Applied Mathematics E-Notes, 1(2001), 130-138 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

On Picard-Fuchs Type Equations Related to Integrable Hamiltonian Systems ^{∗}

Anatoliy K. Prykarpatsky^{†}, Ufuk Taneri^{‡}, Valeriy Samoylenko^{§}

Received 23 April 2001

Abstract

The structure properties of integral submanifold imbedding mapping for a class of algebraically Liouville integrable Hamiltonian systems on cotangent phase spaces are studied in relation with Picard -Fuchs type equations. It is shown that these equations can be constructed by making use of a given a priori set of involutive invariants and proved that their solutions in the Hamilton-Jacobi separable variable case give rise to the integral submanifold imbedding mapping, which is known to be a main ingredient for Liouville-Arnold integrability by quadratures of the Hamiltonian system under consideration.

1 Introduction

We consider a completely integrable Liouville-Arnold Hamiltonian system [1, 2] on a cotangent canonically symplectic manifold (T^{∗}(R^{n}), ω^{(2)}), n ∈ Z_{+}, possessing exactly n functionally independent and Poisson commuting algebraic polynomial invariants H_{j} : T^{∗}(R^{n}) → R, j = \overline{1, n}. Due to the Liouville-Arnold theorem [1, 2], this Hamiltonian system can be completely integrated by quadratures in quasi-periodic functions on its integral submanifold when this submanifold is compact. This is equivalent to the statement that this compact integral submanifold is diffeomorphic to a torus T^{n}, and that makes it possible to formulate the problem of integrating the system by means of searching the corresponding integral submanifold imbedding mapping π_{h} : M_{h}^{n} ⟶ T^{∗}(R^{n}), where

M_{h}^{n} := \{(q, p) ∈ T^{∗}(R^{n}) : H_{j}(q, p) = h_{j} ∈ R, j = \overline{1, n}\} .  (1)

Since M_{h}^{n} ≃ T^{n}, and the integral submanifold (1) is invariant subject to all Hamiltonian flows K_{j} : T^{∗}(R^{n}) → T (T^{∗}(R^{n})), j = \overline{1, n}, where

i_{K_{j}}ω^{(2)} = −dH_{j},  (2)

∗Mathematics Subject Classifications: 35B20, 35J20, 35Q55, 58F08, 70H35.

†Department of Applied Mathematics, University of Mining and Metallurgy, Al. Mickiewicz 30/A4, Krakow 30059, Poland

‡Department of Mathematics, EMU of Gazimagusa, N. Cyprus, Mersin 10, Turkey

§Department of Mechanics and Mathematics, Kyiv State University, Kyiv 00004, Ukaraina

there exist [1, 2] corresponding “action-angle”-coordinates (φ, γ) ∈ (T_{γ}^{n}, R^{n}) on the torus T_{γ}^{n} ⋍ M_{h}^{n}, specifying its imbedding π_{γ} : T_{γ}^{n} → T^{∗}(R^{n}) by means of a set of smooth functions γ ∈ D(R^{n}), where

T_{γ}^{n} := \{(q, p) ∈ T^{∗}(R^{n}) : γ_{j}(H) = γ_{j} ∈ R, j = \overline{1, n}\} .  (3)

The mapping γ : R^{n} ∋ h → R^{n} induced by (3) is of great interest in many applications and was studied earlier by Picard and Fuchs subject to the corresponding differential equations it satisfies:

∂γ_{j}(h)/∂h_{i} = F_{ij}(γ; h),  (4)

where h ∈ R^{n} and F_{ij} : R^{n} × R^{n} → R, i, j = \overline{1, n}, are some almost everywhere smooth functions. In the case where the right hand side of (4) is a set of algebraic functions on C^{n}× C^{n} ∋ (γ; h), all Hamiltonian flows K_{j} : T^{∗}(R^{n}) → T (T^{∗}(R^{n})), j = \overline{1, n}, are said to be algebraically completely integrable in quadratures. Equations such as (4) were studied in [3], a recent example can also be found in [4].

