On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions

Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrow...

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Veröffentlicht in:	Нелінійні коливання
Datum:	2005
Hauptverfasser:	Prykarpatsky, Y.A., Samoilenko, A.M.
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spelling	Prykarpatsky, Y.A. Samoilenko, A.M. 2021-02-17T15:50:08Z 2021-02-17T15:50:08Z 2005 On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions / Y.A. Prykarpatsky, A.M. Samoilenko // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 360-387. — Бібліогр.: 41 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178006 517.9 Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan’s theory of differential systems on the associated jet-manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler – Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integraldifferential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied. Наведено деякi аспекти опису лагранжевого та гамiльтонового формалiзму, який природно виникає iз структури iнварiантностi заданих нелiнiйних динамiчних систем на нескiнченновимiрному функцiональному многовидi. Основнi iдеї, якi використовуються для формування канонiчної симплектичної структури, взято з теорiї Картана диференцiальних систем на вiдповiдних многовидах струмiв. Для диференцiальних та диференцiальних дискретних динамiчних систем наведено детальний опис структури симетрiй, якi редукованi на iнварiантнi пiдмноговиди критичних точок деяких нелокальних ейлерово-лагранжевих функцiоналiв. За допомогою деякого перетворення Беклунда отримано гамiльтонове зображення для iєрархiї рiвнянь лаксового типу на двоїстому до алгебри Лi просторi iнтегрально-диференцiальних операторiв з матричними коефiцiєнтами, яке продовжено еволюцiями власних функцiй та спряжених власних функцiй вiдповiдних спектральних задач. Вивчено зв’язок мiж цiєю iєрархiєю та iнтегровними за Лаксом просторово-двовимiрними системами. en Інститут математики НАН України Нелінійні коливання On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions Про лагранжеві та гамільтонові аспекти нескінченновимірних динамічних систем та їх скінченновимірну редукцію О лагранжевых и гамильтоновых аспектах бесконечномерных динамических систем и их конечномерной редукции Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions
spellingShingle	On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions Prykarpatsky, Y.A. Samoilenko, A.M.
title_short	On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions
title_full	On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions
title_fullStr	On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions
title_full_unstemmed	On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions
title_sort	on the lagrangian and hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions
author	Prykarpatsky, Y.A. Samoilenko, A.M.
author_facet	Prykarpatsky, Y.A. Samoilenko, A.M.
publishDate	2005
language	English
container_title	Нелінійні коливання
publisher	Інститут математики НАН України
format	Article
title_alt	Про лагранжеві та гамільтонові аспекти нескінченновимірних динамічних систем та їх скінченновимірну редукцію О лагранжевых и гамильтоновых аспектах бесконечномерных динамических систем и их конечномерной редукции
description	Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan’s theory of differential systems on the associated jet-manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler – Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integraldifferential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied. Наведено деякi аспекти опису лагранжевого та гамiльтонового формалiзму, який природно виникає iз структури iнварiантностi заданих нелiнiйних динамiчних систем на нескiнченновимiрному функцiональному многовидi. Основнi iдеї, якi використовуються для формування канонiчної симплектичної структури, взято з теорiї Картана диференцiальних систем на вiдповiдних многовидах струмiв. Для диференцiальних та диференцiальних дискретних динамiчних систем наведено детальний опис структури симетрiй, якi редукованi на iнварiантнi пiдмноговиди критичних точок деяких нелокальних ейлерово-лагранжевих функцiоналiв. За допомогою деякого перетворення Беклунда отримано гамiльтонове зображення для iєрархiї рiвнянь лаксового типу на двоїстому до алгебри Лi просторi iнтегрально-диференцiальних операторiв з матричними коефiцiєнтами, яке продовжено еволюцiями власних функцiй та спряжених власних функцiй вiдповiдних спектральних задач. Вивчено зв’язок мiж цiєю iєрархiєю та iнтегровними за Лаксом просторово-двовимiрними системами.
issn	1562-3076
url	https://nasplib.isofts.kiev.ua/handle/123456789/178006
citation_txt	On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions / Y.A. Prykarpatsky, A.M. Samoilenko // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 360-387. — Бібліогр.: 41 назв. — англ.
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fulltext	UDC 517 . 9 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS AND THEIR FINITE-DIMENSIONAL REDUCTIONS ПРО ЛАГРАНЖЕВI ТА ГАМIЛЬТОНОВI АСПЕКТИ НЕСКIНЧЕННОВИМIРНИХ ДИНАМIЧНИХ СИСТЕМ ТА ЇХ СКIНЧЕННОВИМIРНУ РЕДУКЦIЮ Ya. A. Prykarpatsky Inst. Math. Nat. Acad. Sci. Ukraine Tereshchenkivs’ka Str., 3, Kyiv, 01601, Ukraine and AGH Univ. Sci. and Technol. Krakow, 30059, Poland e-mail: yarpry@imath.kiev.ua A. M. Samoilenko Inst. Math. Nat. Acad. Sci. Ukraine Tereshchenkivs’ka Str., 3, Kyiv, 01601, Ukraine e-mail: sam@imath.kiev.ua Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional mani- fold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan’s theory of differential systems on the associated jet-manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler – Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltoni- an representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integral- differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint ei- genfunctions of the corresponding spectral problems, is obtained via some special Backlund transformati- on. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied. Наведено деякi аспекти опису лагранжевого та гамiльтонового формалiзму, який природно ви- никає iз структури iнварiантностi заданих нелiнiйних динамiчних систем на нескiнченновимiр- ному функцiональному многовидi. Основнi iдеї, якi використовуються для формування кано- нiчної симплектичної структури, взято з теорiї Картана диференцiальних систем на вiдпо- вiдних многовидах струмiв. Для диференцiальних та диференцiальних дискретних динамiчних систем наведено детальний опис структури симетрiй, якi редукованi на iнварiантнi пiдмного- види критичних точок деяких нелокальних ейлерово-лагранжевих функцiоналiв. За допомогою деякого перетворення Беклунда отримано гамiльтонове зображення для iєрархiї рiвнянь лак- сового типу на двоїстому до алгебри Лi просторi iнтегрально-диференцiальних операторiв з матричними коефiцiєнтами, яке продовжено еволюцiями власних функцiй та спряжених влас- них функцiй вiдповiдних спектральних задач. Вивчено зв’язок мiж цiєю iєрархiєю та iнтегров- ними за Лаксом просторово-двовимiрними системами. 