HMPHAH 0H3HKA, 2007, moM 70, № 3, c. 588-594
ELEMENTARY PARTICLES AND FIELDS. THEORY
EQUIVALENCE OF SUPERINTEGRABLE SYSTEMS IN TWO DIMENSIONS
© 2007 J. M. Kress*
School of Mathematics, The University of New South Wales, Sydney, Australia
Received May 16, 2006
In two dimensions, all nondegenerate superintegrable systems having constants quadratic in the momenta possess a quadratic algebra. In this paper it is shown how the quadratic algebra can be used to classify all such systems into seven classes that are preserved by coupling constant metamorphosis.
PACS numbers: 02.30.Ik
1. INTRODUCTION
In a recent paper [1] it was shown that all non-degenerate two-dimensional superintegrable systems having constants quadratic in the momenta can be obtained by coupling constant metamorphosis from those on constant curvature spaces. It has also been shown that the Poisson algebras of these systems close quadratically [2]. In this paper the quadratic algebra is used to classify these systems into seven classes on which coupling constant metamorphosis [3] (or the Stackel transform [4]) acts transitively. A similar classification, also reported at this Workshop, has recently been used by Daskaloyannis and Ypsilantis [5] as the basis for calculating the Hamil-tonians and associated integrals for these systems.
We consider the Hamiltonian of a system with two degrees of freedom,
H
J2gij PiPj + v (xi,x2), (1)
i,j=l
having constants quadratic in the momenta and nondegenerate potential V (that is, apart from an additive constant, it is determined by V1, V2, and V11 at a regular point and hence depends on three parameters).
The time evolution of a function L of the position x1, x2 and the momenta p1, p2 is given by
§ = LH}
where { , } is the Poisson bracket:
2
' dx.
t=i
da db dpi
{a, b} = 7--T-T-
da db dpi dxi
and so if {H, L} = 0, the function L is called an integral or constant of the motion.
Given two constants in involution (i.e., having vanishing Poisson bracket) the system is said to be Liouville integrable and when three or more constants polynomial in the momenta are known, the system is said to be superintegrable. A similar situation exists for quantum systems with constants replaced by differential operators and the Poisson bracket replaced by the operator commutator. In n dimensions, n constants in involution are required for Liouville integrability, and a system is said to maximally superintegrable when 2n — 1 polynomial constants are known.
The free particle, Coulomb—Kepler system, and harmonic oscillator (or their quantum counterparts) are well-known superintegrable systems. In 1965 Fris et al. [6] initiated a search for other super-intergable systems and found all such systems in two-dimensional real Euclidean space having three constants quadratic in the momenta.
A similar list of superintegrable potentials has been found in real three-dimensional Euclidean space [7], and recently all two-dimensional nondegenerate Hamiltonians of the form (1) have been found [1].
As an example of the type of superintegrable system considered in this paper, consider one of the four systems found by Fris et al. [6] given by the Hamiltonian
TT 2,2I /2,2\, ß I Y
H = px + py + a(x + y ) + X2 + y2 ■
E-mail: J.Kress@unsw.edu.au
Constants of the motion for this system are ,2 , , 3 R2 = M2 + /3y2
R1 = px + ax +
x
2
x
x
2 + Y72,
2
where M = xpy — ypx, and the Poisson algebra of these constants along with R = {Ri, R2} closes to form a quadratic algebra:
{R, Ri} = 8Rf — 8HR1 + l6aR2, {R, R2} = —I6R1R2 + 8HR2 — — 16(0 + 7)Ri + 160H.
The cubic constant R cannot be functionally independent of Ri, R2, and H and in fact
R2 = — 16R2 R2 + I6HR1R2 — 16(0 + y)R2 —
— 16aR2 + 320HRi — 160H2 + 64a0Y-
This cubic expression for R2 in terms of Ri, R2, and H, contains the complete structure of the quadratic algebra, which can be determined from it by
Rfí\ = 1 dR2 {R,Rl} = - 2 dR
1 dR2
and {r,r2} = 2 ^ ■
H = H0 + aV0 and L = L0 + al0,
such that
{Ho ,Lo} = {H,L} = 0.
It can be shown that
H' = Ho and L = L0 — loH' Vo
are also in involution, that is
{H' ,L'} = 0.
So starting with a superintegrable Hamiltonian, a new conformally related superintegrable Hamiltonian can be constructed. Identities involving integrals associated with H give rise to identities involving
integrals associated with H' by making the replacements
a
-H
H -»■ 0.
While the existence of a quadratic algebra for this type of system has been noted and used, in the quantum case, to determine the spectrum of bound states [8], it has only recently be shown to be a generic feature of all nondegenerate quadratically superintegrable systems in two dimensions [2].
2. COUPLING CONSTANT METAMORPHOSIS
Transformations mapping one integrable system to another have been put to good use in the literature. One such type of transformation, known as coupling constant metamorphosis [3] interchanges a parameter in the potential with the energy. This transformation can be applied to more general systems than those considered here. In the current context it also known as a Stackel transform [4] and is briefly described below.
Consider a Hamiltonian and corresponding constant,
If we allow the addition of a constant to the Hamiltonian and multiplication by a constant then a transformation that simply interchanges H and a can be constructed.
