On the two different bi-Hamiltonian structures for the Toda lattice

0.15 MB·English

by  A V Tsiganov

On the two different bi-Hamiltonian structures for
the Toda lattice.
A V Tsiganov
St.Petersburg State University, St.Petersburg, Russia
7
e–mail: [email protected]
0
0
2

n Abstract
a
J We construct two different incompatible Poisson pencils for the Toda

1 lattice by using known variables of separation proposed by Moser and by
3
Sklyanin.

]
I
S
1 Introduction
.
n
i
l
n A bi-Hamiltonian manifold M is a smooth manifold endowed with two compatible
[ bi-vectors P ,P such that
0 1

1
v [[P ,P ]] = [[P ,P ]] = [[P ,P ]] = 0,
0 0 0 1 1 1
2
6
where [[.,.]] is the Schouten bracket. Such a condition assures that the linear combi-
0
1 nation P0−λP1 is a Poisson pencil, i.e. it is a Poisson bi-vector for each λ ∈ C, and
0
therefore the corresponding bracket {.,.} −λ{.,.} is a pencil of Poisson brackets
7 0 1
0 [5].
/ Dynamical systems on M having enough functionally independent integrals of
n
i motion H ,...H in involution with respect to the both Poisson brackets
l 1 n
n
:
v {Hi,Hj}0 = {Hi,Hj}1 = 0. (1.1)
i
X
will be called bi-integrable systems.
r
a The main aim of this note is to prove that any separable system is bi-integrable
system. As an example we show that the Toda lattice is bi-integrable system with
respect to two different Poisson pencils P +λP and P +λP ⋆, which are related
0 1 0 1
with two known families of the separated variables [6, 7].
2 The separation of variables method
A complete integral S(q,t) of the Hamilton-Jacobi equation
∂S(q,t) ∂S(q,t)
+H q, ,t = 0, (2.1)
∂t ∂q
(cid:18) (cid:19)
1
where q = (q ,...,q ), is a solution S(q,t,α ,...,α ) depending on n parameters
1 n 1 n
α = (α ,...,α ) such that
1 n
∂2S(q,t,α)
det 6= 0. (2.2)
∂q ∂α
(cid:13) i j (cid:13)
(cid:13) (cid:13)
For any complete integral of (2.1)(cid:13)solutions q =(cid:13)q (t,α,β) and p = p (t,α,β) of the
(cid:13) i (cid:13) i i i
Hamilton equations of motion may be found from the Jacobi equations
∂S(q,t,α) ∂S(q,t,α)
β = − , p = , i = 1,...,n. (2.3)
i i
∂α ∂q
i i
According to (2.2) if we resolve a second part of the Jacobi equations with respect
to parameters α ,...,α one gets n independent integrals of motion
1 n
α = H (p,q,t), m = 1,...,n, (2.4)
m m
as functions on the phase space M with coordinates p,q.
Definition 1 A dynamical system is a separable system if the corresponding com-
plete integral S(q,t,α) has an additive form
n
S(q,t,α ,...,α ) = −Ht+ S (q ,α ,...,α ). (2.5)
1 n i i 1 n
i=1
X
Here the i-th component S depends only on the i-th coordinate q and α.
i i
In such a case Hamiltonian H is said to be separable and coordinates q are said
to be separated coordinates for H, in order to stress that the possibility to find an
additive complete integral of (2.5) depends on the choice of the coordinates.
For the separable dynamical system we have
∂2 ∂ ∂
S = S = 0, for all j 6= k, (2.6)
j
∂q ∂q ∂q ∂q
k j k (cid:18) j (cid:19)
such that the second Jacobi equations (2.3) are the separated equations
∂ ∂
p = S (q ,α ,...,α ), or φ (p ,q ,α) = p − S (q ,α) = 0. (2.7)
j j j 1 n j j j j j j
∂q ∂q
j j
Here j-th equation contains a pair of canonical variables p and q only.
j j
Proposition 1 For any separable dynamical system integrals of motion H (p,q,t)
m
(2.4) are in the involution
{H ,H } = 0, k,m = 1,...,n,
k m f
with respect to the following brackets {.,.} on M
f
{p ,q } = δ f (p,q), {p ,p } = {q ,q } = 0, (2.8)
i j f ij j i j f i j f
which depend on n arbitrary functions f (p,q),...,f (p,q).
1 n
2
Proof: In fact we have to repeat the proof of the Jacobi theorem due by Liouville.
Namely, differentiate the relations (2.4) by q and then substitute momenta p from
j k
