗ξ

be fixed. Then the reduced symplectic manifold

(Mξ, ω(2)ξ ) is a symplectic covering of the co-adjoint orbit Or(ξ, η(ξ); G), which symplectically

embeds into the sub-bundle over Nξ := G/Gη(ξ)ξ of (T ∗(Nξ), σ

(2)ξ ), with ω

(2)ξ := d pr∗Nξ

β(1) +

d pr∗Nξα

(1)η(ξ) ∈ Λ2(T ∗(Nξ).

This theorem follows directly from Theorem 1.6 if one to define a connection 1-form A(g) :

Tg(G)→ Gξ as follows:

(1.27) < A(g), ξ >G:= R∗gη(ξ)

for any g ∈ G. The expression (1.27) generates a completely horizontal 2-form d < A(g), ξ >G

on the Lie group G, which gives rise immediately to the symplectic structure σ(2)ξ on the phase

space T ∗(Nξ), where Nξ := G/Gη(ξ)ξ .

1.5. The geometric structure of abelian Yang-Mills type gauge field equations within

the reduction method. If one studies the motion of a charged particle under an abelian Yang-

Mills type gauge field, it is convenient [7, 4, 15] to introduce a special fiber bundle structure

p : M → N . Namely, one can take a fibered space M such that M = N×G N := D ⊂ Rn, with

G := R/0 being the corresponding (abelian) structure Lie group. An analysis similar to that

above gives rise to a reduced, upon the symplectomorphic spaceMξ := l−1(ξ)/G ≃ T ∗(N), ξ ∈G, with the symplectic structure σ

(2)ξ (q, p) =< dp,∧dq > +d < A(q, g), ξ >G , where A(q, g) :=

< A(q), dq > +g−1dg is the usual connection 1-form on M, with (q, p) ∈ T ∗(N) and g ∈ G.

The corresponding canonical Poisson brackets on T ∗(N) are easily found to be

(1.28) qi, qjξ = 0, pj, qiξ = δi

j , pi, pjξ = Fji(q)8

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for all (q, p) ∈ T ∗(N) and the curvature tensor is Fij(q) := ∂Aj/∂qi − ∂Ai/∂qj , i, j = 1, n. If

one introduces a new momentum variable p := p + A(q) on T ∗(N) ∋ (q, p), it is easy to verify

that σ(2)ξ → σ

(2)ξ :=< dp,∧dq >∈ Λ(2)(N), which gives rise to the following Poisson brackets [8]

on T ∗(N):

(1.29) qi, qjξ = 0, pj , qiξ = δi

j , pi, pjξ = 0,

iff for all i, j, k = 1, n the standard abelian Yang-Mills equations

(1.30) ∂Fij/∂qk + ∂Fjk/∂qi + ∂Fki/∂qj = 0

hold on N. This a construction permits a natural generalization to the case of a nonabelian

structure Lie group yielding a description of nonabelian Yang-Mills type field equations covered

by the reduction approach described above.

1.6. The geometric structure of nonabelian Yang-Mills type gauge field equations

within the reduction method. As before, we start with defining a phase space M of a particle

under a nonabelian Yang-Mills type gauge field given on a region D ⊂ R3 as M := D×G, where

G is a (not in general semi-simple) Lie group, acting on M from the right. Over the space M

one can define quite naturally a connection Γ(A) if to consider a trivial principal fiber bundle

p : M → N, where N := D, with the structure group G. Namely, if g ∈ G, q ∈ N, then a

connection 1-form on M ∋ (q, g) can be written [2, 12, 10] as

(1.31) A(q; g) := g−1(d +n∑

i=1

aiA(i)(q))g,

where ai ∈ G : i = 1,m is a basis of the Lie algebra G of the Lie group G, and Ai : D → Λ1(D),

i = 1, n, are the Yang-Mills fields on the physical n-dimensional open space D ⊂ Rn. One can

define the natural left invariant Liouville type form on M as

(1.32) α(1)(q; g) :=< p, dq > + < y, g−1dg >G ,

where y ∈ T ∗(G) and < ·, · >G denotes, as before, the usual Ad-invariant non-degenerate

bilinear form on G∗ × G, as evidently g−1dg ∈ Λ1(G) ⊗ G. The main assumption we need to

assume for further is that the connection 1-form should be compatible with the Lie group G

action on M. This means that the condition

(1.33) R∗hA(q; g) = Adh−1A(q; g)

is satisfied for all (q, g) ∈ M and h ∈ G, where Rh : G → G means the right translation by

an element h ∈ G on the Lie group G. Having stated all preliminary conditions, needed for

applying the reduction Theorem 1.4 to our model, suppose now that the Lie group G canonical

action on M is naturally lifted to that on the cotangent space T ∗(M) endowed, owing to (1.32),

with the following G-invariant canonical symplectic structure:

ω(2)(q, p; g, y) : = d pr∗α(1)(q, p; g, y) =< dp,∧dq >(1.34)

+ < dy,∧g−1dg >G + < ydg−1,∧dg >G

9

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for all (q, p; g, y) ∈ T ∗(M). Take an element ξ ∈ G∗ and assume that its isotropy subgroup

Gξ = G, that is Ad∗hξ = ξ for all h ∈ G. In the general case such an element ξ ∈ G∗ can not

exist but trivial ξ = 0, as it happens to the Lie group G = SL2(R). Then owing to (1.22)

one can construct the reduced phase space Mξ := Mξ/G symplectomorphic to (T ∗(N), σ(2)ξ ),

Mξ := l−1(ξ) ∈ T ∗(M), where for any (q, p) ∈ T ∗(N)

σ(2)ξ (q, p) = < dp,∧dq > +Ω

(2)ξ (q)(1.35)

= < dp,∧dq > +

m∑

s=1

n∑

i,j=1

esF(s)ij (q)dqi ∧ dqj .

In the above we have expanded the element G∗ ∋ ξ =∑n

i=1 eiai with respect to the bi-orthogonal

basis ai ∈ G∗ :< ai, aj >G= δij , i, j = 1,m with ei ∈ R, i = 1, n, being some constants, and

we denoted by F(s)ij (q), i, j = 1, n, s = 1,m, the respectively reduced on N components of the

curvature 2-form Ω(2) ∈ Λ2(N)⊗ Gaswell; that is

(1.36) Ω(2)(q) :=m∑

s=1

n∑

i,j=1

as F(s)ij (q)dqi ∧ dqj

at any point q ∈ N. Summarizing the calculations accomplished similar to these above, we can

formulate instantly the following result.

Theorem 1.7. Suppose that a nonabelian Yang-Mills type field (1.31) on the fiber bundle p :

M → N with M = D × G is invariant with respect to the Lie group G action G ×M → M.

Suppose also that an element ξ ∈ G∗ is chosen so that Ad∗Gξ = ξ. Then for the naturally

constructed momentum mapping l : T ∗(M) → G∗ the reduced phase space Mξ := l−1(ξ)/G,

is symplectomorphic to the space (T ∗(N), σ(2)), and is endowed with the symplectic structure

(1.35) on T ∗(N), having the following component-wise Poisson brackets:

(1.37) pi, qjξ = δj

i , qi, qjξ = 0, pi, pjξ =

n∑

s=1

esF(s)ji (q)

for all i, j = 1, n and (q, p) ∈ T ∗(N).

The respectively extended Poisson bracket on the whole cotangent space T ∗(M) owing to

(1.34) amounts to the following set of Poisson brackets relationships:

ys, ykξ =

n∑

r=1

crsk yr, pi, q

jξ = δji ,(1.38)

ys, pjξ = 0 = qi, qj, pi, pjξ =

n∑

s=1

ys F(s)ji (q),

where i, j = 1, n, crsk ∈ R, s, k, r = 1,m, are the structure constants of the Lie algebra G, and we

made use of the expansion A(s)(q) =∑n

j=1 A(s)j (q) dqj , as well as we introduced alternative fixed

values ei := yi, i = 1, n. The result (1.38) can easily be seen, if one to make a shift σ(2) → σ(2)ext

within the expression (1.34), where σ(2)ext := σ(2)

∣∣A0→A

, A0(g) := g−1dg, g ∈ G. Whence, one10

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can conclude, in virtue of the invariance properties of the connection Γ(A), that

σ(2)ext(q, p;u, y) =< dp,∧dq > +d < y(g), Adg−1A(q; e) >G

=< dp,∧dq > + < d Ad∗g−1y(g),∧A(q; e) >G=< dp,∧dq > +m∑

s=1

dys ∧ dus

+

n∑

j=1

m∑

s=1

A(s)j (q)dys ∧ dq− < Ad∗g−1y(g),A(q, e) ∧A(q, e) >G

(1.39) +

m∑

k≥s=1

m∑

l=1

yl clsk duk ∧ dus +

n∑

s=1

3∑

i≥j=1

ysF(s)ij (q)dqi ∧ dqj ,

where coordinate points (q, p;u, y) ∈ T ∗(M) are defined as follows: A0(e) :=∑m

s=1 dui ai,

Ad∗g−1y(g) = y(e) :=

∑ms=1 ys as for any element g ∈ G. Whence one gets right away the Poisson

brackets 1.38 plus additional brackets connected with conjugated sets of variables us ∈ R :

s = 1,m ∈ G∗ and ys ∈ R : s = 1,m ∈ G :

(1.40) ys, ukξ = δk

s , uk, qjξ = 0, pj , usξ = A

(s)j (q), us, ukξ = 0,

where j = 1, n, k, s = 1,m, and q ∈ N. Note here that the above transition from the symplectic

structure σ(2) on T ∗(N) to its extension σ(2)ext on T ∗(M) just consists formally in adding to the

symplectic structure σ(2) an exact part, which transforms it into equivalent one. Looking now

at expressions (1.39), one can infer immediately that an element ξ :=∑m

s=1 esas ∈ G∗ will be

invariant with respect to the Ad∗-action of the Lie group G iff

(1.41) ys, ykξ |ys=es=

m∑

r=1

crsk er ≡ 0

holds identically for all s, k = 1,m, j = 1, n and q ∈ N. In this and only this case, the reduction

scheme elaborated above will go through. Returning our attention to expression (1.40), one can

easily write down the following exact expression:

(1.42) ω(2)ext(q, p;u, y) = ω(2)(q, p +

n∑

s=1

ys A(s)(q) ;u, y),

on the phase space T ∗(M) ∋ (q, p;u, y), where for brevity we abbreviated < A(s)(q), dq >

as∑n

j=1 A(s)j (q) dqj . The transformation like (1.42) was discussed within somewhat different

context in article [8] containing also a good background for the infinite dimensional generalization

of symplectic structure techniques. Having observed from (1.42) that the simple change of

variable

(1.43) p := p +

m∑

s=1

ys A(s)(q)

11

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of the cotangent space T ∗(N) recasts our symplectic structure (2.34) into the old canonical form

(1.34), one obtains that the following new set of Poisson brackets on T ∗(M) ∋ (q, p;u, y) :

ys, ykξ =n∑

r=1

crsk yr, pi, pjξ = 0, pi, q

j = δji ,(1.44)

ys, qjξ = 0 = qi, qjξ, us, ukξ = 0, ys, pjξ = 0,

us, qiξ = 0, ys, ukξ = δk

s , us, pjξ = 0,

where k, s = 1,m and i, j = 1, n, holds iff the nonabelian Yang-Mills type field equations

(1.45) ∂F(s)ij /∂ql + ∂F

(s)jl /∂qi + ∂F

(s)li /∂qj

+

m∑

k,r=1

cskr(F

(k)ij A

(r)l + F

(k)jl A

(r)i + F

(k)li A

(r)j ) = 0

are fulfilled for all s = 1,m and i, j, l = 1, n on the base manifold N. This method complete

reduction of gauge Yang-Mills type variables from the symplectic structure (1.39) is known in

literature [8, 4, 46] as the principle of minimal interaction and appeared to be useful enough

for studying different interacting systems as in [9, 14]. In part 2 of this work we plan to

continue our study of the geometric properties of reduced symplectic structures connected with

infinite-dimensional coupled dynamical systems like Yang-Mills-Vlasov, Yang-Mills-Bogolubov

and Yang-Mills-Josephson ones [9, 14] as well as their relationships with associated principal

fiber bundles endowed with canonical connection structures.

2. Integrability by quadratures of Hamiltonian systems and Picard-Fuchs type

equations: the modern differential-geometric aspects

2.1. Introduction. As is well known [1, 2], the integrability by quadratures of a differential

equation in space Rn is a method of seeking its solutions by means of finite number of algebraic

operations (together with inversion of functions) and ”quadratures”- calculations of integrals of

known functions. Assume that our differential equation is given as a Hamiltonian dynamical

system on some appropriate symplectic manifold (M2n, ω(2)), n ∈ Z+, in the form

(2.1) du/dt = H,u,

where u ∈M2n, H : M2n →R is a sufficiently smooth Hamiltonian function [1, 2] with respect

to the Poisson bracket ·, · on D(M2n), dual to the symplectic structure ω(2) ∈ Λ2(M2n), and

t ∈R is the evolution parameter. More than one hundred and fifty years ago French mathemati-

cians and physicists, first de E. Bour and next J. Liouville, proved the first ”integrability by

quadratures” theorem which in modern terms [47] can be formulated as follows.