2 Canonical Transformations Properties

It is clear that the Picard-Fuchs equations (4) are related with the associated canonical transformation of the symplectic 2-form ω^{(2)} ∈ Λ^{2}(T^{∗}(R^{n})) in a neighborhood U(M_{h}^{n}) of the integral submanifold M_{h}^{n} ⊂ T^{∗}(R^{n}). More precisely, denote ω^{(2)}(q, p) = dpr^{∗}α^{(1)}(q; p), where for (q, p) ∈ T^{∗}(R^{n}),

α^{(1)}(q; p) := ∑_{j=1}^{n} p_{j}dq_{j} = ⟨p, dq⟩ ∈ Λ^{1}(R^{n})  (5)

is the canonical Liouville 1-form on R^{n}, ⟨·, ·⟩ is the usual scalar product in R^{n}, and pr : T^{∗}(R^{n}) → R^{n} is the bundle projection. We define a mapping dS_{q} : R^{n} → T_{q}^{∗}(R^{n}), such that on M_{h}^{n} the relationship

pr^{∗}α^{(1)}(q; p) + ⟨t, dh⟩ = dS_{q}(h),  (6)

holds, where t ∈ R^{n} is the set of evolution parameters. From (6) one gets right away S_{q}(h) = ∫_{q^{(0)}}^{q} ⟨p, dq⟩|_{M_{h}^{n}} for any q, q^{(0)} ∈ M_{h}^{n}. On the other hand one can define a

generating function S_{μ} : R^{n} → R such that

dS_{μ} : R^{n} → T_{μ}^{∗}(M_{h}^{n}),  (7)

where μ ∈ M_{h}^{n} ⋍ ⊗_{j=1}^{n}S_{j}^{1} are the global separable coordinates existing on M_{h}^{n} owing to the Liouville-Arnold theorem. Thus we have the following canonical relationship

⟨w, dμ⟩ + ⟨t, dh⟩ = dS_{μ}(h),  (8)

where w_{j} := w_{j}(μ_{j}; h) ∈ T_{μ_{j}}^{∗}(S_{j}^{1}) for every j = \overline{1, n}. Whence

S_{μ}(h) = ∑_{j=1}^{n} ∫_{μ_{j}^{(0)}}^{μ_{j}} w(λ; h)dλ,  (9)

satisfies on M_{h}^{n} ⊂ T^{∗}(R^{n}) the following relationship dS_{μ} +dL_{μ} = dS_{q}|_{q=q(μ;h)} for some mapping L_{μ} : R^{n} → R. As a result of (9) we get the following important expressions

t_{i} = ∂S_{μ}(h)/∂h_{i}, ⟨p, ∂q/∂μ_{i}⟩ = w_{i} + ∂L_{μ}/∂μ_{i}, i = \overline{1, n}.  (10)

A construction similar to the above can be done subject to the imbedded torus T_{γ}^{n} ⊂ T^{∗}(R^{n}) : dS̃_{q}(γ) := ∑_{j=1}^{n} p_{j}dq_{j} + ∑_{i=1}^{n} φ_{i}dγ_{i}, where in view of (7), S̃_{q}(γ) := S_{q}(ξ(γ)), ξ(γ) = h, for all (q; γ) ∈ U(M_{h}^{n}). For angle coordinates φ ∈ T_{γ}^{n}, we obtain from dS̃_{q}(γ) that φ_{i} = ∂S̃_{q}(γ)/∂γ_{i} for all i = \overline{1, n}. As φ_{i} ∈ R/2πZ, i = \overline{1, n}, we may easily derive

\frac{1}{2π} ∮_{σ_{j}^{(h)}} dφ_{i} = δ_{ij} = \frac{1}{2π} \frac{∂}{∂γ_{i}} ∮_{σ_{j}^{(h)}} dS̃_{q}(γ) = \frac{1}{2π} \frac{∂}{∂γ_{i}} ∮_{σ_{j}^{(h)}} ⟨p, dq⟩  (11)

for all canonical cycles σ_{j}^{(h)} ⊂ M_{h}^{n} , j = \overline{1, n}, constituting a basis of the one dimensional homology group H^{1}(M_{h}^{n}; Z). It follows that for all i = \overline{1, n}, “action” variables can be found as γ_{i} = \frac{1}{2π} ∮_{σ_{i}^{(h)}} ⟨p, dq⟩ . Recall now that M_{h}^{n} ≃ T_{γ}^{n} are also diffeomorphic to

