Lax representation for two--particle dynamics splitting on two tori

Abstract

Lax representation in terms of 2×22\times 2 matrices is constructed for a separable multiply--periodic system splitting on two tori. Hyperelliptic Kleinian functions and their reduction to elliptic functions are used. Comment: 9 pages, LaTeX

Full text

arXiv:solv-int/9603004v1 12 Mar 1996
Lax representation for two–particle dynamics
splitting on two tori
V Z Enolskii
1)
and M Salerno
2)
1)
Department of Theoretical Physics
Institute of Magnetism
of National Academy of Sciences of Ukraine,
Vernadsky str. 36, Kiev-680, 252142, Ukraine
2)
Istituto Nazionale di Fisica della Materia and
Department of Theoretical Physics Institute of Physics,
University of Salerno, Baronissi I-84100 , Italy
Abstract
Lax representation in terms of 2 × 2 matrices is constructed for sep-
arable multiply–periodic systems splitting on two tori. Hyperelliptic
Kleinian functions and their reduction to elliptic functions are used.
Introduction. Completely integrable systems with two degree of
freedoms and with dynamics splitting on two tori have been largely
investigated during the past years as examples of separable multiply-
periodic systems. The list of such systems includes the well known in-
tegrable cases of the H´enon-Heiles system [1], several integrable cases
of quartic potentials [2], the motion of a particle in the Coulomb po-
tential and in external uniform field, the Chaplygin top [3], etc. A Lax
representation for these sys tems can be readily constructed in terms
of a direct product of Lax operators [1], one for each splitting tori,
as first proposed in ref.[4]. This approach leads for a system of two
1

particles to 4 ×4 Lax representations (see e.g. [1, 2]) this making the
quantization of the above systems much more difficult to perform. In
order to simplify the quantum problem it would be more convenient
to use Lax representations in terms of 2 × 2 m atrices. The problem
of the existence of such representations for the above systems is still
open.
The aim of the present paper is to show how to construct for
dynamics splitting on two tori, Lax representations in terms of 2 × 2
matrices. The main idea is to use hyperelliptic curve of genus two,
which is a N–sheeted cover of two given elliptic curves. Such covers are
known to exist for any N > 1 and for arbitrary tori (see e.g.[5, 6, 7]). It
is clear that if the hyperelliptic curve is associated with an hamiltonian
system for which a 2 ×2 Lax representation is known, one can readily
construct a similar representation for the two tori dynamics simply
by using the transformation induced by the covering. To illustrate
this approach we take as working example the integrable cases of the
H´enon–Heiles system [8 ]. The possibilities of generalization of this
approach to s ystem with more then two degrees of freedom is briefly
discussed at the end of the paper.
Reduction. Consider the hyperelliptic curve V = (y, z) of genus
two,
y
2
= 4z
5
+ λ
4
x
4
+ λ
3
x
3
+ λ
2
x
2
+ λ
1
x + λ
0
(1)
with λ
i
∈ C chosen in such a way that (1) takes th e form
w
2
= z(z − 1)(z − α)(z − β)(z − αβ) . (2)
The curve (2) gives a two-sheeted covering of two tori π
±
: V =
(w, z) → E
±
= (η
±
, ξ
±
),
η
2
±
= ξ
±
(1 − ξ
±
)(1 − k
2
±
ξ
±
) , (3)
with Jacobi moduli
k
2
±
= −
(
√
α ∓
√
β)
2
(1 − α)(1 − β)
. (4)
Equation (4) can be inverted as
α + β = 2
k
2
+
+ k
2
−
(k
′
+
− k
′
+
)
2
, αβ =
k
′
+
+ k
′
−
k
′
+
− k
′
+
!
2
, (5)
2

