HUEPHAH 0H3HKA, 2007, moM 70, № 3, c. 535-541
ELEMENTARY PARTICLES AND FIELDS. THEORY
THREE SUPERINTEGRABLE TWO-DIMENSIONAL OSCILLATORS: SUPERINTEGRABILITY, NONLINEARITY, AND CURVATURE
® 2007 J. F. Cariñena1)*, M. F. Rañada1)**, M. Santander2)***
Received May 16, 2006
The superintegrability of three different two-dimensional oscillators is studied: (i) a nonlinear oscillator dependent of a parameter A (two-dimensional version of the oscillator of Lakshmanan and Mathews), (ii) a nonlinear oscillator related with the Riccati equation, and (iii) the standard harmonic oscillator on constant curvature spaces. They can be considered as nonlinear deformations, or curvature-dependent versions, of the linear harmonic oscillator.
PACS numbers: 02.30.Ik
1. INTRODUCTION
It is well known that the most important and best known of all the superintegrables systems is the harmonic oscillator. On the other side, it is also known that the superposition principle characterizing the linear equations, is no longer valid for nonlinear ones and, because of this, most properties of the harmonic oscillator disappear when nonlinearities are introduced. In the general nonlinear case the associated frequency is amplitude-dependent and the nonlinear oscillators are no longer superintegrable systems. The main objective of this article is to present a report on some very particular two-dimensional oscillators that, in spite of the presence of nonlinearities, are endowed with the property of superintegrability.
The plan of the article is as follows: Section 2 is devoted to the study of the superintegrability of a nonlinear deformation dependent of a parameter A (for A = 0 the linear system is recovered); it represents the two-dimensional version of a nonlinear one-dimensional oscillator previously studied by M. Lakshmanan and P.M. Mathews. Section 3 is devoted to a nonlinear oscillator related with the second-order Riccati equation, and Section 4 is devoted to the standard harmonic oscillator on constant curvature spaces. Finally in Section 5 we make some final comments.
''Departamento de Física Teórica, Facultad de Ciencias,
Universidad de Zaragoza, Spain.
2)Departamento de Física Teórica, Facultad de Ciencias, Universidad de Valladolid, Spain. E-mail: jfc@unizar.es E-mail: mfran@unizar.es E-mail: msn@fta.uva.es
2. A-DEPENDENT n = 2 QUASI-HARMONIC OSCILLATOR
Lakshmanan and Mathews studied in 1974 [1,2], the differential equation
(1 + Ax2)x - (Ax)x2 + a2x = 0, A > 0, (1)
and proved that it admits as solution the following function:
x = A sin(wi + 4>),
but with a restriction linking the frequency with the amplitude
2
a
u
1 + AA2'
That is, the Eq. (1) is therefore an interesting example of a system with nonlinear oscillations having a "quasi-harmonic form". Moreover, it can be obtained from the Lagrangian
L = Ti(x,vx; A) - Vi(x; A),
(2)
T =1 vX 1 2 V 1 + Ax2
V1 =1
a2x2 1 + Ax2
and therefore, it can also be considered as a particular case of a system with a position-dependent effective mass [3].
The properties of the Eq. (1) and the Lagrangi-an (2) have been recently analysed [4] and it has been proved that this A-dependent nonlinear system admits a n-dimensional version and also that, for n = 2, it is related with a nonlinear deformation of the Smorodinsky—Winternitz system [5]. Moreover, a geometric interpretation of this nonlinear system in relation with the dynamics on spaces of constant curvature has been proposed. As a quantum system, the
Schrôdinger equation involving the potential x2/(1 + + gx2) was considered in [6—11], and it has been recently proved [12] that the quantum version of (2) can be studied by using a factorization method and even that it is endowed with the property of shape invariance.
In this section we will analyze the superintegrabi-lity of the A-dependent system described by the two-dimensional version of the Lagrangian (2).
It has been proved [4] that the appropriate n = 2 version of the A-dependent kinetic term T is given by
T2A =
1
1
2 V1 + Xr2
vX + v2y + XJ2
(3)
J = xvy — yvx, r2 = xy + y2,
so that the following X-dependent Lagrangian:
L = T2(X) — V2(r; X), (4)
a
V2(r; X) = —
2 V1 + Xry J '
represents the more direct extension to n = 2 of the Lagrangian (2) (the same kinetic term but with a X-independent potential is studied in [13, 14]). The two-dimensional dynamics is given by the following X-dependent vector field:
d . d . „ , ^ d
dvx
d_
' dv,
Ta = vx~ + vy — + Fx(x,y,vx,vy; X)---h
dx dy
+ Fy(x,y,vx,vy; X)d,
where the two functions Fx and Fy are given by:
Fx = a2
1 + Xr2
+
+ X [vx + v2 + XJ2] (T+XX^
Fy = —a2 (
y 1 1 + Xr2
+
+X v+v2y +XJy](,
in such a way that for X = 0 we recover the dynamics of the standard linear harmonic oscillator:
^ d d , y . d , y , d ro = + Vy—--(a2 x)---(a2 y) —.
dx
dy
dvx
dvv
The Legendre transformation is given by
Px =
py
(1 + Xy2)vx — Xxyvy
1 + Xr2 (1 + Xx2)vy — Xxyvx
1 + Xry
so that the form of the angular momentum is preserved by the Legendre map, in the sense that we have xpy — ypx = xvy — yvx (notice that this fact is consequence of the introduction of the term J in the definition of Ty(X)), and the general expression for a X-dependent Hamiltonian becomes
H (x,y,px ,Py; X) = (5)
1
pi + + X(xpx + ypy )2
+ 2 a2 V (x,y)
+X
and, hence, the associated Hamilton—Jacobi equation takes the form
OS\2 (dS\2
dx J \dy J
dS dS \2 2t„ . x- + y-j + a2V(x,y) = 2E.
