Table Of ContentOn relation between discrete Frenet frames and the bi-Hamiltonian
structure of the discrete nonlinear Schrödinger equation
Theodora Ioannidou1,∗ and Antti J. Niemi2,3,4,5,†
1 Faculty of Civil Engineering, School of Engineering,
Aristotle University of Thessaloniki, 54249, Thessaloniki, Greece
2Department of Physics and Astronomy, Uppsala University, P.O. Box 803, S-75108, Uppsala, Sweden
3Nordita, Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
4 Laboratoire de Mathematiques et Physique Theorique CNRS UMR 6083,
Fédération Denis Poisson, Université de Tours, Parc de Grandmont, F37200, Tours, France
7 5Department of Physics, Beijing Institute of Technology, Haidian District, Beijing 100081, P. R. China
1
ThediscreteFrenetequationentailsalocalframingofadiscrete,piecewiselinearpolygonalchain
0
in terms of its bond and torsion angles. In particular, the tangent vector of a segment is akin the
2
classical O(3) spin variable. Thus there is a relation to the lattice Heisenberg model, that can be
n used to model physical properties of the chain. On the other hand, the Heisenberg model is closely
a related to the discrete nonlinear Schrödinger (DNLS) equation. Here we apply these interrelations
J
to develop a perspective on discrete chains dynamics: We employ the properties of a discrete chain
3 intermsofaspinorialrepresentationofthediscreteFrenetequation,tointroduceabi-hamiltonian
1 structure for the discrete nonlinear Schrödinger equation (DNLSE), which we then use to produce
integrable chain dynamics.
]
h
c
e
m
I. INTRODUCTION
-
t
a
t String-like objects have important ramifications in many areas of physics, from understanding turbulent flows in
s
fluid dynamics [1] to new forms of matter in high energy physics [2]. Similarly, piecewise linear discrete chains have
.
t many applications from the modelling of proteins in terms of its Cα backbone [3–5] to robotics and 3D virtual reality
a
m [6].
Here,thestartingpointistheobservationbyHasimoto[7]thattheonedimensionalnonlinearSchrödingerequation
-
d (NLSE)describesstring-likeobjectssuchasvortexfilaments. Forthis,achangeofvariablesisintroducedthatrelates
n the NLSE to the Frenet frame representation of a space curve. The NLSE is a widely studied integrable model with
o various theoretical [8, 9] and practical [10] applications. Among the properties of the NLSE is the support of solitons
c
as classical solutions. In the continuous case, the energy obtained from the hamiltonian of the NLSE relates to the
[
geometry of a curve by a Hasimoto transformation. Thus, the corresponding solitons describe the buckling of the
1 curve. Similarly, a properly discretized version of the NLSE preserves the integrability [11, 12] and supports soliton
v
solutions. These can be utilized to model the buckling of piecewise linear polygonal chains, via a discretized version
1
of the Hasimoto transformation [13].
3
In [13], a continuous string in R3 is represented by a two component complex spinor. The dreibein Frenet equation
6
8 becomes a two component spinor Frenet equation which relates to the integrable hierarchy of the NLSE. That way
0 the conserved charges of the NLSE hierarchy can be considered as hamiltonians; and can be used to compute the
. energyandgovernthetimeevolutionofastring. Finally,ithasbeenshownthatthestringhamiltoniansintheNLSE
1
hierarchy can be obtained alternatively by using a formalism of projection operators, originally introduced in the
0
7 CPN models [17]. In particular, the connection between the projection operator formalism of the CP1 model and the
1 spinor Frenet equation has been explicitly presented.
: In [13–15] the Frenet frame formalism of [13, 16] has been generalized to the case of polygonal strings that are
v
i piecewise linear. In particular, the Poisson geometry of a discrete string in R3 has been derived. To do so, the Frenet
X frames are converted into a spinorial representation, the discrete spinor Frenet equation is interpreted in terms of a
r transfer matrix formalism, and the Poisson brackets are determined in terms of the spinor components.
a
The present article combines the results of [13] and [15] to study the discrete nonlinear Schrödinger equation
(DNLSE), its mathematical structure and its use in modelling discrete, piecewise linear filamental chains and strings.
