Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

0.37 MB·English

by  Joshua J. Capel

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 038, 17 pages
Invariant Classification and Limits
of Maximally Superintegrable Systems in 3D
Joshua J. CAPEL †, Jonathan M. KRESS † and Sarah POST ‡
† Department of Mathematics, University of New South Wales, Sydney, Australia
E-mail: [e-mail], [e-mail]
‡ Department of Mathematics, University of Hawai‘i at M¯anoa, Honolulu, HI, 96822, USA
E-mail: [e-mail]
5 Received February 03, 2015, in final form April 21, 2015; Published online May 08, 2015
1
http://dx.doi.org/10.3842/SIGMA.2015.038
0
2

y Abstract. The invariant classification of superintegrable systems is reviewed and utilized
a to construct singular limits between the systems. It is shown, by construction, that all
M
superintegrable systems on conformally flat, 3D complex Riemannian manifolds can be
obtained from singular limits of a generic system on the sphere. By using the invariant
8
classification, the limits are geometrically motivated in terms of transformations of roots of

] the classifying polynomials.
h
p Key words: integrablesystems;superintegrablesystems;Liealgebrainvariants;contractions
-
h
2010 Mathematics Subject Classification: 33C45; 33D45; 33D80; 81R05; 81R12
t
a
m
[
1 Introduction

3
v
1 The discovery and classification of superintegrable systems is currently undergoing significant
0 activity, see [22] and citations within. Superintegrable systems are dynamical systems, for
6
example classical or quantum Hamiltonian systems, that have more conserved quantities than
6
0 degreesoffreedom. Oneofthemanyactiveareasofresearchinsuperintegrablesystemsisintheir
1. classification. On this front, there has been significant progress on maximally superintegrable
0 systems in 2 and 3-D complex Riemannian manifolds with constants of the motion that are
5
at most second-order in the momenta, a necessary condition for multi-separability. In two
1
: dimensions, all such systems have been classified [12]. Somewhat surprisingly, all such systems
v
can be obtained through singular limits of a “generic” system defined on the sphere. This fact
i
X
was first recognized by Bˆocher [1] in his search for metrics that admit the maximal number of
r
second-orderKillingtensors. Againinthe2Dcase,thelimitsbetweensuchsystemswererecently
a
givenexplicitlyin[19],wheretheywereshowntoinducecontractionsontheassociatedsymmetry
algebrasaswellaslimitsintherepresentationsofsuchalgebras. Theserepresentationsaregiven
intermsofclassicalhypergeometricorthogonalpolynomialsandthelimitsofthesuperintegrable
systems correspond to the limits within the Askey scheme.
The aim of this paper is to give the corresponding limits of superintegrable systems in 3D.
We will show explicitly that all such systems are limiting cases of a “generic” system defined
on S3 and give the coordinate transformation that generates each limit. In order to describe
the limits we discuss the classification theory of 3D second-order superintegrable systems [2, 3].
The classification makes use of classical invariant theory and associates to each superintegrable
system a 7-dimensional representation of the conformal group. By studying the action of the
conformal group on this representation, we can identify 10 equivalence classes each of which
correspondstoaclassofsuperintegrablesystems. Fromtheinvariantclassification,thenecessary
limits are abstractly motivated as opposed to the constructions of [19], which are given ad hoc.
2 J.J. Capel, J.M. Kress and S. Post
Thestructureofthepaperisasfollows. WebegininSection2withanoverviewoftheresults
in[2,3]givingtheinvariantclassificationofnondegeneratesecond-ordersuperintegrablesystems
in 3D. Section 3 contains the results of the paper; namely we prove that all such superintegrable
systems are limits of a “generic” system by demonstrating the appropriate limits. Section 4
contains a brief discussion of the results and future applications of this research.
2 Invariant classification of superintegrable systems
In this section, we describe the invariant classification of nondegenerate second-order superin-
tegrable systems in 3D, where the metric is assumed to be conformally flat. The dynamics are
given by the following classical Hamiltonian on 6-dimensional phase space (p,x)
3
(cid:88)
H(p,x) = gij(x)p p +V(x).
i j
i,j=1
The conformally flat assumption implies that the metric tensor takes the form gij = λ(x)δi.
j
Any function on the phase space A(p,x) will transform in time as
dA ∂A
= {A,H}+ ,
dt ∂t
where {, } is the standard Poisson bracket
3
(cid:88) ∂A ∂B ∂B ∂A
{A,B} ≡ − .
∂x ∂p ∂x ∂p
i i i i
i=1
Thus, a function on phase-space will be invariant under the Hamiltonian flow if and only if
it Poisson commutes with H, ({A,H} = 0) and is referred to as a constant of the motion.
Conversely, theHamiltonianisalsoinvariantunderflowgeneratedbytheconstantofthemotion
and so the function A is also often referred to as a symmetry of the Hamiltonian.
A Hamiltonian on 2n-dimensional phase space is said to be Liouville integrable if there ex-
ist n mutually commuting constants of the motion that are functionally independent. We say
that a Hamiltonian is superintegrable if there exist more than n constants of the motion and
maximally superintegrable if there exist 2n−1 such symmetries. Not all of the symmetries can
mutually commute and so we drop the requirement that the integrals commute among them-
selves. Thereisacloseconnectionbetweensuperintegrabilityandnoncommutativeintegrability,
notably that maximally superintegrable systems are noncommutatively integrable. Indeed, the
celebrated result of Nekhoroshev [23] concerning foliations of phase space for noncommutative
integrable systems includes, as a special case, the result that, generically, bounded trajectories
of superintegrable systems are closed.
We would like to classify Hamiltonians H on a conformally flat 3D manifold with 5 con-
stants of motion that are second-order in the momentum, i.e., that there exist 5 (including the
Hamiltonian) functionally independent constants of the motion
3
(cid:88)
A = aij(x)p p +W(x), aij = aji,
i j
i,j=1
such that
{A,H} = 0.
Invariant Classification and Limits of Maximally Superintegrable Systems in 3D 3
If such symmetries exist then a direct computation of the Poisson commutators leads to the
following set of differential equations: the Killing equations, which require the second-order
part of A to be a Killing tensor on the manifold, and the Bertrand–Darboux (BD) equations.
The BD equations are the compatibility conditions for the differential equations for W(x) and
comprises a set of four linear second-order PDEs for the potential V(x), one for each integral
that is not the Hamiltonian. Combining the equations into vector form gives
 
