Classical mechanics on GL(n, R) group and Euler-Calogero-Sutherland model

10 Pages·0.12 MB·English

by  A. M. Khvedelidze

Classical mechanics on GL(n,R) group and
Euler-Calogero-Sutherland model
A.M. Khvedelidze∗ and D.M. Mladenov†
Bogoliubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research, 141980 Dubna, Russia
1
0 Relations between the free motion on the GL+(n,R) group manifold and the dy-
0
2 namicsofann-particlesystemwithspindegreesoffreedomonalineinteractingwith

n
a a pairwise 1/sinh2x “potential” (Euler-Calogero-Sutherland model) is discussed in
J

0 the framework of Hamiltonian reduction. Two kinds of reductions of the degrees of
3
freedom are considered: due to the continuous invariance and due to the discrete

]
I
S symmetry. Itisshownthatafterprojectiononthecorrespondinginvariantmanifolds
.
n
the resulting Hamiltonian system represents the Euler-Calogero-Sutherland model
i
l
n
in both cases.
[

2
v
3
3
0 1. INTRODUCTION
1
0
1
In this contribution, we deal with two finite dimensional Hamiltonian systems. The
0
/
n first one is a generalization of the Calogero-Sutherland-Moser [1] model by introducing the
i
l
n internal degrees of freedom [2, 3] described by the Hamiltonian
:
v
Xi H = 1 N p2 + 1 N li2j (1)
r ECS 2 i 8 sinh2(x −x )
a i=1 i6=j i j
X X
with canonical pairs (x ,p ) obeying the nonvanishing Poisson brackets
i i
{x ,p } = δ , (2)
i j ij
and “internal” variables l satisfy the SO(n,R) Poisson bracket algebra
ab
{l ,l } = δ l +δ l −δ l −δ l . (3)
ab cd bc ad ad bc ac bd bd ac
∗Department of Theoretical Physics, A. Razmadze Mathematical Institute, GE-380093Tbilisi, Georgia.
†Electronic address: [email protected]
2
The dynamics of the second system is given in terms of geodesic motion on the GL(n,R)
groupmanifold. ThecorrespondingLagrangianbasedonthebi-invariantmetriconGL(n,R)
is given by [4, 5]
1
L = tr g˙g−1 2 , (4)
GL
2
(cid:0) (cid:1)
where g ∈ GL(n,R), and the dot over the symbols means differentiation with respect to
time. Below we shall represent the Hamiltonian system corresponding to this Lagrangian
(4) in terms of a special parameterization adapted to the action of the symmetry group of
the system. We demonstrate that the resulting Hamiltonian is a generalization of the Euler-
Calogero-Sutherland model (1) with two types of internal degrees of freedom. Performing
the Hamiltonian reduction owing to two types of symmetry: continuous and discrete, we
show how to arrive at the conventional Hamiltonian of the Euler-Calogero-Sutherland model
(1).
2. BI-INVARIANT GEODESIC MOTION ON THE GROUP MANIFOLD
2.1.. Explicit integration of the classical equation of motion
The Euler-Lagrange equation following from the Lagrangian (4) can be represented as
d
g−1g˙ = 0. (5)
dt
(cid:0) (cid:1)
This form demonstrates their explicit integrability
g(t) = g(0)exp(tJ) (6)
with two arbitrary constant matrices g(0) and J.
2.2.. Hamiltonian in terms of special coordinates
The canonical Hamiltonian corresponding to the bi-invariant Lagrangian (4) reads
1
H = tr πTg 2 . (7)
GL
2
(cid:0) (cid:1)
The nonvanishing Poisson brackets between the fundamental phase space variables are
{g ,π } = δ δ . (8)
ab cd ab cd
3
To find out the relation to the conventional Euler-Calogero-Sutherland model (1), it is
convenient to use the polar decomposition [6] for an arbitrary element of GL(n,R). For the
sake of technical simplicity we investigate in details the group GL(3,R) hereinafter, i.e.
g = OS, (9)
whereS isapositive definite3×3symmetric matrix, andO(φ ,φ ,φ ) = eφ1J3eφ2J1eφ3J3 isan
1 2 3
orthogonal matrix with SO(3,R) generators (J ) = ε . Since the matrix g represents an
a ik iak
element of GL(n,R) group, we can treat the polar decomposition (9) as a uniquely invertible
transformation from the configuration variables g to a new set of six Lagrangian coordinates
S and three coordinates φ . The induced transformation of momenta to new canonical
ij i