For example, H = pX + p2y + ax is a flat space superintegrable system with constants
K = py, Ri = Mpy — a y2,
. ^ a
and R2 = PxPy + j y,
and Poisson algebra defined by
a
{K,Ri} = —R2, {K,R2 } = -, {R1R2} = —2K3 + HK,
and R2 + K4 — HK2 + aRi = 0.
Taking V0 = x gives the transformed Hamiltonian H '
and constants
px + py X
K' = py, Ri = Mpy +y-(pX + Pi), and R2 = pxpy — 2x (pX + pI ), with Poisson algebra defined by
{K',Ri} = —R'2, {K',R } = — 2 H', {Ri, R2} = —2K3, and R'2 + K'4 — H'R'i = 0.
In this way, nonflat superintegrable systems can be generated from known quadratically superintegrable systems in two dimensions.
3. HAMILTONIANS WITH TWO ADDITIONAL QUADRATIC CONSTANTS
Koenigs [9] found all two-dimensional surfaces
ds2 = 4f (x,y)(dx2 + dy2)
admitting at least two rank-2 Killing tensors in addition to the metric. This gives us an equivalent list of corresponding Hamiltonians
H
px + py
2f (x,y)
admitting at least two additional quadratic constants. For example, those possessing two quadratic constants and one linear constant:
Di : H
0 —
px + py
4x ,
Da : Ho
D4 : Ho — — -
D2 : Ho —
px + py
4 + x 2 + y2 '
px + py
Px + Py 1 + 1/xy:
4 (a + 2)/x2 + (a - 2)/y2
These have been rewritten in a rational form so that it is apparent that each of the denominators is in fact a superintegrable potential from those known to exist in flat space. Hence we can obtain each of these Hamiltonians from a flat space superintegrable Hamiltonian by coupling constant metamorphosis.
Since each of the demoninators above appears as a term in several nondegenerate superintegrable potenials on Euclidean space, we can generate a nondegenerate superintegrable potentials on these spaces. For example, the potential in each of E2, E9, and E3' (see Appendix or [10]) contains the term x and so dividing throughout by V0 = x gives three distinct nondegenerate superintegrable potential on Di.
Alternatively we can start with Koenigs list and use it as a basis for finding all quadratically superintegrable systems in two dimensions. This approach was taken in [11, 12].
For example, starting with D1 we can look for a Hamiltonian of the form
H = Pi+r + V (x,y) having two additional constants of the form
Xi = aiK2 + biRi + CiR2 + di(x,y), i = 1,2, and find
Pi + P2y , a(4x2 + y2) 3
H1
H2 — 2
4x
P2x + Py
4x 2
+
x
Y
+ - + -Ly + s,
a
H
3
p x + py
4x
+ x + x + a
x xy
Py , Y(x2 + y2)
+ s,
P(2x — iy) y
+ —. . + \ + 2 + s.
x^/x — iy x^/x — iy x
Forms for cubic and quadratic terms of R2 (the coefficient f (ai,H) is a linear function of the Hamiltonian and the parameters in the potential)
Form of cubic in R1 and R2 Label Systems from [10]
R? + f (oi,H)R* R? + f (a.i,H)RiR2 R? + 0 R2R2 + f (ai, h)r2 R2R2 + 0 Ri R2R1 + R2) + + f (a.i,H)RiR2 0 + f (a.i,H)RiR2 [3,2] [3,11] [3,0] [21,2] [21,0] [111,11] [0,11] E2, S1 E9, E10 E15 E1, E16, S2, S4 E7, E8, E17, E19 S7, S8, S9 E3', E11, E20
It is easily seen that each of these is essentially one of E2, E9, and E3' divided by throughout by x. The first two of these were given in [11] and the third was noted in the Appendix of [ 12].
Either approach yields all quadratically superintegrable systems in two dimensions [1].
4. CLASSIFICATION OF THE QUADRATIC ALGEBRA
Each three-parameter potential has an associated quadratic algebra characterised by an identity of the form
R2 = aiR3 + a2 R2 + a3R2R2 + a4RiR2 +
+ biR2 + b2 R2 + b3RiR2 + ciRi + C2R2 + d,
where the ai are numbers, the bi, ci, and d are, respectively, linear, quadratic, and cubic in H and the parameters. Hence the coupling constant metamorphosis preserves the form of this cubic expression, and we can classify the superintegrable systems accordingly.
Since there is no preferred basis for the space spanned by R2 and R2, and we can add multiples of H and the parameters to R2 and R2, and the expression for R2 can always be reduced to one of the forms in the table. The classes are given labels in this table that reflect the form of the cubic and quadratic parts. (Note that there is no system with vanishing cubic and a perfect square for its quadratic part.)
5. GENERATING THE KNOWN SUPERINTEGRABLE SYSTEMS
From one known system we can attempt to generate other systems in two-dimensional Euclidean space with the same type of quadratic algebra.
For example, starting with system E8, we can ask when the transformed Hamiltonian,
H '
p 2x + p 2y Vo
p 2x + p 2y
az/z3 + ¡3/z2 + yzz + S'
x
is a flat space Hamiltonian? This question has been considered in [13] and it amounts to solving
д д л
— — log Vq = 0,
dz dz
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.