In particular, a bi-hamiltonian description of the DNLSE, based on the discrete spinorial Frenet frame, is presented.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
II. THE DISCRETE FRENET FRAME
Ourstartingpointisthedescriptionofadiscretestringintermsofanopenandoriented,piecewiselinearpolygonal
chain (cid:126)r(s) ∈ R3 [16]. The arc length parameter takes values on s ∈ [0,L] where L is the total length of the string.
The vertices D (for i∈Z+) that specify the string are located at the points (cid:126)r =((cid:126)r ,...,(cid:126)r ) with (cid:126)r(s )=(cid:126)r ; while,
i i 0 n i i
the endpoints are(cid:126)r(0)=(cid:126)r and(cid:126)r(L)=(cid:126)r . In addition, the distance of the nearest neighbour vertices D and D is
0 n i i+1
|(cid:126)r −(cid:126)r |=s −s ,
i+1 i i+1 i
which are connected by the line segments
s−s s−s
(cid:126)r(s) = i (cid:126)r − i+1 (cid:126)r , s∈(s ,s ),
s −s i+1 s −s i i i+1
i+1 i i+1 i
and are characterised by the unit length discrete tangent vector(cid:126)t =(t ,t ,t ) that points from D to D
i 1 2 3 i i i+1
(cid:126)r −(cid:126)r
(cid:126)t = i+1 i . (1)
i |(cid:126)r −(cid:126)r |
i+1 i
Then, the discrete Frenet frame at the vertex D at the point(cid:126)r can be introduced assuming that the vertices at the
i i
points (cid:126)r , (cid:126)r and (cid:126)r are not located on a common line (that is, (cid:126)t and(cid:126)t are not parallel): The unit binormal
i+1 i i+1 i i−1
vector is defined by
(cid:126)t ×(cid:126)t
(cid:126)b = i−1 i
i |(cid:126)t ×(cid:126)t |
i−1 i
while the unit normal vector (cid:126)n =(cid:126)b ×(cid:126)t takes the form
i i i
−(cid:126)t +(cid:0)(cid:126)t ·(cid:126)t (cid:1)(cid:126)t
i−1 i−1 i i
(cid:126)n = .
i |(cid:126)t +(cid:0)(cid:126)t ·(cid:126)t (cid:1)(cid:126)t |
i−1 i−1 i i
The orthogonal triplet ((cid:126)t,(cid:126)n,(cid:126)b) constitutes the discrete Frenet frame associated the vertex D for each i ∈ [1,n−1].
i i
The aforementioned formulae are the two-point finite difference variants of the (corresponding) continuous Frenet
frames. That way, the discretised curvature and torsion become geometrical. They correspond to the bond and
torsion angle which define the transfer angle that maps the discrete Frenet frame at D to D through the transfer
i i+1
matrix R
i+1,i
(cid:126)n (cid:126)n
(cid:126)b = Ri+1,i(cid:126)b
(cid:126)t (cid:126)t
i+1 i
cosΘcosΦ cosΘsinΦ −sinΘ (cid:126)n
= −sinΦ cosΦ 0 (cid:126)b . (2)
sinΘcosΦ sinΘsinΦ cosΘ (cid:126)t
i+1 i
Note that, the transfer matrix is parametrized in terms of the two Euler angles (Θ,Φ) and represents an orthogonal
i
rotation between the two consecutive discrete Frenet frames; i.e., is an element of SO(3) as illustrated in Figure [1].
The third Euler angle has been excluded due to the orthogonality of(cid:126)b with(cid:126)t .
i i−1
From the transfer matrix (2) all the discrete Frenet frames can be constructed since
(cid:126)t =sinΘ cosΦ (cid:126)n +sinΘ sinΦ (cid:126)b +cosΘ (cid:126)t
i+1 i+1 i+1 i i+1 i+1 i i+1 i
implying that
(cid:126)t ·(cid:126)t = cosΘ
i+1 i i+1
(cid:126)t ·(cid:126)n = sinΘ cosΦ
i+1 i i+1 i+1
(cid:126)t ·(cid:126)b = sinΘ sinΦ . (3)
i+1 i i+1 i+1
3
(cid:126)t
i
(cid:126)t
i+1
Θ
i+1
(cid:126)n
i
(cid:126)b Φi+1 (cid:126)bi+1
i
(cid:126)n
i+1
Figure 1: The bond and torsion angles of the transfer matrix R .