V −V
,22 ,11
 
V,33−V,11 V,1
B(cid:0)aij(cid:1) V,12  = C(cid:0)λ,λ,k,aij,ai,kj(cid:1)V,2,
 V,13  V,3
V
,23
where the subscript f denotes the partial derivative with respect to x and B and C are
,k k
matrix functions of the given arguments of dimensions 12×5 and 12×3, respectively. Suppose
in addition that the constants of the motion are functionally, linearly independent. This implies
that the BD equations are of rank 5 and it is possible to solve for second-order derivatives of V
as in
V  1 −4S1−R12−R13 2S2+R12 2S3+R13 
,11 2 3 1 1
V,22 1 2S1+R212 −4S2−R112−R323 2S3+R223 
   
V,33 = 1 2S1+R313 2S2+R323 −4S3−R113−R223(cid:126)v, (1)
V,12 0 R112−3S2 R212−3S1 Q123 
V,13 0 R113−3S3 Q123 R313−3S1 
V 0 Q123 R23−3S3 R23−3S2
,23 2 3
 
V
,ee
(cid:126)v ≡ VV,,12, V,ee = 13(V,11+V,22+V,33).
V
,3
HereV isnotasecondderivative, butasymmetryadaptedvariable. TheuseofV introduces
,ee ,ee
redundancy in the equations (6 equations instead of 5), which is preserved for the sake of
symmetry in the variables. If the compatibility conditions of (1) are assumed to hold identically
then the potential depends on 5 parameters, the values of V , V , V , V and V at a generic
,ee ,1 ,2 ,3
point, the last point coming from a trivial additive parameter. Such a system is said to be
non-degenerate. For a non-degenerate system, the potential is uniquely determined by the value
of the matrix in (1) at a given generic point x . The values of V(x ), V (x ), V (x ), V (x )
0 0 ,1 0 ,2 0 ,3 0
and V (x ) are the parameters in the potential. As is clear from the form of the matrix, it
,11 0
depends on 10 functions
(cid:8)Q123,S1,S2,S3,R12,R12,R13,R13,R23,R23(cid:9) ≡ {Q,S,R},
1 2 1 3 2 3
which are assumed to satisfy the compatibility conditions for (1) identically. The compatibility
conditions of (1) are first-order equations in the variables whose compatibility conditions are
themselvesidenticallysatisfied. Therefore,thevaluesoffunctions{Q,S,R}atagenericpointx
0
are enough to uniquely determine the functions themselves along with a superintegrable system.
Example 1. The harmonic oscillator potential
V = a(cid:0)x2+x2+x2(cid:1)+bx +cx +dx +e
OO 2 1 2 3 1 2 3
4 J.J. Capel, J.M. Kress and S. Post
is a non-degenerate second-order superintegrable system. The BD equations for the potential
are
   
V 1 0 0 0
,11
 
V,22 1 0 0 0 V,ee
   
V,33 = 1 0 0 0V,1.
V,12 0 0 0 0V,2
   
V,13 0 0 0 0 V,3
V 0 0 0 0
,23
Taking the origin as a generic point, the coefficients are related to V as a = V (0), b = V (0),
,ee ,1
c = V (0), d = V (0), and e = V(0).
,2 ,3
2.1 Equivalence classes
The next step in the classification is to identify the appropriate equivalence classes. Clearly, we
would like to consider two potentials as equivalent if they are related by a change of parameters.
This is accomplished in the previous section by making a canonical choice of parameters as the
value of (cid:126)v(x ). We would also like systems to be equivalent if they are related by translations
0
in the position variables; such a translation corresponds to moving the regular point. We would
