degenerate poisson pencils on curves; new separability theory - blaszak

8/13/2019 Degenerate Poisson Pencils on Curves; New Separability Theory - Blaszak

1/31

arXiv:nlin/0004002v2

[nlin.SI]

1Apr2000

Journal of Nonlinear Mathematical Physics 2000, V.7, N 2,213243. Review Article

Degenerate Poisson Pencils on Curves:

New Separability Theory

Maciej BLASZAK

Physics Department, A. Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland

e-mail: [email protected]

Received September 29, 1999; Revised February 1, 2000; Accepted February 2, 2000

Abstract

A review of a new separability theory based on degenerated Poisson pencils and the so-called separation curves is presented. This theory can be considered as an alternativeto the Sklyanin theory based on Lax representations and the so-called spectral curves.

1 Introduction

The separation of variables belongs to the basic methods of mathematical physics fromthe previous century. Originating from the early works of DAlambert (the 18th century),Fourier and Jacobi (the first half of the 19th century), for many decades it has beenthe only known method of exact solution of dynamical systems. Let us briefly recall theidea from classical mechanics. Consider a Hamiltonian mechanical system of 2n degreesof freedom and integrable in the sense of Liouville/Arnold theorem. This means thatthere existn linearly independent functionshi(Hamiltonians) which are in involution withrespect to the canonical Poisson bracket

{hj , hk} = 0, j, k= 1,...,n. (1.1)

A system of canonical variables (i, i)ni=1

{i, j} = {i, j} = 0, {i, j} =ij (1.2)

is called separated [1] if there exist n relations of the form

i(i, i, h1,...,hn) = 0, i= 1,...,n (1.3)

joining each pair (i, i) of conjugate coordinates and all Hamiltonians hk, k = 1,...,n.

Fixing the values of Hamiltonians hj = constj = aj one obtains from (1.3) an explicitfactorization of the Liouville tori given by the equations

i(i, i, a1,...,an) = 0, i= 1,...,n. (1.4)

Copyright c 2000 by M. Blaszak
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214 M. Blaszak

In the Hamilton-Jacobi method, for a given Hamiltonian function hr(, ) we are lookingfor a canonical transformation (, ) (a, b) in the form bi = Wai , i = Wi , whereW(, a) is a generating function given by the related Hamilton-Jacobi equation

hr(,W

) =ar. (1.5)

If (, ) are separated coordinates, then

W(, a) =ni=1

Wi(i, a) (1.6)

and the partial differential equation (1.5) splits into n ordinary differential equations

i(i,W

i, a1,...,an) = 0, i= 1,...,n (1.7)

just of the form (1.4). In (a, b) coordinates the flow is trivial

(aj)tr = 0, (bj)tr =jr (1.8)

and the implicit form of the trajectories i(tr) is the following

bj(, a) =W

aj=jrtr+const, j = 1,...,n. (1.9)

So, given a set of separated variables, it is possible to solve a related dynamical system byquadratures. In the 19thcentury and most of the present century, for a number of modelsof classical mechanics the separated variables were either guessed or found by some ad hoc

methods. For example, in the second half of the 19thcentury, Neumanns investigation ofa particle moving on a sphere under the action of a linear force [2] and Jacobis study ofthe geodesics motion on an ellipsoid [3] exploited the separability of the Hamilton-Jacobiequation to solve the equations of motion by quadratures. In 1891 Stackel initiated theprogram dealing the classification of Hamiltonian systems according to their separabilityor nonseparability, presenting conditions for separability of the Hamilton-Jacobi equationin orthogonal coordinates [4]. For three dimensional flat space, the 11 possible coordinatesystems in which separation may take place were deduced in a paper by Eisenhart [5].They were all obtained as degenerations of the confocal ellipsoidal coordinates [6]. Foreach of the coordinates Eisenhart [7] determined the form of the potential that permitteda separation of variables. These potentials, designated Stackel or separable potentials,

played a crucial role in Hamiltonian mechanics before the development of more qualitativegeometric methods for differential equations. Although in all classical papers the trans-formation to separated coordinates was searched in the form of point transformations,nevertheless they can b e produced by an arbitrary canonical transformation involvingboth coordinates and momenta.

A fundamental progress in the theory of separability was made in 1985, when Sklyaninadopted the method of soliton systems, i.e. the Lax representation, to systematic deriva-tion of separated variables (see his review article [1]). It was the first constructive theory


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Degenerate Poisson Pencils on Curves: New Separability Theory 215

of separated coordinates for dynamical systems. In his approach, the appropriate Hamil-tonians appear as coefficients of the spectral curve, i.e. the characteristic equation of Laxmatrix. His method was successfully applied to separation of variables for many old andnew integrable systems [8]-[12].

In this paper we present a new constructive separability theory, which has been re-

cently intensely developed, based on a bi-Hamiltonian property of integrable systems. Inlast decade a considerable progress has been made in construction of new integrable fi-nite dimensional dynamical systems showing bi-Hamiltonian property. The majority ofthem originate from stationary flows, restricted flows or nonlinearization of Lax equationsof underlying soliton systems [13]-[25]. Quite recently a fundamental property of suchsystems has been discovered, i.e. their separability. It was proved [26]-[31] that mostbi-Hamiltonian finite dimensional chains, which start with a Casimir of the first Poissonstructure and terminate with a Casimir of the second Poisson structure, are integrable byquadratures, through the solutions of the appropriate Hamilton-Jacobi equation.

The presented review article is based on results of papers [26]-[31] systematizing andunifying them into a compact separability theory in the frame of the set of canonical

coordinates. The results derived so far are sufficiently promising to consider the theory asan alternative or complement to the Sklyanin ones.