i+1
Similarly, from the other rows one gets
(cid:126)b ·(cid:126)b = cosΦ
i+1 i i+1
(cid:126)b ·(cid:126)n = −sinΦ
i+1 i i+1
(cid:126)b ·(cid:126)t = 0, (4)
i+1 i
and
(cid:126)n ·(cid:126)t = −sinΘ
i+1 i i+1
(cid:126)n ·(cid:126)b = cosΘ sinΦ
i+1 i i+1 i+1
(cid:126)n ·(cid:126)n = cosΘ cosΦ . (5)
i+1 i i+1 i+1
Thus, Θ is the bond angle and Φ is the torsion angle corresponding to the discrete analogue of the continuum
i+1 i+1
curvature and torsion, respectively; i.e., the chain is determined by its bond and torsion angles uniquely up to global
rotation and translation.
III. THE DISCRETE SPINOR FRENET EQUATION
In[15],analternativeapproachtorepresentdiscretepiecewiselinearchainsisintroducedanddevelopedtodescribe
their time evolution, and the two dimensional Riemann surfaces that are swept by time evolution. In particular, the
spinorial formulation of a discrete string in combination with the formalism of the discrete Frenet equations has been
employed as described below.
Each link from vertex D to vertex D is associated to a two component complex spinor
i i+1
(cid:18) (cid:19)
z
ψ = 1 , (6)
i z
2 i
where zi (for a = 1,2) are complex variables with support on the link. The spinors are related to the unit length
a
tangent vectors by the relation
√
g (cid:126)t =<ψ ,σˆψ >
i i i i
where (cid:126)σ =(σ1,σ2,σ3) are the Pauli matrices and
√
g ≡|zi|2+|zi|2
i 1 2
4
is a metric scale factor. Explicitly,
(cid:18)ti +iti(cid:19) 1 (cid:18) 2z¯izi (cid:19)
1 2 = √ 1 2 . (7)
ti g |zi|2−|zi|2
3 i 1 2
Together with (1) this determines the spinor components zi in terms of the vertices D , up to an overall phase. In
α i
addition, for each i the conjugate spinor ψ¯ is defined by introducing the charge conjugation operation C that acts on
i
ψ in the following way
i
(cid:18) (cid:19)
−z¯
Cψ = −iσ ψ(cid:63) =ψ¯ = 2 . (8)
i 2 i i z¯
1 i
SinceC2 =−Ithespinorsareorthogonalbyconstruction,i.e. <ψ ,ψ¯ >=0. Inaddition,theyarecombinedthrough
i i
the SU(2) matrix
(cid:18) (cid:19)
z −z¯
u = 1 2
i z z¯
2 1 i
since
(cid:18) (cid:19) (cid:18) (cid:19)
1 0
ψ =u & ψ¯ =u
i i 0 i i 1
For completeness, note that
u σ u−1 =((cid:126)t·(cid:126)σ)
i 3 i i
which is in agreement with equation (7).
In the continuum case, the parametrisation of the link variables zi in terms of the local angular variables ϑ and
α
ϕ =ϕ ±ϕ gives rise to the classical version of sl which appears in the case of local spin chains [13]. Its discrete
± 2 1 2
analogue is of the form
ϑ ϑ
z1i =cos 2 e−i2ϕ|i & z2i =sin 2 ei2ϕ|i (9)
assuming that ϕ = 0 and ϕ ≡ ϕ (due to the overall phase). Note that, in the local coordinate representation (9)
+ −
the metric equals one.
In [15], it was shown that by combining the two spinors (6) and (8) into a Majorana spinor
(cid:18)−ψ¯(cid:19)
Ψ = ,
i ψ
i
the discrete Frenet euqation (2) is equivalent to the spinorial one
Ψ = U† Ψ . (10)
i+1 i+1 i
U is the ensuing transfer matrix in the chain of spinors Ψ defined by the rotational variables Zi (for a=1,2)
i i a
(cid:18)Z −Z¯ (cid:19)
U = 1 2
i Z Z¯
2 1 i
where
Z1i =cosΘ2i e−iΦ2i & Z2i =sinΘ2i eiΦ2i. (11)
It is straightforward to see that
cosΘ =(cid:0)2|Zi|2−1(cid:1) & cosΦ = 1(cid:8)(cid:2)(Zi)2+(Z¯i)2(cid:3)+cc(cid:9)
i 2 i 2 1 2
in accordance with the Frenet frame obtained from equations (3) and (4). For example,
(cid:126)t ·(cid:126)t =(cid:0)2|Zi|2−1(cid:1)
i+1 i 2
5
and so on.