⊗_{j=1}^{n}S_{j}^{1}, where S_{j}^{1}, j = \overline{1, n}, are some one-dimensional real circles. The evolution along any of the vector fields K_{j} : T^{∗}(R^{n}) → T (T^{∗}(R^{n})), j = \overline{1, n}, on M_{h}^{n} ⊂ T^{∗}(R^{n}) is known [1, 2] to be a linear winding around the torus T_{γ}^{n}. This can also be interpreted in the following manner: the independent global coordinates on circles S_{j}^{1}, j = \overline{1, n}, introduced above are such that the resulting evolution undergoes a quasiperiodic motion. These coordinates may still be called Hamilton-Jacobi and are important for accomplishing the complete integrability by quadratures by solving the corresponding Picard-Fuchs type equations.

Let us denote these separable coordinates on the integral submanifold M_{h}^{n} ≃ ⊗_{j=1}^{n}S_{j}^{1} by μ_{j} ∈ S_{j}^{1}, j = \overline{1, n}, and define the corresponding imbedding mapping π_{h} : M_{h}^{n} → T^{∗}(R^{n}) as

q = q(μ; h), p = p(μ; h).  (12)

There exist two important cases. The first case is related to the integral submanifold M_{h}^{n} ⊂ T^{∗}(R^{n}) which can be parametrized as a manifold by means of the base coordinates q ∈ R^{n} of the cotangent bundle T^{∗}(R^{n}). This can be explained as follows: the canonical Liouville 1-form α^{(1)} ∈ Λ^{1}(R^{n}), in accordance with the diagram

\begin{aligned}T^{∗}( M_{h}^{n}) & ≃ & T^{∗}(⊗_{j=1}^{n}S_{j}^{1}) & ⟵^{π^{∗}} & T^{∗}(R^{n}) \\ pr ↓ & & pr ↓ & & pr ↓ \\ M_{h}^{n} & ≃ & ⊗_{j=1}^{n}S_{j}^{1} & →^{π} & R^{n} \end{aligned}  (13)

is mapped by the imbedding mapping π = pr · π_{h} : M_{h}^{n} → R^{n} not depending on a set of parameters h ∈ R^{n}, into the 1-form

α_{h}^{(1)} = π^{∗}α^{(1)} = ∑_{j=1}^{n} w_{j}(μ_{j}; h)dμ_{j},  (14)

where (μ, w) ∈ T^{∗}(⊗_{j=1}^{n}S_{j}^{1}) ≃ ⊗_{j=1}^{n}T^{∗}(S_{j}^{1}). The imbedding mapping π : M_{h}^{n} → R^{n} due to (14) reduces the function L_{μ} : R^{n} → R to zero and gives rise to the generating function S_{μ} : R^{n} → R which satisfies the condition dS_{μ} = dS_{q}|_{q=q(μ)} ,where as before S_{q}(h) = ∑_{j=1}^{n} p_{j}dq_{j} + ∑_{j=1}^{n} t_{j}dh_{j} and det ||∂q(μ)/∂μ|| ≠ 0 almost everywhere on M_{h}^{n} for all h ∈ R^{n}. Similar to (10), we now have

t_{j} = ∂S_{μ}(h)/∂h_{j}  (15)

for j = \overline{1, n}. From the second part of the imbedding mapping (12) we arrive, in view of (14), at the following simple result: p_{i} = ∑_{j=1}^{n} w_{j}(μ_{j}; h)∂μ_{j}/∂q_{i}, where i = \overline{1, n} and det ||∂μ(q)/∂q|| ≠ 0 almost everywhere on π(M_{h}^{n}) due to the local invertibility of the imbedding mapping π : M_{h}^{n} → R^{n}. Thus, we can assert that the problem of complete integrability in the first case is solved iff the only imbedding mapping π : M_{h}^{n} → R^{n} ⊂ T^{∗}(R^{n}) is constructed. This case was considered in detail in [6] where the corresponding Picard-Fuchs type equations were built based on an extension of the Galisot-Reeb and Francoise results [4, 5]. Namely, similar to (4), these equations are defined as follows:

∂w_{j}(μ_{j}; h)/∂h_{k} = P_{kj}(μ_{j}, w_{j}; h),  (16)

where P_{kj} : T^{∗}(⊗_{j=1}^{n}S_{j}^{1}) × C^{n} → C, k, j = \overline{1, n}, are some algebraic functions of their arguments.