also like systems to be equivalent if they are related by, possibly complex, rotations. This con-
dition will be studied extensively in this section. Finally, we would like the equivalence classes
to include systems that are related via the St¨ackel transform or coupling constant metamor-
phosis [10, 11, 17, 24]. That is, suppose a Hamiltonian can be expressed as H = H + αU
0
then the St¨ackel transformed Hamiltonians Hˆ = 1H will also be superintegrable, perhaps on
U
adifferentconformallyflatmanifold. Underthistransformation,classicaltrajectoriesandquan-
tumwavefunctionsaswellasthecorrespondingsymmetryalgebrasareessentiallypreserved, up
toachangeofparameters. Thus, wewouldliketoconsidertwosuchsystemsasbeingequivalent.
An analogous classification of equivalence classes for superintegrable systems in 2D has already
been performed [20].
In order to consider St¨ackel equivalent systems, we focus our attention on conformally super-
integrable systems [13]. For conformal superintegrable systems, the Hamiltonian is identically 0
on trajectories. This can be accomplished mathematically by including the trivial added pa-
rameter to the system. Thus, along a trajectory the Hamilton–Jacobi equation is expressed as
H −E = 0 but with H −E being the Hamiltonian. A function on phase space will then be
a constant of the motion whenever its Poisson bracket is given by
{A,H} = f(p,x)H.
It is a direct computation to verify (see, e.g., [2, Lemma 4.1.4]) that a conformal integral of
the motion A of H will be a conformal integral of the scaled Hamiltonian U(x)H. Thus it is
possible to transform any conformal superintegrable Hamiltonian on conformally flat space to
a conformally superintegrable Hamiltonian on Euclidean space. Furthermore, since the action
of the St¨ackel transform corresponds to multiplying the Hamiltonian by a function, it is clear
that two Hamiltonians equivalent under this action will be equivalent conformal superintegrable
systems, from the perspective of their symmetry algebras.
Example 2. The Hamiltonian for the simple Harmonic oscillator p2−ω2x2 corresponds to the
following conformal Hamiltonian
H = p2−ω2x2−E.
Invariant Classification and Limits of Maximally Superintegrable Systems in 3D 5
Dividing by x2 gives
1 E
H(cid:98) = p2−ω2− .
x2 x2
Clearly, trajectories that satisfy H = 0 will also satisfy H(cid:98) = 0, however the “energy” of the new
Hamiltonian is given by the parameter ω2.
Conversely,anyconformallysuperintegrablesystemthatdependslinearlyonatleastonearbi-
traryconstantcanbetransformed, see, e.g., [2, Theorem4.1.8], intoasuperintegrablesystemby
couplingconstantmetamorphosis. Therefore,classifyingsuperintegrablesystemsonconformally
flat manifolds up to St¨ackel transform is equivalent to classifying conformally superintegrable
systems on Euclidean space. The determining equations for an integral of a conformal system
are similar to (1) except that the additive constant is no longer trivial but instead essential for
specifying the Hamiltonian. The corresponding structure equations are then
 