Finally, the relation between the link zi and the rotational variables Zi is given by (10) since
a a
1 (cid:16) (cid:17)
U† = √ Ψ Ψ†
i+1 g i+1 i
i
leading to
Zi = √1 (cid:0)z¯i+1zi +z¯i+1zi(cid:1)
1 g 1 1 2 2
i
Zi = √1 (cid:0)zi+1zi −zi+1zi(cid:1). (12)
2 g 1 2 2 1
i
Let us conclude, by discussing the discrete analogue of the equivalence between the projection operator formalism
of the CP1 model and the classic Frenet equation of a continuous string [13]. To do so, let us define the discrete
projection operator
(cid:18) (cid:19)
1 0
P = ψ ⊗ψ† = u u−1 (13)
i i i i 0 0 i
accompanied by the complemental projection operator
(cid:18) (cid:19)
0 0
P¯ = ψ¯ ⊗ψ¯† = u u−1 (14)
i i i i 0 1 i
such that the following relations exist
P +P¯ =I & P2 =P
i i i i i
and so forth; for details, see [13]. Note that, the identity
P2 =P
i i
√
sets the metric equal to one; i.e., g = 1 in accordance with the local angular parametrization (9). This was not
i
the case in [15] where the Poisson algebra of the link ziand rotational Zi variables have been presented due to a
α α
different (than the projection operator formalism) approach. Therefore, the Poisson brackets of the corresponding
link variables zi (given in Appendix B) are different from the ones presented in [15].
α
Finally,usingthefactthattheelementsoftheFrenetframearesu(2)Liealgebrageneratorsthefollowingequivalence
has been derived
P −P¯ (cid:39)(cid:126)t . (15)
i i i
In the continuum [13], it was shown that the time evolution of the spinor (that describes the string) can be obtained
from the integrable hierarchy of NLSE. In addition, it was shown that the conserved charges of the NLSE hierarchy
can be expressed in terms of the projection operator formalism and, in particular, that are equivalent to the energy
density of the CP1 chiral model.
Inthenextsection,weusetheDSLNEtointroduceanenergyfunctionfordiscretepolygonalchains. Equation(10)
implies that the rotational angles are the natural variables for constructing energy functions for discrete piecewise
linear strings. In analogy, with the continuum case, the energy must remain invariant under the local SO(2) frame
rotation. For this we simply identify the spin variables with the tangent vector in accordance with (14). Thus, the
energy function determines a chain dynamics which is manifestly invariant under the local frame rotation.
IV. BI-HAMILTONIAN STRUCTURE
In order to study the dynamics of a curve, Hamilton’s equation of motion is a natural choice. In [13], using the
projection formalism of the CP1 model, it was shown that the string hamiltonians in the NLSE hierarchy are related
to the spinor Frenet equation.
In fact, it was shown that the continuous string hamiltonian via the projection operator formalism
H1 =tr(P P ) (16)
NLSE s s
6
due to (15), is equivalent to the number operator in the NLSE hierarchy (otherwise, the Heisenberg spin chain
hamiltonian)
H2 = 1(cid:0)∂ (cid:126)t·∂ (cid:126)t(cid:1). (17)
NLSE 2 s s
That is,
1
H =H1 =H2 = |κ |2, (18)
NLSE NLSE NLSE 2 c
where κ is the complex valued curvature (for details, see [13]).
c
Inthediscretecase, thepolygonalcurveisasetofsequentiallyconnectedsegmentswhoselengthhereisconsidered
to be locally inextensible and therefore, the dynamics of the curve is essential the rotation of tangent vectors. Since
the degree of freedom for tangent vectors is the angular rotation, the chain can be identified as a spin chain described
by the lattice isotropic Heisenberg magnet model.
Thus, for discretising H given by (18) the forward finite difference scheme (i.e., ∂ f = f −f ) is applied.