1. Introduction. One of the fundamental problems in modern theory of infinite-dimensional dynamical systems is that of their invariant reduction on some invariant submanifolds with c© Ya. A. Prykarpatsky, A. M. Samoilenko, 2005 360 ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 361 enough rich mathematical structures as to treat their properties analytically. The first approaches to these problems were suggested already at the end of the preceding century, in the classi- cal works of S. Lie, J. Liouville, J. Lagrange, V. R. Hamilton, J. Poisson and E. Cartan. They introduced at first the important concepts of symmetry, conservation law, symplectic, Poisson and Hamiltonian structures as well as invariant reduction procedure, which appeared to be extremely useful for the proceeding studies. These notions were widely generalized further by Souriau [1], Marsden and Weinstein [2, 3], Lax [4], Bogoyavlensky and Novikov [5], as well as by many other researchers [6 – 10]. It seems worthwhile to mention here also the recent enough studies in [11 – 18], where special reduction methods were proposed for integrable nonlinear dynamical systems on both functional and operator manifolds. In the present paper we describe in detail the reduction procedure for infinite dimensional dynamical systems on an invariant set of critical points of some global invariant functional. The method uses the Cartan’s differential-geometric treating of differential ideals in Grassmann algebra over the associated jet-manifold. As one of the main results, we show also that both the reduced dynami- cal systems and their symmetries generate Hamiltonian flows on the invariant critical submani- folds of local and nonlocal functionals with respect to the canonical symplectic structure on it. These results are generalized to the case of differential-difference dynamical systems that are gi- ven on discrete infinite-dimensional manifolds. The direct procedure to construct the invariant Lagrangian functionals for a given a priori Lax-type integrable dynamical system is presented for both the differential and the differential-difference cases of the manifold M. Some remarks on the Lagrangian and Hamiltonian formalisms, concerned with infinite-dimensional dynami- cal systems with symmetries are given. The Hamiltonian representation for a hierarchy of Lax-type equations on a space dual to the Lie algebra of integral-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with Lax integrable spatially two-dimensional systems is studied. 2. General setting. We are interested in treating a given nonlinear dynamical system du dt = K[u], (2.1) with respect to an evolution parameter t ∈ R on an infinite-dimensional functional manifold M ⊂ C(∞)(R; Rm), possessing two additional ingredients: a homogeneous and autonomous conservation law L ∈ D(M) and a number of homogeneous autonomous symmetries du/dtj = = Kj [u], j = 1, k, with evolution parameters tj ∈ R. The dynamical system (2.1) is not- supposed to be in general Hamiltonian, all the maps K, Kj : M → T (M), j = 1, k, are considered to be smooth and well-defined on M. To pose the problem to be discussed further more definitely, let us use the jet-manifold J (∞)(R; Rm), locally isomorphic to the functional manifold M mentioned above. This means the following: the vector field (2.1) on M is completely equivalent to that on the jet-manifold J (∞)(R; Rm) via the representation [19, 20] (M 3 u → K[u]) jet−→ ( K(u, u(1), . . . , u(n+1)) ← (x; u, u(1), . . . , u(∞)) ∈ J (∞)(R; Rm) ) , (2.2) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 362 YA. A. PRYKARPATSKY, A. M. SAMOILENKO where n ∈ Z+ is fixed, x ∈ R is the function parameter of the jet-bundle J (∞)(R; Rm) π−→ R, and π is the usual projection on the base R. Let us allow also that the smooth functional L ∈ ∈ D(M) is a conservation law of the dynamical system (2.1), that is, dL/dt = 0 along orbits of (2.1) for all t ∈ R. Due to the jet-representation (2.2) we can write the density of the functional L ∈ D(M) in the following form: L = ∫ R dxL[u], (2.3) with R×Rm 3 [x;u] jet−→ (x;u, u(1), . . . , u(N+1)) ∈ J (N+1)(R; Rm) being the standard jet- mapping and the number N ∈ Z+ fixed. Besides, the functional (2.3) will be assumed to be nondegenerate in the sense that the Hessian of L : J (N+1)(R; Rm) → R has nonvanishing determinant, det ∥∥∥∥∥∂2L ( u, u(1), . . . , u(N+1) ) ∂u(N+1)∂u(N+1) ∥∥∥∥∥ 6= 0. 3. Lagrangian reduction. Consider now the set of critical points Mn ⊂ M of the functional L ∈ D(M), MN = {u ∈ M : grad L[u] = 0}, (3.1) where, due to (2.2), grad L[u] := δL(u, . . . , u(N+1))/δu is the Euler variational derivative. As proved by Lax [4], the manifold MN ⊂ M is smoothly imbedded and well-defined, due to the condition HessL 6= 0. Besides, the manifold MN appears to be invariant with respect to the dynamical system (2.1). This means in particular that the Lie derivative of any vector field X : M → T (M), tangent to the manifold MN , with respect to the vector field (2.1) is again tangent to MN , that is, the implication X[u] ∈ Tu(MN ) ⇒ [K, X][u] ∈ Tu(MN ) (3.2) holds for all u ∈ MN . Here we are in a position to begin with a study of the intrinsic structure of the manifold MN ⊂ M within the geometric Cartan’s theory developed on the jet-manifold J (∞)(R; Rm) [4, 20 – 22]. Let us define an ideal I(ξ) ⊂ Λ(J (∞)), generated by the vector one- forms ξ (1) j = du(j)−u(j+1)dx, j ∈ Z+, which vanish on the vector field d/dx on the jet-manifold J (∞)(R; Rm), i d dx ξ (1) j = 0, j ∈ Z+, (3.3) where x ∈ R belongs to the jet-bundle base, i d dx is the intrinsic derivative along the vector field d dx = ∂ ∂x + ∑ j∈Z+ 〈 u(j+1), ∂ ∂u(j) 〉 , where 〈., .〉 is the standard scalar product in Rm. The vector field (2.1) on the jet-manifold J (∞)(R; R) has an analogous representation, d dt = ∂ ∂t + ∑ j=Z+ 〈 K(j), ∂ ∂u(j) 〉 , (3.4) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 363 where, by definition, K(j) := dj dxj K, j = Z+. There is the following problem: how to build the intrinsic variables on the manifold MN ⊂ M from the jet-manifold coordinates on J (∞)(R; Rm))? To proceed with the solution of the problem above, let us study the 1-form dL = Λ1 ( J (∞)(R; R) ) as one defined on the submanifold MN ⊂ M. We have the following chain of identities in the Grassmann subalgebra Λ(J (2N+2)(R; Rm)) : dL = d ( i d dx Ldx ) = di d dx Ldx + N∑ j=0 〈pj , R〉  = = ( di d dx + i d dx d )L dx + N∑ j=0 〈 pj , ξ (1) j 〉− i d dx d L dx + N∑ j=0 〈 pj , ξ (1) j 〉 , (3.5) where pj : J (2N+2)(R; Rm) → Rm, j = 0, N, are some unknown vector-functions. Requiring now that the 2-form d(Ldx + ∑N j=0〈pj , ξ (1) j 〉) do not depend on the differentials du(j), j = = 1, N + 1, that is i ∂ ∂u(j) ( dL ∧ dx + N∑ k=0 〈 dpk ∧ ξ (1) j 〉) = 0, (3.6) one can thus determine the vector-functions pj = Rm, j = 0, N. As a result we obtain the following simple recurrence relations: dpj dx + pj−1 = ∂L ∂u(j) (3.7) for j = 1, N + 1, setting p−1 = 0 = pN+1 by definition. The unique solution to (3.7) is given by the following expressions, j = 0, N : pj = N∑ k=0 (−1)k dk dxk ∂L ∂u(j+k+1) . (3.8) Thereby we have got, owing to (3.5) and (3.6), the following final representation for the dif- ferential dL : dL = d dx L − N∑ j=0 〈 pj , u (j+1) 〉 dx− 〈 gradL[u], u(1) 〉 dx+ + d dx  N∑ j=0 〈 pj , du(j) 〉+ 〈gradL[u], du〉, (3.9) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 364 YA. A. PRYKARPATSKY, A. M. SAMOILENKO with d dx := di d dx + i d dx d being the Lie derivative along the vector field d dx , and gradL[u] := := δL/δu, as it was mentioned above in Section 2. Below we intend to treat the representati- on (3.9) using the symplectic structure that arises from the above analysis on the invariant submanifold MN ⊂ M. 4. Symplectic analysis and Hamiltonian formulation. Let us put, into the expression (3.9), the condition grad L[u] = 0 for all u = MN . Then the following equality is satisfied: dL = d dx α(1), α(1) = N∑ j=0 〈 pj , du(j) 〉 , (4.1) since the function h(x) := ∑N j=0〈pj , u (j+1)〉−L(u, . . . , u(N+1)) satisfies the condition dh(x)/dx = = −〈gradL[u], u(1)〉 for all x = R, owing to the relations (3.7). Taking now the external deri- vative of (4.1), we obtain that d dx Ω(2) = 0, Ω(2) = dα(1), (4.2) where we have used the well known identity d · d dx = d dx · d. From (4.2) we can conclude that the vector field d/dx on the submanifold MN ⊂ M is Hamiltonian with respect to the canonical symplectic structure Ω(2) = ∑N j=0〈dpj ∧ du(j)〉. It is a very simple exercise to see that the function h(x) : J (2N+2)(R; Rm) → R defined above plays a role of the corresponding Hamiltonian function for the vector field d/dx on MN , i.e., the equation dh(x) = −i d dx Ω(2) (4.3) holds on MN . Therefore, we have got the following theorem. Theorem 4.1. The critical submanifold MN ⊂ M defined by (3.1) for a given non degenerate smooth functional L = D(M) ⊂ D(J (N+1)(R;Rm)), being imbedded into the jet-manifold J (∞)(R;Rm), carries a canonical symplectic structure such that the induced vector field d/dx on MN is Hamiltonian. The theorem analogous to the above was stated before in different terms by many authors [6, 23]. Our derivation presented here is much simpler and constructive, giving rise to all ingredi- ents of symplectic theory, stemming from imbedding the invariant submanifold MN into the jet-manifold. Now we are going to proceed further to studying the vector field (2.1) on the manifold MN ⊂ M endowed with the symplectic structure Ω(2) = Λ2(J (N+1)(R; Rm)), constructed via the formula (4.2). We have the following implications: dL dt = 0 ⇒ 〈gradL[u],K[u]〉 = −dh(t) dx , (4.4) dL dx = 0 ⇒ 〈 gradL[u], du dx 〉 = −dh(x) dx , ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 365 where the functions h(t) and h(x) serve as corresponding Hamiltonian ones for the vector fields d/dt and d/dx. This means in part that the following equations hold: dh(x) = −i d dx Ω(2), dh(t) = −i d dt Ω(2). (4.5) To prove the above statement (4.5), we shall build the following quantities (for the vector field d/dx at first): d · i d dx 〈gradL[u], du〉 = − d dx ( dh(x) ) (4.6) stemming from (4.4), and i d dx · d 〈gradL[u], du〉 = − d dx ( i d dx Ω(2) ) (4.7) stemming from (3.9), where we used the above mentioned evident identity [ i d dx , d dx ] = 0. Adding now the expressions (4.6) and (4.7) entails the following one: d dx 〈gradL[u], du〉 = − d dx (dh(x) + i d dx Ω(2)) (4.8) for all x = R and u = M . Since grad L[u] = 0 for all u = MN , we obtain from (4.8) that the first equality in (4.5) is valid in the case of the vector field d/dx reduced on MN . The analogous procedure fits also for the vector field d/dt reduced on the manifold MN ⊂ M. The even difference of the procedure above stems from the condition on the vector fields d/dt and d/dx to be commutative, [d/dt, d/dx] = 0, which gives the needed identity [ i d dt , d dx ] ≡ i[ d dt , d dx ] = 0 as a simple consequence of the procedure considered above. This completes the proof of equations (4.5). Theorem 4.2. Dynamical systems d/dt and d/dx reduced on the invariant submanifold MN ⊂ M (3.1) are Hamiltonian ones with the corresponding Hamiltonian functions built from the equations (4.4) in a unique way. By the way, we have stated also that the Hamiltonian functions h(x) and h(t) on the submani- fold MN ⊂ M commute with each other, that is, {h(t), h(x)} = 0, where {·, ·} is the Poisson structure on D(MN ) corresponding to the symplectic structure (4.2). This indeed follows from the equalities (4.4), since {h(t), h(x)} = dh(x) dt = −dh(t) dx ≡ 0 on the manifold MN ⊂ M. 5. Symmetry invariance. Let us consider now any vector field Kj : M → T (M), j = 1, k, that are symmetry fields related to the given vector field (2.1), i.e., [K, Kj ] = 0, j = 1, k. Since the conservation law L ∈ D(M) for the vector field (2.1) need not to be such for the vector fields Kj , j = 1, k, the submanifold MN ⊂ M need not to be invariant also with respect to these vector fields. Therefore, if a vector field X ∈ T (MN ), the vector field [Kj , X] /∈ T (MN ) in general, if d dtj , j = 1, k, are chosen as symmetries of (2.1). Let us consider the following identity ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 366 YA. A. PRYKARPATSKY, A. M. SAMOILENKO for some functions h̃j : J (2N+2)(R; Rm) → R, j = 1, k, which follow from the conditions[ d dt , d dtj ] = 0, j = 1, k, on M : d dt i d dtj 〈gradL[u], du〉 = −dh̃j [u] dx . (5.1) Lemma 5.1. The functions h̃j [u], j = 1, k, reduced on the invariant submanifold MN ⊂ M turn into constant ones. These constants can be chosen obviously as zeros. Proof. We have [ d dt , i d dtj ] = 0, j = 1, k, hence i d dtj ( i d dt d + di d dt ) 〈gradL[u], du〉 = −dh̃j dx ⇒ ⇒ i d dtj ( i d dt d 〈gradL[u], du〉+ di d dt 〈gradL[u], du〉 ) = = −i d dtj i d dt d dx Ω(2) − i d dtj d dx ( dh(+) ) = − d dx i d dtj ( i d dt Ω(2) + dh(t) ) = −dh̃j dx . Whence we obtain that on the whole jet-manifold M ⊂ J∞(R; Rm) the following identities hold: i d dtj ( dh(t) + i d dt Ω(2) ) = h̃j (5.2) for all j = 1, k. Since on the submanifold MN ⊂ M one has i d dt Ω(2) = −dh(t), we find that h̃j ≡ 0, j = 1, k, which proves the lemma. Note 5.1. The result above could be stated also using the standard functional operator calculus of [6]. Indeed, d dt ( i d dtj 〈gradL[u], du〉 ) = d dt 〈gradL[u],Kj [u]〉 = = 〈 d dt gradL[u],Kj [u] 〉 + 〈 gradL[u], d dt Kj [u] 〉 = = 〈 −K ′∗gradL[u],Kj [u] 〉 + 〈 gradL[u],K ′ j ·K[u] 〉 = = − 〈 K ′∗gradL[u],Kj [u] 〉 + 〈 gradL[u],K ′ ·Kj [u] 〉 = = − d dx H [gradL[u],Kj [u]] = −dh̃j [u] dx . (5.3) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 367 Here the bilinear form H{·.·} is found via the usual definition of the adjoined operator K ′∗ for a given operator K ′ : L2 → L2 with respect to the natural scalar product (·, ·), ( K ′∗a, b ) := (a,K ′b), (a, b) := ∫ R dx〈a, b〉, (5.4) whence we simply obtain 〈 K ′∗a, b 〉 − 〈 a,K ′b 〉 = dH[a, b] dx (5.5) for all a, b ∈ L2. Therefore, we can identify h̃j [u] = H[gradL[u],Kj [u]] for all u ∈ M, j = 1, k. If u ∈ MN , we thereupon obtain that h̃j [u] ≡ 0, j = 1, k, that was needed to prove. As a result of the lemma proved above one gets the following: the functions h̃j [u], j = = 1, k, can not serve as nontrivial Hamiltonian functions for the dynamical systems d/dtj , j = 1, k, on the submanifold MN ⊂ M. To overcome this difficulty we assume the invariant submanifold MN ⊂ M to possess some additional symmetries d/dtj , j = 1, k, which satisfy the following characteristic criterion: L d dtj gradL[u] = 0, j = 1, k, for all u ∈ MN . This means that for j = 1, k, L d dtj gradL[u] = Gj(gradL[u]), (5.6) where Gj(·), j = 1, k, are some linear vector-valued functionals on T ∗(M). Otherwise, equati- ons (5.6) are equivalent to the following: i d dtj 〈gradL[u], du〉 = −dhj [u] dx + gj(gradL[u]), (5.7) where gj(·), j = 1, k, are some scalar linear functionals on T ∗(M). From (3.9) and (5.7) we hence find that for all j = 1, k L d dtj 〈gradL[u], du〉 = − d dx ( dhj [u] + i d dtj Ω(2) ) + dgj(gradL[u]). (5.8) If we put now u ∈ MN , that is, gradL[u] = 0, we will immediately find the following: for all j = 1, k, dhj [u] + i d dtj Ω(2) = 0. (5.9) Whence we can make a conclusion that the vector fields d/dtj , j = 1, k, are Hamiltonian too on the submanifold MN ⊂ M. Since dhj/dx = {h(x), hj} = 0, j = 1, k, on the manifold MN , we obtain that dh(x)/dtj = 0, j = 1, k. This is also an obvious corollary of the commutativity [d/dtj , d/dx] = 0, j = 1, k, for all x, tj ∈ R on the whole manifold M. Indeed, in the general case we have the identity {h(x), hj} = i[ d dtj , d dx ]Ω (2), whence the equalities {h(x), hj} ≡ 0 hold ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 368 YA. A. PRYKARPATSKY, A. M. SAMOILENKO for all j = 1, k on the submanifold MN ⊂ M, since [ d dtj , d dx ] = 0 on MN due to (5.8). The analysis carried out above makes it possible to treat given vector fields d/dtj , j = 1, k, satisfying either conditions (5.6) or conditions (5.7) on the canonical symplectic jet-submanifold MN ⊂ M analytically as Hamiltonian systems. 