Normalization and the detection of integrability: The generalized Van Der Waals potential

N O R M A L I Z A T I O N AND THE D E T E C T I O N OF INTEGRABILITY: THE

G E N E R A L I Z E D VAN DER WAALS P O T E N T I A L

DAVID FARRELLY* Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for

Astrophysics, 60 Garden Street, Cambridge, MA, 02138, U. S. A.

and

T. UZER School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430, U.S.A.

(Received 13 September 1993; accepted 21 September 1994)

Abstract. Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits.

Key words: Dynamics, Hamiltonian systems, Keplerian systems, degeneracy, integrability, hydrogen atom

1. Introduction

The definition of integrability is clear enough; for an autonomous N degree of freedom Hamiltonian, if N independent global invariants exist and are in involution with each other, integrability is guaranteed (Arnold, 1985). On the one hand, if the prescribed global invariants can be found explicitly then integrability is established. On the other hand, a failure to find such a set of global invariants by no means rules out the possibility that the Hamiltonian in question is integrable. The confirmation or denial of the integrability of a given Hamiltonian can thus prove difficult, often presenting daunting and unforeseen obstacles. A fundamental goal is thus to establish effective tests for integrability, or, failing that, to construct reasonable conjectures or procedures that might provide indicators that a particular system is integrable.

A catalog of procedures has been developed to essay the integrability of Hamil- tonian systems. The most obvious is to check if the Hamiltonian is separable (Arnold, 1985). It turns out that a number of integrable Hamiltonians of physical interest are separable in one or possibly more coordinate systems (Born, 1925). Separability eases the search for integrability and may indeed establish it if enough separation constants (i.e., global invariants) can be found. The St/~ckel conditions

* Permanent address: Department of Chemistry and Biochemistry, Utah State University, Logan, LIT 84322-0300, U. S. A.

Celestial Mechanics and Dynamical Astronomy 61: 71-95, 1995. (~) 1995 Kluwer Academic Publishers. Printed in the Netherlands.

72 DAVID FARRELLY AND T. UZER

may be used to guide one to appropriate coordinate system(s) but it should be noted that finding coordinates that separate a particular problem can itself be a difficult task (Arnold, 1985). Further, for integrability, "absence of proof is not proof of absence"; while non-separable, integrable systems are much rarer than the separable variety, they do exist. More general and systematic approaches than simply seek- ing separability are therefore in order (Hietarinta, 1987). Darboux (1901) probably originated what we might consider a taxonomic approach at the turn of the century by undertaking a systematic assault on the general problem of the determination of integrability. His method has come to be known as the Whittaker program, ever since it was featured in Whittaker's book (1944) on analytical dynamics. In essence, the method proceeds by assuming a particular functional form (i.e. an ansatz) for a presumed invariant; one containing a set of undetermined coefficients. The subsequent evaluation of Poisson brackets then provides the unknown coefficients. In the case of quadratic invariants under certain governing assumptions the Whittaker program is capable of determining the set of integrable potentials. Various extensions of the method have met with mixed success (Hietarinta, 1988). An alternative and more general, analytical method is the Painlev6 test (Tabor, 1988) in which the analytic structure of the equations of motion in the complex time plane is examined. This method has been used to uncover integrability but is somewhat fragile of application. False indicators may surface and this, together with unexpected omissions, can derail progress.

In a different kidney are a variety of numerical approaches to the detection of integrability. The advent of electronic computers has facilitated direct integration of the relevant equations of motion and has thereby popularized numerical studies as a way to unearth integrability. Probably the simplest numerical procedure is to generate and subsequently vet Poincar6 surfaces of section for chaos and thus diagnose, by visual examination, whether or not the motion appears integrable. Prudence is in order, however, whenever digital simulation is involved. Undeniably, numerical inquests can often reveal strong circumstantial evidence for or against integrability [e.g., the Toda lattice, Toda (1970); a good account of the discovery of the integrability of the Toda lattice by Ford, Stoddard and Turner (1973) is given by Gutzwiller (1991)], but they cannot in themselves establish or refute the integrability of a particular Hamiltonian. Cases exist in which numerically based claims of integrability (Ghikas, 1989) or chaoticity (Baumann and Nonnenmacher, 1992) have been disestablished under the weight of further analytical inquiry (Deprit and Ferrer, 1990; Farrelly and Howard, 1993).