In the second case where the integral submanifold M_{h}^{n} ⊂ T^{∗}(R^{n}) cannot be imbedded almost everywhere into the base space R^{n} ⊂ T^{∗}(R^{n}), the relationship like (14) does not take place, and we are forced to consider the usual canonical transformation from T^{∗}(R^{n}) to T^{∗}(R^{n}) based on a mapping dL_{q} : ⊗_{j=1}^{n}S_{j}^{1} → T^{∗}(R^{n}), where L_{q} : ⊗_{j=1}^{n}S_{j}^{1} → R enjoys for all μ ∈ ⊗_{j=1}^{n}S_{j}^{1} ⋍ M_{h}^{n} ∋ q the following relationship : pr^{∗}α^{(1)}(q; p) = ∑_{j=1}^{n} w_{j} dμ_{j} + dL_{q}(μ). In this case we can derive for any μ ∈ ⊗_{j=1}^{n}S_{j}^{1} the hereditary generating function L_{μ} : R^{n} → T^{∗}(⊗_{j=1}^{n}S_{j}^{1}) introduced before as

∑_{j=1}^{n} w_{j}(μ_{j}; h) dμ_{j} + ∑_{j=1}^{n} t_{j}dh_{j} + dL_{μ} = dL_{q}|_{q=q(μ;h)} ,  (17)

satisfying evidently the following canonical transformation condition:

dS_{q(μ;h)}(h) = ∑_{j=1}^{n} w_{j}(μ_{j}; h) dμ_{j} + ∑_{j=1}^{n} t_{j}dh_{j} + dL_{μ}(h),  (18)

for almost all μ ∈ ⊗_{j=1}^{n}S_{j}^{1} and h ∈ R^{n}. Based on (18) and (17) one can derive the following relationships:

∂L_{μ}(h)/∂h_{j} = ⟨p, ∂q/∂h_{j}⟩|_{M_{h}^{n}}  (19)

for all j = \overline{1, 2}, μ ∈ ⊗_{j=1}^{n}S_{j}^{1} and h ∈ R^{n}. Whence the following important analytical results

\begin{matrix}t_{s} = ∑_{j=1}^{n} ∫_{μ_{j}^{(0)}}^{μ_{j}}(∂w_{j}(λ; h)/∂h_{s})dλ, \\ ∑_{j=1}^{n} p_{j}(μ; h)(∂q_{j}/∂μ_{s}) = w_{s} + ∂L_{μ}(h)/∂μ_{s}, \end{matrix}  (20)

hold for all s = \overline{1, 2} and μ, μ^{(0)} ∈ ⊗_{j=1}^{n}S_{j}^{1} with parameters h ∈ R^{n} being fixed. Thereby we have found a natural generalization of the relationships for p-variables subject to the extended integral submanifold imbedding mapping π_{h} : M_{h}^{n} → T^{∗}(R^{n}) in the form (12).

Assume now the functions w_{j} : C×C^{n}→C, j = \overline{1, n}, satisfy the Picard-Fuchs equations (16) and also the following [3, 5] algebraic conditions:

w_{j}^{n_{j}} + ∑_{k=0}^{n_{j}−1} c_{j,k}(λ; h)w_{j}^{k} = 0,  (21)

where c_{j,k} : C×C^{n}→C , k = \overline{0, n_{j} − 1,} j = \overline{1, n}, are some polynomials in λ ∈ C. In view of the Riemann theorem [7, 8], each algebraic curve of (21) is known to be topologically equivalent to some Riemannian surface Γ_{h}^{(j)} of genus g_{j} ∈ Z_{+}, j = \overline{1, n}. Thereby, one can realize the local diffeomorphism ρ : M_{h}^{n} → ⊗_{j=1}^{n}Γ_{h}^{(j)}, by mapping the homology group basis cycles σ_{j}^{(h)} ⊂ M_{h}^{n} , j = \overline{1, n}, into the homology subgroup H_{1}(⊗_{j=1}^{n}Γ_{h}^{(j)}; Z) basis cycles σ_{j}(Γ_{h}) ⊂ Γ_{h}^{(j)}, j = \overline{1, n}, satisfying the following relationships: ρ( σ_{j}^{(h)}) = ∑_{k=1}^{n} n_{jk} σ_{k}(Γ_{h}), where n_{jk} ∈ Z, k = \overline{1, j} and j = \overline{1, n}, are some fixed integers. Based now on (17) one can write down, for instance, action-variables expressions as follows:

γ_{i} = \frac{1}{2π} ∑_{j=1}^{n} n_{ij} ∮_{σ_{j}(Γ_{h})} w_{j}(λ; h)dλ,  (22)

where i = \overline{1, n}. Subject to the evolution on M_{h}^{n} ⊂ T^{∗}(R^{n}) one can easily obtain from (19) that

dt_{i} = ∑_{j=1}^{n}(∂w_{j}(μ_{j}; h)/∂h_{i})dμ_{j}  (23)

at dh_{i} = 0 for all i = \overline{1, n}, giving rise to a global τ-parametrization of the set of circles ⊗_{j=1}^{n}S_{j}^{1} ⊂ ⊗_{j=1}^{n}Γ_{h}^{(j)}. That is, one can define some inverse algebraic functions to Abelian type integrals (22) as μ = μ(τ; h), where as before, τ = (t_{1}, t_{2}, ..., t_{n}) ∈ R^{n} is a vector of evolution parameters. Recall now the expressions (12) for the integral submanifold mapping π_{h} : M_{h}^{n} → T^{∗}(R^{n}), one can at last write down the “quadratures” mappings for the evolutions on M_{h}^{n} ⊂ T^{∗}(R^{n}) as follows: q = q(μ(τ; h)) = q̃(τ; h), p = p(μ(τ; h)) = p̃(τ; h), where obviously, a vector (q̃, p̃) ∈ T^{∗}(R^{n}) is quasiperiodic in each variable t_{i} ∈ τ, i = \overline{1, n}.

THEOREM 1. Every completely integrable Hamiltonian system admitting an algebraic submanifold M_{h}^{n} ⊂ T^{∗}(R^{n}) possesses a separable canonical transformation (18) which is described by differential algebraic Picard-Fuchs type equations whose solutions are algebraic curves (21).

Therefore, the main ingredient of the scheme of integrability by quadratures is finding the Picard-Fuchs type equations (16) corresponding to the integral submanifold imbedding mapping (12) which depends in general on R^{n} ∋ h-parameters, and then integrating them to curves (21) carrying separable variables.

Similar to the differential-geometric approach developed in [6], one can find 1-forms h_{j}^{(1)} ∈ Λ^{1}(T^{∗}(R^{n})), j = \overline{1, n}, enjoying the following identity on T^{∗}(R^{n}) : ω^{(2)}(q, p) := ∑_{j=1}^{n} dp_{j}∧dq_{j} = ∑_{j=1}^{n} dH_{j}∧h_{j}^{(1)}. The 1-forms h_{j}^{(1)} ∈ Λ^{1}(T^{∗}(R^{n})), j = \overline{1, n}, possess the following important properties when they are pulled back to the integral submanifold (1): π_{h}^{∗}h_{j}^{(1)} := h̄_{j}^{(1)} = dt_{j}, where h̄_{j}^{(1)} ∈ Λ^{1}(M_{h}^{n}), and π_{h∗}d/dt_{j} = K_{j} · π_{h} for all j = \overline{1, n}. The above expressions combined with (23) give rise easily to the following set of relationships

h̄_{j}^{(1)} = ∑_{j=1}^{n}(∂w_{j}(μ_{j}; h)/∂h_{i})dμ_{j}  (24)

at dh_{j} = 0 on M_{h}^{n} ⋍ ⊗_{j=1}^{n}S_{j}^{1} ⊂ ⊗_{j=1}^{n}Γ_{h}^{(j)} for all j = \overline{1, n}. Since we are interested in the integral submanifold imbedding mapping (12) being locally diffeomorphic in a neighborhood U(M_{h}^{n}) ⊂ T^{∗}(R^{n}), the Jacobian det ||∂q(μ; h)/∂μ|| ≠ 0 almost every where in U(M_{h}^{n}). On the other hand, as was proved in [4], the set of 1-forms h̄_{j}^{(1)} ∈ Λ^{1}(M_{h}^{n}), j = \overline{1, n}, can be represented in U(M_{h}^{n}) as