2 Preliminaries

Let us re-examine some facts about bi-Hamiltonian systems. We recall some definitions.Let M be a differentiable manifold, T M and TM its tangent and cotangent bundle.At any point u M, the tangent and cotangent spaces are denoted by TuM and TuM,respectively. The pairing between them is given by the map: TuM TuM R.For each smooth function F C(M), dFdenotes the differential ofF. M is said to bea Poisson manifold if it is endowed with a Poisson bracket

{,

} : C(M)

C(M)

C(M),in general degenerate. The related Poisson tensor is defined by {F, G}(u) := (u) =< dG(u), (u)dF(u) >. So, at each point u, (u) is a linear map(u) :TuM TuMwhich is skew-symmetric and fulfils the Jacobi identity. Any functionc C(M), such that dc ker , is called a Casimir of. Let 0,1 : TM T M betwo Poisson tensors onM .A vector field Kis said to be a bi-Hamiltonian with respect to0 and 1 if there exist two smooth functions H,F C(M) such that

K=0 dH=1 dF. (2.1)

Poisson tensors0and1are said to be compatible if the associated pencil =10isitself a Poisson tensor for any R.Moreover, if0 is invertible, the tensorN=1 10 ,

called a recursion operator, is a Nijenhuis (hereditary) tensor of such a property that whenit acts on a given bi-Hamiltonian vector field K, it produces another bi-Hamiltonian vectorfield being a symmetry generator ofK. Hence, having the invariant Nijenhuis tensor, onecan construct a hierarchy of Hamiltonian symmetries and related hierarchy of constantsof motion for an underlying system, so important for its integrability.

Unfortunately, for majority of bi-Hamiltonian finite dimensional systems, both Poissonstructures are degenerate, so one cannot construct the recursion Nijenhuis tensor invertingone of the Poisson structures. Nevertheless, due to the nonuniqueness of Hamiltonian


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216 M. Blaszak

functions, determined up to an appropriate Casimir function, it is always possible toconstruct a finite bi-Hamiltonian chain starting and terminating with Casimirs of0 and1, respectively.

3 One-Casimir chainsLet us consider a Poisson manifold Mof dim M= 2n+ 1 equipped with a linear Poissonpencil

= 1 0 (3.1)of maximal rank, where0and 1are compatible Poisson structures and is a continuousparameter. As was first shown by Gelfand and Zakharevich [32], a Casimir of the pencilis a polynomial in of an order n

h=h0n +h1

n1 +...+hn (3.2)

and generates a bi-Hamiltonian chain

dh= 0

0 dh0 = 00 dh1 = K1 = 1 dh00 dh2 = K2 = 1 dh1

...0 dhn =Kn = 1 dhn1

0 =1 dhn,

(3.3)

whereK K1, H h1andF h0.In the paper we restrict our considerations to a classof canonical coordinates (q,p,c), where q= (q1,...,qn)

T, p = (p1,...,pn)T are generalized

coordinates and c is a Casimir coordinate. Hence, 0 always stays a canonical Poissonmatrix. This restriction simplifies the theory in the sense that it makes a Marsden-Ratiuprojection procedure [33], [34] trivial.

3.1 Darboux-Nijenhuis representation

To understand the theory better, let us start from the end in some sense, i.e. from asystem written in separated coordinates (i, i)

ni=1. An interesting observation is that

such a system can be represented by n different points of some curve. We start fromGelfand and Zakharevich case. Actually, let us consider a curve in (, ) plane, in theparticular form

f(, ) =h

, h

= cn +h1

n1 +...+hn

, (3.4)

wheref(, ) is an arbitrary smooth function. Then, let us take n different points (i, i)from the curve:

f(i, i) =cni +h1

n1i +...+hn, i= 1,...,n, (3.5)

which will define our separated coordinates as they are of the form ( 1.3). The explicitdependence ofhk on (i, i, c)

ni=1is given by the solution ofn linear equations (3.5), while


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for fixed values ofhk =akand i= Wii

the system (3.5) allows us to solve the appropriateHamilton-Jacobi equations.Here, we have to mention that the idea to relate the multi-Hamiltonian property to anm-parameter family of curves comes from P. Vanhaecke [35].

In refs. [26]-[29] the bi-Hamiltonian chain was constructed for a separation curve in

the form (3.4). Actually, the Hamiltonian functionshk found from the system (3.5) takethe following compact form

hk(,,c) =ni=1

ik1()f(i, i)

i() +ck(), k= 1,...,n

h0= c,

(3.6)

where

i() :=j=i

(i j), (3.7)

k() := (1)k

j1, . ., jkj1 < .. < jk

j1 ... jk , k= 1,...,n, (3.8)

are the so-called Viete polynomials (symmetric polynomials) and

ik1() :=k1(i= 0) = k()

i. (3.9)

For example, for n= 2 we have

1 = 1 2, 2= 12 (3.10)and for n= 3

1 = 1 2 3, 2 = 12+13+23, 3= 123. (3.11)The bi-Hamiltonian chain (3.3) is constructed with respect to the following compatiblePoisson matrix

0 =

0 I 0I 0 0

0 0 0

, 1=

0 h1, 0 h1,

(h1,)T (h1,)T 0

, (3.12)

where =diag(1,...,n) and h1, :=h11

,..., h1n

T.Notice that the last column of1

is just the first vector field K1.All Hamiltonians hk are in involution with respect to both

Poisson structures 0 and 1.Applying the following important relations

r() = ni=1

ir1nii

, (3.13)

ni=1

ri

mii

=ni=1

ir1() mi

i=

1, m= n r0, m =n r

, r= 1,..,n (3.14)


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218 M. Blaszak

and the decomposition (1.6), the Hamilton-Jacobi equations

hr(,W

, c) =ar, r= 1,...,n (3.15)

turn into the form

nk=1

kr1()[f(k, Wk/k) cnk ]k

=ar, r= 1,...,n (3.16)

with the solution

f(k, Wk/k, c) =cnk + a1

n1k +...+an, k= 1,...,n. (3.17)

Hence, W(, a) can be obtained by solving n decoupled first-order ODEs (3.17) and thefamily of dynamical systems (3.3) can be solved by quadratures.

Now, let us pass to the projection of the Poisson pencil onto a symplectic leafSof0 (dim S= 2n) fixing the value ofc. Generally, one has to apply the Marsden-Ratiutheorem which in this case is trivial, as obviously = 1 0,where

0=

0 II 0

, 1=

0 0

, (3.18)

is a nondegenerate Poisson pencil on S. Hence, the related Nijenhuis tensor

N=1 10 =

00

(3.19)

is diagonal and this is the reason why we will refer below to the separated coordinates as tothe Darboux-Nijenhuis (DN) coordinates. Notice that i (3.8) are coefficients of minimalpolynomial of the Nijenhuis tensor

(det(N

))1/2 =n +

n

i=1

ini =

n

i=1

(

i). (3.20)

On Sthe chain (3.3) turns into the form

0 dh1 = K1 = 1n

1 dhn

0 dh2 = K2 = 1n

1 dhn+1 dh1...