Theorem 2.1. Let M2n ≃ T ∗(Rn) be a canonically symplectic phase space and a dynamical

system 2.1 with a Hamiltonian function H: M2n×Rt →R, possess a Poissonian Lie algebra Gof n∈ Z+ invariants Hj : M2n×Rt →R, j=1, n, such that

12

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(2.2) Hi,Hj =

n∑

s=1

csijHs,

and for all i,j,k =1, n the csij ∈R are constants on M2n×Rt. Suppose further that

(2.3) Mn+1h =:= (u, t) ∈M ×Rt : h(Hj) = hj , j = 1, n, h ∈ G∗,

the integral submanifold of the set G of invariants at a regular element h∈ G∗, is a well

defined connected submanifold of M×Rt. Then, if : i) all functions of G are functionally

independent on Mn+1h ; ii)

∑ns=1 cs

ijhs = 0 for all i, j =1, n; iii) the Lie algebra G = spanR

Hj : M2n×Rt →R: j= 1, n is solvable, the Hamiltonian system 2.1 on M2n is integrable by

quadratures.

As a simple corollary of the Bour-Liouville theorem one gets the following:

Corollary 2.2. If a Hamiltonian system on M2n =T ∗(Rn) possesses just n∈ Z+ functionally

independent invariants in involution: that is the Lie algebra G is abelian, then it is integrable

by quadratures.

In the autonomous case when a Hamiltonian H = H1, and invariants Hj :M2n → R, j = 1, n,

are independent of the evolution parameter t ∈R, the involution condition Hi,Hj =0 ,

i, j = 1, n, can be replaced by the weaker one H,Hj = cjH for some constants cj ∈ R,

j = 1, n. The first proof of Theorem 2.1 was based on a result of S. Lie, which can be formulated

as follows.

Theorem 2.3. (S. Lie) Let vector fields Kj ∈ Γ(M2n), j = 1, n, be independent in some open

neighborhood Uh ∈M2n, generate a solvable Lie algebra G with respect to the usual commutator

[·, ·] on Γ(M2n) and [Kj,K] =cjK for all j = 1, n, where cj ∈R, j = 1, n, are constants.

Then the dynamical system

(2.4) du/dt = K(u),

where u ∈Uh ⊂ M2n, is integrable by quadratures.

Example 1 Motion of three particles on line R under uniform potential field. The motion

of three particles on the axis R pairwise interacting via a uniform potential field Q(‖·‖) is

described as a Hamiltonian system on the canonically symplectic phase space M = T ∗(R3) with

the following Lie algebra G of invariants on M2n:

(2.5) H = H1 =3∑

j=1

p2j/2mj +

3∑

i<j=1

Q(‖qi − qj‖),

H2 =

3∑

j=1

qjpj , H3 =

3∑

j=1

pj,

13

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where (qj , pj) ∈ T ∗(R), j = 1, 3, are coordinates and momenta of particles on the axis R. The

commutation relations for the Lie algebra G are

(2.6) H1,H3 = 0, H2,H3 = H3, H1,H2 = 2H1,

hence it is clearly solvable. Taking a regular element h ∈ G∗, such that h(Hj) = hj = 0, for

j = 1 and 3, and h(H2) = h2 ∈ R being arbitrary, one obtains the integrability of the problem

above in quadratures.

In 1974 V. Arnold proved [2] the following important result known as the commutative

(abelian) Liouville-Arnold theorem.

Theorem 2.4. (J.Liouville-V. Arnold). Suppose a set G of functions Hj : M2n → R, j = 1, n,

on a symplectic manifold M2n is abelian, that is

(2.7) Hi,Hj = 0

for all i, j = 1, n. If on the compact and connected integral submanifold Mnh=u∈M2n: h(Hj)=hj ∈

R, j=1, n, h∈ G∗ with h∈ G being regular, all functions H : M2n ∈ R, j = 1, n, are func-

tionally independent, then Mnh is diffeomorphic to the n-dimensional torus T

n ≃ M2n, and the

motion on it with respect to the Hamiltonian H=H1 ∈ G is a quasi-periodic function of the

evolution parameter t∈ R.

A dynamical system satisfying the hypotheses of Theorem 2.4 is called completely integrable.

In 1978 Mishchenko and Fomenko [16] proved the following generalization of the Liouville-Arnold

Theorem 2.4:

Theorem 2.5. (A. Mishchenko-A. Fomenko) Assume that on a symplectic manifold (M2n,ω(2))

there is a nonabelian Lie algebra G of invariants Hj : M ∈R, j=1, k, with respect to the dual

Poisson bracket on M2n, that is

(2.8) Hi,Hj =

k∑

s=1

csijHs,

where all values csij ∈ R, i,j,s =1, k, are constants, and the following conditions are satisfied:

i) the integral submanifold Mrh:=u∈M2n:h(Hj)=h ∈ G∗ is compact and connected at a regular

element h ∈ G∗; ii) all functions Hj:M2n → R, j=1, k, are functionally independent on M2n; iii)

the Lie algebra G of invariants satisfies the following relationship:

(2.9) dimG + rankG = dim M2n,

where rankG = dim Gh is the dimension of a Cartan subalgebra Gh ⊂ G. Then the sub-

manifold M rh ⊂ M2n is r = rankG -dimensional, invariant with respect to each vector field

K∈ Γ(M2n),generated by an element H ∈ Gh, and diffeomorphic to the r-dimensional torus

Tr ≃M r

h, on which the motion is a quasiperiodic function of the evolution parameter t ∈ R.

The simplest proof of the Mishchenko-Fomenko Theorem 2.5 can be obtained from the well-

known [17, 29] classical Lie-Cartan theorem.14

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Theorem 2.6. (S. Lie-E. Cartan) Suppose that a point h∈ G∗ for a given Lie algebra G of

invariants Hj :M2n →R, j=1, k, is not critical, and the rank ||Hi,Hj : i, j = 1, k|| = 2(n − r)

is constant in an open neighborhood Uh ∈Rn of the point h(Hj) = hj ∈R: j = 1, k⊂ R

k.

Then in the neighborhood (h H)−1 :Uh ⊂ M2n there exist k∈ Z+ independent functions

fs : G →R, s = 1, k, such that the functions Fs := (fs H) : M2n ∈ R, s = 1, k, satisfy the

following relationships:

(2.10) F1, F2 = F3, F4 = ... = F2(n−r)−1, F2(n−r) = 1,

with all other brackets Fi, Fj = 0, where (i, j) 6= (2s − 1, 2s), s = 1, n− r. In particular,

(k+r−n) ∈ Z+ functions Fj : M2n →R, j = 1, n − r, and Fs : M2n →R, s = 1, k − 2(n − r),

compose an abelian algebra Gτ of new invariants on M2n, independent on (hH)−1(Uh) ⊂M2n.

As a simple corollary of the Lie-Cartan Theorem 2.6 one obtains the following: in the case

of the Mishchenko-Fomenko theorem when rankG + dimG = dimM2n, that is r + k = 2n, the

abelian algebra Gτ (it is not a subalgebra of G !) of invariants on M2n is just n = 1/2dimM2n-

dimensional, giving rise to its local complete integrability in (hH)−1(Uh) ⊂M2n via the abelian

Liouville-Arnold Theorem 2.4. It is also evident that the Mishchenko-Fomenko nonabelian

integrability Theorem 2.5 reduces to the commutative (abelian) Liouville-Arnold case when a

Lie algebra G of invariants is just abelian, since then rankG = dimG = 1/2 dim M2n = n ∈ Z+

is the standard complete integrability condition. All the cases of integrability by quadratures

described above pose the following fundamental question: How can one effectively construct,

by means of algebraic-analytical methods, the corresponding integral submanifold embedding

(2.11) πh : M rh →M2n,

where r = dim rankG, thereby making it possible to express the solutions of an integrable flow

on M rh as some exact quasi-periodic functions on the torus T

r ≃M rh. Below we shall describe an

algebraic-analytical algorithm for resolving this question for the case when a symplectic manifold

M 2n is diffeomorphic to the canonically symplectic cotangent phase space T ∗(R) ≃M2n..

2.2. General setting and preliminaries. Our main object of study will be differential sys-

tems of vector fields on the cotangent phase space M2n = T ∗(Rn), n ∈ Z+, endowed with the

canonical symplectic structure ω(2) ∈ Λ2(M2n), where by ω(2) = d(pr∗α(1)), and

(2.12) α(1) :=< p, dq >=n∑

j=1

pjdqj,

is the canonical 1-form on the base space Rn, lifted naturally to the space Λ1(M2n), (q, p) ∈

M2n are canonical coordinates on T ∗(Rn), pr : T ∗(Rn)→ R is the canonical projection , and

< ·, · > is the usual scalar product in Rn. Assume further that there is also given a Lie subgroup

G (not necessarily compact), acting symplectically via the mapping ϕ : G ×M2n → M2n on15

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M2n, generating a Lie algebra homomorphism ϕ∗ : T (G)→Γ(M2n) via the diagram

(2.13)G × G ≃ T (G) ϕ∗(u)→ T (M2n)

↓ ↓G ϕ(u)→ M2n

where u ∈M2n. Thus, for any a ∈G one can define a vector field Ka ∈ Γ(M2n) as follows:

(2.14) Ka = ϕ∗ · a.

Since the manifold M2n is symplectic, one can naturally define for any a ∈ G a function

Ha ∈ D(M2n) as follows:

(2.15) −iKaω(2) = dHa,

whose existence follows from the invariance property

(2.16) LKaω(2) = 0

for all a ∈ G. The following lemma [1] is useful in many applications.

Lemma 2.7. If the first homology group H1(G; R) of the Lie algebra G vanishes, then the

mapping Φ : G → D(M2n) defined as

(2.17) Φ(a) := Ha

for any a ∈ G, is a Lie algebra homomorphism of G and D(M2n) (endowed with the Lie

structure induced by the symplectic structure ω(2) ∈ Λ2(M2n)).

In this case G is said to be Poissonian. As the mapping Φ : G → D(M2n) is evidently linear

in G, the expression 2.17 naturally defines a momentum mapping l : M2n → G∗ as follows: for

any u ∈M2n and all a ∈ G

(2.18) (l(u), a)G := Ha(u),

where (·, ·)G is the standard scalar product on the dual pair G∗×G. The following characteristic

equivariance [1] lemma holds.

Lemma 2.8. The diagram

(2.19)

M2n l→ G∗ϕg

yyAd∗

g−1

M2n l→ G∗

commutes for all g ∈ G, where Ad∗g−1 : G∗ → G∗ is the corresponding co-adjoint action of

the Lie group G on the dual space G∗.

Take now any vector h ∈ G∗ and consider a subspace Gh ⊂ G , consisting of elements a ∈ G,such that ad∗ah = 0, where ad∗a : G∗ → G∗ is the corresponding Lie algebra G representation

in the dual space G∗. The following lemmas hold.16

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Lemma 2.9. The subspace Gh ⊂ G is a Lie subalgebra of G, called here a Cartan subalgebra.

Lemma 2.10. Assume a vector h ∈ G∗ is chosen in such a way that r = dimGh is minimal.

Then the Cartan Lie subalgebra Gh ⊂ G is abelian.

In Lemma 2.10 the corresponding element h ∈ G∗ is called regular and the number r = dim

Gh is called the rankG of the Lie algebra G.Some twenty years ago, Mishchenko and Fomenko [16] proved the following important non-

commutative (nonabelian) Liouville-Arnold theorem.

Theorem 2.11. On a symplectic space (M2n, ω(2)) let there be given a set of smooth functions

Hj ∈ D(M2n), j = 1, k, whose linear span over R comprises a Lie algebra G with respect to

the corresponding Poisson bracket on M2n. Suppose also that the set

M2n−kh := u ∈M2n : h(Hj) = hj ∈ R, j = 1, k, h ∈ G∗

with h ∈ G∗ regular, is a submanifold of M2n, and on M2n−kh all the functions Hj ∈ D(M2n),

j = 1, k, are functionally independent. Assume also that the Lie algebra G satisfies the following

condition:

(2.20) dimG+rankG = dim M2n.

Then the submanifold M rh := M2n−k

h is rankG = r−dimensional and invariant with respect to

each vector field Ka ∈ Γ(M2n) with a ∈ Gh ⊂ G. Given a vector field K = Ka ∈ Γ(M2n)

with a ∈ Gh or K ∈ Γ(M2n) such that [K,Ka] = 0 for all a ∈ G, then, if the submanifold

M rh is connected and compact, it is diffeomorphic to the r−dimensional torus T

r ≃ M rh and

the motion of the vector field K ∈ Γ(M2n) on it is a quasi-periodic function of the evolution

parameter t ∈ R.

The easiest proof of this result can be obtained from the well-known [17] classical Lie-Cartan

theorem, mentioned in the Introduction. Below we shall only sketch the original Mishchenko-

Fomenko proof which is heavily based on symplectic theory techniques, some of which have been

discussed above.

