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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
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
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
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
(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
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
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
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
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
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
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
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
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
(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
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
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
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
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
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
(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
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
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
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
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
2π
∮
σ(h)j
dϕi = δij =1
2π
∂
∂γi
∮
σ(h)j
dSq(γ) =1
2π
∂
∂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
2π
∮
σ(h)i
< p, dq >
23
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
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
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
2π
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
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
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
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
(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
2π
∮
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
2π
∮
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
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
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
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
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
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
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
2π
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
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
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
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
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
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
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
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
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.
References
[1] R. Abraham and J. Marsden, Foundations of Mechanics, Second Edition, Benjamin Cummings, NY, 1978.[2] V.I. Arnold, Mathematical Methods of Classical Mechanics., Springer, NY, 1978.[3] D. Blackmore, L. Ting and O. Knio, Studies of perturbed three vortex dynamics, J. Math. Phys. 48 (2007),
in press.[4] V. Guillemin and S. Sternberg, On the equations of motion of a classical particle in a Yang-Mills field and
the principle of general covariance, Hadronic Journal, 1, (1978) 1-32.[5] D. Holm and B. Kupershmidt, Superfluid plasmas: multivelocity nonlinear hydrodynamics of superfluid
solutions with charged condensates coupled electromagnetically, Phys. Rev., 36A(8), (1987) 3947-3956.[6] P.I. Holod and A.I. Klimyk, Mathematical foundations of symmetry theory, Kyiv, ”Naukova Dumka”, 1992
(in Ukrainian).[7] J. Kummer, On the construction of the Reduced phase space of a Hamiltonian system with symmetry, Indiana
University Mathem. Journal, 30(2), (1981) 281-281.[8] B.A. Kupershmidt, Infinite-dimensional analogs of the minimal coupling principle and of the Poincare lemma
for differential two-forms, Diff. Geom. and Appl., 2, (1992) 275-293.[9] J. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D, 4,
(1982) 394-406.[10] J.D. Moor, Lectures on Seiberg-Witten invariants, Lect. Notes in Math., N1629, Springer, 1996.[11] A. Perelomov, Integrable systems of classical mechanics and Lie algebras, Birkhauser, 1990.[12] A. Prykarpatsky and I. Mykytiuk, Algebraic integrability of nonlinear dynamical systems on manifolds.
Classical and quantum aspects, Kluwer, Dordrecht, 1998.[13] Y.A. Prykarpatsky and A.M. Samoilenko, Algebraic-analytic aspects of integrable nonlinear dynamical sys-
tems and their perturbations, Kyiv, Inst. Mathematics Publ., v. 41, 2002 (in Ukrainian)[14] A. Prykarpatsky and J. Zagrodzinski, Dynamical aspects of Josephson type media, Ann. of Inst. H.Poincare,
Physique Theorique, 70(5), (1999) 497-524.
44
[15] W.J. Satzer (jr), Canonical reduction of mechanical systems invariant under abelian group actions with anapplication to celestial mechanics, Indiana Univ. Math. Journal, (1977) 26(5), (1977) 951-976.
[16] A.S. Mishchenko, A.T. Fomenko, Generalized Liouville method of integrating Hamiltonian systems. Funct.Analysis and Appl., 1978,12(2),46-56 (in Russian)
[17] E. Cartan, Lecons sur invariants integraux. Hermann, Paris, 1971[18] A.T. Fomenko, Differential Geometry and Topology. Supplementary chapters. Moscow University Publ., 1983.[19] J. Marsden, A. Weinstein, Reduction of symplectic manifolds with symmetry. Reports on Mathem, Physics,
1974, 5(1),121-130.[20] S. Zakrzewski, Induced representations and induced Hamiltonian actions. Journ. of Geometry and Physics,
1986,3(2),211-219.[21] M. Kummer, On the construction of the reduced phase space of a Hamilton system with symmetry. Indiana
Univ. Math. Journal, 1981,30(2), 281-291.[22] W.J. Satzer (jr), Canonical reduction of mechanical systems invariant under abelian group actions with an
application to celestial mechanics. Indiana Univ. Math. Journal, 1977,26(5), 951-976.[23] J.-M. Souriau, Structures des systemes dynamiques. Dunod, Paris, 1970.[24] Ya.A. Prykarpatsky, A.M. Samoilenko, D.L. Blackmore, Embedding of integral submanifolds and associated
adiabatic invariants of slowly perturbed integrable Hamiltonian systems. Reports on Mathem.physics, 1999,37(5).