NLSE s i+1 i
That way, one obtains
H =H1 =H2 . (19)
DNLS DNLSE DNLSE
The first hamiltonian H1 is the discretized version of the CP1 sigma model (16) given by
DNLSE
H1 = (cid:88)tr(P −P )2
DNLSE i+1 i
i
(cid:88)
= 2 |zizi+1−zi+1zi|2 (20)
1 2 1 2
i
√
provided that g =1.
i
The second hamiltonian H2 is the discretized version of NLSE hierarchy (17) given by
DNLSE
H2 = (cid:88)(cid:0)1−(cid:126)t ·(cid:126)t (cid:1)
DNLSE i+1 i
i
(cid:88)
= 2 |Zi|2 (21)
2
i
that is, the isotropic (XXX) Heisenberg model hamiltonian which describes first neighbour spin-spin interactions.
Note that, equation (19) holds due to (12).
Finally, the equation of motion is given by
d(cid:126)ti = (cid:8)H ,(cid:126)t (cid:9)
dt DNLS i
∂H
= (cid:126)t × DNLS.
i ∂(cid:126)t
i
In what follows, two a priori different discrete versions of the continuum NLSE are presented. Recall that, the NLSE
equation is known to be integrable at the finite segment [0,L] when appropriate boundary conditions are imposed.
Theaforementionedintegrabilityisextendedinthediscretizedversionsinceabi-hamiltonianformof(20)ispresented.
The first hamiltonian (20) is given in terms of the local angular variables (9), that is,
(cid:88)
H1 = [1−cosϑ cosϑ −sinϑ sinϑ cos(ϕ −ϕ )] (22)
DNLS i+1 i i+1 i i+1 i
i
associated with the only non-zero Poisson brackets:
{cosϑ ,ϕ }=δ . (23)
i j ij
In terms of the tangent vectors (cid:126)t , the Poisson brackets are given by {ta,tb} = −εabctcδ . The consistency of the
i i j i ij
Poisson brackets between the two systems of variables is discussed in Appendix A.
7
Next, the second hamiltonian (21) in terms of the discrete Euler’s angles (11) is presented. However, in order to
obtain the corresponding Poisson brackets the relation between the two angular systems need to be known explicitly.
In particular, from (12) and (9) it is straightforward to obtain that
cosΘ = (cid:0)2|Zi+1|2−1(cid:1)
i+1 1
= cosϑ cosϑ +sinϑ sinϑ cos(ϕ −ϕ ) (24)
i+1 i i+1 i i+1 i
sinΘ = 2Zi+1Zi+1
i+1 1 2
= sinϑ cosϑ −cosϑ sinϑ cos(ϕ −ϕ ) (25)
i+1 i i+1 i i+1 i
cosΦ = 1(cid:8)(cid:2)(Zi+1)2+(Z¯i+1)2(cid:3)+cc(cid:9)
i+1 2 1 2
= cos(ϕ −ϕ ) (26)
i+1 i
sinΦ = i (cid:8)(cid:2)(Zi+1)2+(Z¯i+1)2(cid:3)−cc(cid:9)
i+1 2 1 2
= cosϑ sin(ϕ −ϕ ), (27)
i i+1 i
associated with the mixed terms
cosΘ cosΦ = 1(cid:8)(cid:2)(Zi+1)2−(Z¯i+1)2(cid:3)+cc(cid:9)
i+1 i+1 2 1 2
= cosϑ cosϑ cos(ϕ −ϕ )+sinϑ sinϑ (28)
i+1 i i+1 i i+1 i
cosΘ sinΦ = i (cid:8)(cid:2)(Zi+1)2−(Z¯i+1)2(cid:3)−cc(cid:9)
i+1 i+1 2 1 2
= cosϑ sin(ϕ −ϕ ) (29)
i+1 i+1 i
sinΘ cosΦ = Zi+1Z¯i+1+cc
i+1 i+1 1 2
= −cosϑ sinϑ +sinϑ cosϑ cos(ϕ −ϕ ) (30)
i+1 i i+1 i i+1 i
sinΘ sinΦ = i(cid:0)Zi+1Z¯i+1−cc(cid:1)
i+1 i+1 1 2
= sinϑ sin(ϕ −ϕ ). (31)
i+1 i+1 i
Also, due to consistency, the following condition is satisfied
sinϑ sin(ϕ −ϕ )=0 (32)
i i+1 i
whichfollowsfromthetrigonometricidentitysin2Φ +cos2Φ =1usingequations(24)and(26). Also,from(12),
i+1 i+1
(32) and (A1), it is straightforward to show the equivalence between the parametrization (3)-(5) and the equations
(24)-(31).