6. Liouville integrability. Now we suppose that the vector field d/dtj , j = 1, k, are indepen- dent and commute both with each other on the jet-submanifold MN ⊂ M and with the vector fields d/dt and d/dx on the manifold M. Besides, the submanifold MN ⊂ M is assumed to be compact and smoothly imbedded into the jet-manifold J (∞)(R; Rm). If the dimension dim MN = 2k + 4, Liouville theorem [6, 23] implies that the dynamical systems d/dx and d/dt are Hamiltonian and integrable by quadratures on the submanifold MN ⊂ M. This is the case for all Lax integrable nonlinear dynamical systems of the Korteweg – de Vries-type [4 – 6, 23] on spatially one-dimensional functional manifolds. 7. Discrete dynamical systems. Let there be given a differential discrete smooth dynamical system dun dt = Kn[u] (7.1) with respect to a continuous evolution parameter t ∈ R on an infinite-dimensional discrete manifold M ⊂ L2(Z; Rm) of infinite vector-sequences under the condition of rapid decrease in n ∈ Z, supn∈Z \|n\| k‖un‖Rm < ∞ for all k ∈ Z+ at each point u = (. . . , un, un+1, . . .) ∈ M, where un ∈ Rm, n ∈ Z. Assume further that the dynamical system (7.1) possesses a conservation law L ∈ D(M), that is, dL/dt = 0 along orbits of (7.1). Via a standard operator analysis one gets, from (3.5), the variational derivative of the functional L := ∑ n∈Z Ln[u] : gradLn := δL[u] δun = L′n ∗[u] · 1, (7.2) where the last right-hand operation of multiplying by unity is to be done by component. Lemma 7.1. Let Λ(M) be the infinite-dimensional Grassmannian algebra on the manifold M. Then the differential dLn[u] ∈ Λ1(M) enjoys the following reduced representation: dLn[u] = 〈gradLn, dun〉+ dα (1) n [u] dn , (7.3) where the one-form α (1) n [u] ∈ Λ1(M) is determined in a unique way, 〈·, ·〉 is the usual scalar product in Rm and d/dn = ∆− 1, ∆ is the usual shift operator. Proof. By definition we obtain for the external differential dLn[u] the following chain of ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 369 representations for each n ∈ Z : dLn[u] = N∑ k=0 〈 ∂Ln[u] ∂un+k , dun+k 〉 = = N∑ k=0 k∑ s=0 d dn 〈 ∂Ln−s[u] ∂un+k−s , dun+k−s 〉 + N∑ k=0 〈 ∂Ln−k[u] ∂un , dun 〉 = = d dn N∑ k=0 k∑ s=0 〈 ∂Ln−s[u] ∂un+k−s , dun+k−s 〉 + N∑ k=−N 4−k 〈 ∂Ln[u] ∂un+k , dun 〉 = = d dn N∑ k=0 k∑ s=0 〈 ∂Ln−s[u] ∂un+k−s , dun+k−s 〉 + 〈 L′n ∗ .1, dun 〉 = = d dn α(1) n [u] + 〈gradLn[u], dun〉 , (7.4) where N ∈ Z+ is a fixed number depending on the jet-form of the functional L ∈ D(M), α(1) n [u] = N∑ k=0 k∑ s=0 〈 ∂Ln−s[u] ∂un+k−s , dun+k−s 〉 = N∑ k=0 k∑ j=0 〈 ∂Ln+j−k[u] ∂un+j , dun+j 〉 , (7.5) and gradLn[u] = L′n ∗ · 1 = N∑ k=0 ∂Ln−k[u] ∂un . The latter equality in (7.4) proves Lemma 7.1 completely. The proved above representation (7.3) gives rise to the following stationary problem being posed on the manifold M : MN = {u ∈ M : gradLn = 0} (7.6) for all n ∈ Z, where by definition det ∥∥∥∥ ∂2Ln[u] ∂uN+1∂uN+1 ∥∥∥∥ = 0. In virtue of (7.3) we obtain the validity of the following theorem. Theorem 7.1. The finite-dimensional Lagrangian submanifold MN ⊂ M defined by (7.6), is a symplectic one with the canonical symplectic structure Ω(2) n = dα (1) n that is independent of the discrete variable n ∈ Z. Proof. From (7.3) we have that on the manifold MN ⊂ M, dLn[u] = dα (1) n [u]/dn, whence for all n ∈ Z, dΩ(2) n /dn = 0. This obviously means that Ω(2) n+1 = Ω(2) n for all n ∈ Z, or, equivalently, the 2-form Ω(2) n does not depend on the discrete variable n ∈ Z. As the 2-form Ω(2) n := dα (1) n by definition, this form is chosen to be symplectic on the manifold MN ⊂ M. For this 2-form to be ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 370 YA. A. PRYKARPATSKY, A. M. SAMOILENKO nondegenerate on MN , we assume that the Hessian of Ln equals det ∥∥∥∥ ∂2Ln[u] ∂un+N+1∂un+N+1 ∥∥∥∥ 6= 0 on MN . The latter proves the theorem. Let us consider now the dynamical system (7.1) reduced on the submanifold MN ⊂ M. To present it as the vector field d/dt on MN , we need at first to represent it as a Hamiltonian flow on MN . To do this, let us write the following identities on M : i d dt d 〈gradLn, dun〉 = − d dn i d dt Ω(2) n [u], (7.7) di d dt 〈gradLn, dun〉 = − d dn (dh(t) n [u]), which are valid for all n ∈ Z. Adding the last identities in (7.7), we come to the following one for all n ∈ Z : d dt 〈gradLn, dun〉 = − d dn ( i d dt Ω(2) n [u] + h(t) n [u] ) . (7.8) Having reduced the identity (7.8) on the manifold MN ⊂ M, we obtain the needed expression for all u ∈ MN , N ∈ Z, i d dt Ω(2) n [u] + h(t) n [u] = 0. (7.9) The latter means that the dynamical system (7.1) on the manifold MN is a Hamiltonian one, with the function h (t) n [u] being a Hamiltonian function defined explicitly by the second identity in (7.7). We assume now that the symplectic structure Ω(2) n [u] on MN can be represented as follows: Ω(2) n [u] = N∑ j=0 〈dpj+n ∧ duj+n〉 , (7.10) where the generalized coordinates pj+n ∈ Rm, j = 0, N, are determined from the following discrete jet-expression Ln[u] := L(un, un+1, . . . , un+N+1), n ∈ Z, α(1) n [u] := N∑ j=0 〈pj+n, duj+n〉 = N∑ k=0 k∑ j=0 〈 ∂Ln+j−k ∂un+j , dun+j 〉 = = N∑ j=0 N∑ k=j 〈 ∂Ln+j−k ∂un+j , dun+j 〉 , whence we get the final expression pj+n := N∑ k=j ∂Ln+j−k[u] ∂un+j , (7.11) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 371 where j = 0, N, u ∈ MN ⊂ M. Now we are in a position to reformulate the given dynamical system (7.1) as that on the reduced manifold MN ⊂ M, dun+j dt = {h(t) n , un+j} = ∂h (t) n ∂pn+j , dpn+j dt = {h(t) n , pn+j} = − ∂h (t) n ∂un+j (7.12) for all n ∈ Z, j = 0, N. Thereby the problem of embedding a given discrete dynamical system (7.1) into a vector field flow on the manifold MN ⊂ M is solved completely with the final result (7.12). 8. Invariant Lagrangians construction: functional manifold case. In the case where the gi- ven nonlinear dynamical system (2.1) is integrable one of Lax-type, we can proceed effecti- vely to find a commuting infinite hierarchy of conservation laws that can serve as the invariant Lagrangians considered above. At first we have to use the important property [4] of the complexified gradient functional ϕ = grad γ ∈ T ∗(M) ⊗ C generated by an arbitrary conservation law γ ∈ D(M), i.e., the following Lax-type equation: dϕ dt + K ′∗ϕ = 0, (8.1) where the prime sign denotes the usual Frechet derivative of the local functional K : M → → T (M) on the manifold M, the star ”*” denotes its conjugation operator with respect to the nondegenerate standard convolution functional (·, ·) = ∫ R dx〈·, ·〉 on T ∗(M) × T (M). The equation (8.1) admits, which follows from [24 – 26], the special asymptotic solution, ϕ(x, t;λ) ∼= (1, a(x, t;λ))τ exp[ω(x, t;λ) + ∂−1σ(x, t;λ)], (8.2) where a(x, t;λ) ∈ Cm−1, σ(x, t;λ) ∈ C, ω(x, t;λ) is some dispersive function. The sign ”τ ” denotes here the transposition used in the matrix analysis. For any complex parameter λ ∈ C, at \|λ\| → ∞, the following expansions take place: a(x, t;λ) ' ∑ j∈Z+ aj [x, t;u]λ−j+s(a), σ(x, t;λ) ' ∑ j∈Z+ σj [x, t;u]λ−j+s(σ). Here s(a) and s(σ) ∈ Z+ are