0 dhn = Kn = n1n

1 dhn+1 dhn1.

(3.21)

The last equation terminates the sequence of vector fields Kr in the hierarchy as for thenext equation from the chain we have

0 dhn+1= Kn+1= 1 dhn+1 dhn = 0. (3.22)ObviouslyN is not a recursion operator for the hierarchy (3.21). Because of the form ofthe first equation, the vector field K1 is called a quasi-bi-Hamiltonian [36]-[39] and thechain (3.21) could be treated alternatively as a starting point of the separability theoryfor the case ofc= 0.


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3.2 Arbitrary canonical representation

Now, let us consider an arbitrary canonical transformation

(q, p) (, ) (3.23)

independent of a Casimir coordinate c (not necessarily a point transformation!). Theadvantage of staying inside such a class of transformations is that the clear structureof a pencil is preserved and the Marsden-Ratiu projection of the Poisson pencil is stilltrivial. Of course, the most general case of multi-Hamiltonian separability theory takesplace when one goes beyond the set of canonical coordinates. i.e. when one tries to findDN coordinates starting from the pencil written in a non-canonical representation. Butthen the simple structure of degenerated Poisson pencil is lost and the nontrivial problemof the Marsden-Ratiu projection for such pencil appears (see for example ref. [41]).

Applying the transformation (3.23) to Hamiltonian functions (3.6) and Poisson matrices(3.12) one finds that

hk(q,p,c) =hk(q, p) +cbk(q, p), k = 1,...,n (3.24)

and

0 =

0 0

0 0

, 0=

0 II 0

,

1 =

1 K1

KT1 0

, 1=

D(q, p) A(q, p)AT(q, p) B(q, p)

,

(3.25)

where A, B and D are n n matrices. The nondegenerate Poisson p encil on S givesrise to the related Nijenhuis tensor Nand its adjoint N in (q, p) coordinates in the form

N=1 10 = A DB AT , N =10 1=

AT BD A . (3.26)

Obviously, in a real situation we start from a given bi-Hamiltonian chain (3.24)-(3.26)in canonical coordinates (q,p,c), derived by some method (see for example [13]-[25]), andtrying to find the DN coordinates which diagonalize the appropriate Nijenhuis tensor andare separated coordinates for the considered system. So now we pass to a systematicderivation of the inverse of transformation (3.23).

The important intermediate step of the construction of DN coordinates are the so calledHankel-Frobenious (HF) non-canonical coordinates (u, v) [35], [40], related to the DN onesthrough the following transformation

ui= i(1,...,n),

i=

nk=1

vknki , i= 1,...,n.

(3.27)

In (u, v) coordinates one finds

0=

0 UUT 0

, U=

0 0 10 1 u1

1 u1 un1

,


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220 M. Blaszak

N=

F 0

0 F

, F=

u1 1 0u2 0 1 0

un1 0 1

un 0

0

,

1= N 0 =

0 F UF UT 0

, N =NT.

Moreover the differentials dui, dvi satisfy the following recursion relations [40], [41]

N du1 = du2 u1du1,N du2 = du3 u2du1,

...N dun1 = dun un1du1,N dun = undu1,

N

dv1 = dv2 u1dv1,N dv2 = dv3 u2dv1,

...N dvn1 = dvn un1dv1,N dvn = undv1.

(3.28)

Note that vector fields 0 du1 and 0 dv1 are quasi-bi-Hamiltonian.Now we relate the canonical coordinates (q,p,c) to the DN separated coordinates

(,,c). From the minimal polynomial ofN (3.26) we get

uk =k(q, p), k= 1,...,n. (3.29)

Conjugate coordinates vk = vk(q, p), k = 2,...,n are found from the recursion formula(3.28) while v1=v1(q, p) coordinate from relations

{uj, v1}k =j,nk, j= 1, . .., n , k= 0,...n 1. (3.30)

Hence we get

vk =k(q, p), k = 1,...,n. (3.31)

Eliminating (u, v) coordinates from (3.27), (3.29) and (3.31) we derive the desired relations

i(q, p; , ) = 0, i= 1,..., 2n. (3.32)

Now let us concentrate on a special but important case of point transformation between(q,p,c) and (,,c) variables. Then

1=

0 A(q)

AT(q) B(q, p)

,

N=

A(q) 0B(q, p) AT(q)

,

(3.33)


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where matrix elements ofB are at most linear in p coordinates, so coefficients of minimalpolynomial ofN are equal to coefficients of the characteristic polynomial ofA. Hence, wefind the first part of the canonical transformation in the form

i() =i(q), i= 1,...,n = qk =k(), k= 1,...,n. (3.34)The complementary part of the transformation we get from the generating function

G(p, ) =

ni=1

pii(). (3.35)

Then,

i= G

i, i= 1,...,n = pk =k(, ), k= 1,...,n. (3.36)

At the end of this subsection we introduce the notion of an inverse bi-Hamiltonianseparable chains. In ref. [29] it was demonstrated that for each separable bi-Hamiltonian

chain (3.3), (3.24), (3.25), (3.26) in canonical coordinates (q,p,c) there exists a relatedinverse bi-Hamiltonian separable chain

0 dhn+1= 00 dhn =Kn=1 dhn+1

...0 dhr =Kr=1 dhr+1

...0 dh1 =K1=1 dh2

0 =1 dh1

(3.37)

where h

n+1= h0= c,

1=

1 K

n

KTn 0

, 1= 0 11 0 = N2 1, (3.38)

Kn = 0 dhn and

hr(q,p ,c) =hr(q, p) +cbr(q), b

r(q) =

br1(q)

bn(q) , r= 1,...,n. (3.39)

Notice that in both chains the respective Hamiltonians (3.24), (3.39) and related vectorfields differ only by the cdependent parts. In the case of a point transformation to theDN coordinates, when 1 takes the form (3.33), we get

11 =

A1

T D A1 A1TA1 0

(3.40)

and

0 11 0=

0 A1

A1T A1T D A1

. (3.41)


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222 M. Blaszak

Notice that both chains (3.3) and (3.37) given in natural coordinates can be transformedto the Nijenhuis bi-Hamiltonian form (3.6)-(3.12), where the canonical transformation canbe derived from the relation r() = br(q), r = 1,...,n in the first case and from the

relations r() = bnr(q)bn(q)

, r= 1,...,nin the second case.Consider the set of extended functions

hr(q, p; c, c) =hr(q, p) +cbr(q) +c

b(q). (3.42)

They can be simultaneously put into the bi-Hamiltonian and the inverse bi-Hamiltonianhierarchies. In the first case c is treated as the Casimir variable and c as the parameterand in the second case c is treated as the parameter and c as the Casimir variable.