separated equations (2.7) to obtain
∂H n ∂H ∂p ∂H ∂H ∂2S
m m i m m j
+ = + = 0.
∂q ∂p ∂q ∂q ∂p ∂q2
j i=1 i j (cid:18) j j j (cid:19)
X
It follows that for any H and H
k m
n ∂H ∂H ∂H ∂2S n ∂H ∂H n ∂H ∂H ∂2S
k m m j k m k m j
f + = f + f = 0.
j ∂p ∂q ∂p ∂q2 j ∂p ∂q j ∂p ∂p ∂q2
j=1 j (cid:18) j j j (cid:19) j=1 j j j=1 j j j
X X X
Permute indexes k and m and subtract the resulting equation from the previous one
we get
n
∂H ∂H ∂H ∂H
k m m k
f − = 0.
j
∂p ∂q ∂p ∂q
j=1 (cid:18) j j j j (cid:19)
X
The final assertion easily follows.
Brackets {.,.} (2.8) are the Poisson brackets if and only if
f
[[P ,P ]] = 0, (2.9)
f f
where
0 diag(f ,...,f )
P = 1 n . (2.10)
f −diag(f ,...,f ) 0
1 n
(cid:18) (cid:19)
Proposition 2 If j-th function
f (p,q) = f (p ,q ), j = 1,...,n, (2.11)
j j j j
depends only on the j-th pair of coordinates p ,q then brackets {.,.} (2.8) are the
j j f
Poisson brackets, which are compatible with canonical ones.
Proof: Substitute tensor P (2.10) into the equations
f
0 I
[[P ,P ]] = [[P ,P ]] = 0, where P = ,
0 f f f 0 −I 0
(cid:18) (cid:19)
one gets the following system of partial differential equations
∂f ∂f ∂f ∂f
j j j j
= = 0, f = f = 0,
i i
∂q ∂p ∂q ∂p
k k k k
for all j 6= k 6= i. The separable functions f (p ,q ) (2.11) satisfy this system of
j j j
equations. So, the corresponding tensor P (2.10) is the Poisson tensor, which is
f
compatible with canonical tensor P .
0
3
Remark 1 Separated variables (p,q) are defined up to canonical transformations
p = X(p ,q ), q = Y(p ,q ),
j j j j j j
which have to preserve canonical tensor P and would change the second tensor P .
0 f
It is clear, that freedom in the choice of the functions f (p ,q ) is related with this
j j j
freedom in the definition of the separated variables.
According to the following proposition, the separation of variables method is
closely related with the bi-Hamiltonian geometry.
Proposition 3 Any separable dynamical system is bi-integrable system.
Proof: In order to get a pair of compatible Poisson brackets on the phase space
M it is enough to postulate brackets (2.8)
{p ,q } = δ f (p ,q ), {p ,p } = {q ,q } = 0, (2.12)
i j f ij j j j i j f i j f
between the known separated variables q and p for a given dynamical system.
j j
According to the Proposition 1 the corresponding integrals of motion H (p,q,t)
m
(2.4) are in the bi-involution with respect to brackets {.,.} and {.,.} .
0 f
In the next sections we demonstrate how this proposition works for the open and
periodic Toda lattice.
Remark 2 The separated variables (p,q) are canonical variables, which put the
corresponding recursion operator N = P P−1 in diagonal form. So, according to [2,
f 0
3], a set ofthe separated variables (p,q)issaid to beDarboux-Nijenhuis coordinates.
Remark 3 For the stationary systems differentiate the separated equations (2.7)
n
∂φ ∂φ ∂φ
j j j
dq + dp + dH = 0, (2.13)
j j i
∂q ∂p ∂H
(cid:18) j j (cid:19) i=1 i
X
then apply N∗ = P−1P and substitute (2.13) in the resulting equation one gets
0 f
n
∂φ ∂φ ∂φ
f jdq + jdp + j N∗dH =
j j j i
∂q ∂p ∂H
(cid:18) j j (cid:19) i=1 i
X
n n
∂φ ∂φ
= −f j dH + j N∗dH = 0.
j i i
∂H ∂H
i=1 i i=1 i
X X
It follows that
n n
∂φ ∂φ
j N∗dH = f j dH , j = 1,...,n,
i j i
∂H ∂H
i=1 i i=1 i
X X
that is, in matrix form,
N∗dH = dHF. (2.14)
4
Here n×n control matrix F with eigenvalues f ,...,f is defined by
1 n
∂φ
F = J−1diag(f ,...,f )J, J = j
1 n ji
∂H
i
and dH is 2n×n matrix with entries dH = ∂H /∂z , where z = (q,p).
ij j i
Equation (2.14) means that the subspace spanned by covectors dH ,...,dH is
1 n
invariant with respect to N∗ [3].
Example 1 For further use we introduce the special control matrix F, which is the
Fr¨obenius matrix
c 1 0 ··· 0
1
c 0 1 ··· 0
2
F =  . . (2.15)
f .
. ··· 0 1
 