some appropriate nonnegative integers, the operation ∂−1 means the inverse to the differentiation d/dx, that is, d/dx · ∂−1 = 1 for all x ∈ R. To find an explicit form of the representation (8.2) in the case when the associated Lax-type representation [6] depends parametrically on the spectral parameter λ(t; z) ∈ C, satisfying the following nonisospectral condition: dλ(t; z) dt = g(t;λ(t; z)), λ(t; z)\|t=0+ = z ∈ C, (8.3) for some meromorphic function g(t; ·) : C → C, t ∈ R+, we must analyze more carefully the asymptotic solutions to the Lax equation (8.1). Namely, we are going to treat more exactly the ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 372 YA. A. PRYKARPATSKY, A. M. SAMOILENKO case when the solution ϕ ∈ T ∗(M) to (8.1) is represented as an appropriate trace-functional of a Lax spectral problem at the moment τ = t ∈ R+ with the spectral parameter λ(t;λ) ∈ C satisfying the condition (8.3), the evolution of the given dynamical system (2.1) is considered with respect to the introduced above parameter τ ∈ R, that is, du dτ = K[x, τ ;u], (8.4) u\|τ=0 = ū ∈ M is some Cauchy data on M. This means that the functional ϕ̃(x, τ ; λ̃) := reg grad SpS(x, τ ; λ̃), λ̃ = λ̃(τ ;λ(t; z)) ∈ C, (8.5) where S(x, τ ; λ̃) is the monodrony matrix corresponding to a Lax-type spectral problem assumed to exist, has to satisfy the corresponding Lax equation at any point u ∈ M subject to (8.4), dϕ̃ dτ + K ′∗[u] · ϕ̃ = 0 (8.6) for all τ ∈ R+. Under the above assumption it is obvious that the spectral parameter λ̃ = = λ̃(τ ;λ(t; z)), where dλ̃ dτ = g̃(τ ; λ̃), λ̃ ∣∣∣ τ=0 = λ(t; z) ∈ C, (8.7) g̃(t; ·) : C → C is some meromorphic function found simply from (8.6) for instance at u = 0, the Cauchy data λ(t; z) ∈ C, for all t ∈ R+, corresponds to (8.3), the parameter z ∈ C is a spectrum value of the associate Lax-type spectral problem at a moment t ∈ R+. Now we are in a position to formulate the following lemma. Lemma 8.1. The Lax equation (8.6), as the parameter τ = t ∈ R+, admits an asymptotic solution in the form ϕ̃ ( x, τ ; λ̃ ) ∼= ( 1, ã ( x, τ ; λ̃ ))τ exp [ ω̃ ( x, τ ; λ̃ ) + ∂−1σ̃ ( x, τ ; λ̃ )] , (8.8) where ã(x, τ ; λ̃) ∈ Cm−1, σ̃(x, τ ; λ̃) ∈ C, are some local functionals on M, ω̃(x, τ ; λ̃) ∈ C is some dispersion function for all x ∈ R, τ ∈ R+, and if for \|λ\| → ∞ the property \|λ̃\| → ∞ as τ = t ∈ R+ holds, the following expansions follows: ã(x, τ ; λ̃) ' ∑ j∈Z+ ãj [x, τ ;u]λ̃−j+s(ã), σ̃(x, τ ; λ̃) ' ∑ j∈Z+ σ̃j [x, τ ;u]λ̃−j+s(σ̃), (8.9) with s(ã) and s(σ̃) ∈ Z+ being some integers. Proof. In virtue of the theory of asymptotic expansions for arbitrary differential spectral problems, the result (8.8) will hold provided the representation (8.5) is valid and the spectral parameter λ(t; z) ∈ C is taken subject to (8.7). But this is the case because of the Lax-type integrability of the dynamical system (8.4). Further, due to the mentioned above integrability of (8.4), as well as to the well known Stokes property of asymptotic solutions to linear equations ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 373 like (8.1), the condition (8.3) holds for some meromorphic function g(t; ·) : C → C, t ∈ R+, enjoing the determining property d dt ∫ R σ̃(x, t; λ̃(t;λ(t; z)))dx = 0 for all t ∈ R+. The latter proves the lemma completely. As a result of Lemma 8.1 one can formulate the following important theorem. Theorem 8.1. The Lax integrable parametrically isospectral dynamical system (8.4), as τ = = t ∈ R+, admits an infinite hierarchy of conservation laws, in general nonuniform with respect to the variables x ∈ R, τ ∈ R+, which can be represented in an exact form in virtue of the asymptotic expansion (8.8) and (8.9). Proof. Indeed, due to the expansion (8.8), we can obtain right away that the functional γ̃(t;λ(t; z)) = ∫ R dxσ̃ ( x, t; λ̃ (t;λ(t; z)) ) (8.10) does not depend on the parameter t ∈ R+ at τ = t ∈ R+, that is, dγ̃ dτ ∣∣∣ τ=t∈R+ = 0 (8.11) for all t ∈ R+. If we also make the parameter τ ∈ R+ tend to t ∈ R+, due to (8.5) we obtain that ϕ̃(x, τ ; λ̃) ∣∣ τ=t∈R+ → ϕ(x, t;λ) for all x ∈ R, t ∈ R+, and λ(t; z) ∈ C. This means that a complexified local functional ϕ(x, t; z) ∈ T ∗(M) ⊗ C satisfies the equation (8.1) at each point u ∈ M. As an obvious result, the following identifications hold: ω̃(x, τ ; λ̃) ∣∣ τ=t∈R+ → ω(x, t; z), σ̃(x, τ ; λ̃) ∣∣ τ=t∈R+ → σ(x, t; z) for all z ∈ C. Hence, the functional γ(z) := γ̃(τ ;λ(t; z))\|τ=t∈R+ = ∫ R dxσ(x, t; z) ∈ D(M) doesn’t depend on the evolution parameter t ∈ R+ and, due to equation (8.1), is a conserved quantity for the nonlinear dynamical system (2.1) under consideration, i.e., dγ(t; z) dt = 0 (8.12) for all t ∈ R+ and z ∈ C. Therefore, this makes it possible to use the equation (8.12) jointly with (8.7) to find the asymptotic expansions (8.9) and (8.3) in an exact form. To do this, we first need to substitute the asymptotic expansion (8.8) in the determining equation (8.6) for the asymptotic expansions (8.9) to be found explicitly at the moment τ = t ∈ R+. Keeping in mind that, at τ = t ∈ R+, \|λ\| → ∞ if \|λ̃\| → ∞, and solving step by step the resulting recurrence relations for the coefficients in (8.9), we will get the functional γ(z) := γ̃(τ ;λ(t; z))\|τ=t∈R+ , z ∈ C, in the form fitting for the use the criteria equation (8.12). As the second step, we need to use the differential equation (8.7) as to satisfy the criteria equation (8.12) pointwise for all ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 374 YA. A. PRYKARPATSKY, A. M. SAMOILENKO t ∈ R+. This means, in particular, that dγ(z) dt = d dt ∑ j∈Z+ ∫ R dxσ̃j [x, τ ;u]λ̃−j+s(σ̃) ∣∣∣ τ=t∈R+  = = ∫ R dx ∑ j∈Z+ [ dσ̃j [x, τ ;u] dt λ̃−j+s(σ̃) + σ̃j [x, τ ;u]λ̃−j+s(σ̃)−1(s(σ̃)− j) dλ̃ dt ]∣∣∣∣∣ τ=t∈R+ ⇒ ⇒ ∫ R dx ∑ j∈Z+ [(dσ̃j dt ) λ̃−j+s(σ̃) + ∑ k>>−∞ (s(σ̃)− k)σ̃kg̃j−k−1(t)λ̃−j+s(σ̃) ∣∣∣ τ=t∈R+ + + ∑ j∈Z+ σ̃j [x, t;u]λ̃−j+s(σ̃)−1(s(σ̃)− j) ∂λ̃ ∂λ g(t;λ) ∣∣∣ τ=t∈R+ ] ≡ 0, (8.13) where we have put by definition g̃(τ ; λ̃) :' ∑ k>>−∞ g̃k(τ)λ̃−k for τ ∈ R+ and \|λ̃\| → ∞. Since the spectral parameter λ = λ(t; z), at the moment t = 0+, coincides with an arbitrary complex value z ∈ C, the condition \|z\| → ∞ together with (8.13) at the moment t = 0+ gives rise to the following recurrent relations: ∑ j∈Z+ [ ∂σ̃j dt + σ̃′j ·K[t;u] + ∑ k>>−∞ (s(σ̃)− k)σ̃k · g̃j−k−1 ] λ̃−j+s(σ̃) ∣∣∣ τ=t∈R+ = = ∑ j∈Z+ σ̃j(s(σ̃)− j) ∂λ̃ ∂λ g(t;λ)λ̃−j+s(σ̃)−1 ∣∣∣ τ=t∈R+ ≡ 0 (mod d/dx) (8.14) for all j ∈ Z+, x ∈ R, t ∈ R+, and u ∈ M. Having solved the algebraic relations (8.14) for the unknown function g(t;λ), t ∈ R+, we will obtain the generating functional γ(z), z ∈ C, of conservation laws for (2.1) in an exact form. This completes the constructive part of the proof of the theorem above. From the practial point of view we need first to get the differential equation (8.7) in an exact, maybe in an asymptotic form, and find further the dispersive function ω̃(x, t; λ̃) and the local generating functional σ̃(x, τ ; λ̃) defined via (8.8) and (8.9) for all x ∈ R, τ ∈ R+ and \|λ̃\| → ∞, and next one can find the equation (8.3) using the algorithm based on the relations (8.14). This, together with the possibility of applying the general scheme of the gradient-holonomic algo- rithm [27], gives rise to determining in many cases the above mentioned Lax-type representati- on completely in an exact form, which successfully solves the pretty complex direct problem of the integrability theory of nonlinear dynamical systems on functional manifolds. Having obtained the generation function γ(z) ∈ D(M), z ∈ C, of an infinite hierarchy of conservation laws of the dynamical system (2.1) on the manifold M, we can build appropriately ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 375 a general Lagrangian functional LN ∈ D(M) as follows: LN = −γN+1 + N∑ j=0 cjγj , (8.15) where, by definition, γ(z) = ∫ R dxσ(x, t; z) and, for \|z\| → ∞, the functionals γj = ∫ R dxσj [x, t; z], j ∈ Z+, are conservation laws due to expansion (8.2), with cj ∈ R, j = 0, N, are some arbitrary constants and N ∈ Z+ is an arbitrary but fixed nonnegative integer. If the differential order of the functional γN+1 ∈ D(M) is the highest among the orders of the functionals γj ∈ D(M), j = 0, N, and additionally, this Lagrangian is not degenerate, that is, det (Hess γN+1) 6= 0, we can apply in general amost all the theory developed before for proving that the critical submani- fold MN = {u ∈ M : grad LN = 0} is a finite-dimensional invariant manifold inserted into the standard jet-manifold J (∞)(R; Rm) with the canonical symplectic structure subject to which our dynamical system is a finite-dimensional Hamiltonian flow on the invariant submani- fold MN . 