3.3 Examples

We shall illustrate the theory presented by a few representative examples. More examplescan be found in refs. [26]-[29]. In all examples from this section canonical transformationsbetween natural and separated coordinates are point like. An example of a nonpointtransformation will be given at the end of this paper.

Example 1. The one-Casimir extension of the Henon-Heiles system.

Let us consider the integrable case of the Henon-Heiles system generated by the Hamilto-nian H= 12p

21+

12p

22+q

31+

12q1q

22. Its one-Casimir extension reads [22]

(q1)tt= 3q21 1

2q22+ c, (q2)tt= q1q2 (3.43)

and belongs to the bi-Hamiltonian chain

0

dh0= 0

0 dh1=K1= 1 dh00 dh2=K2= 1 dh1

0 =1 dh2,(3.44)

where

h0= c,

h1= h1(q, p) +cb1(q) =1

2p21+

1

2p22+q

31+

1

2q1q

22 cq1,

h2= h2(q, p) +cb2(q) =1

2q2p1p2 1

2q1p

22+

1

16q42+

1

4q21q

22

1

4cq22,

(3.45)

1 =

0 0 q1 1

2q2 p10 0 12q2 0 p2

q1 12q2 0 12p2 h1,q112q2 0 12p2 0 h1,q2p1 p2 h1,q1 h1,q2 0

. (3.46)


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The construction of the related inverse bi-Hamiltonian chain, according to the results fromthe previous subsection, gives the following results:

0 dh3 = 00

dh2 = K

2 =1

dh3

0 dh1 = K1 =1 dh20 =1 dh1,

(3.47)

where

h1= h1(q, p) 4

q22c,

h2= h2(q, p) +4q1

q22c,

h3= c,

(3.48)

1=

0 0 0 2/q2 12q2p20 0 2/q2 4q1/q22 12q2p1 q1p20 2/q2 0 2p2/q22 h2,q1

2/q2 4q1/q22 2p2/q22 0 h2,q212q2p2 12q2p1+q1p2 h2,q1 h2,q2 0

(3.49)

and Newton equations related with the natural Hamiltonian h1 are:

(q1)tt= 3q21 1

2q22, (q2)tt= q1q2

2

q2

3c, (3.50)

being just the second well known one-Casimir extension [16] of the Henon-Heiles systemconsidered. Both systems (3.43) and (3.50) can be transformed to the Nijenhuis chain(3.6)(3.12) through the respective transformations

q1=1+2, p1= 111 2 +

222 1 ,

q2= 2

12, p2=

12

11 2 +

22 1

,

f(i, i) =1

2i

2i +

4i ,

(3.51)

and

q1= 11

+ 12

, p1 = 12

111 2 +

222 1

,

q2= 212

, p2=

12

2111 2 +

2222 1

,

f(i, i) =1

24i

2i +

3i .

(3.52)


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Example 3. One-Casimir extension of elliptic separable potentials.

In refs.[26],[28] it was proved that every natural Hamiltonian system

H=1

2

n

i=1

p2i +V(q) +1

2c(q, q), (3.60)

where (., .) means the scalar product, which admits in extended phase space M (q,p,c)the bi-Hamiltonian formulation

qp

c

t

=0 dh1 = 1 dh0, (3.61)

where 0 is a canonical Poisson structure,

1 =

0 A

1

2q

q h

1,pA+ 12q q 12p q 12qp h1,q (h1,p)T (h1,q)T 0

, (3.62)

A = diag(1,...,N), idifferent positive constants, h0 = c, h1 = H+c1(), is sep-arable in generalized elliptic coordinates. This bi-Hamiltonian formulation generates thechain (1.2) [25] of commuting bi-Hamiltonian vector fields, where

hr(q,p ,c; ) =hr(q, p; ) +cbr(q; ),

br(q; ) =r() +1

2

r1

k=1 k()(q, Ark1q), r= 1,...,n 1,

bn() =n()[1 1

2(q, A1q)],

hr(q, p; ) =r

k=0

k()hrk, hs =1

2

ni=1

s1i Ri,

Ri=i=j

qipj qjpii j +p

2i +Vi(q),

ni=1

Vi(q) =V(q),

(3.63)

{H, Ri}0 = 0, i= 1,...,n and r() are Viete polynomials of.On the other hand, according to our procedure, the inverse bi-Hamiltonian chain (3.47)

for one-Casimir extension of potentials separable in elliptic coordinates reads

hr(q,p ,c; ) =hr(q, p; ) +cbr1(q; )

bn(q; ) , (3.64)

1=

0 B1 hn,p

B1 B1(12p q 12qp)B1 hn,q (hn,p)T (hn,p)T 0

, (3.65)


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226 M. Blaszak

where

B= A 12

q q,

B1 = 1

|B

| |B| +1

2(q q) |A| ,

=diag

1,...,

n

,

|B| = |A| 12

(q, |A| q).

(3.66)

Notice that now, the natural Hamiltonian in the inverse hierarchy is the last one of theform

H =h1(q,p,c) =1

2

ni=1

p2i +V(q) + c

n()[1 (q, A1q)]. (3.67)

The basic example here is the Garnier system with the potentialV(q) = 14 (q, q)2 12(q,Aq).

This potential is a member of an infinite family of permutationally symmetric potentialsseparable in generalized elliptic coordinates [17]. The point transformation to DN sepa-rated coordinates can be constructed from the relations

r() =r() +1

2

r1k=1

k()(q, Ark1q), r= 1,...,n. (3.68)

But so defined DN coordinates are just the generalized elliptic coordinates1,...,ndefinedby the relation

1 +1

2

nk=1

q2kz k =

nj=1(z j)nk=1(z k)

. (3.69)

The proof is given in refs. [26] and [28].