h̄_{j}^{(1)} = ∑_{k=1}^{n} h̄_{jk}^{(1)}(q, p)dq_{k}|_{M_{h}^{n}} ,  (25)

where h̄_{jk}^{(1)} : T^{∗}(R^{n}) → R, k, j = \overline{1, n}, are some algebraic expressions of their arguments. Thereby, one easily finds from (25) and (24) that

∂w_{i}(μ_{i}; h)/∂h_{j} = ∑_{k=1}^{n} h̄_{jk}^{(1)}(q(μ; h), p(μ; h))(∂q_{k}(μ; h)/∂μ_{i})  (26)

for all i, j = \overline{1, n}. Subject to the p-variables in (26) we must, in view of (20), use the expressions

\begin{matrix}∑_{j=1}^{n} p_{j}(μ; h)(∂q_{j}/∂μ_{s}) = w_{s} + ∂L_{μ}(h)/∂μ_{s}, \\ ∂L_{μ}(h)/∂h_{j} = ⟨p, ∂q/∂h_{j}⟩|_{M_{h}^{n}} , \end{matrix}  (27)

being true for s = \overline{1, n} and all μ ∈ ⊗_{j=1}^{n}S_{j}, h ∈ R^{n} in the neighborhood U(M_{h}^{n}) ⊂ T^{∗}(R^{n}) chosen before. Thereby, we arrive at the following

∂w_{i}(μ_{i}; h)/∂h_{j} = P̄_{ji}(μ, w; h),  (28)

where the expressions P̄_{ji}(μ, w; h) := ∑_{k=1}^{n} h̄_{jk}^{(1)}(q(μ; h), p(μ; h))∂q_{k}/∂μ_{i}), i, j = \overline{1, n},

depend only on Γ_{h}^{(i)} ∋ (μ_{i}, w_{i})-variables for each i ∈ \{\overline{1, n}\} and all j = \overline{1, n}. This condition can be written down as follows:

∂P̄_{ji}(μ, w; h)/∂μ_{k} = 0,  (29)

for j, i ≠ k ∈ \{\overline{1, n}\} at almost all μ ∈ ⊗_{j=1}^{n}S_{j}^{1}.

THEOREM 2. Let there be given a completely integrable Hamiltonian system on the coadjoint manifold T^{∗}(R^{n}) whose integral submanifold M_{h}^{n} ⊂ T^{∗}(R^{n}) is described by the Picard-Fuchs type algebraic equations (28). The corresponding imbedding mapping π_{h} : M_{h}^{n} → T^{∗}(R^{n}) defined in (12) is a solution of a compatibility condition subject to the differential-algebraic relationships (29) on the canonical transformations generating function (17).

To show that the scheme described above really leads to an algorithmic procedure for constructing the Picard-Fuchs type equations (28) and the corresponding integral submanifold imbedding mapping π_{h} : M_{h}^{n} → T^{∗}(R^{n}) in the form (12), we apply it to a so called truncated Focker-Plank Hamiltonian system on the canonically symplectic cotangent space T^{∗}(R^{n}).

Consider the following dynamical system on the canonically symplectic phase space T^{∗}(R^{2}) :

\begin{matrix}dq_{1}/dt = p_{1} + α(q_{1} + p_{2})(q_{2} + p_{1}), \\ dq_{2}/dt = p_{2}, \\ dp_{1}/dt = −(q_{1} + p_{2}) − α[q_{2}p_{1} + 1/2(p_{1}^{2} + p_{2}^{2} + q_{2}^{2})], \\ dp_{2}/dt = −(q_{2} + p_{1}), \end{matrix} \} = K_{1}(q, p),  (30)

where K_{1} : T^{∗}(R^{2}) → T (T^{∗}(R^{2})) is the corresponding vector field on T^{∗}(R^{2}) ∋ (q, p), and t ∈ R is an evolution parameter, called a truncated four-dimensional Focker-Plank flow. It is easy to verify that the functions H_{j} : T^{∗}(R^{2}) → R, j = \overline{1, 2}, where