where k
′
±
are additional Jacobian moduli, k
2
±
+ k
′
±
2
= 1. Explicitly,
the covers π
±
are given by
η
±
= −
q
(1 − α)(1 − β)
z ∓
√
αβ
(z − α)
2
(z − β)
2
w , (6)
ξ
+
= ξ
−
=
(1 − α)(1 − β)z
(z − α)(z − β)
. (7)
Let (y
1
, x
1
), (y
2
, x
2
) be arbitrary points on a symmetric degree V ×V .
The Jacobi inversion problem is the problem of finding this point as
a fun ction u = (u
1
, u
2
) from the equations
Z
x
1
x
0
dz
w
+
Z
x
2
x
0
dz
w
= u
1
, (8)
Z
x
1
x
0
zdz
w
+
Z
x
2
x
0
zdz
w
= u
2
. (9)
We write
Z
x
1
x
0
z −
√
αβ
w
dz +
Z
x
2
x
0
z −
√
αβ
w
dz = u
+
, (10)
Z
x
1
x
0
z +
√
αβ
w
dz +
Z
x
2
x
0
z +
√
αβ
w
dz = u
−
. (11)
with
u
±
= −
q
(1 − α)(1 − β) (u
2
∓
p
αβu
1
) . (12)
We can reduce the hyperelliptic integrals in (10,11) to elliptic ones by
using the formula
dξ
±
η
±
= −
q
(1 − α)(1 − β) (z ∓
p
αβ)
dz
w
. (13)
Let us introdu ce the coordinates (see [9])
X
1
= sn(u
+
, k
+
)sn(u
−
, k
−
) ,
X
2
= cn(u
+
, k
+
)cn(u
−
, k
−
) , (14)
X
3
= dn(u
+
, k
+
)dn(u
−
, k
−
) ,
where sn(u
±
, k
±
), cn(u
±
, k
±
), and dn(u
±
, k
±
) denote usual Jacobi el-
liptic functions[10]. Applying th e addition theorem for Jacobi elliptic
functions,
sn(u
1
+ u
2
, k) =
s
2
1
− s
2
2
s
1
c
2
d
2
− s
2
c
1
d
1
,
3

cn(u
1
+ u
2
, k) =
s
1
c
1
d
2
− s
2
c
2
d
1
s
1
c
2
d
2
− s
2
c
1
d
1
,
dn(u
1
+ u
2
, k) =
s
1
d
1
c
2
− s
2
d
2
s
1
s
1
c
2
d
2
− s
2
c
1
d
1
,
where s
i
= sn(u
i
, k), c
i
= cn(u
i
, k), d
i
= dn(u
i
, k), i = 1, 2, we can
write Eq.s (14) in the form
X
1
= −
(1 − α)(1 − β)(αβ + ℘
12
)
(α + β)(℘
12
− αβ) + αβ℘
22
+ ℘
11
, (15)
X
2
= −
(1 + αβ)(αβ − ℘
12
) − αβ℘
22
− ℘
11
(α + β)(℘
12
− αβ) + αβ℘
22
+ ℘
11
, (16)
X
3
=
αβ℘
22
− ℘
11
(α + β)(℘
12
− αβ) + αβ℘
22
+ ℘
11
. (17)
Here ℘
ij
are Kleinian ℘–functions which solve the Jacobi inversion
problem and are expr essed in terms of (y
1
, x
1
), (y
2
, x
2
) as follows
℘
22
= x
1
+ x
2
, ℘
12
= −x
1
x
2
, ℘
11
=
F (x
1
, x
2
) − 2y
1
y
2
4(x
1
− x
2
)
2
and
F (x
1
, x
2
) =
k=2
X
k=0
x
k
1
x
k
2
(2λ
2k
+ λ
2k+1
(x
1
+ x
2
)) (18)
with λ’s calculated from (2). The Kleinian ℘–fun ctions are known to
be a natural generalization of the Weierstrass elliptic f unctions and
can then be exp ressed through second logarithmic derivative of the
Kleinian σ–function,
℘
ij
(u) = −
∂
2
lnσ(u)
∂u
i
∂u
j
, i, j = 1, 2,
(for details see [5, 11]). The three functions ℘
22
, ℘
12
, ℘
11
are alge-
braically d ependent and are coordinates for the so called K u mmer
surface which is a quartic surface in C
3
. For later convenience we
remark that the formulae (15-17) can be inverted as
℘
11
= (B − 1)
A(X
2
+ X
3
) − B(X
3
+ 1)
X
1
+ X
2
− 1
, (19)
℘
12
= (B − 1)
1 + X
1
− X
2
X
1
+ X
2
− 1
, (20)
℘
22
=
A(X
2
− X
3
) + B(X
3
− 1)
X
1
+ X
2
− 1
, (21)
4