9. Invariant Lagrangian construction : discrete manifold case. Let us consider the discrete Lax integrable dynamical system on a discrete manifold M without an a priory given Lax-type representation. The problem arises of how to get the corresponding conservation laws via the gradient-holonomic algorithm [6]. To realize this, let us study solutions to the Lax equation dϕn dt + K ′ n[τ, u] · ϕn = 0, (9.1) where the local functionals ϕn[u] ∈ T ∗un (M) at the point un ∈ M, n ∈ Z. In analogy with the approach of Section 7, we assert that equation (9.1) admits a comlexified generating solution ϕn = ϕn(t;λ) ∈ T ∗un (M)⊗ C, n ∈ Z, with z ∈ C a complex parameter in the form ϕn(t; z) ∼= (1, an(t; z))τexp [ω(t; z)]  n∏ j=−∞ σj(t; z)  , (9.2) where ω(t; z) is some dispersive function for t ∈ R+, an(t; z) ∈ Cm−1, σn(t; z) ∈ R are local functionals on M with the following asymptotic expansions at \|z\| → ∞ : an(t; z) ' ∑ j∈Z+ an[t;u]z−j+s(a), σ(j) n (t; z) ' ∑ j∈Z+ σn[t;u]z−j+s(σ). (9.3) To find an explicit form of the asymptotic representation (9.2), we need to additionally study the asymptotic solutions to the following attached Lax equation with respect to the new evolution parameter τ ∈ R+, dϕ̃n dτ + K ′ n ∗[τ, u] · ϕ̃n = 0, (9.4) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 376 YA. A. PRYKARPATSKY, A. M. SAMOILENKO where ϕ̃n ∈ T ∗un (M) ⊗ C, and the point u ∈ M evolves subject to the following dynamical system: dun dτ = Kn[τ ;u] (9.5) for all n ∈ Z. Having made the assumption above we can assert, based on the general theory of asymptotic solutions to linear equations like (9.4), that it also admits, in general, another asymptotic solution in a similar form, ϕ̃n(τ ; λ̃) ∼= (1, ãn(τ ; λ̃))τexp[ω̃(τ ; λ̃)] n∏ j=−∞ σ̃j(τ ; λ̃), (9.6) where for all n ∈ Z and at τ ∈ R+, the asymptotic expansions ãn(τ ; λ̃) ' ∑ j∈Z+ ã(j) n [x, τ ;u]λ̃−j+s(ã), (9.7) σ̃n(τ ; λ̃) ' ∑ j∈Z+ σ̃(j) n [τ ;u]λ̃−j+s(σ̃) hold. The expansions above are valid if \|λ̃\| → ∞ as \|λ(t; z)\| → ∞, z ∈ C. The latter is the case because of the Lax integrability of the dynamical system (9.5). The evolution dλ̃ dτ = g̃(τ ; λ̃), λ̃ ∣∣ t=0 = λ(t; z) ∈ C, (9.8) where g̃(τ ; ·) : C → C is some meromorphic mapping for all τ ∈ R+, is, in general, found by making use of the corresponding solution to (9.4) at u = 0. Substituting the expansions (9.6) and (9.7) into (9.4), we obtain some recurrence relations that give rise to a possibility of finding exact expressions for local functionals σ̃j [t;un], j ∈ Z+. Having done this, we assert that the functional γ(t; z) = ∑ n∈Z ln σ̃n(τ ; λ̃) ∣∣ τ=t∈R+ ⇒ ∑ n∈Z lnσn(t; z), (9.9) where λ̃ = λ̃(τ ;λ), τ ∈ R+, and λ(t; z) ∈ C, is a meromorphic solution to the equation dλ dt = g(t;λ), λ\|t=0+ = z ∈ C (9.10) with still independent meromorphic function g(t; ·) for almost all t ∈ R+. The latter can be found by making use of the following determining condition: the local functional ϕ̃n(τ ; λ̃) ∣∣ τ=t∈R+ → ϕn(t; z) ∈ T ∗(M)⊗ C ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 377 for all t ∈ R+ and z ∈ C. Hence, the following equality holds immediately: d dt (∑ n∈Z+ ln σ̃n(τ ; λ̃) ∣∣ τ=t∈R+ ) = = ∑ n∈Z+ σ̃−1 n (t; λ̃) [ ∂σ̃n ∂t + σ̃′n ·Kn[u] + ∂σ̃n ∂λ̃ g(t; λ̃) ∣∣ τ=t∈R+ + + ∂σ̃n ∂λ̃ ∂λ̃ ∂λ g(t;λ) ∣∣ τ=t∈R+ ] = 0 (9.11) for all t ∈ R+. Equating the coefficients of (9.11) at all powers of the spectral parameter λ(t; z) ∈ C to zero modulus d/dn, n ∈ Z, we will find recurrence relations for the function g(t;λ) of (9.8). Thereby, using the equation (9.10) and the expansion σ(t; z) ' ∑ j∈Z+ σj [t;un]× ×z−j+s(γ) for \|z\| → ∞, where s(σ) ∈ Z+ is some integer, we obtain an infinite hierarchy of discrete conservation laws of the initially given nonlinear dynamical system (2.1) on the mani- fold M. But because the conservation laws built above parametrically depend on the evolution parameter t ∈ R+, we cannot use right now the theory developed before to prove the Hami- ltonian properties of the corresponding vector fields on the invariant submanifolds. To do this in an appropriate way, it is necessary to augment the theory developed before with some important details. 10. The reduction procedure on nonlocal Lagrangian submanifolds. 10.1. The general algeb- raic scheme. Let G̃ := C∞(S1;G) be a Lie algebra of loops, taking values in a matrix Lie algebra G. By means of G̃ one constructs the Lie algebra Ĝ of matrix integral-differential operators [28], â := ∑ j�∞ ajξ j , (10.1) where the symbol ξ := ∂/∂x denotes the differentiation with respect to the independent vari- able x ∈ R/2πZ ' S1. The usual Lie commutator on Ĝ is defined as:[ â, b̂ ] := â ◦ b̂− b̂ ◦ â (10.2) for all â, b̂ ∈ Ĝ, where "◦"is the product of integral-differential operators and takes the form â ◦ b̂ := ∑ α∈Z+ 1 α! ∂αâ ∂ξα ∂αb̂ ∂xα . (10.3) On the Lie algebra Ĝ there exists the ad-invariant nondegenerate symmetric bilinear form ( â, b̂ ) := 2π∫ 0 Tr (â ◦ b̂) dx, (10.4) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 378 YA. A. PRYKARPATSKY, A. M. SAMOILENKO where Tr-operation for all â ∈ Ĝ is given by the expression Tr â := resξ Sp â = Sp a−1, (10.5) with Sp being the usual matrix trace. With the scalar product (10.4) the Lie algebra Ĝ is trans- formed into a metrizable one. As a consequence, the linear space, dual to Ĝ, of the matrix integral-differential operators Ĝ∗ is naturally identified with the Lie algebra Ĝ, that is Ĝ∗ ' Ĝ. The linear subspaces Ĝ+ ⊂ Ĝ and Ĝ− ⊂ Ĝ such as Ĝ+ := â := n(â)�∞∑ j=0 ajξ j : aj ∈ G̃, j = 0, n(â)  , (10.6) Ĝ− := b̂ := ∞∑ j=0 ξ−(j+1)bj : bj ∈ G̃, j ∈ Z+  , are Lie subalgebras in Ĝ and Ĝ = Ĝ+ ⊕ Ĝ−. Because of the splitting of Ĝ into the direct sum of its Lie subalgebras one can construct the so-called Lie – Poisson structure [14, 27, 29, 30] on Ĝ∗, using a special linear endomorphismR of Ĝ : R := P+ − P− 2 , P±Ĝ := Ĝ±, P±Ĝ∓ = 0. (10.7) For any Frechet smooth functionals γ, µ ∈ D(Ĝ∗), the Lie – Poisson bracket on Ĝ∗ is given by the expression {γ, µ}R (l̂) = ( l̂, [∇γ(l̂),∇µ(l̂)]R ) , (10.8) where l̂ ∈ Ĝ∗ and for all â, b̂ ∈ Ĝ theR-commutator in (10.8) has the form [14, 27][ â, b̂ ] R := [ Râ, b̂ ] + [ â,Rb̂ ] , (10.9) subject to which the linear space Ĝ becomes a Lie algebra too. The gradient ∇γ(l̂) ∈ Ĝ of a functional γ ∈ D(Ĝ∗) at a point l̂ ∈ Ĝ∗ with respect to the scalar product (10.4) is defined as δγ(l̂) := ( ∇γ(l̂), δl̂ ) , (10.10) where the linear space isomorphism Ĝ ' Ĝ∗ is taken into account. The Lie – Poisson bracket (10.8) generates Hamiltonian dynamical systems on Ĝ∗ related to Casimir invariants γ ∈ I(G∗) and satisfying the condition[ ∇γ(l̂), l̂ ] = 0, (10.11) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 379 as the corresponding Hamiltonian functions. Due to the expressions (10.8) and (10.11) the Hamiltonian system mentioned above takes the form dl̂ dt := [ R∇γ(l̂), l̂ ] = [ ∇γ+(l̂), l̂ ] , (10.12) being equivalent to the usual commutator Lax-type representation [27, 31]. The relation (10.12) is a compatibility condition for the linear integral-differential equations l̂f = λf, (10.13) df dt = ∇γ+(l̂)f, where λ ∈ C is a spectral parameter and the vector-function f ∈ W (S1;H) is an element of some matrix representation for the Lie algebra Ĝ in some functional Banach space H. Algebraic properties of the equation (10.12) together with (10.14) and the associated dynami- cal system on the space of adjoint functions f∗ ∈ W ∗(S1;H), df∗ dt = − ( ∇γ(l̂) )∗ + f∗, (10.14) where f∗ ∈ W ∗ is a solution to the adjoint spectral problem l̂∗f∗ = νf∗, (10.15) considered as some coupled evolution equations on the space Ĝ∗⊕W ⊕W ∗, is an object of our further investigation. 