More examples of separated systems by the method presented the reader can find inrefs. [26]-[29],[37]-[39].

4 Multi-Casimir unsplit chains

In the previous section a separability theory of one-Casimir bi-Hamiltonian chains wasreviewed. Here we pass to the generalization of the theory and include multi-Casimircases. This procedure considerably extends the class of separable systems and in generalcovers a new class of chains, i.e. the so-called split chains. In the following section weconsider the simplest generalization of unsplit multi-Casimir chains related to the extension

of the separation curve (3.4) of the formf(, ) =h, h = cn

2n1 +...+c1n +h1

n1 +...+hn. (4.1)

Choosing other admissible forms of the separation curve with more then one Casimir,one can construct split bi(multi)-Hamiltonian chain, i.e. the chain which splits onto a fewbi(multi)-Hamiltonian sub-chains, each starting and terminating with some Casimir of theappropriate Poisson structure. The work on split cases is still in progress but some resultsare presented in the next section.


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4.1 Multi-Hamiltonian Darboux-Nijenhuis chains

In ref. [30] the multi-Hamiltonian chain was constructed for the separation curve in theform (4.1). Actually, the Hamiltonian functions hk found from the system

f(i, i) =cn

2n1

i +...+c1

n

i +h1

n1

i +...+hn, i= 1,...,n, (4.2)

take the following form

hr(,,c) = ni=1

ri

f(i, i)

i+

nj=1

cjj,r(), r= 1,...,n, (4.3)

where 1,r() r() and m,r, m= 2,...,n,are defined by the recursive formula

m,r =m1,r+1 m1,1 1,r. (4.4)

Notice that Hamiltonians (4.3) are just Hamiltonians (3.6) supplemented with extra terms,

linear with respect to additional Casimirs ci, i = 2,...,n. On the extended phase spaceM (1,...,n, 1,...,n, c1,...,cn) functions (4.3) and h1r(c) = cr, r = 1,...,n form

n+ 12

bi-Hamiltonian chains, each generated by a Poisson pencil k1 =k kii

of order 1 (k i) n,

ki dh =

i dhi = 0i dhi+1 = K1 =k dhk+1i dhi+2 = K2 =k dhk+2

...i

dhi+j = Kj =k

dhk+j

...i dhi+n = Kn =k dhk+n

0 =k dhk+n+1

0 i < k n, (4.5)

with respect to (n+ 1) compatible 3n 3n Poisson structures of rank 2n

0 =

0 0 00... 00

, 1=

1 K1 0 0KT1

0... 0

0

,

2 =

2 K2 K1 0 0KT2KT1

0 0...0

, ....., (4.6)


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228 M. Blaszak

n =

n Kn Kn1 K1KTn

KTn1... 0

KT1

,

where

m= Nm 0 =

0 m

m 0

, m=diag(m1 , ...,

mn) (4.7)

Km = (hm,, hm,)T and each Poisson structure m has n Casimir functions:cm+1, cm+2, . . . , cn, hn, . . . , hnm+1. Moreover all functions hr(,,c), r = 1,...,n are ininvolution with respect to an arbitrary Poisson tensor k, k= 1,...,n.

Now, we integrate equations of motion from the hierarchy (4.5) solving the Hamilton-Jacobi equation for Hamiltonians (4.3)

hr(,W

, c) =

nk=1

kr1()fk(k,W/k)

k+

ni=1

cii,r() =ar. (4.8)

First we demonstrate the separability of this equation. Taking the generating functionW(, a) in the formW(, a) =

ni=1Wi(i, a) and the following representation ofk,r

k,r() =

ni=1

ri

n+k1ii

= ni=1

ir1n+k1i

i, (4.9)

eq.(4.8) turns into the form

nk=1

kr1()[f(k, Wk/k)

ni=1ci

n1+ik ]

k=ar. (4.10)

Applying relation (3.14) we get the solution of eq.(4.10) in the form

f(k, Wk/k) =g(k), k= 1,...,n, (4.11)

where

g() =cn2n1 +...+c1

n +a1n1 +...+an1+an. (4.12)

Hence, W(, a) can be obtained by solving n decoupled first-order ODEs (4.11). For

example, if

f(i, i) =(i)f(i) +(i), (4.13)

then we obtain

W(, a) =

nk=1

kf1

g() ()

()

d. (4.14)


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In new canonical variables ai, bi= Wai

,the Hamiltonians hr(,,c) become hr =ar with

bi =W

ai=

nk=1

k f1

ni()

d, (4.15)

where

f1 means the derivative off1. As in the new coordinates each hr generates a

trivial flow

(aj)tr = hrbj

= 0, (bj)tr =hraj

=j,r, (cj)tr = 0, (4.16)

hence

bi = ti+const. (4.17)

Combining (4.15) with (4.17) we arrive at implicit solutions for the trajectories i(tr),with respect to the evolution parameter tr in the form

nk=1

k f1

ni()

d= i,rtr+const, i= 1,...,n. (4.18)

4.2 Multi-Hamiltonian chains in arbitrary canonical coordinates

Let us introduce arbitrary canonical coordinates (q,p,c) related to the Darboux-Nijenhuiscoordinates (,,c) trough some canonical transformation

qk =k(, ), pk =k(, ), k = 1,...,n. (4.19)

Applying the inverse of this transformation to Hamiltonian functions (4.3) and Poisson

matrices (4.6) one finds that

hr(q,p ,c) =hr(q, p) +ni=1

cibi,r(q) (4.20)

and the nondegenerate partm of rank 2nof each m(also implectic) takes now the form

m= Nm 0 =

Dm(q, p) Am(q, p)ATm(q, p) Bm(q, p)

, m= 1,...,n. (4.21)

Conversely, if we have a multi-Hamiltonian chain in (q,p ,c) coordinates and the Nijen-huis tensor

N(q, p) =

D1(q, p) A1(q, p)AT1 (q, p) B1(q, p)

0 II 0

1=

A1(q, p) D1(q, p)B1(q, p) A

T1(q, p)

(4.22)

is nondegenerate and hasndistinct eigenvaluesi each of multiplicity 2, then the canonicaltransformation (4.19) transforms a given chain to the one considered in the previoussubsection.