Sketch of the proof. Define a Lie group G naturally as G = expG, where G is the Lie algebra

of functions Hj ∈ D(M2n), j = 1, k, in the theorem , with respect to the Poisson bracket ·, ·on M2n. Then for an element h ∈ G∗ and any a =

∑kj=1 cjHj ∈ G , where cj ∈ R, j = 1, k,

the following equality

(2.21) (h, a)G :=

k∑

j=1

cjh(Hj) =

k∑

j=1

cjhj

holds. Since all functions Hj ∈ D(M2n), j = 1, k, are independent on the level submanifold

M rh ⊂M2n, this implies that the element h ∈ G∗ is regular for the Lie algebra G. Consequently,

the Cartan Lie subalgebra Gh ⊂ G is abelian. The latter is proved by means of simple17

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straightforward calculations. Moreover, the corresponding momentum mapping l : M2n → G∗

is constant on M rh and satisfies the following relation:

(2.22) l(M rh) = h ∈ G∗.

From this it can be shown that all vector fields K a ∈ Γ(M2n), a ∈ Gh, are tangent to the

submanifold M rh ⊂ M2n. Thus the corresponding Lie subgroup Gh := expGh acts naturally

and invariantly on M rh . If the submanifold M r

h ⊂ M2n is connected and compact, it follows

from (2.20) that dimM rh = dim M2n − dimG = rankG =r, and one obtains via the Arnold

theorem [2], that M rh ≃ T

r and the motion of the vector field K ∈ Γ(M2n) is a quasi-periodic

function of the evolution parameter t ∈ R, thus proving the theorem. As a nontrivial consequence

of Theorem 2.11, one can prove the following dual theorem about abelian Liouville-Arnold

integrability.

Theorem 2.12. Let a vector field K ∈ Γ(M2n) be completely integrable via the nonabelian

scheme of Theorem 1.5. Then it is also Liouville-Arnold integrable on M2n and possesses, under

some additional conditions, yet another abelian Lie algebra Gh of functionally independent

invariants on M2n, for which dimGh = n = 1/2 dim M2n.

The available proof of the theorem above is quite complicated, and we shall comment on it in

detail later on. We mention here only that some analogue of the reduction Theorem 2.11 for the

case where M2n ≃ G∗, so that an arbitrary Lie group G acts symplectically on the manifold ,

were proved also in [19, 23, 48]. Note here, that in case when the equality (2.20) is not satisfied,

one can then construct in the usual way, the reduced manifold M2n−k−rh := M2n−k

h /Gh on which

there exists a symplectic structure ω(2)h ∈ Λ2(M

2n−k−rh ), defined as

(2.23) r∗hω(2)h = π∗

hω(2)

with respect to the following compatible reduction-embedding diagram:

(2.24) M2n−k−rh

rh←−M2n−kh

πh→M2n,

where rh : M2n−kh → M

2n−k−rh and πh : M2n−k

h → M2n are, respectively, the corresponding

reductions and embedding mappings. The nondegeneracy of the 2-form ω(2)h ∈ Λ2(Mh) defined

by (2.23), follows simply from

(2.25) ker(π∗hω(2)(u)) = Tu(M2n−k

h ) ∩ T⊥u (M2n−k

h ) =

spanRKa(u) ∈ Tu(M2n−k−rh := M2n−k

h /Gh) : a ∈ Gh

for any u ∈ M2n−kh , since all vector fields Ka ∈ Γ(M2n), a ∈ Gh, are tangent to M

2n−k−rh :=

M2n−kh /Gh. Thus, the reduced space M

2n−k−rh := M2n−k

h /Gh with respect to the orbits of the

Lie subgroup Gh action on M2n−kh will be a (2n−k−r)-dimensional symplectic manifold. The

latter evidently means that the number 2n− k− r = 2s ∈ Z+ is even as there is no symplectic

structure on odd-dimensional manifolds. This obviously is closely connected with the problem

of existence of a symplectic group action of a Lie group G on a given symplectic manifold18

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(M2n, ω(2)) with a symplectic structure ω(2) ∈ Λ(2)(M2n) being a′priori fixed. From this point

of view one can consider the inverse problem of constructing symplectic structures on a manifold

M2n , admitting a Lie group G action. Namely, owing to the equivariance property (2.19) of the

momentum mapping l : M2n → G∗, one can obtain the induced symplectic structure l∗Ω(2)h ∈

Λ2(M2n−k−rh ) on M

2n−k−rh from the canonical symplectic structure Ω

(2)h ∈ Λ(2)(Or(h;G))

on the orbit Or(h;G) ⊂ G∗ of a regular element h ∈ G∗. Since the symplectic structure

l∗Ω(2)h ∈ Λ2(Mh) can be naturally lifted to the 2-form ω(2) = (r∗h l∗)Ω

(2)h ∈ Λ2(M2n−k

h ),

the latter being degenerate on M2n−kh can apparently be non-uniquely extended on the whole

manifold M2n to a symplectic structure ω(2) ∈ Λ2(M2n), for which the action of the Lie

group G is a′priori symplectic. Thus, many properties of a given dynamical system with

a Lie algebra G of invariants on M2n are deeply connected with the symplectic structure

ω(2) ∈ Λ2(M2n) the manifold M2n is endowed with, and in particular, with the corresponding

integral submanifold embedding mapping πh : M2n−kh → M2n at a regular element h ∈ G∗.

The problem of direct algebraic-analytical construction of this mapping was in part solved

in [24] in the case where n = 2 for an abelian algebra G on the manifold M4 = T ∗(R2).

The treatment of this problem in [24] makes extensive use both on the classical Cartan studies

of integral submanifolds of ideals in Grassmann algebras and on the modern Galissot-Reeb-

Francoise results for a symplectic manifold (M2n, ω(2)) structure, on which there exists an

involutive set G of functionally independent invariants Hj ∈ D(M2n), j = 1, n. In what

follows below we generalize the Galissot-Reeb-Francoise results to the case of a nonabelian set

of functionally independent functions Hj ∈ D(M2n), j = 1, k, comprising a Lie algebra G and

satisfying the Mishchenko-Fomenko condition (2.20): dimG+rankG =dimM2n. This makes

it possible to devise an effective algebraic-analytical method of constructing the corresponding

integral submanifold embedding and reduction mappings, giving rise to a wide class of exact,

integrable by quadratures solutions of a given integrable vector field on M2n.

2.3. Integral submanifold embedding problem for an abelian Lie algebra of invari-

ants. We shall consider here only a set G of commuting polynomial functions Hj ∈ D(M2n),

j = 1, n, on the canonically symplectic phase space M2n = T ∗(Rn). Due to the Liouville-Arnold

theorem [2], any dynamical system K ∈ Γ(M2n) commuting with corresponding Hamiltonian

vector fields Ka for all a ∈ G , will be integrable by quadratures in case of a regular element

h ∈ G∗, which defines the corresponding integral submanifold Mnh := u ∈ M2n : h(Hj ) =

hj ∈ R, j = 1, n which is diffeomorphic (when compact and connected) to the n−dimensional

torus Tn ≃ Mn

h . This in particular means that there exists an algebraic-analytical expression

for the integral submanifold embedding mapping πh : Mnh → M2n into the ambient phase

space M2n, which one should find in order to properly demonstrate integrability by quadra-

tures. The problem formulated above was posed and in part solved (as was mentioned above)

for n = 2 in [24] and in [26] for a Henon-Heiles dynamical system which had previously been19

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integrated [27, 28] using other tools. Here we generalize the approach of [24] for the general case

n ∈ Z+ and proceed further to solve this problem in the case of a nonabelian Lie algebra G of

polynomial invariants on M2n = T ∗(Rn), satisfying all the conditions of Mishchenko-Fomenko

Theorem 2.11.

Define now the basic vector fields Kj ∈ Γ(M2n), j = 1, n, generated by basic elements Hj ∈ Gof an abelian Lie algebra G of invariants on M2n, as follows:

(2.26) −iKjω(2) = dHj

for all j = 1, n. It is easy to see that the condition Hj ,Hi = 0 for all i, j = 1, n , yields

also [Ki,Kj] = 0 for all i, j = 1, n. Taking into account that dimM2n = 2n one obtains

the equality (ω(2))n = 0 identically on M2n. This makes it possible to formulate the following

Galissot-Reeb result.

Theorem 2.13. Assume that an element h ∈ G∗ is chosen to be regular and a Lie algebra Gof invariants on M2n is abelian. Then there exist differential 1-forms h

(1)j ∈ Λ1(U(Mn

h )),

j = 1, n, where U(Mnh ) is some open neighborhood of the integral submanifold Mn

h ⊂ M2n,

satisfying the following properties: i) ω(2)∣∣U(Mn

h)=

∑nj=1 dHj∧h

(1)j ; ii) the exterior differentials

dh(1)j ∈ Λ2(U(Mn

h )) belong to the ideal I(G) in the Grassmann algebra Λ(U(Mnh )), generated

by 1-forms dHj ∈ Λ1(U(Mnh )), j = 1, n.

Proof. Consider the following identity on M2n :

(2.27) (⊗nj=1iKj

)(ω(2))n+1 = 0 = ±(n + 1)!(∧nj=1dHj) ∧ ω(2),

which implies that the 2-form ω(2) ∈ I(G) . Whence, one can find 1-forms h(1)j ∈Λ1(U(Mn

h )),

j = 1, n, satisfying the condition

(2.28) ω(2)∣∣∣U(Mn

h)=

n∑

j=1

dHj ∧ h(1)j .

Since ω(2) ∈ Λ2(U(Mnh )) is nondegenerate on M2n, it follows that all 1-forms h

(1)j , j = 1, n,

in (2.28) are independent on U(Mnh ), proving part i) of the theorem. As dω(2) = 0 on M2n,

from (2.28) one finds that

(2.29)

n∑

j=1

dHj ∧ dh(1)j = 0

on U(Mnh ). Hence it is obvious that dh

(1)j ∈ I(G) ⊂ Λ(U(Mn

h )) for all j = 1, n, proving part

ii) of the theorem. Now we proceed to study properties of the integral submanifold Mnh ⊂M2n

of the ideal I(G) in the Grassmann algebra Λ(U(Mnh )). In general, the integral submanifold

Mnh is completely described [29] by means of the embedding

(2.30) πh : Mnh →M2n

20

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and using this, one can reduce all vector fields Kj ∈ Γ(M2n), j = 1, n, on the submanifold

Mnh ⊂M2n, since they are all evidently in its tangent space. If Kj ∈ Γ(Mn

h ), j = 1, n, are the

corresponding pulled-back vector fields Kj ∈ Γ(M2n), j = 1, n, then by definition, the equality

(2.31) πh∗ Kj = Kj πh

holds for all j = 1, n. Similarly one can construct 1-forms h(1)j := π∗

h h(1)j ∈ Λ1(Mn

h ), j = 1, n,

which are characterized by the following Cartan-Jost [29] theorem.

Theorem 2.14. The following assertions are true: i) the 1-forms h(1)j ∈ Λ1(Mn

h ), j = 1, n, are

independent on Mnh ; ii) the 1-forms h

(1)j ∈ Λ1(Mn

h ), j = 1, n, are exact on Mnh and satisfy

h(1)j (Kj) = δij , i, j = 1, n.

Proof. As the ideal I(G) is by definition vanishing on Mnh ⊂M2n and closed on U(Mn

h ) , the

integral submanifold Mnh is well defined in the case of a regular element h ∈ G∗. This implies

that the embedding (2.30) is non-degenerate on Mnh ⊂ M2n, or the 1-forms h

(1)j := π∗

h h(1)j ,

j = 1, n, will persist in being independent if they are 1-forms h(1)j ∈ Λ1(U(Mn

h )), j = 1, n,

proving part i) of the theorem. Using property ii) of Theorem 2.1, one sees that on the integral

submanifold Mnh ⊂M2n all 2-forms dh

(1)j = 0, j = 1, n. Consequently, owing to the Poincare

lemma [1, 1], the 1-forms h(1)j = dtj ∈ Λ1(Mn

h ), j = 1, n, for some mappings tj : Mnh → R,

j = 1, n, defining global coordinates on an appropriate universal covering of Mnh . Consider now

the following identity based on the representation 2.28:

(2.32) iKjω(2)

∣∣∣U(Mn

h)= −

n∑

i=1

h(1)i (Kj) dHi := −dHj,

which holds for any j = 1, n. As all dHj ∈ Λ1(U(Mnh )), j = 1, n, are independent, from (2.32)

one infers that h(1)i (Kj) = δij for all i, j = 1, n. Recalling now that for any i = 1, n ,

Ki πh = πh∗ Ki, one readily computes that h(1)i (Kj) = π∗

hh(1)i (Kj) := h

(1)i (πh∗ Kj) :=

h(1)i (Kj πh) = δij for all i, j = 1, n, proving part ii) of the theorem. The following is a

simple consequence of Theorem 2.14:

Corollary 2.15. Suppose that the vector fields Kj ∈ Γ(M2n), j = 1, n, are parameterized globally

along their trajectories by means of the corresponding parameters tj : M2n → R, j = 1, n, that

is, on the phase space M2n

(2.33) d/dtj := Kj

for all j = 1, n. Then the following important equalities hold (up to normalization constants)

on the integral submanifold Mnh ⊂M2n:

(2.34) tj|Mnh

= tj ,

where 1 ≤ j ≤ n.21

Page 23: THE DIFFERENTIAL-GEOMETRIC ASPECTS OF ...users.ictp.it/~pub_off/preprints-sources/2007/IC2007030P.pdfstructure group G and base manifold N, on which the Lie group G acts [2, 1, 6]

We consider a completely integrable via Liouville-Arnold Hamiltonian system on the cotan-

gent canonically symplectic manifold (T ∗(Rn), ω(2)), n ∈ Z+, possessing exactly n ∈ Z+ func-

tionally independent and Poisson commuting algebraic polynomial invariants Hj : T ∗(Rn)→ R,

j = 1, n. Owing to the Liouville-Arnold theorem, this Hamiltonian system can be completely

integrated by quadratures in quasi-periodic functions on its integral submanifold when taken

compact. It is equivalent to the statement that this compact integral submanifold is diffeo-

morphic to a torus Tn, which makes it possible to formulate the problem of integrating the

system by means of investigating the corresponding integral submanifold embedding mapping

πh : Mnh −→ T ∗(Rn), where by definition

(2.35) Mnh := (q, p) ∈ T ∗(Rn) : Hj(q, p) = hj ∈ R, j = 1, n.