Deprit and co-workers in their examination of this fundamental problem (Deprit and Miller, 1988; Deprit and Ferre, 1990; Deprit and Elipe, 1991; Miller, 1991) have discovered a new symptom of integrability in Hamiltonian systems of interest in celestial mechanics and atomic physics. Basically, they note, repeatedly, that a correlation seems to exist between the persistence of degeneracy in the normal form to extremely high order and integrability of the pre-normalized Hamiltonian. (It is important to note, however, that while Deprit and Miller note that this type

NORMALIZATION AND THE DETECTION OF INTEGRABILITY 73

of behavior in the normal form may be a symptom of integrability, they make no claims that normalization of any particular integrable system will necessarily result in a degenerate normal form.) In our own studies of the use of normal forms in atomic and molecular physics and motivated partly by the work of Deprit we have also noticed a similar circumstance (Milligan and Farrelly, 1993; Uzer et al., 1991). In particular, we observe that the known cases of integrability of a hydrogen atom subjected to a generalized van der Waals potential, in the particular limit that the magnetic quantum number (axial component of angular momentum) m = 0, do result in high order degenerate equilibria in the normal form. This system is currently of considerable experimental and theoretical interest (Alhassid et al., 1987; Ganesan and Lakshmanan, 1989, 1990, 1992, 1993; Deprit and Ferrer, 1990; Milligan and Farrelly, 1992; Farrelly and Howard, 1993; Howard and Farrelly, 1993). Although of intrinsic interest mainly in atomic physics, this Hamiltonian is an excellent paradigm for the general class of axially symmetric perturbed Kep- lerian systems. We suppose that the reader, who is presumably concerned mainly with dynamics on a celestial scale, will find this model of interest regardless of its origins in the microscopic world. Using the Kustaanheimo-Stiefel transformation we study high order normalization of this system. This transformation enables us to treat both the cases where m = 0 and m ~ 0 in a consistent fashion.

Apart from the details, our study differs in at least two important respects from those of Deprit and co-workers (Deprit and Miller, 1988; Deprit and Ferrer, 1990; Miller, 1991); first of all, the Hamiltonian we study is one of the rare variety that has an integrable but non-separable limit. Previous investigations of the possible connection between normalization and integrability have ultimately uncovered integrable cases in which the original Hamiltonian was separable. A notable excep- tion is the study of Deprit and Miller (1988) of the Toda Lattice which is likely the most well known example of a non-separable yet integrable Hamiltonian (Ford, 1973; Yoshida, 1984; van Moerbecke, 1976). This was apparently the origin of Deprit and Miller's original conjecture. However, normalization of the Toda system requires that the lattice first be Taylor expanded. Having done so, one notices that the expanded lattice is non-integrable if truncated at any finite order [Deprit and Miller (1988)]. Thus, integrability can only be implied by considering the asymptotic behavior of the normal form. The second difference is that most previous studies have centered on a reduced phase space having the structure of the sphere SU(2). In the present study, the reduced phase space is SO(4), albeit with constraints that allow a reduction in dimensionality to be achieved. Despite these differences, several integrable limits of the system are found to result in degenerate equilibria after normalization. We are also able to predict further integrable cases of a modified version of the Hamiltonian. However, we also find that an integrable limit that is non-separable fails to produce degenerate equilibria. Thus it seems that while normalization may provide an indication of integrability it is not an infallible detector of integrable systems.