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230 M. Blaszak

The admissible reductions of the number of Casimir variables are the following. Forarbitrary 1 m < n, let ci= 0, 1 i m and ci = 0 for m < i n. The firstm Poisson structures i, i m survive the projection (,,c1,...,cn) M M (,,c1,...,cm) and we have still

m+12

bi-Hamiltonian chains (4.5). In the limit c1 =

... = cn = 0, the systems considered lose the bi-Hamiltonian property, turning into the

quasi-bi-Hamiltonian systems on a symplectic manifold M (q, p), being still separableand integrable by quadratures. Moreover, because of the property (4.9), each of themulti-Hamiltonian systems considered on a Poisson manifold M (q,p,c) has a quasi-bi-Hamiltonian representation on a symplectic leafSof0 (dim S= 2n) fixing the values ofall ci.

4.3 Examples

The theory extended in this section will be illustrated by several representative examplesof already known as well as new multi-Hamiltonian systems.

Example 4. Stationary t2

flow of dispersive water waves.

The Hamiltonian functions and Poisson structures in Ostrogradsky variables are as follows[42]:

h1(q,p,c) = 4p1p2+ 5q2p21 5

8q1q

32

3

4q21q2

7

64q52+

1

2q2c1+ (

1

2q1+

1

8q22)c2,

h2(q,p,c) =q1p21+ 4q2p1p2

5

4q22p

21 2p22+

5

64q1q

42

3

16q21q

22

1

4q31+

45

6 128 q62

+ (1

2q1+

3

8q22)c1 (

1

4q1q2 3

16q32)c2,

0 =

0 0 1 0 0 0

0 0 0 1 0 01 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0

,

1 =

0 0 32q2 12q1 158q22 h1,p1 00 0 1 q2 h1,p2 032q2 1 0 p1 h1,q1 0

12q1+

158q

22 q2 p1 0 h1,q2 0

h1,p1 h1,p2 h1,q1 h1,q2 0 00 0 0 0 0 0

,

2 =

0 0 38q22 12q1 14q1q2 1516q32 h2,p1 h1,p1

0 0 12q2 12q1 78q22 h2,p2 h1,p238q22+ 12q1 12q2 0 12q2p1 h2,q1 h1,q114q1q2+

1516q

32

12q1+

78q

22 12q2p1 0 h2,q2 h1,q2

h2,p1 h2,p2 h2,q1 h2,q2 0 0h1,p1 h1,p2 h1,q1 h1,q2 0 0

.


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Hence, we have three bi-Hamiltonian chains (4.5)

0 dc1 = 00 dh1= K1 =1 dc10 dh2= K2 =1 dh1

0 =1 dh2,

0 dc1= 00 dh1 = K1 =2 dc20 dh2 = K2 =2 dc1

0 =2 dh1,1 dc2 = 01 dc1 = K1 =2 dc21 dh1= K2 =2 dc1

0 =2 dh1.

(4.23)

The canonical transformation to the Darboux-Nijenhuis coordinates reads

q1= (321+ 322+ 412),q2= 2(1+2),

p1=

1

2

2

11 2 ,

p2= 12

1(32 21) 2(31 22)1 2 ,

where nowhr, r= 1, 2 take the form (4.3) with

fi(i, i) = 26i

1

22i ,

1,1 = 1, 1,2= 2 (3.10) and

2,1= 21 22 12,2,2

= 12(1+2).(4.24)

Example 5. Two-Casimir extension of the Henon-Heiles system.

In Example 1 one-Casimir extension of the Henon-Heiles system was considered in theform

(q1)tt= 3q21 1

2q22+ c, (q2)tt= q1q2,

with two constants of motion (3.45) and the related transformation to Darboux-Nijenhuiscoordinates (3.51). For a two-Casimir extension we get immediately

2,1=

[(1+2)

2

12] =

(q21+

1

4

q22),

2,2= 12(1+2) = 1

4q1q

22.

(4.25)

Hence

h1= H=1

2p21+

1

2p22+q

31+

1

2q1q

22 c1q1 (q21+

1

4q22)c2,

h2=1

2q2p1p2 1

2q1p

22+

1

16q42+

1

4q21q

22

1

4q22c1

1

4q1qc2,

(4.26)


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232 M. Blaszak

where the Newtons equations related to the energy H are

(q1)tt= 3q21 1

2q22+ c1+ 2q1c2, (q2)tt= q1q2+

1

2q2c2. (4.27)

This is tri-Hamiltonian system with the following Poisson structures

0 =

0 0 1 0 0 00 0 0 1 0 0

1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0

,

1 =

0 0 q112q2 h1,p1 0

0 0 12q2 0 h1,p2 0q1 12q2 0 12p2 h1,q1 012q2 0 12p2 0 h1,q2 0h1,p1 h1,p2 h1,q1 h1,q2 0 0

0 0 0 0 0 0

,

2 =

0 0 q21+ 14q

22

12q1q2 h2,p1 h1,p1

0 0 12q1q214q

22 h2,p2 h1,p2

q21 14q22 12q1q2 0 12q1p2 h2,q1 h1,q112q1q2 14q22 12q1p2 0 h2,q2 h1,q2h2,p1 h2,p2 h2,q1 h2,q2 0 0h1,p1 h1,p2 h1,q1 h1,q2 0 0

.

The first two of them come from one-Casimir extension and the last one was constructedaccording to formula (4.21). Notice that again we have three bi-Hamiltonian chains (4.23).

Also in Example 1 the inverse one-Casimir extension of the Henon-Heiles system wasconsidered in the form

(q1)tt= 3q21 1

2q22, (q2)tt= q1q2

8

q32c,

with two constants of motion (3.48) and the respective transformation to the DN coordi-nates (3.52). The two-Casimir extension we get by adding new terms

2,1= [(1+2)2 12] = 4

q22 16q

21

q42,

2,2= 12(1+2) =

16q1

q42 ,

to the constants of motion (3.48)

h1(q,p,c1, c2) =h1(q,p,c = c1) +

16q1q42

c2,

h2(q,p,c1, c2) =h2(q,p,c = c1) +

4

q22 16q

21

q42

c2,


21/31


22/31

234 M. Blaszak

consequence, system (4.30) is quasi-bi-Hamiltonian and separable, with the solution givenby the implicit formulae

1

a

n

k=1

k ni

g()d= i,1 t+const, i= 1,...,n, (4.31)

whereg() =an+ an1+ ... + a1n1.This fact was noticed for the first time by Morosi

and Tondo [38]. Notice, that trajectories i(t) do not depend on the i(i) factors, asexpected, so without a loss of generality one can put i(i) = 1.