Since Mnh ≃ T

n, and the integral submanifold (2.35) is invariant subject to all Hamiltonian flows

Kj : T ∗(Rn)→ T (T ∗(Rn)), j = 1, n, where

(2.36) iKjω(2) = −dHj,

there exist corresponding action-angle - coordinates (ϕ, γ) ∈ (Tnγ , Rn) on the torus T

nγ ⋍ Mn

h ,

specifying its embedding πγ : Tnγ → T ∗(Rn) by means of a set of smooth functions γ ∈ D(Rn),

where

(2.37) Tnγ := (q, p) ∈ T ∗(Rn) : γj(H) = γj ∈ R, j = 1, n.

The mapping γ : Rn ∋ h→ R

n, induced by (2.37), is of great interest for many applications and

was studied still earlier by Picard and Fuchs subject to the corresponding differential equations:

(2.38) ∂γj(h)/∂hi = Fij(γ;h),

where h ∈ Rn and Fij : R

n × Rn → R, i, j = 1, n, are some smooth almost everywhere

functions. In the case when the right-hand side of (2.38) is a set of algebraic functions on

Cn× Cn ∋ (γ;h), all Hamiltonian flows Kj : T ∗(Rn) → T (T ∗(Rn)), j = 1, n, are said to

be algebraically completely integrable in quadratures. In general, equations like (2.38) were

studied in [32, 44], a recent example can be found in [31]. It is clear enough that Picard-Fuchs

equations (2.38) are related to the associated canonical transformation of the symplectic 2-form

ω(2) ∈ Λ2(T ∗(Rn)) in a neighborhood U(Mnh ) of the integral submanifold Mn

h ⊂ T ∗(Rn). To

be more precise, denote ω(2)(q, p) = dpr∗α(1)(q; p), where for (q, p) ∈ T ∗(Rn)

(2.39) α(1)(q; p) :=n∑

j=1

pjdqj =< p, dq >∈ Λ1(Rn)

is the canonical Liouville 1-form on Rn, < ·, · > is the usual scalar product in R

n, pr : T ∗(Rn)→R

n is the bundle projection. One can now define a mapping

(2.40) dSq : Rn → T ∗

q (Rn),

such that dSq(h) ∈ T ∗q (Rn) is an exact 1-form for all q ∈ Mn

h and h ∈ Rn, yielding

(2.41) (dSq)∗(dpr∗α(1)) = (dSq)

∗ω(2) := d2Sq ≡ 0.22

Page 24: THE DIFFERENTIAL-GEOMETRIC ASPECTS OF ...users.ictp.it/~pub_off/preprints-sources/2007/IC2007030P.pdfstructure group G and base manifold N, on which the Lie group G acts [2, 1, 6]

Thereby, the mapping (2.40) defines a so-called generating function [1, 16] Sq : Rn → R,

satisfying on Mnh the relationship

(2.42) pr∗α(1)(q; p)+ < t, dh >= dSq(h),

where t ∈ Rn is the set of evolution parameters. From (2.42) it is evident that equality

(2.43) Sq(h) =

∫ q

q(0)< p, dq >

∣∣∣∣Mn

h

holds for any q, q(0) ∈ Mnh . On the other hand, one can define another generating function

Sµ : Rn → R, such that

(2.44) dSµ : Rn → T ∗

µ(Mnh ),

where µ ∈Mnh ⋍ ⊗n

j=1S1j are global separable coordinates existing on Mn

h owing to the Liouville-

Arnold theorem. Thus one has the following canonical relationship:

(2.45) < w, dµ > + < t, dh >= dSµ(h),

where wj := wj(µj;h) ∈ T ∗µj

(S1j) for every j = 1, n. Whence, is follows readily that

(2.46) Sµ(h) =

n∑

j=1

∫ µj

µ(0)j

w(λ;h)dλ,

satisfying on Mnh ⊂ T ∗(Rn) the following relationship

(2.47) dSµ + dLµ = dSq|q=q(µ;h)

for some mapping Lµ : Rn → R. As a result of (2.46) and (2.47) one obtains the following

important expressions

(2.48) ti = ∂Sµ(h)/∂hi, < p, ∂q/∂µi >= wi + ∂Lµ/∂µi,

holding for all i = 1, n. A construction similar to the above can be done on the embedded torus

Tnγ ⊂ T ∗(Rn) :

(2.49) dSq(γ) :=

n∑

j=1

pjdqj +

n∑

i=1

ϕidγi,

where, owing to (2.40) Sq(γ) := Sq(ξ · γ), ξ · γ(h) = h, for all (q; γ) ∈ U(Mnh ). For angle

coordinates ϕ ∈ Tnγ one obtains from (2.49) that

(2.50) ϕi = ∂Sq(γ)/∂γi

for all i = 1, n. As ϕi ∈ R/2πZ, i = 1, n, from (2.51) it follows that

(2.51)1

σ(h)j

dϕi = δij =1

∂γi

σ(h)j

dSq(γ) =1

∂γi

σ(h)j

< p, dq >

for all canonical cycles σ(h)j ⊂ Mn

h , j = 1, n, constituting a basis of the one dimensional

homology group H1(Mnh ; Z). Hence, owing to (2.51), it follows that for all i = 1, n ”action”

variables can be found as

(2.52) γi =1

σ(h)i

< p, dq >

23

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Recall now that Mnh ≃ T

nγ are diffeomorphic also to ⊗n

j=1S1j , where S

1j , j = 1, n, are some one-

dimensional real circles. The evolution along any of the vector fields Kj : T ∗(Rn)→ T (T ∗(Rn)),

j = 1, n, on Mnh ⊂ T ∗(Rn) is known [1, 16] to be a linear winding around the torus T

nγ , which

also can be interpreted in this way: the above introduced independent global coordinates on

circles S1j , j = 1, n, are such that the resulting evolution undergoes a quasi-periodic motion.

These coordinates are called Hamilton-Jacobi coordinates, and prove to be very important for

demonstrating the complete integrability by quadratures via solving the corresponding Picard-

Fuchs type equations. Let us denote these separable coordinates on the integral submanifold

Mnh ≃ ⊗n

j=1S1j by µj ∈ S

1j , j = 1, n, and define the corresponding embedding mapping

πh : Mnh → T ∗(Rn) as

(2.53) q = q(µ;h), p = p(µ;h).

There exist two important cases subject to the embedding (2.53).

The first case is related to the integral submanifold Mnh ⊂ T ∗(Rn) which can be parameter-

ized as a manifold using the base coordinates q ∈ Rn of the cotangent bundle T ∗(Rn). This

can be explained as follows: the canonical Liouville 1-form α(1) ∈ Λ1(Rn), in accordance with

the diagram

(2.54)T ∗( Mn

h ) ≃ T ∗(⊗nj=1S

1j)

π∗

←− T ∗(Rn)

pr ↓ pr ↓ pr ↓Mn

h ≃ ⊗nj=1S

1j

π→ Rn

is mapped by the embedding mapping π = pr · πh : Mnh → R

n not depending on a set of

parameters h ∈ Rn, into the 1-form

(2.55) α(1)h = π∗α(1) =

n∑

j=1

wj(µj ;h)dµj ,

where (µ,w) ∈ T ∗(⊗nj=1S

1j) ≃ ⊗n

j=1T∗(S1

j). The embedding mapping π : Mnh → R

n due to the

equality (2.55) makes the function Lµ : Rn → R to be zero giving rise to the generating function

Sµ : Rn → R, enjoying the condition

(2.56) dSµ = dSq|q=q(µ;h) ,

where as before

(2.57) Sµ(h) =

n∑

j=1

pjdqj +

n∑

j=1

tjdhj

and det ||∂q(µ;h)/∂µ|| 6= 0 almost everywhere on Mnh for all h ∈ R

n. Similarly to (2.48), one

gets from (2.57) that

(2.58) tj = ∂Sµ(h)/∂hj

24

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for j = 1, n. Concerning the second part of the embedding mapping (2.53) we arrive due to the

equality (2.55) at the following simple result:

(2.59) pi =

n∑

j=1

wj(µj ;h)∂µj/∂qi,

where i = 1, n and det ||∂µ/∂q|| 6= 0 almost everywhere on π(Mnh ) due to the local invertibility

of the embedding mapping π : Mnh → R

n. Thus, we can claim that the problem of complete

integrability in the first case is solved iff the only embedding mapping π : Mnh → R

n ⊂ T ∗(Rn)

is constructed. This case was considered in detail in [24], where the corresponding Picard-Fuchs

type equations were built based on a one extension of Galissot-Reeb and Francoise results [31].

Namely, similarly to (2.38), these equations are defined as follows:

(2.60) ∂wj(µj;h)/∂hk = Pkj(µj , wj ;h),

where Pkj : T ∗(⊗nj=1S

1j)×C

n → C, k, j = 1, n, are some algebraic functions of their arguments.

Concerning the second case when the integral submanifold Mnh ⊂ T ∗(Rn) cannot be embedded

almost everywhere into the base space Rn ⊂ T ∗(Rn), the relationship like (2.57) does not

take place, and we are forced to consider the usual canonical transformation from T ∗(Rn) to

T ∗(Rn) based on a mapping dLq : ⊗nj=1S

1j → T ∗(Rn), where Lq : ⊗n

j=1S1j → R enjoys for all

µ ∈ ⊗nj=1S

1j ⋍ Mn

h ∋ q the following relationship:

(2.61) pr∗α(1)(q; p) =n∑

j=1

wj dµj + dLq(µ).

In this case we can derive for any µ ∈ ⊗nj=1S

1j the hereditary generating function Lµ : R

n →T ∗(⊗n

j=1S1j) introduced before as

(2.62) dLµ = dLq|q=q(µ;h) ,

satisfying evidently the following canonical transformation condition:

(2.63) dSq(h) =n∑

j=1

wj( µj;h) dµj +n∑

j=1

tjdhj + dLµ(h),

for almost all µ ∈ ⊗nj=1S

1j and h ∈ R

n . Based on (2.63) one can derive the following relationships:

(2.64) ∂Lµ(h)/∂hj = < p, ∂q/∂hj >|Mn

h

for all j = 1, 2, µ ∈ ⊗nj=1S

1j and h ∈ R

n. Whence the following important analytical result

ts =

n∑

j=1

∫ µj

µ(0)j

(∂wj(λ;h)/∂hs)dλ,

n∑

j=1

pj(µ;h)(∂qj/∂µs) = ws + ∂Lµ(h)/∂µs(2.65)

holds for all s = 1, 2 and µ, µ(0) ∈ ⊗nj=1S

1j with parameters h ∈ Rn being fixed. Thereby, we

have found a natural generalization of the relationships (2.59) subject to the extended integral25

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submanifold embedding mapping πh : Mnh → T ∗(Rn) in the form (2.53). Assume now that

functions wj : C× Cn→ C , j = 1, n, satisfy, in general, Picard-Fuchs equations like (2.60),

having the following [17] algebraic solutions:

(2.66) wnj

j +

nj−1∑

k=0

cj,k(λ;h)wkj = 0,

where cj,k : C× Cn→ C , k = 0, nj − 1, j = 1, n, are some polynomials in λ ∈ C. Each

algebraic curve of (2.66) is known to be, in general, topologically equivalent due to the Riemann

theorem [33] to some Riemannian surface Γ(j)h , of genus gj ∈ Z+, j = 1, n. Thereby, one can

realize the local diffeomorphism ρ : Mnh → ⊗n

j=1Γ(j)h , mapping homology group basis cycles

σ(h)j ⊂ Mn

h , j = 1, n, into homology subgroup H1(⊗nj=1Γ

(j)h ; Z) basis cycles σj(Γh) ⊂ Γ

(j)h ,

j = 1, n, satisfying the following relationships:

(2.67) ρ( σ(h)j ) =

n∑

k=1

njk σk(Γh),

where njk ∈ Z, k = 1, j and j = 1, n, are some fixed integers. Based on (2.67) and (2.62)

one can write down, for instance, expressions (2.52) as follows:

(2.68) γi =1

n∑

j=1

nij

σj(Γh)wj(λ;h)dλ,

where i = 1, n. Subject to the evolution on Mnh ⊂ T ∗(Rn) one can easily obtain from (2.65)

that

(2.69) dti =n∑

j=1

(∂wj(µj ;h)/∂hi)dµj

at dhi = 0 for all i = 1, n, giving rise evidently to a global τ−parametrization of the set of

circles ⊗nj=1S

1j ⊂ ⊗n

j=1Γ(j)h , that is, one can define some inverse algebraic functions to Abelian

type integrals (2.62) as

(2.70) µ = µ(τ ;h),

where as before, τ = (t1, t2, ..., tn) ∈ Rn is a vector of evolution parameters. Recalling now

expressions (2.53) for integral submanifold mapping πh : Mnh → T ∗(Rn), one can write down a

mapping for evolutions on Mnh ⊂ T ∗(Rn), expressed by means of quadratures, as follows:

(2.71) q = q(µ(τ ;h)) = q(τ ;h), p = p(µ(τ ;h)) = p(τ ;h),

where obviously, a vector (q, p) ∈ T ∗(Rn) is quasi-periodic in each variable ti ∈ τ, i = 1, n.