10.2. The tensor product of Poisson structures and its Backlund transformation. To compacti- fy the description below we will use the following notation for the gradient vector: ∇γ ( l̃, f̃ , f̃∗ ) := ( δγ δl̃ , δγ δf̃ , δγ δf̃∗ )T for any smooth functional γ ∈ D(Ĝ∗ ⊕W ⊕W ∗). On the spaces Ĝ∗ and W ⊕W ∗ there exist canonical Poisson structures [27, 30, 32] δγ δl̃ : θ̃→ [( δγ δl̃ ) + , l̃ ] − [ δγ δl̃ , l̃ ] + (10.16) at a point l̃ ∈ Ĝ∗ and ( δγ δf̃ , δγ δf̃∗ )T : J̃→ ( δγ δf̃∗ , −δγ δf̃ )T (10.17) at a point (f̃ , f̃∗) ∈ W ⊕ W ∗ correspondingly. It should be noted that the Poisson structure (10.17) is transformed into (10.12) for any Casimir functional γ ∈ I(Ĝ∗). Thus, on the extended ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 380 YA. A. PRYKARPATSKY, A. M. SAMOILENKO space Ĝ∗⊕W ⊕W ∗ one can obtain a Poisson structure as the tensor product Θ̃ := θ̃⊗ J̃ of the structures (10.17) and (10.18). Let us consider the following Backlund transformation [27, 32, 33]:( l̂, f, f∗ ) : B→ ( l̃(l̂, f, f∗), f̃ = f, f̃∗ = f∗ ) , (10.18) generating on Ĝ∗ ⊕W ⊕W ∗ a Poisson structure Θ with respect to the variables (l̂, f, f∗) of the coupled evolution equations (10.12), (10.14), (10.15). The main condition for the mapping (10.19) to be defined is the coincidence of the dynami- cal system ( dl̂ dt , df dt , df∗ dt )T := −Θ∇γ(l̂, f, f∗) (10.19) with (10.12), (10.14), (10.15) in the case of γ ∈ I(Ĝ∗), i.e., if this functional is taken to be not dependent of the variables (f, f∗) ∈ W ⊕W ∗. To satisfy this condition, one has to find a variation of any smooth Casimir functional γ ∈ I(Ĝ∗) as δl̃ = 0, considered as a functional on Ĝ∗ ⊕W ⊕W ∗, taking into account flows (10.14), (10.15) and the Backlund transformation (10.19), δγ(l̃, f̃ , f̃∗) ∣∣ δl̃=0 = (〈δγ δf̃ , δf̃ 〉) + (〈 δγ δf̃∗ , δf̃∗ 〉) = = (〈 −df̃∗ dt , δf̃ 〉) + (〈df̃ dt , δf̃∗ 〉)∣∣∣∣∣ f̃=f, f̃∗=f∗ = = (〈(δγ δl̂ )∗ + f∗, δf 〉) + (〈(δγ δl̂ ) + f, δf∗ 〉) = = (〈 f∗, (δγ δl̂ ) + δf 〉) + (〈(δγ δl̂ ) + f, δf∗ 〉) = = ( δγ δl̂ , δfξ−1 ⊗ f∗ ) + ( δγ δl̂ , fξ−1 ⊗ δf∗ ) = = ( δγ δl̂ , δ(fξ−1 ⊗ f∗) ) := ( δγ δl̂ , δl̂ ) . (10.20) As a result of the expression (10.21) one obtains the relations δl̂ ∣∣ δl̃=0 = δ(fξ−1 ⊗ f∗), (10.21) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 381 or, having assumed the linear dependence of l̂ and l̃ ∈ Ĝ∗, one gets right away that l̂ = l̃ + fξ−1 ⊗ f∗. (10.22) Thus, the Backlund transformation (10.19) can now be written as (l̂, f, f∗) : B→ ( l̃ = l̂ − fξ−1 ⊗ f∗, f̃ = f, f̃∗ = f̃∗ ) . (10.23) The expression (10.24) generalizes the result obtained in the papers [27, 33] for the Lie algebra Ĝ of integral-differential operators with scalar coefficients. The existence of the Backlund trans- formation (10.19) makes it possible to formulate the following theorem. Theorem 10.1. A dynamical system on Ĝ∗ ⊕W ⊕W ∗, being Hamiltonian with respect to the canonical Poisson structure Θ̃ : T ∗(Ĝ∗ ⊕W ⊕W ∗) → T (Ĝ∗ ⊕W ⊕W ∗), and generated by the evolution equations: dl̃ dt = [ ∇γ+(l̃), l̃ ] − [ ∇γ(l̃), l̃ ] + , df̃ dt = δγ δf̃∗ , df̃∗ dt = −δγ δf̃ , (10.24) with γ ∈ I(G∗) being the Casimir functional at l̂ ∈ Ĝ∗ connected with l̃ ∈ Ĝ∗ by (10.23), is equi- valent to the system (10.12), (10.14) and (10.15) via the constructed above Backlund transformati- on (10.24). By means of simple calculations via the formula (see e. g. [27, 30]) Θ̃ = B ′ ΘB ′∗, where B ′ : T (Ĝ∗ ⊕W ⊕W ∗) → T (Ĝ∗ ⊕W ⊕W ∗) is the Frechet derivative of (10.24), one brings about the following form of the Poisson structure Θ on Ĝ∗ ⊕W ⊕W ∗ 3 (l̂, f, f∗) : ∇γ(l̂, f, f∗) : Θ→  [ l̂, (δγ δl̂ ) + ] − [ l̂, δγ δl̂ ] + − −fξ−1 ⊗ δγ δf + δγ δf∗ ξ−1 ⊗ f∗ δγ δf∗ − (δγ δl̂ ) + f − δγ δf + (δγ δl̂ )∗ + f  . This permits to formulate the next theorem. Theorem 10.2. The dynamical system (10.20), being Hamiltonian with respect to the Poisson structure Θ in the form (10.26) and a functional γ ∈ I(Ĝ∗), gives the inherited Hamiltonian representation for the coupled evolution equations (10.12), (10.14), (10.15). By means of the expression (10.23) one can construct Hamiltonian evolution equations describe commutative flows on the extended space Ĝ∗ ⊕W ⊕W ∗ at a fixed element l̃ ∈ Ĝ∗. ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 382 YA. A. PRYKARPATSKY, A. M. SAMOILENKO Due to (10.24) every equation of such a type is equivalent to the system dl̂ dτn = [ l̂n+, l̂ ] , df dτn = l̂n+f, (10.25) df∗ dτn = −(l̂∗)n +f∗, generated by the Casimir invariants γn ∈ I(Ĝ∗), n ∈ N, involutive with respect to the Poisson bracket (10.17) and taking here the standard form γn = 1/(n + 1)(l̂n, l̂) at l̂ ∈ Ĝ∗. The compatibility conditions for the Hamiltonian systems (10.25) for different n ∈ Z+ can be used for obtaining Lax integrable equations on usual spaces of smooth 2π-periodic multi- variable functions that will be done in the next section. 10.3. The Lax-type integrable Davey – Stewartson equation and its triple linear representati- on. Choose the element l̃ ∈ Ĝ∗ in an exact form such as l̃ = ( 1 0 0 −1 ) ξ − ( 0 u ū 0 ) , where u, ū ∈ C∞(S1; C) and G = gl (2; C). Then l̂ = l̃ + ( f1ξ −1f∗1 f1ξ −1f∗2 + u ū + f2ξ −1f∗1 f2ξ −1f∗2 ) , (10.26) where f = (f1, f2)T and f∗ = (f∗1 , f∗2 )T , "−"can denote the complex conjugation. Below we will study the evolutions (10.25) of vector-functions (f, f∗) ∈ W (S1; C2) ⊕ W ∗(S1; C2) with respect to the variables y = τ1 and t = τ2 at the point (10.26). They can be obtained from the second and the third equations in (10.25) by putting n = 1 and n = 2, as well as from the first one. The latter is the compatibility condition of the spectral problem l̂Φ = λΦ, (10.27) where Φ = (Φ1,Φ2)T ∈ W (S1; C2), λ ∈ C is some parameter, with the following linear equati- ons: dΦ dy = l̂+Φ, (10.28) dΦ dt = l̂2+Φ, (10.29) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 383 arising from (10.26) at n = 1 and n = 2 correspondingly. The compatibility of equations (10.28) and (10.29) leads to the relations ∂u ∂y = −2f1f ∗ 2 , ∂ū ∂y = −2f∗1 f2, ∂f1 ∂y = ∂f1 ∂x − uf2, ∂f∗1 ∂y = ∂f∗1 ∂x − ūf∗2 , (10.30) ∂f2 ∂y = −∂f2 ∂x + ūf1, ∂f∗2 ∂y = −∂f∗2 ∂x + uf∗1 . Analogously, replacing t ∈ C by it ∈ iR, i2 = −1, one gets from (10.29) and (10.30): du dt = i ( ∂2u ∂x∂y + 2u(f1f ∗ 1 + f2f ∗ 2 ) ) , dū dt = −i ( ∂2ū ∂x∂y + 2ū(f1f ∗ 1 + f2f ∗ 2 ) ) , ∂(f1f ∗ 1 ) ∂y − ∂(f1f ∗ 1 ) ∂x = 1 2 ∂(uū) ∂y = − ( ∂(f2f ∗ 2 ) ∂x + ∂(f2f ∗ 2 ) ∂y ) , df1 dt = i ( ∂2f1 ∂x2 + (2f1f ∗ 1 − uū)f1 − ∂u ∂x f2 ) , (10.31) df∗1 dt = −i ( ∂2f∗1 ∂x2 + (2f1f ∗ 1 − uū)f∗1 − ∂ū ∂x