V
11 V 
VVVV,,,,23112323 = AVVVV,,,,e123e,
V
,23
with
1 −4S1−R12−R13 2S2+R12 2S3+R13 A11
2 3 1 1 0
1 2S1+R212 −4S2−R112−R323 2S3+R223 A202
 
1 2S1+R13 2S2+R23 −4S3−R13−R23 A33
A =  3 3 1 2 0 .
0 R12−3S2 R12−3S1 Q123 A12
 1 2 0 
0 R13−3S3 Q123 R13−3S1 A13
1 3 0
0 Q123 R23−3S3 R23−3S2 A23
2 3 0
The functions Ajk are quadratic functions in the {Q,S,R} given in [2].
0
2.2 Action of the conformal group
So far, we have determined that superintegrable systems are uniquely determined by the value
of the 10 functions {Q,R,S} at a regular point. We are interested in determining equivalence
classes of systems that are stable under the action of the conformal group so we look at the
induced action of this group on these 10 functions. Again, the exact formulas and derivations
can be found in [2, 3] and here we state only the results necessary to understand the limits
obtained in Section 3.
A conformal change of variables can be generated by translations, which act trivially on the
functions{Q,R,S}andinversionsinthespheresofvaryingradii,whichdecomposethefunctions
into a 7-dimensional representation {Q,R} and a 3-dimensional representation carried by {S}.
Significantly, we can use a group transformation to set the values of {S} to any desired value,
described in the following theorem.
Theorem 1 (re-statement of [2, Theorem 4.2.18]). Given a conformally superintegrable system
with function values {Q ,R ,S } at a regular point x , there exists a conformal group motion
0 0 0 0
that maps the function values to {Q ,R ,0} at the transformed regular point xˆ .
0 0 0
6 J.J. Capel, J.M. Kress and S. Post
This theorem allows us restrict our attention to the action of the conformal group on the
7-dimensional space {Q,R}. To understand the action, we consider the continuous generators
of the conformal group, namely translations, scaling, rotations and M¨obius transformations.
As discussed before, translations act trivially on our representations. Scaling the coordinates
correspondstoascalingofthefunctions. Moreinterestingistheactionofrotations. Torepresent
this action, we form weight vectors from the functions
Y = R12+ 1R23±i(cid:18)R12+ 1R13(cid:19), Y = 1√6(cid:0)i(cid:0)R13−R23(cid:1)∓2Q123(cid:1),
±3 1 4 3 2 4 3 ±2 4 1 2
Y = 1√15(cid:0)R23∓iR13(cid:1), Y = −1i√5(cid:0)R13+R23(cid:1),
±1 4 3 3 0 2 1 2
sothattheactionofrotationsaregivenbythestandardraisingandloweringoperatorsforso(3).
Using the isomorphism between so (C) and sl (C), we obtain a covariant representation of the
3 2
action via the polynomial
(cid:115)
6 (cid:18) (cid:19)
(cid:88) 6
q(z) = (−1)j Y z6−j.
3−j
j
j=0
Here the action of SO (C) is represented, via the isomorphism, as the standard action of SL (C)
3 2
as M¨obius transformations
(cid:18) (cid:19)
azˆ+b
qˆ(zˆ) = (czˆ+d)6q . (2)
czˆ+d
Thus, the action of the conformal group on the functions {Q,R} can be represented by the
action of GL (C) = C × SL (C) on the polynomials q(z) via scaling and (2). Furthermore,
2 2
the action of M¨obius transformations on the coordinates can be represented locally as rotation