pairs (S ,P ) and (φ ,P ) is
ab ab a a
π = O(P −k J ) , (10)
a a
where
k = γ−1 ηL −ε (SP) . (11)
a ab b bmn mn
Here ηL are three left-invariant vector fi(cid:0)elds on SO(3,R) (cid:1)
a
sinφ
ηL = − 3 P −cosφ P +cotφ sinφ P ,
1 sinφ 1 3 2 2 3 3
2
cosφ
ηL = − 3 P +sinφ P +cotφ cosφ P , (12)
2 sinφ 1 3 2 2 3 3
2
ηL = −P
3 3
and γ = S −δ trS. In terms of the new variables, the canonical Hamiltonian takes the
ik ik ik
form
1 1
H = tr(PS)2 + tr(J SJ S)k k . (13)
GL a b a b
2 2
2.2.1.. Restriction of the Hamiltonian to the Principal orbit
The system (13) is invariant under the orthogonal transformations S′ = RT SR, and the
orbit space is given as a quotient space S/SO(3,R). The quotient space S/SO(3,R) is a
stratifiedmanifold; orbitswiththesameisotropygrouparecollectedinto strataanduniquely
parameterized by the set of ordered eigenvalues of the matrix S x ≤ x ≤ x . The strata
1 2 3
are classified according to the isotropy groups which are determined by the degeneracies of
the matrix eigenvalues:
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1. Principal orbit-type stratum, when all eigenvalues are unequal x < x < x , with the
1 2 3
smallest isotropy group Z ⊗Z .
2 2
2. Singular orbit-type strata forming the boundaries of orbit space with
(a) two coinciding eigenvalues (e.g. x = x ), when the isotropy group is SO(2)⊗Z .
1 2 2
(b) all three eigenvalues are equal (x = x = x ), here the isotropy group coincides
1 2 3
with the isometry group SO(3,R).
Now we shall at first restrict ourselves to the investigation of dynamics which takes
place on the principal orbits. To write down the Hamiltonian describing the motion on
the principal orbit stratum, we introduce the coordinates along the slices x and along the
i
orbitsχ. Namely, sincethematrixS ispositivedefiniteandsymmetric, weusethemain-axes
decomposition in the form
S = RT(χ)e2XR(χ), (14)
where R(χ) ∈ SO(3,R) is an orthogonal matrix parameterized by three Euler angles χ =
(χ ,χ ,χ ), and the matrix e2X is a diagonal e2X = diagke2x1,e2x2,e2x3k. The original
1 2 3
physical momenta P are expressed in terms of the new canonical pairs (x ,p ) and (χ ,p )
ik i i i χi
as
3 3
P = RTe−X P¯ α¯ + P α e−XR, (15)
a a a a
!
a=1 a=1
X X
with
1
P¯ = p , (16)
a a
2
1 ξR
P = − a ,(cyclic permutation a 6= b 6= c). (17)
a
4sinh(x −x )
b c
Intherepresentation (15),weintroducetheorthogonalbasisforthesymmetric 3×3matrices
α = (α , α ) i = 1,2,3 with the scalar product
A i i
tr(α¯ α¯ ) = δ , tr(α α ) = 2δ , tr(α¯ α ) = 0
a b ab a b ab a b
and the SO(3,R) right-invariant Killing vectors
ξR = −p , (18)
1 χ1
sinχ
ξR = sinχ cotχ p −cosχ p − 1 p , (19)
2 1 2 χ1 1 χ2 sinχ χ3
2
cosχ
ξR = − cosχ cotχ p −sinχ p + 1 p . (20)
3 1 2 χ1 1 χ2 sinχ χ3
2
5
After passing to these main-axes variables, the canonical Hamiltonian reads
1 3 1 (ξR)2 1 R ηL + 1ξR 2
H = p2 + a − ab b 2 a . (21)
GL 8 a 16 sinh2(x −x ) 4 cosh2(x −x )
a=1 (abc) b c (abc) (cid:0) b c(cid:1)
X X X
Here (abc) means cyclic permutations a 6= b 6= c. Thus, the integrable dynamical system de-
scribing the free motion on principal orbits represents, in the adapted basis, the Generalized
Euler-Calogero-Sutherland model. The generalization consists in the introduction of two
types of internal dynamical variables ξ and η — “spin” and “isospin” degrees of freedom,
interacting with each other. Below, the relations to the standard Euler-Calogero-Sutherland
model (1) are demonstrated.
2.2.2.. Restriction of the Hamiltonian to the Singular orbit
The motion on the Singular orbit is modified due to the presence of a continuous isotropy
group. In the case of GL(3,R), it is SO(2)⊗Z . Applying the same machinery as for the
2
Principal orbits to the two-dimensional orbit (x = x = x ,x = y) , one can derive the
1 2 3
Hamiltonian
1 g2 g2
H(2) = (p2 +p2)+ − (22)
GL 8 x y sinh2(x−y) cosh2(x−y)
with two arbitrary constants g2 and g2 related to the value of the spin ξ and isospin η. Due