The paper is organized as follows: In Section 2 the generalized van der Waals Hamiltonian (GvdWH) is introduced, and as a preliminary demonstration, we study a particular limit of that Hamiltonian. We admit at the outset that this limit has been investigated by Deprit and Ferrer (1990). A more detailed examination is reserved for Section 3. There the scene is set for normalization of the full problem. In order to normalize the GvdWH, regularization must first be imposed on the Hamiltonian. This is accomplished using the Kustaanheimo-Stiefel (KS) regularization procedure (Stiefel and Scheifele, 1971). Section 3 proceeds to establish a connection between the KS coordinates and the generators of the Lie algebra of SO(4) which constitutes the reduced phase space for the problem. Section 4 is given over to a discussion of normalization of the problem paying special attention to the known integrable limits of the generalized van der Waals potential. In Section 5 a modification of the Hamiltonian studied in Section 4 is checked for integrability. In each case (presumably) all of the separable integrable limits are found using the conjecture. Finally, brief conclusions are presented in Section 6.

2. The Hamiltonian

In Cartesian coordinates and atomic units me = e = h = 1 the Hamiltonian for the hydrogen atom in a generalized van der Waals potential is,

H = l ( p2 + p~ + p2) + 7(x2 y2 1 z 2" + + 3 2 z z ) - - r (1)

where E is energy and 7 and 3 are dimensionless physical parameters. Scaling the coordinates by 7 -1/3 and the momenta by 71/6 and converting to cylindrical coordinates (x = p cos ~b, y = p sin ~b, z) gives the Hamiltonian,

7-i = 7 -1 /3H = l (p) + p}) + (p2 + 32z2)_ 1 + _ _ (2) 2 r 2p 2"

where r = x,/p 2 + z 2. In (2) m is the z-component of the angular momentum vector and is conserved due to axial symmetry. Various physically interesting limits of this problem exist; when 3 = 0 the Hamiltonian is that of the quadratic Zeeman effect (QZE) (Krantzman et aI., 1992), when 3 = + l it is the spherical QZE and when 3 = q-v~ the system corresponds to the instantaneous van der Waals interaction between an atom and a metal surface (Alhassid et al., 1987). Interestingly a reversal of the sign of the term in 1 / r in Equation (2) produces the Hamiltonian of the Paul trap (Paul, 1990; Bliimel et al., 1989); normalization of that particular problem will not be attempted here, but it is striking that the integrable limits of the Paul trap are in complete concord with those of the GvdWH (Farrelly and Howard, 1993; Howard and Farrelly, 1993; Ganesan and Lakshmanan, 1993).

The integrability of Equation (2) has been analyzed by a variety of methods, including Lie group and Painlev6 methods. In particular, it has been shown that the equations of motion derived from (2) in semiparabolic coordinates with m = 0


have the Painlev6 property for/3 = 4-1, 4-2, and 4-1/2 (Ganesan and Lakshmanan, 1989, 1990, 1992). Global invariants were obtained for these cases using Noether's theorem, but only for m = 0. A simpler and more direct resolution of the issue of the integrability of Equation (2) has recently been put forward; the invariants when m = 0 have been shown to be connected in al l cases to separability of Equation (2) in appropriate coordinates (Farrelly and Howard, 1993). These limits are now summarized:

(i) /3 = +1 When/3 = +1 is substituted into Equation (1) it is apparent that the system is separable in spherical polar coordinates (for abitrary m) and the problem reduces to a radial equation. As an aside, note that replacement of/32 with _/32 in Equation (2) also provides an integrable limit when/3 = + 1, as a consequence of spherical symmetry. The integral of the motion in cylindrical coordinates is

m 2 z 2 I1 = (pz -- z/)) 2 + p-----2-- (3)

(ii) /3 = + 2 In the event that/3 = -4-2 the transformation to parabolic coordinates,

Z = ( U 2 - - v2)/2, p = u v (4)

leads to a complete separation for arbitrary m. In these coordinates Equation (2) becomes,

-¢ _ 1 [p U6 V6 m2(u 2 "-[- '0 2) 2( u2 + v2) 2 + p2 + + + 4 + uZv2 (5)

which is separable upon multiplication by (u 2 + V2). In this limit the integral of motion in cylindrical coordinates is given by