combined with a scaling and so, if we find invariants that are closed under translation, scaling
and rotations they will be automatically closed under the entire action of the conformal group.
Finally, we have the following theorem.
Theorem 2 (Theorem4.2.21of[2]innon-homogeneouscoordinates). Given a conformal super-
integrable system and a regular point, there is a local conformal transformation (i.e., excluding
translation of the regular point) taking it to the regular point of another superintegrable system
if and only if the roots of the corresponding covariant polynomial at the corresponding regular
points are equivalent up to a general linear transform.
Therefore, a key to understanding the classification of conformal superintegrable systems is
to understand the classification of invariants of the roots of degree 6 polynomials. Furthermore,
sincewewouldliketheequivalenceclassestobestableundertranslationoftheregularpoint, we
require that any invariants obtained be closed under derivations. This significantly reduces the
number of equivalence classes to 10, exactly the expected number of equivalent superintegrable
systems. For more details on the classification, we refer the reader to the original papers [2, 3].
3 Contractions
For each of the contractions, we need only record the following data. The initial regular point
x , the rotation angles t , t , t , the scaling parameter c and the final regular point y . We
0 1 2 3 0
note that the rotations can be represented in SL (C) by the following matrices:
2
ρ(R ) = (cid:20)eit3/2 0 (cid:21), ρ(R ) = (cid:20)cos(t2/2) −sin(t2/2)(cid:21),
3 0 e−it3/2 2 sin(t2/2) cos(t2/2)
Invariant Classification and Limits of Maximally Superintegrable Systems in 3D 7
(cid:20) (cid:21)
cos(t /2) −isin(t /2)
ρ(R ) = 1 1 .
1 −isin(t /2) cos(t /2)
1 1
The scaling c can be encoded in the same manner by the matrix
(cid:20)c−1/6 0 (cid:21)
ρ(c) = .
0 c−1/6
In the following sections, unless otherwise mentioned, the angles are assumed to be set to 0 and
the scale factor c to be set to 1. The action of the matrices on the polynomials are given by
(cid:20) (cid:21)
a b
ρ(A)◦q(z) = qˆ(zˆ), ρ(A) = ,
c d
as in (2), which transform the roots as
da−b
aˆ = .
−ca+a
To emphasize the dependence of the polynomials on the regular point, we write q(x ,x ,x )(z)
1 2 3
where necessary.
The action of the SL (C) matrices can also be interpreted in terms of their action on a ste-
2
reographic projection of the complex plane onto the unit sphere using the formula
(cid:18) 2x 2y −1+x2+y2(cid:19)
(X,Y,Z) = , , .
1+x2+y2 1+x2+y2 1+x2+y2
For real t , the action of R (t ) is to rotate the roots around the X axis by an angle t counter-
1 1 1 1
clockwise (i.e., from Y to Z). Similarly, the action of R (t ) is to rotate the roots around the Y
2 2
axis clockwise (i.e., from X to Z). Finally, the action of R (t ) is to rotate the roots around
3 3
the Z axis clockwise (i.e., from Y to X).
The action of these rotations on R3 can be recovered as
   
1 0 0 cos(t ) 0 sin(t )
2 2
R1 = 0 cos(t1) −sin(t1), R2 =  0 1 0 ,
0 sin(t ) cos(t ) −sin(t ) 0 cos(t )
1 1 2 2
 