c 0 ··· 0
n
 
 
Here c are coefficients of the characteristic polynomial ∆ (λ) of the recursion
k N
operator N = P P−1:
f 0
n
1/2
∆ (λ) = det(N −λI) = λn −(c λn−1 +...+c ) = (λ−λ ). (2.16)
N 1 n j
(cid:16) (cid:17) Yj=1
3 Open Toda lattice
LetusconsideropenTodaassociatedwiththerootsystemofA type. TheHamilton
n
function is equal to
n n−1
1
H = p 2 + eqi−qi+1,
i
2
i=1 i=1
X X
where q denotes the position of i-th particle and p is its momenta, such that
i i
{q ,p } = δ , {p ,p } = {q ,q } = 0. (3.17)
i j 0 ij i j 0 i j 0
Consequently, the equations of motion read
q˙ = p , p˙ = −eq1−q2, p˙ = eqn−1−qn
i i 1 n
and
p˙ = eqi−1−qi −eqi−qi+1, i = 2,...,n−1.
i
The exact solution is due to existence of a Lax matrix. Consider the L-operator
λ−p −eqi
L = i , (3.18)
i e−qi 0
(cid:18) (cid:19)
and the monodromy matrix
A(λ) B(λ)
T(λ) = L (λ)···L (λ)L (λ) = , detT(λ) = 1, (3.19)
1 n−1 n C(λ) D(λ)
(cid:18) (cid:19)
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which depends polynomially on the parameter λ
λn +A λn−1 +...+A B λn−1 +...+B
T(λ) = 1 n 1 n .
C λn−1 +...+C D λn−2 +...+D
1 n 2 n
(cid:18) (cid:19)
The monodromy matrix satisfies Sklyanin’s Poisson brackets:
{T(λ)⊗, T(µ)} = [r(λ−µ), T(λ)⊗T(µ)], (3.20)
0
where r(λ−µ) is the 4×4 rational r-matrix
1 0 0 0
−1 0 0 1 0
r(λ−µ) = Π, Π =   , η ∈ C. (3.21)
λ−µ 0 1 0 0
 0 0 0 1 
 