1 m 2 z - - - p 2 z -k- p 2 (6) 12 zb) r

(iii) /3 = -t-1/2 For /3 = + 1 / 2 the Hamiltonian might again be expected to separate. Suitable candidates are the parabolic coordinates,

p = (11, 2 -- V 2 ) / 2 , Z = UV, (7)

which yields

1 ( u6v6) 2m2 7-/-- 2 ( u Z + v 2 ) P ~ + P 2 + - 4 - + - 4 - + 4 + (u 2 _ v 2 ) 2 . (8)


The term in m 2 in the Hamiltonian ruins any hopes of separability. Hence, the problem separates only when m = 0 for/3 = + 1/2, so that the integral obtained upon separation is valid only in the limit that m = 0. Nevertheless, the problem is integrable for arbitrary m when/3 = 4-1/2 and the integral of the motion expressed in cylindrical coordinates is the following (Howard and Farrelly, 1993),

I1/2 = 12 + I~ + m 2 r 2 (9)

where

Ip = 2,(p~ - z~) P -~pzl 2 +__m2 r p

and I , is given by

I , = + P

This last case represents the only known integrable and (apparently) non-separable limit of Equation (2).

In addition to exact global invariants in particular limits, an adiabatic invariant also exists for the GvdWH problem. It is given explicitly by Alhassid et aL (1987)

A/~ = (4 -/32)A2 + 5(/32 - 1)A 2 (10)

where

A = ~ p x L - (11)

is the scaled Runge-Lenz vector and/-/0 = E0 is the unperturbed Kepler energy. Of course, in the integrable limits the adiabatic invariant does not necessarily reduce to the actual invariant relevant to that particular limit; however, Alhassid et al. (1987) have shown that dynamical symmetries exist in the various integrable limits through lowest order in the perturbation. In fact, the adiabatic invariant (10) is essentially the lowest non-zero order term in a normal form expansion of Equation (2). We calculate the normal form for the system to higher order in order to determine if it shows any special behavior in the integrable limits. The normal form is a power series representation having the following structure,

= ~ enT-gn. (12) n_>0

In (12) 7-/o is a quadratic form in the coordinates and momenta and ~ n is a homogeneous polynomial of degree n + 2 in the coordinates and momenta. Note that a polynomial of degree p enters in at order p - 2. The reader could be directed to numerous sources but a quartet of papers by Deprit and co-workers provides an excellent account of normal form theory and further details of the generation and properties of the normal form (Deprit, 1991; Deprit and Elipe, 1991; Deprit and Williams, 1991; Miller, 1991). The utility of normalization together with advanced computer graphical techniques in uncovering details of the flow in phase space is demonstrated in an article by Coffey et al. (1990).


2.1. THE LIMIT m = 0

It is illuminating to consider a particular limit of (2) when m = 0. In the semiparabolic coordinates of Equation (4) the Hamiltonian (2) with m = 0 and after regularization (Stiefer and Scheifele, 1971; Edmonds and Pullen, 1979) and a scaling becomes,

2 V2) __4 + + (u 2 + + +

7-g - ~t - z

4 2 q- C°'flz (U 2 - - V2)(U 4 - - V4). ( 1 3 )

4 Q 4

Here w = ~ and the parameter e serves to order the expansion; it can subsequently be set to unity without loss of generality. Equation (13) is a particular limit of the case studied by Deprit and Ferrer (1991) in an investigation of a set of integrable limits proposed by Ghikas (1990). In the vicinity of the equilibrium at the origin, the normal form can be generated to high order using computer algebra [e.g., Math- ematica (Wolfram, 1988)]. We elect to do this v ia the Birkhoff-Gustavson method (Raines and Uzer, 1992) although Lie Transform methods (Deprit and Elipe, 1991) might be quicken The normal form is generated in the (u, v) coordinates but is most conveniently and compactly re-expressed in terms of the generators of the Lie algebra of the group SU(2). To fourth order in the perturbation parameter e it is given by,