Theorem 2.16. Every completely integrable Hamiltonian system admitting an algebraic sub-

manifold Mnh ⊂ T ∗(Rn) possesses a separable canonical transformation (2.63) which is described

by differential algebraic Picard-Fuchs type equations whose solution is a set of algebraic curves

(2.66).26

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Therefore, the main ingredient of this scheme of integrability by quadratures is finding the

Picard-Fuchs type equations (2.60) corresponding to the integral submanifold embedding map-

ping (2.53) depending in general on Rn ∋ h-parameters for the case when the integral subman-

ifold Mnh ⊂ T ∗(Rn) cannot be embedded into the base space R

n ⊂ T ∗(Rn) of the phase space

T ∗(Rn). From Theorem 2.13 one can find 1-forms h(1)j ∈ Λ1(T ∗(Rn)), j = 1, n, satisfying the

following identity on T ∗(Rn) :

(2.72) ω(2)(q, p) :=n∑

j=1

dpj ∧ dqj =n∑

j=1

dHj ∧ h(1)j .

The 1-forms h(1)j ∈ Λ1(T ∗(Rn)), j = 1, n, possess the important, previously stated property:

pulled-back to the integral submanifold (2.35) gives rise to the global linearization

(2.73) π∗hh

(1)j := h

(1)j = dtj

where h(1)j ∈ Λ1(Mn

h ), and πh∗d/dtj = Kj · πh for all j = 1, n. The expressions (2.73)

combined with (2.69) give rise directly to the following set of relationships

(2.74) h(1)j =

n∑

j=1

(∂wj(µj;h)/∂hi)dµj

at dhj = 0 for all j = 1, n on Mnh ⋍ ⊗n

j=1S1j ⊂ ⊗n

j=1Γ(j)h for all j = 1, n. Since we are

interested in the integral submanifold embedding mapping (2.50) being locally diffeomorphic in

a neighborhood U(Mnh ) ⊂ T ∗(Rn), the Jacobian det ||∂q(µ;h)/∂µ|| 6= 0 almost every where

in U(Mnh ). On the other hand, as it was proved in [2], the set of 1-forms h

(1)j ∈ Λ1(Mn

h ),

j = 1, n, can, in general, be represented in U(Mnh ) as

(2.75) h(1)j =

n∑

k=1

h(1)jk (q, p)dqk

∣∣∣Mn

h

,

where h(1)jk : T ∗(Rn) → R, k, j = 1, n, are some algebraic expressions of their arguments.

Thereby, one easily finds from (2.75) and (2.74) that

(2.76) ∂wi(µi;h)/∂hj =n∑

k=1

h(1)jk (q(µ;h), p(µ;h))(∂qk(µ;h)/∂µi)

for all i, j = 1, n. Since we have p-variables in (2.76), we must, owing to (2.65), use the

expressions

n∑

j=1

pj(µ;h)(∂qj/∂µs) = ws + ∂Lµ(h)/∂µs,(2.77)

∂Lµ(h)/∂hj = < p, ∂q/∂hj >|Mnh

,

which hold for s = 1, n and all µ ∈ ⊗nj=1Sj, h ∈ R

n in the neighborhood U(Mnh ) ⊂ T ∗(Rn)

chosen above. Thereby, we arrived at the following form of equations (2.76):

(2.78) ∂wi(µi;h)/∂hj = Pji(µ,w;h),27

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where for all i, j = 1, n expressions

(2.79) Pji(µ,w;h) :=n∑

k=1

h(1)jk (q(µ;h), p(µ;h))∂qk/∂µi)

depend correspondingly only on ⊗nj=1Γ

(j)h ∋ (µi, wi)- variables for each i ∈ 1, n and all

j = 1, n. This condition can obviously be expressed as follows:

(2.80) ∂Pji(µ,w;h)/∂µk = 0, ∂Pji(µ,w;h)/∂wk = 0

for j, i 6= k ∈ 1, n at almost all µ ∈ ⊗nj=1S

1j and h ∈ R

n. The set of conditions (2.76) gives

rise, in general, to a system of algebraic-differential equations subject to the embedding mapping

prπh : Mnh → R

n defined analytically by (2.53) and the generating function (2.62). As a result

of solving these equations, we obtain, evidently owing to (2.78) and (2.80), the following system

of Picard-Fuchs type equations:

(2.81) ∂wi(µi;h)/∂hj = Pji(µi, wi;h)

where, in general, mappings

(2.82) Pji : Γ(i)h × R

n → C

are algebraic expressions. Since the set of algebraic curves (2.66) must satisfy system (2.81), we

can retrieve it by means of solving the Picard-Fuchs type equations (2.81). These curves give

rise, owing to expressions (2.64) and (2.53), to the integrability of all flows on Mnh ⊂ T ∗(Rn) by

quadratures as described in what follows.

Theorem 2.17. Let there be given a completely integrable Hamiltonian system on the co-adjoint

manifold T ∗(Rn) whose integral submanifold Mnh ⊂ T ∗(Rn) is described by Picard-Fuchs type

algebraic equations (2.81). The corresponding embedding mapping πh : Mnh → T ∗(Rn) (2.53)

is a solution of a compatibility condition constrained by the differential-algebraic relationships

(2.80) on the canonical transformations generating function (2.62).

To show that the scheme described above really leads to an algorithmic procedure for con-

structing the Picard-Fuchs type equations (2.81) and the corresponding integral submanifold

embedding mapping πh : Mnh → T ∗(Rn) in the form (2.53), we apply it below to some Hamil-

tonian systems including a so-called truncated [43] Focker-Plank Hamiltonian system on the

canonically symplectic cotangent space T ∗(Rn). Making use of the representations (2.76) and

(2.77) and equation (2.80), we have shown above that the set of 1-forms (2.75) is reduced to

the following purely differential-algebraic relationships on M2nh,τ :

(2.83) ∂wi(µi;h)/∂hj = Pji(µi, wi;h),

generalizing similar ones of [31, 44], where the characteristic functions Pji : T ∗(Mnh ) → R,

i, j = 1, n, are defined as follows:

(2.84) Pji(µi, wi;h) := Pji(µ,w;h)∣∣Mn

h

.

28

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It is clear that the above set of purely differential-algebraic relationships (2.83) and (2.84) makes

it possible to write down explicitly some first order compatible differential-algebraic equations,

whose solutions yield the first half of the desired embedding (2.76) for the integral submanifold

Mnh ⊂M2n in an open neighborhood M2n

h,τ ⊂M2n. As a result of the above computations, one

can formulate the following main theorem.

Theorem 2.18. The embedding (2.76) for the integral submanifold Mnh ⊂ M2n (compact

and connected), parameterized by a regular parameter h ∈ G∗, is an algebraic solution (up to

diffeomorphism) to the set of characteristic Picard-Fuchs type equations (2.83 on T ∗(Mnh ), and

can be represented in general [32] in the following algebraic-geometric form:

(2.85) wnj

j +

n∑

s=1

cjs(λ;h)wnj−s

j = 0,

where cjs : R×G∗ → R, s, j = 1, n are algebraic expressions, depending only on the functional

structure of the original abelian Lie algebra G of invariants on M2n. In particular, if the right-

hand side of the characteristic equations (2.83) is independent of h ∈ G∗, then this dependence

will be linear in h ∈ G∗.

It should be noted here that some ten years ago an attempt was made in [31, 32] to describe

the explicit algebraic form of the Picard-Fuchs type equations (2.83) by means of straightforward

calculations for the well-known completely integrable Kowalewskaya top Hamiltonian system.

The idea suggested in [31, 32] was in some aspects very close to that devised independently and

thoroughly analyzed in [24] which did not consider the explicit form of the algebraic curves (2.85)

starting from an abelian Lie algebra G of invariants on a canonically symplectic phase space

M2n. As is well known, a set of algebraic curves (2.85), prescribed via the above algorithm, to

a given a priori abelian Lie algebra G of invariants on the canonically symplectic phase space

M2n = T ∗(Rn) can be realized by means of a set of nj−sheeted Riemannian surfaces Γnj

h , j =

1, n, covering the corresponding real-valued cycles S1j , j = 1, n, generating the corresponding

homology group H1(Tn; Z) of the Arnold torus T

n ≃ ⊗nj=1S

1j , which is diffeomorphic to the

integral submanifold Mnh ⊂M2n. Thus, upon solving the set of algebraic equations (2.83) with

respect to functions wj : S1j × G∗ → R, j = 1, n, from (2.65) one obtains a vector parameter

τ = (t1, ..., tn) ∈ Rn on Mn

h explicitly described by means of the following Abelian type

equations:

(2.86) tj =

n∑

s=1

∫ µs

µ0s

dλ ∂ws(λ;h)/∂hj =

=

n∑

s=1

∫ µs

µ0s

dλ Pjs(λ,ws;h),

where j = 1, n, (µ0;h) ∈ (⊗nj=1Γ

nj

h ) × G∗. Using expression (2.86) and recalling that the

generating function S : Mnh ×R

n → R is a one-valued mapping on an appropriate covering space29

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(Mnh ;H1(M

nh ; Z)), one can construct via the method of Arnold [2], the so-called action-angle

coordinates on Mnh . Denote the basic oriented cycles on Mn

h by σj ⊂Mnh , j = 1, n. These cycles

together with their duals generate homology group H1(Mnh ; Z) ≃ H1(T

n; Z) =⊕nj=1Zj. In virtue

of the diffeomorphism Mnh ≃ ⊗n

j=1S1j described above, there is a one-to-one correspondence

between the basic cycles of H1(Mnh ; Z) and those on the algebraic curves Γ

nj

h , j = 1, n, given

by (2.85):

(2.87) ρ : H1(Mnh ; Z)→ ⊕n

j=1Zjσh,j ,

where σh,j ⊂ Γnj

h , j = 1, n are the corresponding real-valued cycles on the Riemann surfaces

Γnj

h , j = 1, n. Assume that the following meanings of the mapping (2.87) are prescribed:

(2.88) ρ(σi) := ⊕nj=1nij σh,j

for each i = 1, n, where nij ∈ Z, i, j = 1, n - some fixed integers. Then following the Arnold

construction [2, 31], one obtains the set of so-called action-variables on Mnh ⊂M2n :

(2.89) γj: :=1

σj

dS =

n∑

s=1

njs

σh,s

dλ ws(λ;h),

where j = 1, n . It is easy to show [2, 29], that expressions (2.89) naturally define an a.e.

differentiable invertible mapping

(2.90) ξ : G∗ → Rn,

which enables one to treat the integral submanifold Mnh as a submanifold Mn

γ ⊂M2n, where

(2.91) Mnγ := u ∈M2n : ξ(h) = γ ∈ R

n.

But, as was demonstrated in [31, 45] the functions (2.89) do not, in general, generate a global

foliation of the phase space M2n, as they are connected with both topological and analytical

constraints. Since the functions (2.89) are evidently also commuting invariants on M2n, one can

define a further canonical transformation of the phase space M2n, generated by the following

relationship on M2nh,τ :

(2.92)n∑

j=1

wj dµj +n∑

j=1

ϕj dγj = dS(µ; γ),

where ϕ = (ϕ1, ..., ϕn) ∈ Tn are the so-called angle-variables on the torus T

n ≃ Mnh and

S : Mnγ × R

n → R is the corresponding generating function. Whence, it follows easily from

(2.89) and (2.92) that

(2.93) ϕj :=∂S(µ; γ)/∂γj =n∑

s=1

∂S(µ; γ(h))/∂hs ∂hs/∂γj =

=

n∑

s=1

tsωsj(γ),1

σj

dϕk =δjk ,

where Ω := ωsj : Rn → R, s, j = 1, n is the so-called [2] frequency matrix, which is

invertible on the integral submanifold Mnγ ⊂ M2n. As an evident result of (2.93), we claim

30

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that the evolution of any vector field Ka ∈ Γ(M2n) for a ∈ G on the integral submanifold

Mnγ ⊂M2n is quasi-periodic with a set of frequencies generated by the matrix Ω

a.e.∈ Aut(Rn),

defined above. As examples showing the effectiveness of the above method of construction of

integral submanifold embeddings for abelian integrable Hamiltonian systems, one can verify

the Liouville-Arnold integrability of all Henon-Heiles and Neumann type systems described in

detail in [34, 35]; however, we shall not dwell on this here.