So, in terms of the discrete Euler’s angles, the second hamiltonian (21) takes the simple form
(cid:88)
H2 = (1−cosΘ ). (33)
DNLS i+1
i
associated with the Poisson brackets
{Θ ,Θ } = −sinΦ (34)
i+1 i i+1
1
{Φ ,Φ } = (sinΦ cotΘ +sinΦ cotΘ ) (35)
i+1 i sinΘ i+1 i+1 i i−1
i
cosΦ
{Θ ,Φ } = i+1 (36)
i+1 i sinΘ
i
Θ
{Θ ,Φ } = −cot i −cosΦ cotΘ (37)
i i 2 i i−1
Θ
{Θ ,Φ } = cot i−1 +cosΦ cotΘ (38)
i−1 i 2 i i
cosΦ
{Θ ,Φ } = − i−1 (39)
i−2 i sinΘ
i−1
8
in addition with the shifted ones to the left and right; i.e., i → i−1 and i → i+1 respectively. For completeness,
in Appendix C the consistence of the two algebras (23) and (34)-(39) is presented. Note that, the Poisson brackets
(34)-(39) was first introduce in [14] but in another context.
Let us conclude by stating that two different hamiltonians for the DNLS have been obtained. The first H1
DNLS
(22) has a "complicate" form associated by a "simple algebra" (23) while the second hamiltonian H2 (33) has a
DNLS
"simple" form associated with a "complicate" algebra (34)-(39).
V. CONCLUSION
Insummary,wehaveaddressedtheproblemhowtoutilizeextrinsicgeometryinordertoderivehamiltonianenergy
functions for discrete strings that move in three dimensional space. We utilize the transfer matrix formalism to
consecutively map the discrete Frenet frame from one vertex of discrete curve to its neighbour. In this manner we
arrive naturally to energy densities that relate to the integrable hierarchy of the NLSE defined on the lattice. In
particular, under this frame the tangent vectors are considered to be classical spins and thus, the lattice Heisenberg
model comes to be a natural choice of the chain’s hamiltonian.
We have shown that the introduced discrete version of the NLSE is integrable since it holds a bi-hamiltonian
structure. That way, the equation of motion of the tangent vectors and bond and torsion angles Poisson brackets are
constructed and therefore, the dynamics of the lattice chain can be investigated. We will report on these issues in a
future publication.
VI. ACKNOWLEDGMENTS
T.I. thanks the School of Physics at Beijing Institute of Technology for hospitality during the completion of this
research.
T.I. acknowledges support from FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme
(IRSES-606096); and from The Hellenic Ministry of Education: Education and Lifelong Learning Affairs, and Euro-
pean Social Fund: NSRF 2007-2013, Aristeia (Excellence) II (TS-3647). A.J.N. acknowledges support from CNRS
PEPS grant, Region Centre Recherche d(cid:48)Initiative Academique grant; Sino-French Cai Yuanpei Exchange Program
(PartenariatHubertCurien),Vetenskapsrådet,CarlTrygger’sStiftelseförvetenskapligforskning;andQianRenGrant
at BIT.
Appendix A
In the discrete case, the unit length tangent vector(cid:126)t= √1 (cid:0)ψ†(cid:126)σψ(cid:1) and the binormal and normal vectors (cid:126)n+i(cid:126)b=
√1 (cid:0)ψ¯†(cid:126)σψ(cid:1) in terms of the bond and torsion angles (9) areggiiven by
gi
z¯ z +z¯z sinϑcosϕ cosϑcosϕ −sinϕ
1 1 2 2 1
(cid:126)ti = √g i(z¯2z1−z¯1z2) =sinϑsinϕ (cid:126)ni+1 =cosϑsinϕ , (cid:126)bi+1 = cosϕ , (A1)
i |z |2−|z |2 cosϑ −sinϑ 0
1 2
i i i+1 i+1
√
where ψ is given by (6) and g = 1. Then the continuum hamiltonian (18), using the forward finite difference
i
scheme, becomes
(cid:88)
H =2 (1−(cid:126)t ·(cid:126)t ) (A2)
NLSE i+1 i
i
accompanied by the Poisson bracket
(cid:8)ta,tb(cid:9)=−εabctcδ (A3)
i j i ij
in consistence with (20).