\begin{matrix}H_{1} = 1/2(p_{1}^{2} + p_{2}^{2} + q_{1}^{2}) + q_{1}p_{2} + α(q_{1} + p_{2})[q_{2}p_{1} + 1/2(p_{1}^{2} + p_{2}^{2} + q_{2}^{2})], \\ H_{2} = 1/2(p_{1}^{2} + p_{2}^{2} + q_{2}^{2}) + q_{2}p_{1}, \end{matrix}  (31)

are functionally independent invariants with respect to the flow (30). Moreover, the invariant (31) is the Hamiltonian function for (30), that is, the relationship i_{K_{1}}ω^{(2)} = −dH_{1} holds on T^{∗}(R^{2}), where the symplectic structure ω^{(2)} ∈ Λ^{2}(T^{∗}(R^{2})) is given as follows: ω^{(2)} := d(pr^{∗}α^{(1)}) = ∑_{j=1}^{2} dp_{j} ∧ dq_{j}, with α^{(1)} ∈ Λ^{1}(R^{2}) to be the canonical Liouville form on R^{2} : α^{(1)}(q; p) = ∑_{j=1}^{2} p_{j}dq_{j} for any (q, p) ∈ T^{∗}(R^{2}) ≃ Λ^{1}(R^{2}).

The invariants (31) commute with each other subject to the associated Poisson bracket on T^{∗}(R^{2}) : {H_{1}, H_{2}} = 0. Thereby, in view of the abelian Liouville-Arnold theorem [1, 2], the dynamical system (30) is completely integrable by quadratures on T^{∗}(R^{2}), and we can apply our scheme to the commuting invariants (31) subject to the symplectic structure ω^{(2)} ∈ Λ^{2}(R^{2}). One easily states that ω^{(2)} = ∑_{i=1}^{2} dH_{i} ∧ h_{i}^{(1)}, where the corresponding 1-forms π_{h}^{∗}h_{i}^{(1)} := h̄_{i}^{(1)} ∈ Λ^{1}(M_{h}^{2}), i = \overline{1, 2}, are given as

\begin{matrix}h̄_{1}^{(1)} = \frac{p_{2}dq_{1}−(p_{1}+q_{2})dq_{2}}{p_{1}p_{2}−(p_{1}+q_{2})(q_{1}+p_{2})−αh_{2}(p_{1}+q_{2})}, \\ h̄_{2}^{(1)} = \frac{−[(q_{1}+p_{2})(1+αp_{2})+αh_{2}]dq_{1}+(p_{1}+α[h_{2}+(q_{2}+p_{1})( q_{1}+p_{2})])dq_{2}}{p_{1}p_{2}−(q_{2}+p_{1})(αh_{2}+q_{1}+p_{2})} , \end{matrix}  (32)

and an invariant submanifold M_{h}^{2} ⊂ T^{∗}(R^{2}) is defined as

M_{h}^{2} := \{(q, p) ∈ T^{∗}(R^{2}) : H_{i}(q, p) = h_{i} ∈ R,i = \overline{1, 2}\}

for some parameters h ∈ R^{2}. Based now on expressions (32) and (18), we can easily construct functions P̄_{ij}(w; h), i, j = \overline{1, 2}, in (28), defined on T^{∗}(M_{h}^{2}) ≃ T^{∗}(⊗_{j=1}^{2}S_{j}^{1}) subject to the integral submanifold imbedding mapping π_{h} : M_{h}^{2} → T^{∗}(R^{2}) in coordinates μ ∈ ⊗_{j=1}^{2}S_{j}^{1} ⊂ ⊗_{j=1}^{2}Γ_{h}^{(j)}, which we will not write down in detail due to their cumbersome form. Having applied then the criterion (29), we arrive at the following compatibility relationships subject to the mappings q : (⊗_{j=1}^{2}S_{j}^{1}) × R^{2} → R^{2} and p : (⊗_{j=1}^{2}S_{j}^{1}) × R^{2} → T_{q}^{∗}(R^{2}) :