to the translation invariance, the equations of motion are equivalent to the corresponding
equations for a one-dimensional problem and thus the system (22) is integrable.
3. REDUCTION TO EULER-CALOGERO-SUTHERLAND MODEL
3.1.. Reduction using discrete symmetries
Now we shall demonstrate how the IIA Euler-Calogero-Sutherland model arises from the
3
canonical Hamiltonian (7) after projection onto a certain invariant submanifold determined
by discrete symmetries. Let us impose the condition of symmetry of the matrices g ∈
GL(n,R)
χ(1) = ε g = 0. (23)
a abc bc
6
In order to find an invariant submanifold, it is necessary to supplement the constraints (23)
with the new ones
χ(2) = ε π = 0. (24)
a abc bc
One can check that the surface defined by both constraints (23) and (24) represents an
invariant submanifold in the GL(3,R) phase space, and the dynamics of the corresponding
induced system is governed by the reduced Hamiltonian
1
H | = tr(PS)2 . (25)
GL χ(a1)=0, χ(a2)=0 2
The matrices S andP arenow symmetric nondegenerate matrices, andone canbeconvinced
that this expression leads to the Hamiltonian of the IIA Euler-Calogero- Sutherland model.
3
To verify this statement, it is necessary to note that after projection on the invariant sub-
manifold, the canonical Poisson structure is changed. We have to deal with the new Dirac
brackets
{F,G} = {F,G} −{F,χ }C−1{χ ,G} (26)
D PB a ab b
(1) (2)
for arbitrary functions on the phase space. In our case, because C = k{χ ,χ }k = 2δ ,
ab a b ab
the fundamamental Dirac brackets between the main-axes variables are
1 1
{x ,p } = δ , {χ ,p } = δ
a b D 2 ab a χb D 2 ab
and the Dirac bracket algebra for the right-invariant vector fields on SO(3,R) reduces to
1
{ξR,ξR} = ε ξR.
a b D 2 abc c
Thus, after rescaling of the canonical variables, one can be convinced that the reduction via
discrete symmetry indeed leads to the IIA Euler-Calogero- Sutherland model.
3
3.2.. Reduction due to the continuous symmetry
The integrals of motion corresponding to the geodesic motion with respect to the bi-
invariant metric on GL(n,R) group are
J = (πTg) . (27)
ab ab
7
The algebra of this integrals realizes on the symplectic level the GL(n,R) algebra
{J ,J } = δ J −δ J . (28)
ab cd bc ad ad cb
After transformation to the scalar and rotational variables, the expressions for J reads
3
J = RT (p α¯ −i α −j J )R, (29)
a a a a a a
a=1
X
where
1 1
i = ξRcoth(x −x )+ R ηL + ξR tanh(x −x ) (30)
a 2 a b c ab b 2 a b c
(cid:18) (cid:19)
and
j = R ηL +ξR. (31)
a ab b a
When these integrals are used, there appear several ways to choose an invariant manifold
and to derive the corresponding reduced system. Let us consider the surface on phase space
defined by the constraints
ηR = 0. (32)
a
These constraints, in the Dirac’s terminology [7, 8] are first class constraints {ηR,ηR} =
a b
−ǫ ηR, and the surface (32) is invariant under the evolution governed by the Hamiltonian
abc c
{ηR,H } = 0.
a GL
Using the relation between left and right-invariant Killing fields ηR = O ηL, we find out
a ab b
that after projection to the constraint surface (32), the Hamiltonian reduces to
1 3 1 (ξR)2
H (ηR = 0) = p2 + a . (33)
GL a 8 a 4 sinh22(x −x )
a (abc) b c
X X
Afterrescalingofthevariables2x → x ,oneisconvincedthatthederivedHamiltoniancoin-
a a
cides with the Euler-Calogero-Sutherland Hamiltonian (1), where the intrinsic spin variables
are l = ǫ ξR. Note that performing the reduction to the surface defined by the vanishing
ij ijk k
integrals j = 0, we again arrive at the same Euler-Calogero-Sutherland system.
a
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3.3.. Lax-pair for Generalized Euler-Calogero-Sutherland model
The expressions (29) for the integrals of motion allow us to rewrite the classical equation
of motion for the Generalized Euler-Calogero-Sutherland model in the Lax form
L˙ = [A,L], (34)
where 3×3 matrices are given explicitly as
p L+, L−
1 3 2
L = L−, p L+ 
3 2 1
L+, L−, p 
 2 1 3 
 