 
Monodromy matrix T(λ) is the Lax matrix for periodic Toda lattice, whereas
the Lax matrix for open Toda lattice is equal to
A B 1 0
T (λ) = KT(λ) = (λ), K = .
o 0 0 0 0
(cid:18) (cid:19) (cid:18) (cid:19)
The trace of the Lax matrix
trT (λ) = A(λ) = λn +H λn−1 +···H , {H ,H } = 0. (3.22)
o 1 n i j
generates n independent integrals of motion H in the involution providing complete
i
integrability of the system [7].
3.1 The Moser variables
According to [6] we introduce the n pairs of separated variables λ , µ , i = 1,...,n,
i i
having the standard Poisson brackets,
{λ ,λ } = {µ ,µ } = 0, {λ ,µ } = δ , (3.23)
i j 0 i j 0 i j 0 ij
with the λ-variables being n zeros of the polynomial A(λ) and the µ-variables being
values of the polynomial B(λ) at those zeros,
A(λ ) = 0, µ = η−1lnB(λ ), i = 1,...,n. (3.24)
i i i
The Moser variables λ may be identified with poles of the Baker-Akhiezer function
j
Ψ~ associated with the Lax matrix T(λ) (3.19)
~ ~ ~
T(λ)Ψ = λΨ, (Ψ,α~) = const,
having some very special normalization α~ [8].
The interpolation data (3.24) plus n identities
B(λ )C(λ ) = detT(λ) = 1
i i
6
allow us to construct the separation representation for the whole monodromy matrix
T(λ):
A(λ) = (λ−λ )(λ−λ )···(λ−λ ),
1 2 n
eµi
B(λ) = A(λ) n ,
i=1 (λ−λ )A′(λ )
i i
P
(3.25)
e−µi
C(λ) = −A(λ) n ,
i=1 (λ−λ )A′(λ )
i i
P
1+B(λ)C(λ)
D(λ) = .
A(λ)
If we postulate the following second Poisson brackets (2.12)
{λ ,µ } = λ δ , {λ ,λ } = {µ ,µ } = 0
i j 1 i ij i j 1 i j 1
one gets [9]
{A(λ),A(µ)} = {B(λ),B(µ)} = 0,
1 1
(3.26)
1
{A(λ),B(µ)} = µA(λ)B(µ)−λA(µ)B(λ) .
1
λ−µ
(cid:16) (cid:17)
The first bracket in (3.26) guaranties that integrals of motion H (3.22) from A(λ)
i
are in the bi-involution.
Substitute polynomials A(λ) and B(λ) in initial (p,q)-variables into the brackets
(3.26) and solve the resulting equations to obtain the known second Poisson tensor
[1, 4]
n−1 n n
∂ ∂ ∂ ∂ ∂ ∂
Popen = eqi−qi+1 ∧ + p ∧ + ∧ . (3.27)
1 i
∂p ∂p ∂q ∂p ∂q ∂q
i=1 i+1 i i=1 i i i<j j i
X X X
TheminimalcharacteristicpolynomialofthecorrespondingrecursionoperatorN =
M
PopenP−1 is equal to
1 0
n n
∆ (λ) = A(λ) = (λ−λ ) = λn + H λn−j.
NM j j
j=1 j=1
Y X
So, (λ,µ)-coordinatesare variables of separation of the action-angle type [3], i.e. the
corresponding separated equations are trivial
{H ,λ } = 0, i,j = 1,...,n.
i j
The Hamiltonians H (3.22) from A(λ) satisfy the Fr¨obenius recursion relations
i
N∗ dH = dH −H dH , (3.28)
M i i+1 i 1
where N∗ = P−1Popen and H = 0, i.e. Hamiltonians H satisfy equation (2.14)
M 0 1 n+1 i
with the Fr¨obenius matrix F (2.15).
f
7
Example 2 For the 3-particles open Toda lattice Hamiltonians H (3.22) are
i
H = −(p +p +p ),
1 1 2 3
H = p p +p p +p p −eq1−q2 −eq2−q3,
2 1 2 1 3 2 3
H = −p p p +p eq2−q3 +p eq1−q2.
3 1 2 3 1 3
It’s obvious that H = H2/2−H . The second Poisson tensor P (3.27) in the matrix
1 2 1
form reads as
0 −1 −1 p 0 0
1
1 0 −1 0 p 0
2
 
1 1 0 0 0 p
Popen = 3 . (3.29)
1  −p1 0 0 0 −eq1−q2 0 
 
 0 −p2 0 eq1−q2 0 −eq2−q3 
 
 0 0 −p 0 eq2−q3 0 
3
 
 
The control matrix F in (2.14) is the Fr¨obenius matrix
−H 1 0
1
Fopen = −H 0 1 , (3.30)
M  2 
−H 0 0
3
 