2.4. Integral submanifold embedding problem for a nonabelian Lie algebra of in-

variants. We shall assume that there is given a Hamiltonian vector field K ∈ Γ(M2n) on the

canonically symplectic phase space M2n = T ∗(Rn), n ∈ Z+, which is endowed with a nonabelian

Lie algebra G of invariants, satisfying all the conditions of the Mishchenko-Fomenko Theorem

1.5, that is

(2.94) dimG+rankG = dimM2n .

Then, as was proved above, an integral submanifold M rh ⊂M2n at a regular element h ∈ G∗

is rankG =r−dimensional and diffeomorphic (when compact and connected) to the standard

r-dimensional torus Tr ≃ ⊗r

j=1S1j . It is natural to ask the following question: How does one

construct the corresponding integral submanifold embedding

(2.95) πh : M rh →M2n ,

which characterizes all possible orbits of the dynamical system K ∈ Γ(M2n)? Having gained

some experience in constructing the embedding (2.95) in the case of the abelian Liouville-Arnold

theorem on integrability by quadratures, we proceed below to study the integral submanifold

M rh ⊂M2n by means of Cartan’s theory [17, 25, 29, 35] of the integrable ideals in the Grassmann

algebra Λ(M2n). Let I(G∗) be an ideal in Λ(M2n),generated by independent differentials

dHj ∈ Λ1(M2n) , j = 1, k, on an open neighborhood U(M rh), where by definition, r = dimG. The

ideal I(G∗) is obviously Cartan integrable [36, 29] with the integral submanifold M rh ⊂ M2n

(at a regular element h ∈ G∗), on which it vanishes, that is π∗h I(G

∗) = 0. The dimension

dim M rh = dim M2n- dimG = r = rankG due to the condition (2.94) imposed on the Lie

algebra G. It is useful to note here that owing to the inequality r ≤ k for the rank G , one

readily obtains from (2.94) that dimG =k ≥ n. Since each base element Hj ∈ G, j = 1, k,

generates a symplectically dual vector field Kj ∈ Γ(M2n), j = 1, k, one can try to study

the corresponding differential system K(G), which is also Cartan integrable on the entire open

neighborhood U(M rh) ⊂M2n. Denote the corresponding dimension of the integral submanifold

by dim Mkh = dim K(G) =k. Consider now an abelian differential system K(Gh) ⊂ K(G),

generated by the Cartan subalgebra Gh ⊂ G and its integral submanifold M rh ⊂ U(M r

h). Since

the Lie subgroup Gh = expGh acts on the integral submanifold M rh invariantly and dim M r

h

= rank G =r, it follows that M rh = M r

h. On the other hand, the system K(Gh) ⊂ K(G)by definition, meaning that the integral submanifold M r

h is an invariant part of the integral31

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submanifold Mkh ⊂ U(M r

h) with respect to the Lie group G = expG - action on Mkh . In this

case, one has the following result.

Lemma 2.19. There exist just (n-r)∈ Z+ vector fields Fj ∈ K(G)/K(Gh), j = 1, n − r, for

which

(2.96) ω(2)(Fi, Fj) = 0

on U(M rh) for all i, j = 1, n − r.

Proof. Obviously, the matrix ω(K) := ω(2)(Ki, Kj) : i, j = 1, k has on U(M rh) the rank

ω(K) = k− r, since dimR ker(π∗hω(2)) = dimR(πh∗ K(Gh)) = r on M r

h at the regular element

h ∈ G∗. Now complexify the tangent space T (U(M rh)) using its even dimensionality. Whence,

one can easily deduce that on U(M rh) there exist just (n−r) ∈ Z+ vectors (not vector fields!)

KCj ∈ KC(G)/KC(Gh), j = 1, n− r, from the complexified [37] factor space KC(G)/KC(Gh).

To show this, let us reduce the skew-symmetric matrix ω(K) ∈ Hom(Rk−r) to its selfadjoint

equivalent ω(KC) ∈ Hom(Cn−r), having taken into account that dimR Rk−r = dimR R

k+r−2r =

dimR R2(n−r) = dimC C

n−r. Let fCj ∈ C

n−r, j = 1, n − r, be eigenvectors of the nondegenerate

selfadjoint matrix ω(KC) ∈ Hom(Cn−r), that is

(2.97) ω(KC) fCj = λjf

Cj ,

where λj ∈ R, j = 1, n− r, and for all i, j = 1, n − r, < fCi , fC

j >= δi,j . The above

obviously means that with respect to the basis fCj ∈ KC(G)/KC(Gh) : j = 1, n − r the

matrix ω(KC) ∈ Hom(Cn−r) is strictly diagonal and representable as

(2.98) ω(KC) =n−r∑

j=1

λjfCj ⊗C fC

j ,

where ⊗C is the usual Kronecker tensor product of vectors from Cn−r. Owing to the construction

of the complexified matrix ω(KC) ∈ Hom(Cn−r), one sees that the space KC(G)/KC(Gh) ≃C

n−r carries a Kahler structure [37] with respect to which the following expressions

(2.99) ω(K) = Imω(K), < ·, · >R= Re < ·, · >

hold. Making use now of the representation (2.98) and expressions (2.99), one can find vector

fields Fj ∈ K(G)/K(Gh), j = 1, n− r, such that

(2.100) ω(F ) = Imω(FC) = J,

holds on U(M rh) , where J ∈ Sp(Cn−r) is the standard symplectic matrix, satisfying the complex

structure [37] identity J2 = −I. In virtue of the normalization conditions < fCj , fC

j >= δi,j , for

all i, j = 1, n− r, one readily infers from (2.100) that ω(2)(Fi, Fj) = 0 for all i, j = 1, n− r,

where by definition

(2.101) Fj := ReFCj

for all j = 1, n− r, and this proves the lemma.32

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Assume now that the Lie algebra G of invariants on M2n has been split into a direct sum

of subspaces as

(2.102) G = Gh ⊕ Gh ,

where Gh is the Cartan subalgebra at a regular element h ∈ G∗ (being abelian) and Gh ≃ G/Gh is the corresponding complement to Gh. Denote a basis of Gh as Hi ∈ Gh : i = 1, r,where dimGh = rankG = k ∈ Z+, and correspondingly, a basis of Gh as Hj ∈ Gh ≃ G/Gh :

j = 1, k − r. Then, owing to the results obtained above, the following hold:

(2.103) Hi, Hj = 0, h(Hi, Hs) = 0

on the open neighborhood U(M rh) ⊂M2n for all i, j = 1, r and s = 1, k − r. As yet we have

not mentioned anything about expressions h(Hs, Hm) for s,m = 1, k − r. Making use of the

representation (2.101) for our vector fields (if they exist) Fj ∈ K(G)/K(Gh), j = 1, n − r, one

can write down the following expansion:

(2.104) Fi =k−r∑

j=1

cji(h)Kj ,

where iKjω(2) := −dHj, cji : G∗ → R, i = 1, n− r, j = 1, k − r, are real-valued functions

on G∗, being defined uniquely as a result of (2.104). Whence it clearly follows that there exist

invariants fs : U(M rh)→ R, s = 1, n− r, such that

(2.105) −iFsω(2) =

k−r∑

j=1

cjs(h) dHj := dfs ,

where fs =∑k−r

j=1 cjs(h)Hj , s = 1, n − r, holds on U(M rh).

To proceed further, let us look at the following identity which is similar to (2.27):

(2.106) (⊗rj=1iKj

)(⊗n−rs=1 i

Fs)(ω(2))n+1 = 0 = ±(n + 1)!(∧r

j=1dHj)(∧n−rs=1d fs) ∧ ω(2),

on U(M rh). Consequently, the following result is easily obtained using Cartan theory [17, 29]:

Lemma 2.20. The symplectic structure ω(2) ∈ Λ2(U(M rh)) has the following canonical repre-

sentation:

(2.107) ω(2)∣∣∣U(Mr

h)=

r∑

j=1

dHj ∧ h(1)j +

n−r∑

s=1

dfs ∧ h(1)s ,

where h(1)j , h

(1)s ∈ Λ1(U(M r

h)), j = 1, r, s = 1, n − r.

The expression (2.107) obviously means, that on U(M rh) ⊂ M2n, the differential 1-forms

h(1)j , h

(1)s ∈ Λ1(U(M r

h)), j = 1, r, s = 1, n− r, are independent together with exact 1-forms

dHj , j = 1, r, and dfs, s = 1, n − r. Since dω(2) = 0 on M2n identically, from (2.107) one

obtains that the differentials dh(1)j , dh

(1)s ∈ Λ2(U(M r

h)), j = 1, r, s = 1, n − r, belong to the

ideal I( Gh) ⊂ I(G∗), generated by exact forms dfs, s = 1, n− r, and dHj , j = 1, r, for all

regular h ∈ G∗. Consequently, one obtains the following analogue of the Galissot-Reeb Theorem

2.13.33

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Theorem 2.21. Let a Lie algebra G of invariants on the symplectic space M2n be nonabelian

and satisfy the Mishchenko-Fomenko condition (2.94). At a regular element h ∈ G∗ on some

open neighborhood U(M rh) of the integral submanifold M r

h ⊂ M2n there exist differential

1-forms h(1)j , j = 1, n, and h

(1)s , s = 1, n − r, satisfying the following properties: i)

ω(2)∣∣U(Mr

h)=

∑rj=1 dHj∧ h

(1)j +

∑n−rs=1 dfs∧ h

(1)s where Hj ∈ G, j = 1, r, is a basis of the Cartan

subalgebra Gh ⊂ G (being abelian), and fs ∈ G, s = 1, n − r, are invariants from the

complementary space Gh ≃ G/Gh; ii) 1-forms h(1)j ∈ Λ1(U(M r

h)), j = 1, r, and h(1)s

∈ Λ1(U(M rh)), s = 1, n − r, are exact on Mr

h and satisfy the equations: h(1)j (Ki) = δi,j

for all i, j = 1, r, h(1)j (Fs) = 0 and h

(1)s (Kj) = 0 for all j = 1, r, s = 1, n− r, and

h(1)s (Fm) = δs,m for all s,m = 1, n− r.

Proof. Obviously we need to prove only the last statement ii). Making use of Theorem 2.21,

one finds on the integral submanifold M rh ⊂M2n the differential 2-forms dh

(1)j ∈ Λ2(U(M r

h)),

j = 1, r, and dh(1)s ∈ Λ2(U(M r

h)), s = 1, n− r, are identically vanishing. This means in

particular, owing to the classical Poincare lemma [1, 2, 29], that there exist some exact 1-forms

dth,j ∈ Λ1(U(M rh)), j = 1, r, and dth,s ∈ Λ1(U(M r

h)), s = 1, n − r, where th,j : M rh → R,

j = 1, r, and th,s : M rh → R, s = 1, n − r, are smooth independent almost everywhere

functions on M rh; they are one-valued on an appropriate covering of the manifold M r

h ⊂ M2n

and supply global coordinates on the integral submanifold M rh. Using the representation (2.107),

one can easily obtain that

(2.108) − i Kiω(2)

∣∣∣U(Mr

h)=

r∑

j=1

dHj h(1)j ( Ki) +

n−r∑

s=1

dfs h(1)s (Ki) = dHi

for all i = 1, r and

(2.109) − iFmω(2)

∣∣∣U(Mr

h)=

r∑

j=1

dHj h(1)j ( Fm) +

n−r∑

s=1

dfs h(1)s (Fm) = dfm

for all m = 1, n − r. Whence, from (2.108) it follows on that on U(M rh) ,

(2.110) h(1)j ( Ki) = δi,j, h(1)

s (Ki) = 0

for all i, j = 1, r and s = 1, n − r, and similarly, from (2.109) it follows that on U(M rh),

(2.111) h(1)j ( Fm) = 0, h(1)

s (Fm) = 0

for all j = 1, r and s,m = 1, n − r. Thus the theorem is proved. Having now defined global

evolution parameters tj : M2n → R, j = 1, r, of the corresponding vector fields Kj = d/dtj ,

j = 1, r, and local evolution parameters ts : M2n ∩ U(M rh) → R, s = 1, n − r, of the

corresponding vector fields Fs

∣∣∣U(Mr

h)

:= d/dts, s = 1, n− r, one can easily see from (2.111)

that the equalities

(2.112) tj|U(Mrh) = tj, ts

∣∣U(Mr

h)

= th,s

34

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hold for all j = 1, r, s = 1, n− r, up to constant normalizations. Thereby, one can develop a

new method, similar to that discussed above, for studying the integral submanifold embedding

problem in the case of the nonabelian Liouville-Arnold integrability theorem. Before starting,

it is interesting to note that the system of invariants

Gτ := Gh ⊕ spanRfs ∈ G/Gh : s = 1, n− r

constructed above, comprise of a new involutive (abelian) complete algebra Gτ , to which one

can apply the abelian Liouville-Arnold theorem on integrability by quadratures and the integral

submanifold embedding theory devised above, in order to obtain exact solutions by means of

algebraic-analytical expressions. Namely, the following corollary holds.