where A = α + β, B = 1 + αβ.
Lax representation. Let us consider the following equations for
the four–indexed functions ℘:
℘
2222
= 6℘
2
22
+ 4℘
12
+ λ
4
℘
22
+
1
2
λ
3
, (22)
℘
1222
= 6℘
22
℘
12
− 2℘
11
+ λ
4
℘
12
, (23)
with λ
3
and λ
4
arbitrary . The first equation, after u
2
differentiation,
is the standard K dV equation w ritten with respect to the function ℘
22
while the second equation represents the stationary flow for the two
gap KdV–solution (℘
22
) of the third vector field of the KdV hierarchy.
As well known equations (22,23) can be w ritten in the Lax form [12],
∂L
∂t
= [M, L], L =
V U
W −V
!
, M =
0 1
Q 0
!
. (24)
Here we take the elements of the matrices L and M to be polynomials
in x of th e f orm
U = x
2
− ℘
22
x − ℘
12
, (25)
V = −
1
2
∂U
∂u
2
, (26)
W = −
1
2
∂
2
U
∂u
2
2
+ UQ , (27)
Q = x + 2℘
22
+
1
4
λ
4
. (28)
The discriminant curve det (L −yE) = 0, (E is the 2 ×2 unit matrix)
has th en the form of Eq. (1) with λ
4
, λ
3
, λ
0
arbitrary and λ
2
, λ
1
chosen as the level set of the integrals of motion:
− λ
2
= −℘
2
222
+ 4℘
11
+ λ
3
℘
22
+ 4℘
3
22
+ 4℘
12
℘
22
+ λ
4
℘
2
22
, (29)
−
1
2
λ
1
= −℘
222
℘
221
+ 2℘
2
12
− 2℘
11
℘
22
+
1
2
λ
3
℘
12
(30)
+ 4℘
12
℘
2
22
+ λ
4
℘
12
℘
22
.
The following proposition represents the main result of the p aper.
Prop osition Let
U = x
2
−
A(X
2
− X
3
) + B(X
3
− 1)
X
1
+ X
2
− 1
x
5

+ (B − 1)
X
1
− X
2
+ 1
X
1
+ X
2
− 1
, (31)
Q = x + 2
A(X
2
− X
3
) + B(X
3
− 1)
X
1
+ X
2
− 1
+ A + B , (32)
where X
i
are the coordinates giv en in (14) and A = α +β, B = αβ + 1
are expressed in terms of Jacobian moduli k
±
according to (5). Then
the Lax equation (24) is equivalent to the equations f or Jacobi elliptic
functions,
d
du
±
sn(u
±
; k
±
) =
q
(1 − sn
2
(u
±
; k
±
)(1 − k
2
±
sn
2
(u
±
; k
±
)) . (33)
To prove this statement one can expand sn(u
±
; k
±
) around u
±
= 0
to obtain from (24) the equation (33) with the superscripts ’±’. We
remark that a d irect substitution of (19,20,21) into the equations of
motion (22,23) would be quite involved even for symbolic calculations
on a computer.
An example: Lax representation for the integrable cases
of the H´enon–Heiles system. Let us apply the above result to the
integrable cases of the H´enon–Heiles system (see e.g.[8]). One of them
(the case (ii) in the terminology of [8]) is isomorphic to the fi fth–order
stationary KdV flow, this giving a Lax representation in terms of 2×2
matrices. Th e other two cases– the cases (i) and (iii)– are isomorphic
to fifth stationary flows of respectively Sawada–Kotera and Kaup-
Kupershmidt equations. They both lead to a Lax representations in
terms of 3 × 3 matrices[8]. The 4 × 4 Lax representation is derived
in [1]. Let us show how to construct 2 × 2 Lax repr esentation for the
cases (i),(iii).
Consider first the integrable case (i). The Hamiltonian H and
second integral of motion K have the form
H =
1
2
p
2
1
+
1
2
p
2
2
+ q
1
q
2
2
+
1
3
q
3
1
+ a(q
2
1
+ q
2
2
) , (34)
K = p
1
p
2
+
1
3
q
3
2
+ q
2
q
2
1
+ 2aq
1
q
2
. (35)
The hamiltonian system is separated in Cartesian coordinates, q
1,2
=
˜
Q
1
±
˜
Q
2
, p
1,2
=
˜
P
1
±
˜
P
2
and the dynamics is splitting to two tori
˜
P
2
1
= −
4
3
˜
Q
3
1
− 2a
˜
Q
2
1
+
1
2
(
˜
H +
˜
K) ,
˜
P
2
2
= −
4
3
˜
Q
3
2
− 2a
˜
Q
2
2
+
1
2
(
˜
H −
˜
K) , (36)
6