\begin{matrix}∂q_{1}/∂μ_{1} − ∂q_{2}/∂μ_{2} = 0, \\ w_{1}∂L_{μ}/∂w_{1} − w_{2}∂L_{μ}/∂w_{2} = 0, \\ ∂^{2}q_{1}/∂μ_{2}∂h_{2} + ∂^{2}w_{2}/∂μ_{2}∂h_{2} = 0, \\ w_{1}∂w_{1}/∂h_{1} − w_{2}∂w_{2}/∂h_{2} = 0, \\ ∂w_{1}/∂h_{1}(∂q_{1}/∂h_{1}) = ∂w_{2}/∂h_{1}(∂q_{2}/∂h_{1}), \\ ∂(w_{1}∂w_{1}/∂h_{2})/∂h_{2} − α^{2}∂q_{1}/∂μ_{1} = 0, \end{matrix}  (33)

and so on, subject to variables μ ∈ ⊗_{j=1}^{2}S_{j}^{1} and h ∈ R^{2}. Solving all equations like (33), one can find right away that the expressions

\begin{matrix}p_{1} = w_{1}, p_{2} = w_{2}, q_{1} = c_{1} + μ_{1} − w_{2}(μ_{2}; h), \\ q_{2} = c_{2} + μ_{2} − w_{1}(μ_{1}; h), L_{μ}(h) = −w_{1}w_{2}, \end{matrix}  (34)

where c_{j}(h_{1}, h_{2}) ∈ R^{1}, j = \overline{1, 2}, are constant, hold on T^{∗}(M_{h}^{2}), giving rise to the following Picard-Fuchs type equations in the form (28):

\begin{matrix}∂w_{1}(μ_{1}; h)/∂h_{1} = 1/w_{1}, \\ ∂w_{1}(μ_{1}; h)/∂h_{2} = α^{2}h_{2}/w_{1}, \\ ∂w_{2}(μ_{2}; h)/∂h_{1} = 0, \\ ∂w_{2}(μ_{2}; h)/∂h_{2} = 1/w_{2}. \end{matrix}  (35)

The Picard-Fuchs equations (35) can be easily integrated by quadratures as follows:

w_{1}^{2} + k_{1}(μ_{1}) − α^{2}h_{2} − 2h_{1} = 0, w_{2}^{2} + k_{2}(μ_{2}) − 2h_{2} = 0,  (36)

where k_{j} : S_{j}^{1} → C, j = \overline{1, 2}, are still unknown functions. To determine these functions, it is necessary to substitute (34) into expressions (31), and make use of (36) to reach the following results: k_{1} = μ_{1}^{2}, k_{2} = μ_{2}^{2} under the condition that c_{1} = −αh_{2}, c_{2} = 0. Thereby, we have constructed the corresponding algebraic curves Γ_{h}^{(j)}, j = \overline{1, 2}, in the form (21):

Γ_{h}^{(1)} := {(λ, w_{1}) : w_{1}^{2}+λ^{2}−α^{2}h_{2}^{2}−2h_{1}) = 0}, Γ_{h}^{(2)} := {(λ, w_{2}) : w_{2}^{2}+λ^{2}−2h_{2} = 0}, (37)

where (λ, w_{j}) ∈ C×C, j = \overline{1, 2}, and h ∈ R^{2} are arbitrary parameters. Making use now expressions (37) and (34), we can construct in explicit form the integral submanifold imbedding mapping π_{h} : M_{h}^{2} → T^{∗}(R^{2}) for the flow (30):

\begin{matrix}q_{1} = μ_{1} − \sqrt{2h_{2} − μ_{2}^{2}} − αh_{2}^{2}, p_{1} = w_{1}(μ_{1}; h), \\ q_{2} = μ_{2} − \sqrt{2h_{1} − α^{2}h_{2}^{2} − μ_{1}^{2}}, p_{2} = w_{2}(μ_{2}; h), \end{matrix}  (38)

where (μ, w) ∈ ⊗_{j=1}^{2}Γ_{h}^{(j)}. As was mentioned before, the formulas in (38) together with the explicit expressions (20) make it possible right away to find solutions to the truncated Focker-Plank flow (30) by quadratures, thereby completing its integrability.

Acknowledgment. The first author is thankful to Professors D. Blackmore, J. Ombach and A. Pelczar for helpful discussions.

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