Corollary 2.22. Assume that a nonabelian Lie algebra of invariants G satisfies the Mishchenko-

Fomenko (2.94) with M rh ⊂ M2n being its integral submanifold (compact and connected) at a

regular element h ∈ G∗, diffeomorphic to the standard torus Tr ≃ Mn

h,τ . Assume also that

the dual complete abelian algebra Gτ (dimGτ = n = 1/2 dim M2n) of independent invariants

constructed above is globally defined. Then its integral submanifold Mnh,τ ⊂M2n is diffeomorphic

to the standard torus Tn ≃ Mn

h,τ , and contains the torus Tr ≃ M r

h as a direct product with

some completely degenerate torus Tn−r, that is, Mn

h,τ ≃M rh × T

n−r.

Having successfully applied the algorithm developed above to the algebraic-analytical char-

acterization of integral submanifolds of a nonabelian Liouville -Arnold integrable Lie algebra Gof invariants on the canonically symplectic manifold M2n ≃ T ∗(Rn), one can produce a wide

class of exact solutions represented by quadratures - which is just what we set out to find. At

this point it is necessary to note that up to now the (dual to G) abelian complete algebra Gτ of

invariants at a regular h ∈ G ∗ was constructed only on some open neighborhood U(M rh) of

the integral submanifold M rh ⊂M2n. As was mentioned before, the global existence of the alge-

bra Gτ strongly depends on the possibility of extending these invariants to the entire manifold

M2n. This possibility is in one-to-one correspondence with the existence of a global complex

structure [37] on the reduced integral submanifold M2(n−r)h,τ := Mk

h/Gh, induced by the reduced

symplectic structure π∗τ ω(2) ∈ Λ2(Mk

h/Gh), where πτ : Mkh → M2n is the embedding for the

integrable differential system K(G) ⊂ Γ(M2n), introduced above. If this is the case, the resulting

complexified manifold CMn−rh,τ ≃ M

2(n−r)h,τ will be endowed with a Kahlerian structure, which

makes it possible to produce the dual abelian algebra Gτ as a globally defined set of invariants

on M2n. This problem will be analyzed in more detail below.

2.5. Examples. We now consider some examples of nonabelian Liouville-Arnold integrability

by quadratures covered by Theorem 1.7.

Example 2. Point vortices in the plane.35

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Consider n ∈ Z+ point vortices on the plane R2, which were thoroughly analyzed recently

in [3] and are described by the Hamiltonian function

(2.113) H = − 1

n∑

i6=j=1

ξiξj ln ‖qi − pj‖

with respect to the following partially canonical symplectic structure on M2n ≃ T ∗(Rn) :

(2.114) ω(2) =n∑

j=1

ξjdpj ∧ dqj ,

where (pj , qj) ∈ R2, j = 1, n, are coordinates of the vortices in R

2. There exist three additional

invariants

(2.115) P1 =

n∑

j=1

ξjqj, P2 =

n∑

j=1

ξjpj,

P =1

2

n∑

j=1

ξj(q2j + p2

j),

satisfying the following Poisson bracket s:

(2.116) P1, P2 = −n∑

j=1

ξj , P1, P = −P2, P2, P = P1 ,

P,H = 0 = Pj ,H .

It is evident, that invariants (2.113) and (2.115) comprise on∑n

j=1 ξj = 0 a four-dimensional

Lie algebra G , whose rank G = 2. Indeed, assume a regular vector h ∈ G∗ is chosen, and

parameterized by real values hj ∈ R, j = 1, 4, where

(2.117) h(Pi) = hi, h(P ) = h3, h(H) = h4 ,

and i = 1, 2. Then, one can easily verify that the element

(2.118) Qh = (n∑

j=1

ξj)P −n∑

i=1

hiPi

belongs to the Cartan Lie subalgebra Gh ⊂ G, that is

(2.119) h(Qh, Pi) = 0, h(Qh, P) = 0 .

Since Qh,H = 0 for all values h ∈ G∗, we claim that Gh = spanRH,Qh is the Cartan

subalgebra of G. Thus, rankG = dimGh = 2, and one immediately has that the (2.94)

(2.120) dim M2n = 2n = rankG+ dimG = 6

holds only if n = 3. Thus, the following theorem is proved.

Theorem 2.23. The three-vortex problem (2.113) on the plane R2 is nonabelian Liouville-

Arnold integrable by quadratures on the phase space M6 ≃ T ∗(R3) with the symplectic structure

(2.114).36

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As a result, the corresponding integral submanifold M2h ⊂ M6 is two-dimensional and

diffeomorphic (when compact and connected) to the torus T2 ≃M2

h , on which the motions are

quasi-periodic functions of the evolution parameter. Concerning Corollary 2.22, the dynamical

system (2.113) is also abelian Liouville-Arnold integrable with an extended integral submanifold

M3h,τ ⊂M6, which can be found via the scheme suggested above. Using simple calculations, one

obtains an additional invariant Q = (∑3

j=1 ξj)P −∑3

i=1 P 2i /∈ G, which commutes with H and

P of Gh. Therefore, there exists a new complete dual abelian algebra Gτ = spanR Q,P,Hof independent invariants on M6 with dimGτ = 3 = 1/2 dim M6, whose integral submanifold

M3h,τ ⊂ M6 (when compact and connected) is diffeomorphic to the torus T

3 ≃ M2h × S

1. Also

note here, that the above additional invariant Q ∈ Gτ can be naturally extended to the case of

an arbitrary number n ∈ Z+ of vortices as follows: Q = (∑n

j=1 ξj)P −∑n

i=1 P 2i ∈ Gτ , which

obviously also commutes with invariants (2.113) and (2.115) on the entire phase space M2n.

Example 3. A material point motion in a central field.

Consider the motion of a material point in the space R3 under a central potential field

whose Hamiltonian

(2.121) H =1

2

3∑

j=1

p2j + Q(‖q‖),

contains a central field Q : R+ → R . The motion takes place in the canonical phase space

M6 = T ∗(R3), and possesses three additional invariants:

(2.122) P1 = p2q3 − pq, P2 = p3q1 − p1q3, P3 = p1q2 − p2q1,

satisfying the following Poisson bracket relations:

(2.123) P1, P2 = P3 , P3, P1 = P2 , P2, P3 = P1.

Since H,Pj = 0 for all j = 1, 3, one sees that the problem under consideration has a

four-dimensional Lie algebra G of invariants, isomorphic to the classical rotation Lie algebra

so(3) × R ≃ G. Let us show that at a regular element h ∈ G∗ the Cartan subalgebra Gh⊂ Ghas the dimension dimGh = 2 = rankG . Indeed, one easily verifies that the invariant

(2.124) Ph =3∑

j=1

hjPj

belongs to the Cartan subalgebra Gh, that is

(2.125) H,Ph = 0, h(Ph, Pj) = 0

for all j = 1, 3. Thus, as the Cartan subalgebra Gh = spanRH and Ph ⊂ G, one gets dimGh

= 2=rankGh, and the Mishchenko-Fomenko (2.9)

(2.126) dimM6 = 6 = rankG + dimG = 4 + 2

holds. Hence, one can prove its integrability by quadratures via the nonabelian Liouville-Arnold

Theorem 1.7 and obtain the following theorem.37

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Theorem 2.24. It follows from Theorem 1.5 that the free material point motion in R3 is a

completely integrable by quadratures dynamical system on the canonical symplectic phase space

M6 = T ∗(R3). The corresponding integral submanifold M2h ⊂ M6 at a regular element

h ∈ G∗ (if compact and connected) is two-dimensional and diffeomorphic to the standard torus

T2 ≃M2

h .

Making use of the integration algorithm devised above, one can readily obtain the correspond-

ing integral submanifold embedding mapping πh : M2h →M6 by means of algebraic-analytical

expressions and transformations. There are clearly many other interesting nonabelian Liouville-

Arnold integrable Hamiltonian systems on canonically symplectic phase spaces that arise in

applications, which can similarly be integrated using algebraic-analytical means. We hope to

study several of these systems in detail elsewhere.

2.6. Existence problem for a global set of invariants. It was proved above, that locally,

in some open neighborhood U(M rh) ⊂ M2n of the integral submanifold M r

h ⊂ M2n one

can find by algebraic-analytical means just n − r ∈ Z+ independent vector fields F j ∈

K(G)/K(Gh) ∩ Γ(U(M rh), j = 1, n − r, satisfying the (2.96). Since each vector field Fj ∈

K(G)/K(Gh), j = 1, n− r, is generated by an invariant Hj∈ D( U(M rh)), j = 1, n − r, it

follows readily from (2.96) that

(2.127) Hi, Hj = 0

for all i, j = 1, n − r. Thus, on an open neighborhood U(M rh) there exist just n−r invariants in

addition to Hj∈ D( U(M rh)), j = 1, n − r, all of which are in involution. Denote, as before, this

new set of invariants as Gτ , keeping in mind that dimGτ = r+(n−r) = n ∈ Z+. Whence, on an

open neighborhood U(M rh) ⊂M2n we have constructed the set Gτ of just n = 1/2 dim M2n

invariants commuting with each other, thereby guaranteeing via the abelian Liouville-Arnold

theorem, its local complete integrability by quadratures. Consequently, there exists locally a

mapping πτ : Mkh,τ → M2n, where Mk

h,τ := U(M rh) ∩Mk

τ is the integral submanifold of the

differential system K(G), and one can therefore describe the behavior of integrable vector fields

on the reduced manifold M2(n−r)h,τ := Mk−r

h,τ /Gh. For global integrability properties of a given

set G of invariants on (M2n, ω(2)), satisfying the Mishchenko-Fomenko (2.9), it is necessary

to have the additional set of invariants Hj∈ D( U(M rh)), j = 1, n− r, extended from U(M r

h)

to the entire phase space M2n. This problem clearly depends on the existence of extensions of

vector fields Fj ∈ Γ(U(M rh)), j = 1, n − r, from the neighborhood U(M r

h) ⊂M2n to the whole

phase space M2n . On the other hand, as stated above, the existence of such a continuation

depends intimately on the properties of the complexified differential system KC(G)/KC(Gh),

which has a nondegenerate complex metric ω(KC) : T (M2(n−r)h,τ )C×T (M

2(n−r)h,τ )C → C, induced

by the symplectic structure ω(2) ∈ Λ2(M2n). This point can be clarified by using the notion

[37, 38, 39, 40] of a Kahler manifold and some of the associated constructions presented above.38

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Namely, consider the local isomorphism T (M2(n−r)h,τ )C ≃ T (CMn−r

h,τ ), where CMn−rh,τ is the

complex (n − r)-dimensional local integral submanifold of the complexified differential system

KC(G)/KC(Gh). This means that the space T (M2(n−r)h,τ ) is endowed with the standard almost

complex structure

(2.128) J : T (M2(n−r)h,τ )→ T (M

2(n−r)h,τ ), J2 = −1,

such that the 2-form ω(K) := Imω(KC) ∈ Λ2(M2(n−r)h,τ ) induced from the above metric on

T (CMn−rh,τ ) is closed, that is, dω(K) = 0. If this is the case, the almost complex structure on the

manifold T (M2(n−r)h,τ ) is said to be integrable. Define the proper complex manifold CMn−r

h,τ ,

on which one can then define global vector fields Fj ∈ K(G)/K(Gh), j = 1, n − r, which are

being sought for the involutive algebra Gτ of invariants on M2n to be integrable by quadratures

via the abelian Liouville-Arnold theorem. Whence, the following theorem can be obtained.

Theorem 2.25. A nonabelian set G of invariants on the symplectic space M2n ≃ T ∗(Rn),

satisfying the Mishchenko-Fomenko (2.9), admits algebraic-analytical integration by quadratures

for the integral submanifold embedding πh : M rh → M2n, if the corresponding complexified

reduced manifold CMn−rh,τ ≃ M

2(n−r)h,τ = Mk−r

h,τ /Gh of the differential system KC(G)/KC(Gh)

is Kahlerian with respect to the standard almost complex structure (2.128) and the nondegenerate

complex metric ω(KC) : T (M2(n−r)h,τ )C×T (M

2(n−r)h,τ )C → C, induced by the symplectic structure

ω(2) ∈ Λ2(M2n) is integrable, that is dImω(KC) = 0.

Theorem 2.25 shows, in particular, that nonabelian Liouville-Arnold integrability by quadra-

tures does not in general imply integrability via the abelian Liouville-Arnold theorem; it actually

depends on certain topological obstructions associated with the Lie algebra structure of invari-

ants G on the phase space M2n. We hope to explore this intriguing problem in another work.

2.7. Supplement. Here we consider some examples of investigations of integral submanifold

embedding mappings for abelian Liouville-Arnold integrable Hamiltonian systems on T ∗(R2).

The Henon-Heiles system.