Another way to obtain (A3) is to use the continuum XXX chain formulation. That is, assume that the canonical
coordinates are p = cosϑ and q = ϕ where {p ,q } = δ and {q ,q } = 0 = {p ,q } (in consistence with (23)).
i i i i i j ij i j i j
Also, assume that t±i = t1i ±it2i = sinϑie±iφi and tzi = t3i = cosϑi due to (A1). Thus, in terms of the canonical
coordinates this implies that t±i =(cid:112)1−p2i e±iqi and tzi =pi and so it is easy to obtain that
{t+,t−}=−2itzδ , {tz,t±}=∓it±δ (A4)
i j i ij i j ij
9
that is, (A3).
Appendix B
In terms of the link variables zi variables the associated Poisson brackets are non-trivial. In fact, from (9) using
a
(23) it can be shown that the only non-vanishing Poisson brackets are the following:
i
{zi,z¯j} = δ ,
a a 4 ij
i (cid:18)zi zi(cid:19)
{zi,zj} = − 1 − 2 δ
1 2 8 z¯i z¯i ij
2 1
i (cid:18)zi z¯i(cid:19)
{zi,z¯j} = − 1 + 2 δ .
1 2 8 zi z¯i ij
2 1
Recall that, |zi|2+|zi|2 =1.
1 2
Appendix C
Let us conclude by presented some explicit examples of how the Poisson brackets of the Euler’s angles (Θ,Φ) can
i
be obtained directly from the corresponding ones of the local angular variables (ϑ,ϕ) .
i
Let us start by considering the first bracket (34). Substituting equation (24) in the right hand side of (34) and
applying the properties of the Poisson bracket one gets
∂ ∂ ∂ ∂
{cosΘ ,cosΘ } = (cosΘ ) {ϑ ,ϕ } (cosΘ )− (cosΘ ) {ϑ ,ϕ } (cosΘ )
i+1 i ∂ϑ i+1 i i ∂ϕ i ∂ϕ i+1 i i ∂ϑ i
i i i i
(cid:18) (cid:19)
−1
= sinΘ cosΦ [−sinϑ sinϑ cos(ϕ −ϕ )]
i+1 i+1 sinϑ i i−1 i i−1
i
(cid:18) (cid:19)
−1
− sinϑ sinϑ sin(ϕ −ϕ ) (−sinΘ )
i+1 i i+1 i sinϑ i
i
= 0−sinϑ sin(ϕ −ϕ )sinΘ
i+1 i+1 i i
= −sinΘ sinΦ sinΘ . (C1)
i+1 i+1 i
Note that, the condition (32) and the equation (31) is (also) been used.
However, the right-hand side of the above equation gives
{cosΘ ,cosΘ } = sinΘ sinΘ {Θ ,Θ }. (C2)
i+1 i i+1 i i+1 i
Thus, equation (34) is obtained.
For completeness, let us (also) show how the Poisson bracket (36) can be derived. Substituting equation (24) and
(27) in the right hand side of (36) and applying the properties of the Poisson bracket one gets
∂ ∂ ∂ ∂
{cosΘ ,sinΦ } = (cosΘ ) {ϑ ,ϕ } (sinΦ )− (cosΘ ) {ϑ ,ϕ } (sinΦ )
i+1 i ∂ϑ i+1 i i ∂ϕ i ∂ϕ i+1 i i ∂ϑ i
i i i i
(cid:18) (cid:19)
−1
= sinΘ cosΦ cosϑ cos(ϕ −ϕ )
i+1 i+1 sinϑ i−1 i i−1
i
(cid:18) (cid:19)
−1
= sinΘ cosΦ cosϑ cosΦ
i+1 i+1 sinϑ i−1 i
i
= sinΘ cosΦ sinΘ cosΦ . (C3)
i+1 i+1 i i
Equation (26) has (also) been used as well as the identity sinΘi ≡ cossinϑϑi−i1 obtained when (31) is divided by (27).
However, the right-hand side of the above equation gives
{cosΘ ,sinΦ } = −sinΘ cosΦ {Θ ,Φ }. (C4)
i+1 i i+1 i i+1 i
10
Therefore, equation (36) is derived.
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