£471"0 - 2 7-(NF = 2rr0 + --~-[5rr o -- 5rr 2 - - 4 % 2 +/52(5rr 2 + 7r2)1 (14)

where

71" 0 =

71" 1 =

71- 2 =

1 2 1 ~(Pu + u 2) + ~(P~ + v 2)

~(p2 + u2)_ + /4

(vPu - uPv)

1 7r 3 = - ~ ( P u P v n t- u v ) . ( 1 5 )

The {Tri, i = 1,2, 3} are sometimes called the H o p f v a r i a b l e s and satisfy the same Poisson bracket relations as angular momentum, namely,

{rc j , rck } = ejkZrrl (16)

where ejkZ is equal to 1 for even permutation of its subscripts, to - 1 for odd permutations and to 0 otherwise. Together with the relation,

%2 = 7r 2 + 7r2 2 + 7r 2 (17)


the {Tri, i = 1, 2, 3} generate the Lie algebra of the group SU(2). Actually, the {Tri, i = 1, 2, 3} constitute the Lie algebra for both SO(3) and SU(2) which are locally isomorphic, but SU(2) is the appropriate symmetry group of the isotropic oscillator. Equation (17) illustrates an important geometric property of these quantities; they define a constant energy sphere whose radius is half the unperturbed energy (27r0). The other 7r's act like Cartesian coordinates on the surface of the sphere. Thus the orbits of the normalized GvdWH problem in the 4-dimensional phase space can be mapped into points on the surface of a 3-dimensional sphere.

Use of the {Tri, i = 1,2, 3} provides a ready transformation to action-angle variables that exploits the connection between the Poisson brackets in Equation (16) and angular momentum. For the isotropic 2D harmonic oscillator, the transformation from the 7r's to action-angle variables has been described elsewhere (Farrelly, 1986) and will not be repeated in detail here. First a transformation to the action-angle variables j~, j~, eu, ¢~ is made as follows,

u = V ~ u sin ¢~; v = V ~ v sin C u ;

Pu = 2V/~ cos Cu; P . = Y ~ cos ¢, (18)

followed

J l - -

¢ 1 =

by a second transformation to a new set of action variables,

(ju + j v ) . J2 - (ju - j r ) 2 ' 2

Cu +¢v; ¢2=¢u-¢v (19)

in terms of which the {Tri} become,

7to = J1

71-1 = J 2

: c o s ¢ 2

71"3 ~--" C - j 2 sin ¢2 (20)

where ¢2 is the angle conjugate to the action J2. The rotational invariance of angular momentum means that one could just have well have defined 7c2 or ~3 (or linear combinations) to be the second action J2. Tedious canonical transformations between the several possible sets of apt action-angle variables can thus be avoided by making rotations directly in 7rl, ~2, ~3 space (apt action-angle variables are discussed by Martens and Ezra, 1987).

Even superficial perusal of Equation (14) reveals that interesting limits exist; when /3 = + l , d:2, ±1 /2 the normal form depends only on one of the Hopf variables. What is remarkable, and this will be demonstrated shortly, is that this behavior persists to extremely high (probably all) orders in the normal form. Consequently no angular dependence appears in the normal form when written in the action-angle variables of Equation (20); despite the unperturbed problem


exhibiting a 1 : 1 resonance, the normal form is expressible entirely in terms of action variables, and gives rise to degenerate equilibria.

Deprit and Ferrer (1991) computed the normal form symbolically for a Hamil- tonian which has (13) as a special limit. In the more general case they argued, based on an examination of the normal form, that a case put forward by Ghikas (1990) as being integrable was mischaracterized. Neither Ghikas nor Deprit and Ferrer presented surfaces of section; however, a quick numerical calculation reveals strongly chaotic looking surfaces of section, supporting the ability of normalization to single out integrable cases. In the particular limit of interest to us, namely that corresponding to the GvdWH with m = 0, Deprit and Ferrer (1991) calculated the normal form through 8th order. We extend the calculation to 20th order in c without being obligated to modify the conclusions of Deprit and Ferrer regarding integrability. Dynamically, dependence of the normal form on a single Hopf variable implies a particular kind of motion that we label local, normal or precessional (Milligan and Farrelly, 1993; Farrelly, 1986; Sahm and Uzer, 1989; Martens and Ezra, 1987). This classification is based partly on visual inspection of the topologies of the trajectories in the three limits and partly in analogy with the accepted spectro- scopic designation of degenerate molecular vibrational modes. The normal forms in each of the three limits through 20th order were derived using Mathematica implemented on a NeXT machine, and are the following:

(i) Normal mode limit,"/3 = +2

5EaTr0(Tr02 + 37r~)-393E87r0(Tr 4 + 107r27r 2 + 57r 4) 7-~NF = 27ro +

+

W 4 8Cd 8

14745cleTro(Tr~ + 217r47r 2 + 357r27r 4 + 77r 6)

16co 12 11451165~167r°(Tr2 + 37rZ)(Tr6 + 337r47r2 + 277rZTr4 + 37r6) (21)

512~ 16 639784665c2°Tro(Tr~ ° + 557r87r 2 + 3307r67r 4 + 4627r47r 6 + 1657roZTr 8 + llTr~ °)

1024~ 2o +

(ii) Precessional mode limit,"/3 = ± 1

E 4 7/Np ---- 27r0 + ~q-TrO(57r 2 -- 37r 2)

+ 3,8 o/-131 -4 + 23 4) 8~08

+ 3E127ro(49157ro 6 -- 66577r47r2 z + 23857r27r24 -- 2117r 6) 16w12

+ 3c167ro(--38170557ro 8 + 65209807ro67r 2 - 35093947r47r 4) 512w 16


+

+

+

3e167ro(6775407r27r 6 - 368077r28) 512w 16

9e2°Tro(710871857r l ° - 1463916857ro87r 2 + 1048887307r% 4) 1024w 20

9e2°Tr0(-a1842202%47r 6 + 39320057rETr28 -1464817r 10)

1024~020 (22)

(iii) Local mode limit;/3 = 4-1/2

56471"0(71-2 + 371 "2) 7-~NF = 27ro +

4004

-393,8 o % + + 128w 8

14745e12rro(Tr 6 + 21~-~rr~ + 35rrZTr 4 + 7~-36)

1024w 12 -11451165e16rro(% 2 + 37rZ)(rc 6 + 33rr4rr 2 + 27rr2rr 4 + 3rr36)

131072w 16

+

(23)

639784665e2°Tro(% 1° + 557r87r 2 + 3307r67r 4 + 4627r47r 6 + 1657r27r 8 + llTr 1°)

1048576w 2°

In Equation (23) if e is replaced by x / ~ and 7r3 by 71" 1 then Equation (21) is recovered. This is as expected because the local and normal mode limits are simply related to each other by a coordinate rotation.

2.2. DEGENERATE EQUILIBRIA IN SU(2)

The averaged dynamics of the system can be obtained by finding the equations of motion for the normal form, making use of the SU(2) properties of the generators. Specifically,

~i = - - {~-~NF, 7ri}, i = 1, 2, 3. (24)

Through 4th order the equations of motion based on the normal form (14) become (after neglecting inessential factors and constants),

~q = (4 -/32)71"2713 (25)

~2 = 5(fl 2 -- 1)TrlTr3 (26)

/r3 = (1 - 4f12)TrlTr2 (27)

The equilibria are determined by the condition that #i = 0 which yields the three integrable limits, namely;/3 = 4-2, +1,4-1/2 . It is an easy calculation to verify that these equilibria are degenerate, i.e., the equilibria are not isolated. For example, when/3 = + 2 and 7rl ---- 0 then all points in the 7r2 - 7r3 plane are fixed points and therefore degenerate equilibria. Degeneracy persists through all computed orders