The system (4.30) can be naturally extended to an m-Casimir one with the Hamiltonian

H=ni=1

i(i)

ieai +

mj=1

cjj,1(), 1 m n (4.32)

and the related Newton equations of motion

(i)tt= 2k=i

(i)t(k)ti k a

mj=1

cjj,1i

(i)t. (4.33)

The dynamical system (4.33) has n constants of motion

hr(,,c) = ni=1

r()

i

eai

i+

mj=1

cjj,r(), r= 1,...,n,

(m+ 1) Poisson structures (4.6) and the solution given by implicit formula (4.31), wherenow

g() =an+an1+...+a1n1 +c1n +...+cmn+m1.

The one-Casimir extension, which is bi-Hamiltonian, has the following Newton equationsof motion

(i)tt= 2k=i

(i)t(k)ti k +ac(i)t, i= 1,...,n.

The two-Casimir extension, which is tri-Hamiltonian, has the Newton equations in theform

(i)tt= 2k=i

(i)t(k)ti k +ac1(i)t+ac2

i+

nk=1

k

(i)t, i= 1,...,n.

Example 7. (m+ 1)-Hamiltonian formulation for elliptic separable potentials.In Example 3 we presented bi-Hamiltonian systems separable in generalized elliptic coordi-nates. In the following example we extend the result onto appropriate multi-Hamiltonianchains.


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According to the theory presented in this section, let us generalize the Hamiltoniansystem (3.60) to the form

H(q,p ,c) =1

2(p, p) +V(q) +

m

k=1

ckbk,1(q), m= 1,...,n (4.34)

being multi-Hamiltonian and separable. The few first bk,1(q) functions are as follows

b1,1(q) =1

2(q, q),

b2,1(q) =1

2(q,Aq) 1

4(q, q)2,

b3,1(q) =1

2(q, q)(q,Aq) 1

2(q, A2q) 1

8(q, q)3,... .

(4.35)

For example, the three Poisson structures of the system (4.34) with two Casimirs read

0 =

0 I 0 0I 0 0 00 0 0 00 0 0 0

,

1 =

0 A 12q q h1,p 0A+ 12q q 12p q 12qp h1,q 0

(h1,p)T (h1,q)T 0 00 0 0 0

,

2 =

0 (A 12q q)2 h2,p h1,p

(A 1

2q q)2 (A

12q

q)(12p

q

12q

p)

+(12p q 12qp)(A 12q q) h2,q h1,q(h2,p)T (h2,q)T 0 0(h1,p)T (h1,q)T 0 0

.

The functions hr(q,p,c), forming three admissible bi-Hamiltonian chains, are given byformulas (3.63), where now

hr =1

2

ni=1

r1i Ki+c1b1,r(q) +c2b2,r(q).

5 Multi-Casimir split chainsIn the following section we extend the results from previous sections onto the so-calledsplit bi(multi)-Hamiltonian chains [31]. Potentially, the variety of such systems is muchricher than the one of unsplit chains, but still less recognized. The reason is that systemsfrom this family are generally non-physical in the sense that are either non-Stackel, oreven if are of Stackel type, the underlying Stackel space is never flat (at most conformallyflat).


24/31

236 M. Blaszak

LetMbe a (2n+k) dimensional manifold endowed with a linear Poisson pencil=1 0. We suppose that it admits k polynomial Casimir functions

h() =

n

j=0

h()j

nj, = 1,...,k (5.1)

withn = n1+ ... +nk.From beeing Casimir of a p encil it follows that the set {h()0 }k=1is aset of Casimirs of0,while{h()n }k=1 is a set of Casimirs of1,respectively. In canonicalcoordinates (q,p,c) h

()0 =c, = 1,...,k,

0 =

0 0 ... 00... 00

, 1=

1 K(1)1 K(k)1

K(1)1

T... 0

K(k)

1T

, (5.2)

where K(i)1 =0dh

(i)1 and 0, 1 are given by (3.25). Now, we are looking for a separation

curve in the form

f(, ) =

h() =h, (5.3)

where are admissible functions of and . Further on we concentrate on particulartwo-Casimir cases, but it will be enough to make some insight into the theory.

Let us start from a separation curve for the unsplit two-Casimir case

f(, ) =h, h = c2n+1 +c1n +h1n1 +...+hn. (5.4)

An arbitrary shift of c1 Casimir variable along the polynomial h leads to a separationcurve

f(, ) =c2n+1 +h1

n +...+hini+1 +c1

ni +hi+1ni1 +...+hn (5.5)

of some split bi-Hamiltonian chain. This is the case (5.3) with k = 2, n1 = n i, n2 =i, 1 = 1, 2 =

n+1i, h(2)l =hl, l= 1,...,i and h

(1)j =hi+j , j = 1,...,n i.We illustrate

the situation for i = 1.The following results were obtained. For a separation curve in theform

f(, ) =c2n+1 +h1

n +c1n1 +h2

n2 +...+hn h (5.6)

Hamiltonian functions hr, r= 1,...,n read

hr(,,c) =hr(, ) +1,r()c1+2,r()c2, (5.7)

where


25/31


hr(, ) =ni=1

ir1()f(i, i)

i() ,

i() = nk=1

k

k=i

(i k),

1,1() = 1

1, 1,r() =

r1

, r= 2,...,n,

2,1() =21 2

1, 2,r() =

1r+1 2r1

, r= 2,...,n,

ir() =r(i= 0), r() 2,r() =r+1 1r.