This flow is governed by the Hamiltonian

(2.129) H1 =1

2p21 +

1

2p22 + q1q

22 +

1

3q31

on the canonically symplectic phase space M4 = T ∗(R2) with the symplectic structure

(2.130) ω(2) =

2∑

j=1

dpj ∧ dqj.

As is well known, there exists the following additional invariant that commutes with that of

(2.129):

(2.131) H2 = p1p2 + 1/3q32 + q2

1q2,39

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that is H1,H2 = 0 on the entire space M4. Take a regular element h ∈ G := Hj : M4 → R:

j = 1, 2, with fixed values h(Hj) = hj ∈ R, j = 1, 2. Then the integral submanifold

(2.132) M2h := (q, p) ∈M4 : h(Hj) = hj ∈ R, j = 1, 2,

if compact and connected, is diffeomorphic to the standard torus T2 ≃ S

1 × S1 owing to the

Liouville-Arnold theorem, and one can find cyclic (separable) coordinates µj ∈ S1, j = 1, 2, on

the torus such that the symplectic structure (2.130) takes the form:

(2.133) ω(2) =

2∑

j=1

dwj ∧ dµj ,

where the conjugate variables wj ∈ T ∗(S1), j = 1, 2, on M2h depend only on the corresponding

variables µj ∈ S1j , j = 1, 2. In this case it is evident that the evolution along M2

h will be

separable and representable by means of quasi-periodic functions of the evolution parameters.

To show this, recall that the fundamental determining equations (2.72) based on the 1-forms

h(1)j ∈ Λ(M2

h), j = 1, 2, satisfy the identity

(2.134)

2∑

j=1

dHj ∧j h(1)j =

2∑

j=1

dpj ∧ dqj .

Here

(2.135) h(1)j =

2∑

k=1

hjk(q, p)dqk,

where j = 1, 2. Substituting (2.135) into (2.134), one obtains

(2.136) h(1)1 =

p1dq1

p21 − p2

2

+p2dq2

p21 − p2

2

, h(1)2 =

p2dq1

p221 − p2

1

+p1dq2

p21 − p2

2

.

On the other hand, the following implication holds on M2h ⊂M4 :

(2.137) α(1)h =

2∑

j=1

wj(µj;h)dµj ⇒2∑

j=1

pjdqj : = α(1),

where we have assumed that the integral submanifold M2h admits the local coordinates in the

base manifold R2 endowed with the canonical 1-form α

(1)h ∈ Λ(M2

h) as given in (2.137). Thus,

making use of the embedding πh : M2h → T ∗(R2) in the form

(2.138) qj = qj(µ;h) , pj = pj(µ;h) ,

j = 1, 2, one readily finds that

(2.139) pj =

2∑

k=1

wk(µk;h) ∂µk/∂qj

hold for j = 1, 2 on the entire integral submanifold M2h . Substituting (2.139) into (2.136) and

using the characteristic relationships (2.83), one obtains after simple, albeit tedious, calculations

the following differential-algebraic expressions:

(2.140) ∂q1/∂µ1 − ∂q2/∂µ1 = 0, ∂q1/∂µ2 + ∂q2/∂µ2 = 0,40

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whose simplest solutions are

(2.141) q1 = (µ1 + µ2)/2 , q2 = (µ1 − µ2)/2 .

Using expressions (2.139), one finds that

(2.142) p1 = w1 + w2 , p2 = w1 − w2 ,

where

(2.143) w1 =√

h1 + h2 − 4/3µ31, w2 =

√h1 − h2 − 4/3µ3

2.

Consequently, one obtains the separable [28] Hamiltonian functions (2.129) and (2.131) in a

neighborhood of the cotangent space T ∗(M2h) :

(2.144) h1 =1

2w2

1 +1

2w2

2 +2

3(µ3

1 + µ32), h2 =

1

2w2

1 −1

2w2

2 +2

3(µ3

1 − µ32),

which generate the following separable motions on M2h ⊂ T ∗(R2) :

(2.145) dµ1/dt := ∂h1/∂w1 =√

h1 + h2 − 4/3µ31,

dµ2/dt := ∂h1/∂w2 =√

h1 − h2 − 4/3µ32

for the Hamiltonian (2.129), and

dµ1/dx := ∂h2/∂w1 =√

h1 + h2 − 4/3µ31,

dµ2/dt := ∂h1/∂w2 = −√

h1 − h2 − 4/3µ32

for the Hamiltonian (2.131), where x, t ∈ R are the corresponding evolution parameters.

Analogously, one can show that there exists [41, 42] an integral submanifold embedding similar

to (2.141) and (2.142) for the following integrable modified Henon-Heiles involutive system:

(2.146) H1 =1

2p21 +

1

2p22 + q1q

22 +

16

3q31,

H2 = 9p42 + 36q1p

22q

22 − 12p1p2q

32 − 2q4

2(q22 + 6q2

1) ,

where H1,H1 = 0 on the entire phase space M4 = T ∗(R2). Based on considerations similar

to the above, one can deduce the following [42] expressions:

(2.147) q1 = −1

4(µ1 + µ2)−

3

8(w1 + w2

µ1 − µ2)2,

q22 = −2

√h2/(µ1 − µ2), w1 =

√2/3µ3

1 − 4/3√

h2 − 8h1 ,

p1 =1

2√−6(µ1 + µ2 + 4q1)

[−2√

h2

µ1 − µ2− µ1µ2 + 4(µ1 + µ2)q1 + 32q2

1 ],

p2 =√

h2(µ1+µ2 + 4q1)/(3(µ1−µ2)) , w2 =

√2/3µ3

2 + 4/3√

h2 − 8h1 ,

thereby solving explicitly the problem of finding the corresponding integral submanifold embed-

ding πh : M2h → T ∗(R2) that generates separable flows in the variables (µ ,w) ∈ T ∗(M2

h).

41

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A truncated four-dimensional Focker-Plank Hamiltonian system on T ∗(R2) and

its integrability by quadratures.

Consider the following dynamical system on the canonically symplectic phase space T ∗(R2) :

(2.148)dq1/dt = p1 + α(q1 + p2)(q2 + p1), dq2/dt = p2,dp1/dt = −(q1 + p2)− α[q2p1 + 1/2(p2

1 + p22 + q2

2)],dp2/dt = −(q2 + p1),

= K1(q, p),

where K1 : T ∗(R2) → T (T ∗(R2)) is the corresponding vector field on T ∗(R2) ∋ (q, p), t ∈ R

is an evolution parameter, called a truncated four-dimensional Focker-Plank flow. It is easy to

verify that functions Hj : T ∗(R2)→ R, j = 1, 2, where

(2.149) H1 = 1/2(p21 + p2

2 + q21) + q1p2 + α(q1 + p2)[q2p1 + 1/2(p2

1 + p22 + q2

2)]

and

(2.150) H2 = 1/2(p21 + p2

2 + q22) + q2p1

are functionally independent invariants with respect to the flow (2.94). Moreover, the invariant

(2.149) is the Hamiltonian function for (2.148); that is, the relationship

(2.151) iK1ω(2) = −dH1

holds on T ∗(R2), where the symplectic structure ω(2) ∈ Λ2(T ∗(R2)) is given as follows:

(2.152) ω(2) := d(pr∗α(1)) =

2∑

j=1

dpj ∧ dqj ,

with α(1) ∈ Λ1(R2) the canonical Liouville form on R2 :

(2.153) α(1)(q; p) =

2∑

j=1

pjdqj

for any (q, p) ∈ T ∗(R2) ≃ Λ1(R2). The invariants (2.149) and (2.150) obviously commute with

respect to the associated Poisson bracket on T ∗(R2) :

(2.154) H1,H2 = 0.

Therefore, owing to the abelian Liouville-Arnold theorem [1, 16], the dynamical system (2.148)

is completely integrable by quadratures on T ∗(R2), and we can apply the scheme devised above

to the commuting invariants (2.149) and (2.150) subject to the symplectic structure (2.152).

One easily calculates that

(2.155) ω(2) =2∑

i=1

dHi ∧ h(1)i ,

where the corresponding 1-forms π∗hh

(1)i := h

(1)i ∈ Λ1(M2

h), i = 1, 2, are given as

(2.156)

h(1)1 = p2dq1−(p1+q2)dq2

p1p2−(p1+q2)(q1+p2)−αh2(p1+q2),

h(1)2 = −[(q1+p2)(1+αp2)+αh2]dq1+(p1+α[h2+(q2+p1)( q1+p2)])dq2

p1p2−(q2+p1)(αh2+ q1+p2),

42

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and an invariant submanifold M2h ⊂ T ∗(R2) is defined as

(2.157) M2h := (q, p) ∈ T ∗(R2) : Hi(q, p) = hi ∈ R,i = 1, 2

for some parameters h ∈ R2. Based now on expressions (2.157) and (2.63), one can easily

construct functions Pij(w;h), i, j = 1, 2, in (2.78), defined on T ∗(M2h) ≃ T ∗(⊗2

j=1S1j) subject

to the integral submanifold embedding mapping πh : M2h → T ∗(R2) in coordinates µ ∈

⊗2j=1S

1j ⊂ ⊗2

j=1Γ(j)h , which we shall not write down in detail due to their length and cumbersome

form. Then applying the criterion (2.34), we obtain the following compatibility relationships

subject to the mappings q : (⊗2j=1S

1j)× R

2 → R2 and p : (⊗2

j=1S1j)× R

2 → T ∗q (R2) :

(2.158)

∂q1/∂µ1 − ∂q2/∂µ2 = 0, w1∂Lµ/∂w1 − w2∂Lµ/∂w2 = 0,

∂2q1/∂µ2∂h2 + ∂2w2/∂µ2∂h2 = 0,

∂w1/∂h1(∂q1/∂h1) = ∂w2/∂h1(∂q2/∂h1),

w1∂w1/∂h1 −w2∂w2/∂h2 = 0,

∂(w1∂w1/∂h2)/∂h2 − α2∂q1/∂µ1 = 0, ...

and so on, subject to variables µ ∈ ⊗2j=1S

1j and h ∈ R

2. Solving equations like (2.158), one

immediately obtains the expressions

(2.159)

p1 = w1, p2 = w2,q1 = c1 + µ1 −w2(µ2;h),q2 = c2 + µ2 −w1(µ1;h),Lµ(h) = −w1w2,

where cj(h1, h2) ∈ R1, j = 1, 2, are constant, hold on T ∗(M2

h), giving rise to the following

Picard- Fuchs type equations:

(2.160)

∂w1(µ1;h)/∂h1 = 1/w1,∂w1(µ1;h)/∂h2 = α2h2/w1,∂w2(µ2;h)/∂h1 = 0∂w2(µ2;h)/∂h2 = 1/w2.

The Picard-Fuchs equations (2.160) can easily be integrated by quadratures as follows:

(2.161)w2

1 + k1(µ1)− α2h2 − 2h1 = 0,

w22 + k2(µ2)− 2h2 = 0,

where kj : S1j → C, j = 1, 2, are still unknown functions. For them to be determined explicitly,

it is necessary to substitute (2.159) into expressions (2.149) and (2.150), making use of (2.161)

that amounts to the following results:

(2.162) k1 = µ21, k2 = µ2

2

under the that c1 = −αh2, c2 = 0. Thereby we have constructed, owing to (2.161), the corre-

sponding algebraic curves Γ(j)h , j = 1.2, (2.66) in the explicit form:

(2.163)Γ

(1)h := (λ,w1) : w2

1 + λ2 − α2h22 − 2h1) = 0,

Γ(2)h := (λ,w2) : w2

2 + λ2 − 2h2 = 0,43

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where (λ,wj) ∈ C×C , j = 1, 2, and h ∈ R2 are arbitrary parameters. Making use now of

expressions (2.163) and (2.159), one can construct in an explicit form the integral submanifold

embedding mapping πh : M2h → T ∗(R2) for the flow (??):

(2.164)q1 = µ1 −

√2h2 − µ2

2 − αh22, p1 = w1(µ1;h),

q2 = µ2 −√

2h1 − α2h22 − µ2

1, p2 = w2(µ2;h),

where (µ,w) ∈ ⊗2j=1Γ

(j)h . As was mentioned before, formulas (2.164) together with explicit

expressions (??) make it possible to directly find solutions to the truncated Focker-Plank flow

(2.148) by quadratures, thereby verifying its integrability.

Acknowledgments

Two of authors (A.P. and N.B.) are cordially indebted to the Abdus Salam International

Center for Theoretical Physics, Trieste, Italy, for the kind hospitality during their ICTP-2007

research scholarship. They wish to express their warm thanks to Professor Le Dung Trang,

Head of the Mathematical Department, for invitation to visit the ICTP, where the creative

atmosphere played a significant role in the completion of this work. The authors also wish to

thank Profs. P.I. Holod, M.O. Perestiuk, I.O. Parasiuk and V.G. Samoylenko for remarks and

valuable discussions of problems under study. D. Blackmore completed this work while holding a

visiting professorship at the Institute of Mathematics, Technical University of Denmark (DTU),

and he thanks the Istitute, and especially his host, Morten Brøns, for their gracious and generous

support.

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