[25] P. Basarab-Horwath, Integrability by quadratures for systems of involutive vector fields. Ukrainian Math.Journal, (Plenum Pres Publ., USA),1992,43(10), 1236-1242.
[26] M. Kopych, Ya. Prykarpatsky, R. Samulyak, Proceedings of the NAS of Ukraina (Mathematics),1997, 2,32-38.
[27] A. Ankiewicz, C. Pask, Journal of Physics,A, 1983,16,4203-4208.[28] S. Rauch-Wojciechowski, A.V. Tsiganov, Quasi-point separation of variables for the Henon-Heiles systems
and a system with quadratic potential. J.Phys. 1996 A29, 7769-7778.[29] M. Prytula, A. Prykarpatsky, I. Mykytiuk, Fundamentals of the Theory of Differential-Geometric Structures
and Dynamical Systems. Kiev, the Ministry of Educ. Publ., 1988, (in Ukrainian)[30] A. Prykarpatsky, I. Mykytiuk, Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Clas-
sical and Quantum Aspects. Kluwer, Dordrecht, 1998.[31] J.P. Francoise, Monodromy and the Kowalewskaya top. Asterisque, 1987,150/151,87-108; Arnold’s formula
for algebraically completely integrable systems. Bull. AMS, 1987,17, 301-303.[32] P. Deligne, Equations differentielles a points singulairs. Springer Lect. Notes in Mathem., 1970,163.[33] E. Zverovich, Boundary problems of the theory of analytical functions in Holder classes on Riemanniann
surfaces. Russian Math. Surveys, 1971,26(1),113-176. (in Russian)[34] M. Blaszak, Multi-Hamiltonian Dynamical Systems. Springer, 1998.[35] A. Prykarpatsky, I. Mykytiuk, Algebraic Aspects of Integrable Dynamical Systems on Manifolds. Kiev,
Naukova Dumka Publ., 1987 (in Russian).[36] D. Blackmore, Ya. Prykarpatsky, R. Samulyak, On the Lie invariant geometric objects, generated by inte-
grable ideals in Grassmann algebra. Journ.of Nonl.Math.Physics, 1998, 5(1), 54-67.[37] R.O. Wells, Differential Analysis on Complex Manifolds. Prentice-Hall, Englewood Cliff., NJ, 1973.[38] A. Newlander, L. Nirenberg, Complex analytical coordinates in almost-complex manifolds. Ann. of Mathe-
matics, 1957, 65, 391-404.[39] L. Hormander, An Introduction to Complex Analysis in Several Variables. Van Nostrand Reinhold Publ. Co.,
N4, 1986.[40] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry. John Wiley and Sons, NY, v.1,1963; v.2,
1969.[41] M. Salerno, V.Z. Enolski, D.V. Leykin, Canonical transformation between integrable Henon-Heiles systems.
Phys. Rev.E,1994, 49(6),5897-58899.[42] V. Ravoson, L. Gavrilov, R. Caloz, Separability and Lax pairs for Henon-Heiles system. J. Mathem. Physics,
1993, 34(6), 2385-2393.[43] D. Roekaerts, F. Scwarz, Painleve analysis, Yoshida’s theorems and direct methods in the search for integrable
Hamiltonians. J. Phys. A: Mat. Gen., 1987, 20, L127-L133.[44] L. Hormander, Course at Lund University,1974.[45] J.J. Duistermaat, On global action-angle coordinates . Comm. Pure and Appl. Math., 1980, 33, 687-706.[46] B.A. Dubrovin, A.T. Fomenko, S.P. Novikov and R.G. Burns, Modern Geometry: Methods and Applications:
The Geometry of Surfaces, Transformations Groups and Fields Pt. 1 (Graduate Texts in Mathematics),Publisher: Springer-Verlag Berlin and Heidelberg GmbH and Co. 1991
[47] V.V. Kozlov, N.N. Kolesnikov, On dynamics theorems. Prykl. Mathem. and Mechanics, 1978, 42(1), 28-33.[48] N.N. Niekhoroshev, Action-angle variables and their generalization. Trudy mosk. Matem. Obshchestva, 1972,
26, 181-198.
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