(5.8)

Two compatible Poisson structures

0 =

0 I 0 0I 0 0 00 0 0 00 0 0 0

,

1 =

0 h2, h1, 0 h2, h1,

(h2,)T (h2,)T 0 0(h1,)T (h1,)T 0 0

,

(5.9)

give rise to a linear Poisson pencil= 10, which acting on its Casimirh generatesa bi-Hamiltonian chain

dh =

0 dc2= 00 dh1= K1 =1 dc2

0 =1 dh10 dc1= 00 dh2= K2 =1 dc1

...0 dhn = Kn =1 dhn1

0 =1 dhn

(5.10)

which splits onto two sub-chains, each starting and terminating with a Casimir of an

appropriate Poisson structure.

Example 8. The case of three degrees of freedom.

We construct a system comparable with the Newton representation of the 7thorder KdV[26], [28]. Thus let us take

f(i, i) =1

82i + 16

7i , i= 1, 2, 3 (5.11)


26/31


27/31


It has the form (5.3) with k = 2, n1 = i, n2 = n i, 1 = 1, 2 = , h(2)l = hl, l =1,...,n i,and h(1)j =hni+j, j = 1,...,i. Here we concentrate on the case i= n 1

f(, ) =(c2+h1) +c1n1 +h2

n2 +...+hn. (5.13)

The Poisson pencil and the chain are the same as in the previous split case, i.e. are of theform (5.9) and (5.10), buthr(,,c), being the solution of the system

f(i, i) =i(c2i+h1) +c1n1i +h2

n2i +...+hn, i= 1,...,n,

are of more complicated form and we omit here the explicit formulas.

We illustrate the case (5.13) and n = 3 with the example where the transformation toDN coordinates will be of non-point nature.

Example 9. The 4th order Boussinesq stationary flow.

The Hamiltonians and compatible Poisson structures for the 4th order Boussinesq sta-tionary flow, written in generalized Ostrogradsky canonical coordinates, are the following[44]

h1= 27p2p3 27p22 9p23+ 2q21p3+ 3q21p2+q2p1 2

9q1q

23

1

9q1q

22

8

81q41

+1

3q1c1+

1

3q3c2,

h2= 9p1p3+q21p1+ 2q1(q2 2q3)p2+4

3q1q2p3 4

27q33+

8

81q31q3

16

81q31q2

+ 1

27q32+

2

9q2q

23

4

27q22q3+

1

3(2q3 q2)c1+ (3p2 1

9q21)c2,

h3= 54p32 27p2p23 81p22p3+ 3q1p21+ 18q21p22+ 2q21p23+ 3q2p1p2+ 3(q2 q3)p1p3+ 15q21p2p3 q21(q3+

2

3q2)p1 ( 44

27q41+

2

3q1q2q3)p2+ (

2

3q1q

23

1

9q1q

22

49

q1q2q3 1627

q41)p3+ 32

243q31q2q3+

32

729q61+

16

243q31q

33+

4

243q31q

22

481

q22q23+

1

81q32q3+

2

27q2q

33

1

27q43+ (

4

27q31+

1

9q23

1

9q2q3 2q1p2

q1p3)c1+ (q1p1+q3p2 527

q21q3 2

27q21q2)c2,

0 =

0 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 0

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

,


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240 M. Blaszak

1 =

0 0 0 13q3 13q1 23q1 h2,p1 h1,p10 0 3q1 A 13q2 13q3 13q2 h2,p2 h1,p20 3q1 0

29q

21 0 13q3 h2,p3 h1,p3

13q3 A 29q21 0 B C h2,q1 h1,q1

1

3q1 1

3q2+ 1

3q3 0 B 0 8

81q2

1 h2,q2 h1,q223q1

13q2

13q3 C 881q21 0 h2,q3 h1,q3

h2,p1 h2,p2 h2,p3 h2,q1 h2,q2 h2,q3 0 0h1,p1 h1,p2 h1,p3 h1,q1 h1,q2 h1,q3 0 0

where A = 9p2+ 3p3 q21, B= 227q1q2+ 881q1q3 13p1, C= 881q1q2 427q1q3+ 13p1.From the minimal polynomial (3.20) of related Nijenhuis tensor None finds the first

part of transformation to the HF coordinates (u, v)

u1= q3 13

q2,

u2

=1

3q23

+ 5

27q31

3q1p2

q1p3

2

9q2

q3

,

u3= 1

27q33+

1

9q31q3 q1q3p2

1

3q1q3p3+

2

81q31q2

1

3q21p1

1

27q2q3,

(5.14)

while the system (3.30) with the second chain (3.28) give the second part of the transfor-mation

v1= 3q1

,

v2= q2 2q3

q1,

v3= 3p3+ 6p2

1

3

q23

q1+

1

3

q2q3

q1 4

9

q21.

(5.15)

On the other hand the relation between HF and DN coordinates reads

u1= 1 2 3,u2= 12+13+23,

u3= 123,i= v1

2i +v2i+v3, i= 1, 2, 3.

(5.16)

Hence, eliminating the HF coordinates (u, v) from the system (5.14)-(5.16) we arrive atthe explicit relation between DN coordinates (, ) and natural coordinates (q, p). Unfor-tunately, the formulas are too long to quote them in the text. In DN coordinates the twoPoisson structures take the form (5.9) and the related separated curve

3 4 =(c2+h1) +c12 +h2+h3. (5.17)The implicit solutions of the system can be obtained by solving the three decoupled firstorder ODEs

Wii

3=

Wii

(c2i+a1) + (4i +c1

2i +a2i+a3), i= 1, 2, 3. (5.18)


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6 Concluding remarks

In this review paper we presented a multi-Hamiltonian separability theory of finite dimen-sional dynamical systems, in the frame of the set of canonical coordinates. It reveals theimportant fact that the multi-Hamiltonian property of considered system is closely related

to its separability. Actually, we presented the constructive method of finding a separationcoordinates once having a bi(multi)-Hamiltonian representation of the underlying dynam-ical system, written down in some natural canonical coordinates. There is still an openquestion: whether each bi-Hamiltonian chain with sufficient number of constants of mo-tion guarantees a separability of underlying Hamiltonian systems? Or, in other words:whether the arbitrary degenerate Poisson pencil, written in noncanonical coordinates, canbe restricted to the nondegenerate one on a symplectic leaf of one of the Poisson ten-sors from the pencil? The second important problem is how to perform effectively sucha Marden-Ratiu projection if it is possible? Some progress in this direction was maderecently [41] but this part of separability theory still requires further investigations.

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degenerate poisson pencils on curves; new separability theory - blaszak

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