where
˜
H =
˜
P
2
1
+
˜
P
2
2
+
4
3
˜
Q
3
1
+
4
3
˜
Q
3
3
+ 2a(
˜
Q
2
1
+
˜
Q
2
2
),
˜
K =
˜
P
2
1
−
˜
P
2
2
+
4
3
˜
Q
3
1
−
4
3
˜
Q
3
3
+ 2a(
˜
Q
2
1
−
˜
Q
2
2
). By passing from (36) to the standard f orm
of the elliptic curve (33) we find
℘
±

it
√
3

=
1
2
(q
1
(t) ± q
2
(t) + a) (37)
with ℘
±
standard Weierstrass elliptic functions with moduli e
±
i
, i =
1, 2, 3 satisfying the equations
4e
±
1
e
±
2
e
±
3
= a
3
−
3
2
(
˜
H ±
˜
K) , 8(e
±
1
e
±
2
+e
±
1
e
±
3
+e
±
2
e
±
3
)+3
a
2
2
= 0 . (38)
The Lax representation (24) is then valid for the system with
X
1
=
s
2e
+
1
− 2e
+
3
q
1
+ q
2
+ a − 2e
+
3
s
2e
−
1
− 2e
−
3
q
1
− q
2
+ a − 2e
−
3
.
X
2
=
s
q
1
+ q
2
+ a − 2e
+
1
q
1
+ q
2
+ a − 2e
+
3
s
q
1
− q
2
+ a − 2e
−
1
q
1
− q
2
− 2e
−
3
.
X
3
=
s
q
1
+ q
2
+ a − 2e
+
2
q
1
+ q
2
+ a − 2e
+
3
s
q
1
− q
2
+ a − 2e
−
2
q
1
− q
2
+ a − 2e
−
3
.
As shown in ref.[13], the integrable case (iii) is linked to case (i) by
means of a canonical transformation. The corresponding 2 × 2 Lax
representation can be then derived from the one of case (i) by means
of this transform ation.
Concluding remarks. In closing this paper we make the fol-
lowing remark. Equip the curve by the canonical basis of cycles
A
1
, A
2
, B
1
, B
2
and normalize the holomorphic differentials dv
i
= (c
i1
+
zc
i2
)dz/w(z), i = 1, 2 in such a way that the Riemann matrix Ω has
the following form
Ω =
H
A
1
dv
1
H
A
2
dv
1
H
B
1
dv
1
H
B
2
dv
1
H
A
1
dv
2
H
A
2
dv
2
H
B
1
dv
2
H
B
2
dv
2
!
=
1 0 τ
11
τ
12
0 1 τ
12
τ
22
!
(39)
It is known (see e.g. [6, 9, 7]), that the curve (1) cover N–sheetedly
two tori if and only if the Riemann matrix Ω can be transformed by
some linear transformation of the basis cycles to the form
τ =
τ
11
1
N
1
N
τ
22
!
,
7

where the positive integer N is also called a Picard number. The
condition for the matrix τ to be transformed to the form given above
is that τ belongs to the Hu mbert surface H
N
H
N
=
n
ατ
11
+ βτ
12
+ γτ
22
+ δ(τ
2
12
− τ
11
τ
22
) + ε = 0,
α, β, γ, δ, ε ∈ Z , β
2
− 4(αγ + εδ) = N
2
o
.
The case considered in th is paper corresponds, among the infinite
transformations of N–th order which permit to reduce a dynamics of
two particle system associated with N –sheeted covering of tori, just
to the case N = 2. It is clear, however, that the above analysis can
be extended to cur ves of high genus.
These arguments were used in [7] to describe elliptic potentials
of the Schr¨odinger equation, which also studied in the frameworks of
spectral theory [14, 15, 16 ].
Acknowledgement. The authors are grateful to V.Kuznetsov
who attracted their attention to the problem discuss ed . The researches
described in this publication were supported in part by grant no.
U44000 (VZE) from the ISF and INTAS grant no. 93–1324 (VZE
and MS).
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