 
and
p −A , A
1 3 2
A = 1 e+X  A , p , −A e−X ,
4 3 2 1
 
−A , A , p
 2 1 3 
 
 
where
1 ξR R ηL + 1ξR
L± = − 1 ± 1m m 2 1 , (35)
1 2sinh(x −x ) cosh(x −x )
2 3 2 3
1 ξR R ηL + 1ξR
L± = − 2 ± 2m m 2 2 , (36)
2 2sinh(x −x ) cosh(x −x )
3 1 3 1
1 ξR R ηL + 1ξR
L± = − 3 ± 3m m 2 3 (37)
3 2sinh(x −x ) cosh(x −x )
1 2 1 2
and
1 ξR R ηL + 1ξR
A = 1 − 1m m 2 1 , (38)
1 2sinh2(x −x ) cosh2(x −x )
2 3 2 3
1 ξR R ηL + 1ξR
A = 2 − 2m m 2 2 , (39)
2 2 sinh2(x −x ) cosh2(x −x )
3 1 3 1
1 ξR R ηL + 1ξR
A = 3 − 3m m 2 3 . (40)
3 2 sinh2(x −x ) cosh2(x −x )
1 2 1 2
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4. CONCLUDING REMARKS
In this talk, we have discussed the generalization of the Euler-Calogero-Sutherland model
by introducing two internal variables “spin” and “isospin”, using the integrable model based
on the general matrix group GL(n,R). We outline its relation to the well-known integrable
model. Our consideration confirms once more that the clue to an integrability of a model is
often hidden in the possibility to connect it with a known higher-dimensional exactly solv-
able system by its symplectic reduction to its invariant submanifold [4, 5]. A rich spectrum
of these types of finite-dimensional models, obtained by the generalized “momentum map”
is well-known (see e.g. [9]). Over the last decade it has been recognized that the same
happens in the infinite-dimensional case. Integrable two-dimensional field theories have
been found from the so-called WZNW theory applying the Hamiltonian reduction method
[10]. An important class of finite-dimensional systems was discovered by the Hamiltonian
reduction method from the so-called matrix models (for a recent review see e.g.[11]). The
interest to this type of models has a long history starting with the Wigner study of the
statistical theory of energy levels of complex nuclear system [12]. Nowadays we have revival
of the interest to a matrix models connected with the search of relations between the su-
persymmetric Yang-Mills theory and integrable systems (for a modern review see e.g. [13]).
The relation between the Euler-Calogero-Moser model and the SU(2) Yang-Mills theory in
long-wavelength approximation was obtained in ([14]).
Acknowledgments
It is a pleasure to thank the local organizers of the XXIII International Colloquium on
Group Theoretical Methods in Physics and B. Dimitrov, V.I. Inozemtsev, A. Kvinikhidze,
M.D. Mateev, and P. Sorba for illuminating discussions.
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10
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