where coefficients c = −H coincide with integrals of motion.
i i
3.2 The Sklyanin variables
According to [7, 8] we can consider another set of the separated coordinates, which
are poles of the Baker-Akhiezer function Ψ~ associated with the Lax matrix T(λ)
(3.19) having the standard normalization α~ = (0,1).
In this case the first half of variables is coming from (n − 1) finite roots and
logarithm of leading coefficient of the non-diagonal entry of the monodromy matrix
n−1
B(λ) = −eun (λ−v ), (3.31)
j
j=1
Y
Another half is given by
n
u = −lnA(v ), j = 1,...,n−1, and v = p . (3.32)
j j n i
i=1
X
In these separated variables other entries of T(λ) read as
n n−1 n−1 n−1
λ−v
A(λ) = λ+ v (λ−v )+ e−uj i . (3.33)
j j
v −v
j=1 !j=1 j=1 i6=j j i
X Y X Y
and
n−1 n−1
λ−v A(λ)D(λ)−1
D(λ) = − euj i , C(λ) = .
v −v B(λ)
j=1 i6=j j i
X Y
8
If we postulate the second Poisson brackets (2.12)
{v ,u }⋆ = v δ , {v ,v }⋆ = {u ,u }⋆ = 0
i j 1 i ij i j 1 i j 1
one gets
{A(λ),A(µ)}⋆ = {B(λ),B(µ)}⋆ = 0, (3.34)
1 1
and
1
{A(λ),B(µ)}⋆ = λA(λ)B(µ)−µA(µ)B(λ)
1
λ−µ
(cid:16) (cid:17)
n−1
+ e−un λ+µ+ v B(λ)B(µ). (3.35)
i
!
i=1
X
In initial (p,q)-variables the last bracket looks like
1
{A(λ),B(µ)}⋆ = λA(λ)B(µ)−µA(µ)B(λ) (3.36)
1
λ−µ
(cid:16) (cid:17)
n−1
+ e−qn λ+µ+ p B(λ)B(µ).
i
!
i=1
X
The first bracket in (3.34) guaranties that integrals of motion H (3.22) from A(λ)
i
are in the bi-involution.
Substitute into the brackets (3.34)-(3.36) polynomials A(λ) and B(λ) in initial
(p,q)-variables and solve the resulting equations to obtain the following Poisson
tensor
n−2 n−1 n−1
∂ ∂ ∂ ∂ ∂ ∂
P ⋆ = eqi−qi+1 ∧ + p ∧ + ∧ (3.37)
1 i
∂p ∂p ∂q ∂p ∂q ∂q
i=1 i+1 i i=1 i i i<j j i
X X X
n−1 n−1
∂ ∂ ∂ ∂
+ (p +p) ∧ + eqi−qi+1 −eqi−1−qi ∧
i
∂p ∂q ∂p ∂p
i=1 n i i=2 n i
X X(cid:0) (cid:1)
∂ ∂ ∂ ∂ ∂ ∂
+ p ∧ +eq1−q2 ∧ +eqn−2−qn−1 ∧ ,
∂p ∂q ∂p ∂p ∂p ∂p
n n 1 n n−1 n
where p = n p is a total momentum.
i=1 i
The tensor P ⋆ is independent on q and the minimal characteristic polynomial
1 n
of the correPsponding recursion operator N = P ⋆P−1 is equal to
S 1 0
∆ = −eqn(λ+p)B(λ).
NS
The normalized traces of the powers of N are integrals of motion for n−1 particle
S
open Toda lattice. As consequence the Hamiltonians H (3.22) from A(λ) satisfy
i
equation (2.14) with the following control matrix
−p 0
Fopen = , (3.38)
S 0 Fopen
(cid:18) M (cid:19)
9
where Fopen is the Fr¨obenius matrix (2.15) associated with the recursion operator
M
N for n−1 particle open Toda lattice.
M
The corresponding separated equations follow directly from the definitions of
(u,v)-variables (3.32)
e−uj −A(v ) = 0, j = 1,...,n−1 and Φ = v −α = 0, (3.39)
j n n 1
where A(λ) = λn + n−1α λn−i and α = H are the values of integrals of motion.
i=1 i i i
The first (n−1) separated equations give rise to the equations of motion
P
n−1
λ−v
i
{A(λ),v } = A(v ) , j = 1,...,n−1, (3.40)
j j
v −v
j i
i6=j
Y
which are linearized by the Abel transformation [7]
n−1 vk λj−1dλ
A(λ), σ = −λj−1, σ = , j = 1,...,n−1,
j j
A(λ)
( )
k=1Z
X
where {σ } is a basis of abelian differentials of first order on an algebraic curve
j
z = A(λ) corresponding to the separated equations (3.39).
Remark 4 From the factorization of the monodromy matrix T(λ) (3.19) one gets
T (λ) = T (λ)L (λ), ⇒ B (λ) = −eqnA (λ),
n n−1 n n n−1
where B (λ) is entry of the monodromy matrix T (λ) of n particles Toda lattice and
n n
A (λ) is entry of the monodromy matrix T (λ) of n−1 particles Toda lattice.
n−1 n−1
This implies that for the (n−1)-particle chain the Moser variables λ coincide
j
with the n−1 Sklyanin variables u , i = 1,...,n−1 for the n-particle chain.
j
Example 3 At n = 3 the Poisson tensor P ⋆ in the matrix form reads as
1
0 −1 0 p 0 −p −p
1 1
1 0 0 0 p −p −p
2 2
 
0 0 0 0 0 p
P ⋆ = .
1  −p1 0 0 0 −eq1−q2 eq1−q2 
 
 0 −p 0 eq1−q2 0 −eq1−q2 
2
 
 p +p p +p p −eq1−q2 eq1−q2 0 
1 2
 
 
and at n=4 it looks like
0 −1 −1 0 p1 0 0 −p1−p
1 0 −1 0 0 p2 0 −p2−p
1 1 0 0 0 0 p3 −p3−p
 0 0 0 0 0 0 0 p 
P1⋆ = −p1 0 0 0 0 −eq1−q2 0 eq1−q2 .
 0 −p2 0 0 eq1−q2 0 −eq2−q3 eq2−q3−eq1−q2 
 0 0 −p3 0 0 eq2−q3 0 eq2−q3 
p1+p p2+p p3+p p −eq1−q2 −eq2−q3+eq1−q2 −eq2−q3 0 
 
10