cos(t ) −sin(t ) 0
3 3
R3 = sin(t3) cos(t3) 0.
0 0 1
In total, the change of coordinates for the potential is given by
y = cR (t )R (t )R (t )(x−x )+y .
3 3 2 2 1 1 0 0
Notice that moving the regular point corresponds to translating the coordinates.
For the remainder of this section, we give the limits between the systems. The first few case
involve three or fewer roots, or (as in the case of [3111b]) four roots with a fixed cross ratio. The
limits between these cases can be achieved solely with the action of GL (C), that is, without
2
appealing to translation the regular point. For most of these case the limits can be achieved by
fixing one root and collapsing the rest together at infinity (or another prescribed point in C∗).
Several of the limits are elaborated to give a more complete understanding of the process.
8 J.J. Capel, J.M. Kress and S. Post
3.1 [6] V to [0] V
A O
The covariant polynomial for the [0] equivalence class is simply 0, so to contract down to this
equivalenceclass, weneedonlyscalebyafactorof(cid:15)−1 andmovetheregularpointappropriately.
For V , the regular point is already x = (0,0,0) so it is unaffected by the scaling. The
A 0
contraction is then
t = t = t = 0, c = (cid:15)−1, x = (0,0,0), y = (0,0,0),
1 2 3 0 0
q (0,0,0)(z) = −iz6 → q (0,0,0)(z) = 0.
A O
We start with the potential
(cid:18) (cid:19) (cid:18) (cid:19)
V = a 1(cid:0)x 2+x 2+x 2(cid:1)+ 1 (x −ix )3 +b x + 1(x −ix )2
A 1 2 3 1 2 1 1 2
2 12 4
(cid:18) (cid:19)
i
+c x − (x −ix )2 +dx +e.
2 1 2 3
4
The conformal change of coordinates y = (cid:15)−1x introduces a factor of (cid:15)−2 to the second order
terms in the Hamiltonian (due to the conformal scaling of the metric) and so we consider the
rescaled potential Vˆ = (cid:15)2V . Adjusting the parameters to be
A A
aˆ = Vˆ (y ) = (cid:15)4V (x ) = (cid:15)4a, ˆb = Vˆ (y ) = (cid:15)3V (x ) = (cid:15)3b,
A,ee 0 A,ee 0 A,1 0 A,1 0
cˆ= Vˆ (y ) = (cid:15)3V (x ) = (cid:15)3c, dˆ= Vˆ (y ) = (cid:15)3V (x ) = (cid:15)3d,
A,2 0 A,2 0 A,3 0 A,3 0
eˆ= Vˆ (y ) = (cid:15)2V (x ) = (cid:15)2e,
A 0 A 0
gives the potential
(cid:18) (cid:19)
Vˆ = (cid:15)2V = aˆ 1(cid:0)y 2+y 2+y 2(cid:1)+ (cid:15) (y −iy )3 +ˆb(cid:16)y + (cid:15)(y −iy )2(cid:17)
A A 1 2 3 1 2 1 1 2
2 12 4
(cid:18) (cid:19)
i(cid:15)
+cˆ y − (y −iy )2 +dˆy +eˆ,
2 1 2 3
4
which tends in the (cid:15) → 0 limit to the potential
V = aˆ(cid:0)y2+y2+y2(cid:1)+ˆby +cˆy +dˆy +eˆ.
O 2 1 2 3 1 2 3
3.2 [51] V to [6] V
VII A
The regular points for V and V are the origins, x = 0 and y = 0. The change of variables
VII A 0 0
is given by
π
c = 9/2(cid:15)−3, t = 0, t = , t = iln((cid:15)),
1 2 3
2
which transforms the polynomial
q (0,0,0)(z) = −36iz → q (0,0,0)(z) = −iz6.
VII A
Consider first the roots of the (projective) sextic q (0,0,0) = −36iz. There are five at
VII
z = ∞ and one at z = 0.
(cid:39)∞(cid:114) (cid:36)
(cid:45)
Re(z)
(cid:38)(cid:114) (cid:37)
0
Invariant Classification and Limits of Maximally Superintegrable Systems in 3D 9
In terms of a stereographic projection of the complex plane z = x + iy (with the plane of
projection intersecting the equator) the action of R (t ) is to rotate the sphere around the y-
2 2
axis (i.e., the imaginary axis which is directed into the plane) by the angle t . So consider
2
(cid:32)√1 −√1 (cid:33)
ρ(R (π)) = 2 2 .