f∗2 ) , df2 dt = i ( ∂2f2 ∂x2 − (2f2f ∗ 2 + uū)f2 − ∂ū ∂x f1 ) , df∗2 dt = −i ( ∂2f∗2 ∂x2 − (2f2f ∗ 2 + uū)f∗2 − ∂u ∂x f∗1 ) . The relations (10.31), (10.32) take the well known form of the Davey – Stewartson equation [30, 34] at ū ∈ C∞(S1; C), which is the complex conjugate of u ∈ C∞(S1; C). The compatibility for every pair of equations (10.28), (10.29) and (10.30), which can be rewritten as the first order ordinary linear differential equations as dΦ dx =  λ u −f1 ū −λ f2 f∗1 f∗2 0 Φ, (10.32) ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 384 YA. A. PRYKARPATSKY, A. M. SAMOILENKO dΦ dy =  λ 0 −f1 0 λ −f2 f∗1 f∗2 0 Φ, (10.33) dΦ dt = i  λ2 + f1f ∗ 1 1 2 ∂u ∂y −λf1 − ∂f1 ∂y −1 2 ∂ū ∂y λ2 − f2f ∗ 2 −λf2 − ∂f1 ∂y λf∗1 + ∂f∗1 ∂y λf∗2 + ∂f∗2 ∂y 0  Φ, (10.34) where Φ = (Φ1,Φ2,Φ3)T ∈ W (S1; C3), providing its Lax-type integrability. Thus, the following theorem holds. Theorem 10.3. The Davey – Stewartson equation (10.32), (10.33) possesses the Lax represen- tation as a compatibility condition for equations (10.34) under the additional natural constraint (10.27). In fact, one has found above a triple linearization for a (2+1)-dimensional dynamical system, that is a new important ingredient of the Lie algebraic approach to Lax-type integrable flows, based on the Backlund-type transformation (10.23) developed in this work. It is clear that a similar construction of a triple linearization like (10.4) can be done for many other both old and new (2 + 1)-dimensional dynamical systems, on what we plant to stop in detail in another work under preparation. 11. Conclusion. The developed above theory of parametrically Lax-type integrable dynami- cal systems allows to widen to a great extent the class of exactly treated nonlinear models in many fields of science. It is to be noted here the following important mathematical fact obtai- ned in the paper: almost every nonlinear dynamical system admits a parametrically isospectral Lax-type representation but a given dynamical system is the Lax-type integrable if an evoluti- on of the spectrum parameter doesn’t depend on a point u ∈ M at all Cauchy data. This result has allowed us to develop a very effective direct criterion for the following problem: whether a given nonlinear dynamical system on the functional manifold M is parametrically Lax-type integrable or not. Having the problem above solved, we have suggested the reducti- on procedure for the associated nonlinear dynamical systems to be descended on the invariant submanifold MN ⊂ M built before inheriting the canonical Hamiltonian structure and the Liouville complete integrability. Thereby, the powerful techniques of perturbation theory can be successfully used for dynamical systems under consideration, as well as the relationships between the full Hamiltonian theory and various Hamiltonian truncations could be now got understandable more deeply. The imbedding problem for infinite-dimensional dynamical systems with additional structu- res such as invariants and symmetries is as old as the Newton – Lagrange mechanics, having been treated by many researches, using both analytical and algebraic methods. The powerful differential-geometric tools used here were created mainly in works by E. Cartan at the begi- nning of the twentieth century. The great impact in the development of imbedding the methods was done in recent time, especially owing to the theory of isospectral deformations for some linear structures built on special vector bundles over the space M as the base of a given nonli- ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 ON THE LAGRANGIAN AND HAMILTONIAN ASPECTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . 385 near dynamical system. Among them there are such structures as the moment map l : M → G∗ into the adjoint space to the Lie algebra G of symmetries, acting on the symplectic phase space M equivariantly [6, 7], the connection of the Cartan – Eresman structures appearing via the Wahlquist – Estabrook approach [8], and many others. For the last years the general structure of Lagrangian and Hamiltonian formalisms was studied thoroughly using both geometrical and algebraical methods [9, 10]. The special attenti- on was paid to the theory of differential-difference dynamical systems on the infinite-dimen- sional manifolds [10, 35]. Some number of articles was devoted to the theory of pure discrete dynamical systems [36 – 39], as well treating the interesting examples [39] appeared to be impor- tant for applications. In future work we intend to treat further imbedding problems for infinite-dimensional both continuous and discrete dynamical systems basing on the differential-geometric Cartan’s theory of differential ideals in Grassmann algebras over jet-manifolds, intimately connected with the problem under consideration. As it is well known, there existed by now only two regular enough algorithmic approaches [27, 28, 33] to constructing integrable multidimensional (mainly 2 + 1) dynamical systems on infinite-dimensional functional spaces. Our approach, devised in this work, is substantially based on the results previously done in [27, 33], explains completely the computational properties of multidimensional flows before delivered in works [30, 34]. As the key points of our approach there used the canonical Hamiltonian structures naturally existing on the extended phase space and the related with them Backlund transformation which saves Casimir invariants of a chosen matrix integral-differential Lie algebra. The latter gives rise to some additional Hamiltonian properties of the considered extended evolution flows studied before in [27, 30] making use of the standard inverse scattering transform [27, 30, 31] and the formal symmetry reduction for the KP-hierarchy [32, 33] of commuting operator flows. As one can convince ourselves analyzing the structure of the Backlund-type transformation (10.24), it strongly depends on the type of an ad-invariant scalar product chosen on an operator Lie algebra Ĝ and its Lie algebra decomposition like (10.6). Since there exist in general other possibilities of choosing such decompositions and ad-invariant scalar products on Ĝ, they gi- ve rise naturally to another resulting types of the corresponding Backlund transformations, which can be a subject of another special investigation. Let us here only mention the choice of a scalar product related with the operator Lie algebra Ĝ centrally extended by means of the standard Maurer – Cartan two-cocycle [14, 27, 28], bringing about new types of multidimensio- nal integrable flows. The last aspect of the Backlund approach to constructing Lax-type integrable flows and their partial solutions which is worth of mention is related with Darboux – Backlund-type transfor- mations [30, 40] and their new generalization recently developed in [33, 41]. They give rise to very effective procedures of constructing multidimensional integrable flows on functional spaces with arbitrary number of independent variables simultaniously delivering a wide class of their exact analytical solutions, depending on many constant parameters, which can appear to be useful for diverse applications in applied sciences. All the mentioned above Backlund-type transformations aspects can be studied as special investigations, giving rise to new directions in the theory of multidimensional evolution flows and their integrability. 12. Acknowledgements. The authors thank Profs D. L. Blackmore (NJIT, Newark, NJ, USA), M. O. 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