This equation is not separable in (x, y) coordinates because of the X-dependent term; nevertheless it admits separability in at least three different systems. (i) In terms of the new coordinates
Z,y) Zx = V1 + Xy2 ,
a potential V is separable if it can be written on the form
Wi(Zx )
V=
1 + Xy2
+ W2(y).
(6)
Then V is integrable with two quadratic integrals of motion
h(x) = (1 + Xr2)pX + a2Wi(zx), /2(A) = (1 + Xr2)p2 - XJ2 +
+ a2 W2 (y) —
Xy2
1 + Xy'
:Wi(Zx )
(ii) In terms of
(x, zy), Zy =
y
We see that for X< 0 the potential Vy(r; X) is a well with a boundless wall at ry = 1/|X|; therefore, all the trajectories will be bounded. For X > 0 we have that Vy(r; X) ^ (1/2)(ay/X) for r ^to; so for small energies the trajectories will be bounded but for E(X) > (1/2)(ay/X) the trajectories will be unbounded.
V1 + Xx2 '
a potential V is separable if it can be written on the form
W2(Zy )
V = Wi(x) +
(7)
1 + Xx2'
Then V is integrable with two quadratic integrals of motion
Ii(X) = (1 + Xr2)pX - XJ2 +
2
r
x
+ a2 Wi(x) -
THREE SUPERINTEGRABLE TWO-DIMENSIONAL OSCILLATORS
Ax2
Wi(zy )
1 + Ax2
12(A) = (1 + Ar2)p2y + a2W2(zy )•
(iii) In polar coordinates (r,4>) a potential V that takes the form:
V = F (r) +
G(0)
(8)
is separable and integrable with the following two quadratic integrals of motion:
/1 _ rX\
Ii(A) = (1 + ArX)pX + ' ' » -X
Pi +
+ a2
F (r) +
1 - r2
G(®
12(A) = pi + a2G(0).
The potential V2(A) can alternatively be written as follows:
a2
V2(A) = y
1
a
"2
1 + Ay2 1
1 + Az2
+ y2
1 + Ax2
= a! = 2
x2 +
1 + Az2
1 + Ar2
Therefore, it is superseparable [15—21] since it is separable in three different systems of coordinates (zx ,y), (x, zy), and (r, 0). Because of this the Hamiltonian
H = 1 [vl + vl + A(x'px + ypy )2] +
(9)
x2 + y2
a!
+ 2 V 1 + A(x2 + y2)
H = 1
2 2
(1 + Ar2)pl2 + a2
1
1 + Ar2
y2
1 + Ar2
So the total Hamiltonian can be written as a sum, not of two, but of three integrals of motion. The third one, which represents the contribution of the angular momentum J to H, has the parameter A as coefficient; so it vanishes in the linear limit A ^ 0.
3. A NONLINEAR OSCILLATOR RELATED WITH THE RICCATI EQUATION
Leach et al. studied in [22, 23] the equation q + + qq + /3q3 = 0 and point out that for /3 = 1/9 it is completely integrable and possesses the maximum number of symmetries. Recently Chandrasekar et al. [24] have analyzed, making use of the so-called Prelle—Singer method, a generalization of this equation and have interpreted it as an "unusual Lienard-type oscillator with properties of a linear harmonic oscillator". The important point is that, in these two cases, the corresponding equations are second-order Riccati equations [25].
In this section we will analyze the superintegrabi-lity of a two-dimensional nonlinear Riccati oscillator using the Lagrangian formalism as an approach.
3.1. Second-Order Riccati Equations and Lagrangian Formalism
In this section we shall study a nonlinear system related with the following equation:
y" + [bo(t) + bi(t)y]y' + ao(t)+ (10)
+ ai(t)y + a2(t)y2 + as(t)y3 = 0.
with the three coefficients a3, b0, and b1 satisfying the following restrictions
(13 > 0, bo =
a2 a3 -
- 7;—, bi = 3V03.
VÔ3 2(3
admits the following decomposition: H = Hi + Hy — AH3,
where the three partial functions H1, H2, and H3 are given by
Hi = 1 [(1 + Ar2)p2x + ay( x2
This nonlinear equation is important for two reasons: Firstly, because it can be transformed into a third-order linear equation by the substitution
1 v' (t)
y(t) =
v((t) v(t).
H3 = 2 (xPy — yPx) ,
each one of these three terms has a vanishing Poisson bracket with H for any value of the parameter A:
{H,Hi} =0, {H,H2} =0, {H, H3} =0.
Secondly, because it can be obtained from a Lagrangian function. This can be proved as follows: the one-degree-of-freedom Lagrangian
L = vx + kU(x,t). (11)
leads to the following second-order nonlinear equation:
ix + 2kU (f) +1 k2uux + kU = 0, (12)
2
r
2
z
z
2
r
so, that in the particular case of the function U = = U(x, t) is given by
U = co (t)+ci(t)x + cy(t)x2,
then the above Eq. (12) reduces to
$ + <"» + K f> + <13>
2 3
+ a0 + a1x + a2x + a3x = 0, with the four functions ao, a1, a2, a3 given by
1 , 1 2 ,
ao = 2 coci + co, ai = coC2 + ^ Ci + Ci,
3 » 2
a2 = 2 cic2 + c2, a3 = c2,
and the two functions bo and b1 satisfying the appropriate restrictions.
Thus the nonlinear equation (13) is the Euler—Lagrange equation of the Lagrangian function (11) in the p
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