2 2 √1 √1
2 2
Under the action described above the covariant polynomial becomes
9
ρ(R (π))◦q = − i(z+1)5(z−1),
2 2 VII 2
which has brought the fifth root down from infinity down to z = −1 and moved the root at zero
up to z = 1.
(cid:39)(cid:36)
∞ˆ (cid:114) (cid:114)ˆ0 (cid:45)
Re(z)
(cid:38)(cid:37)
Setting t = iln((cid:15)) gives the matrix
3
(cid:18)(cid:15)−1/2 0 (cid:19)
ρ(R (iln((cid:15)))) = .
3 0 (cid:15)1/2
On the polynomial this induces the change
(cid:0)ρ(R (π))ρ(R (iln(cid:15)))(cid:1)◦q = −9i(cid:15)−3(z+(cid:15))5(z−(cid:15)).
2 2 3 VII 2
So finally, scaling by c = 9(cid:15)−3 gives
2
(cid:0)cρ(R (π))ρ(R (iln(cid:15)))(cid:1)◦q = −i(z+(cid:15))5(z−(cid:15)) → q = −iz6 as (cid:15) → 0.
2 2 3 VII A
Thislimitcanbevisualizedasallthepointsonthesphere(except∞)beingdrawndowntowards
the origin.
(cid:39)(cid:36)
(cid:4)(cid:68)
(cid:4) (cid:68) (cid:45)
Re(z)
(cid:4) (cid:68)
∞ˆ(cid:38)(cid:4)(cid:114) (cid:68)(cid:114)ˆ0(cid:37)
(cid:4) (cid:68)
(cid:4) (cid:68)
Or equivalently the roots can remain stationary while the sphere descends into the plane of
projection. From this point of view, the image of any point on the sphere (expect ∞) end
up eventually being projected inside the circle where the plane and sphere intersect, which is
shrinking to the origin in the limit.
(cid:39)(cid:36)
(cid:45)
(cid:0)(cid:64) Re(z)
∞∗(cid:0)(cid:114) (cid:64)(cid:114)0∗
(cid:0) (cid:64)
(cid:0)(cid:0) (cid:38)(cid:37)(cid:64)(cid:64)
10 J.J. Capel, J.M. Kress and S. Post
3.3 [33] V to [6] V
OO A
Here, we would like to take the two roots of q , namely 0 and ∞, and move them both to 0.
OO
This is going to be essentially the same contraction as the previous one. The contraction is
π
c = 3/4(cid:15)−3, t = iln((cid:15)), t = − , t = 0.
1 2 3
2
The regular points are
x = (0,0,1), y = (0,0,0).
0 0
The covariant polynomial is transformed as
q (0,0,1)(z) = 6iz3 → q (0,0,0)(z) = −iz6.
OO A
3.4 [411] V to [51] V
III VII
The covariant polynomial for V is given by q = −3(1 + 3z2). In order to transform this
III III
polynomial into q = −36iz we need to take one of the finite roots and send it to zero and the
VII
other to infinity. The contraction that accomplishes this is
−9 2
c = √ (cid:15)−2, t = − π, t = π, t = −iln((cid:15)).
1 2 3
32 3 3
The regular points for each system do not move and they are
x = (0,1,0), y = (0,0,0).
0 0
The limit of the polynomial is then
q (0,1,0)(z) = −9z2−3 → q (0,0,0)(z) = −36iz.
III VII
3.5 [3111b] V to [51] V
VI VII
The necessary contraction is given by
i π i
c = (cid:15)−1, t = , t = 0, t = − ln((cid:15)),
1 2 3
16 2 2
x = (0,0,2i), y = (0,0,0).
0 0
In this limit, all of the roots coalesce to infinity except for a single root at zero. The limit of
the polynomial is then
q (0,0,2i)(z) = 3iz6+3z3 → q (0,0,0)(z) = −36iz.
VI VII
3.6 [3111b] V to [33] V
VI OO
For this contraction, we would like to move the roots of the polynomial q by sending the three
VI
non-zero roots to infinity and leaving the 0 root fixed. This can be accomplished by choosing
an imaginary, singular value for t which has the effect of scaling the variable z. The required
3
contraction is then
1
t = −iln((cid:15)), c = , x = (0,0,2), y = (0,0,1).
3 0 0
2
The polynomials transform as
q (0,0,2)(z) = 3iz3(cid:0)1+z3(cid:1) → q (0,0,1)(z) = 6iz3.
VI OO