of the normal form because, in each case, the normal form depends on a single Hopf variable which occurs only in even powers. This conclusion can be verified either by explicit calculation or by using the algebraic properties of the Poisson bracket. More physically, for arbitrary values of/3 two of the three topologically distinct types of trajectory co-exist, e.g., if/3 = 0 the dynamics as generated by the normal form is made up of local and normal mode trajectories, whereas if/3 -- the motion consists of precessional and normal mode trajectories (Milligan and Farrelly, 1993; Sahm et al., 1990). In each of the integrable limits of Equation (2), however, only a single type of topology is encountered. As/3 is increased from 0-2 the system passes through the three integrable limits. Changing/3, in the language of semiclassical mechanics, corresponds to a gradual re-orientation of the axis of quantization. In particular, in each of the integrable limits the orbits correspond to simple circulations about the axis associated with one of the three Hopf variables.

3. Regularization and Normalization

In the previous section a special case of Equation (2) with m = 0 was examined in semi-parabolic coordinates. Study of the full GvdWH necessitates a considerable re-analysis; in particular the reduced phase space is SO(4). Our plan of attack is to regularize first and then normalize Equation (1). Faced with a perturbed Keplerian system one is immediately tempted to eliminate the mean anomaly using the Delaunay elements (Coffey et al., 1986). Admittedly the Delaunay elements allow for a particularly expeditious route to the normal form and also provide direct contact with physical orbital variables such as angular momentum. Nevertheless, it is equally true that performing perturbation expansions in the Delaunay elements is not always satisfying. In fact, one may be plagued by awkward singularities that arise precisely because of electing to use the Delaunay elements. For example, in the context of semiclassical quantization, use of the Delaunay variables can lead to quantization rules that fail catastrophically at the separatrix between topologically distinct families of trajectories. Perhaps more seriously, the cases m = 0 and m # 0 must be treated as separate cases. The issue of finding replacements for the Delaunay elements in certain applications has been addressed by several workers in atomic physics and celestial mechanics and dynamical astronomy (Grozdanov and Rakovic, 1990; Farrelly et al., 1992; Ferrer and Miller, 1992). These workers concur that an alternative set of coordinates in which to execute the normalization procedure are based on the generators of SO(4). The coordinates can be obtained by considering the connection between the Kepler (or hydrogen) problem and the isotropic 4-dimensional oscillator (Boiteux, 1973; Iwai, 1982, 1982a,b). We elect to normalize in the KS coordinates chiefly because the case m --- 0 does not have to be treated separately from m non-zero (Deprit et al., 1994). Thus semiclassical quantization of the normal form can be effected in a single consistent treatment valid for all values of m.


The simplest way to make contact with the oscillator picture is to acknowledge that the components of L(Lx, Ly, Lz) and A(Ax, Ay, Az) generate the Lie algebra of the group SO(4) [isomorphic to SU(2) ® SU(2)]. Using the usual notation for the generators of SO(4), we write L = L($23, S13, S12) and A = A(S14, $24, S 3 4 ) .

The generators of SU(2) ® SU(2) are two angular momenta J and K which are related to the SO(4) generators by (Baym, 1969).

j _ ( L + A ) K - ( L - A ) 2 ' 2 (28)

where,

{Jj, Jk} = ejmJt, {Kj , Kk} = ejmKl (29)

and (i, j, k) = (x, y, z). Equation (29) makes clear the correspondence between SO(4) and SU(2) ® SU(2). The transformation of the unperturbed problem proceeds by writing the components of J and K in terms of the classical equivalents of boson operators, {ai} and {a~ } viz.,

(+2 + a,4) (ala2- al4) ( + , - Yl = , J 2 = , J 3 = ( 3 0 )

2 2i 2

K1 = (a~a4 + a3a~) , K 2 = (a~a4 - a3a~) , K 3 = (a~a3 - a~4a4) (31) 2 2i 2

Passage to the harmonic oscillator picture is then most simply accomplished by associating Cartesian coordinates with the (classical) bosonesque quantities (Kibler and Negadi, 1983),

ai = (qi + iPi)/v/~ a~ = (qi - ip i ) /v~. (32)

The Kepler Hamiltonian is thereby converted into that of an isotropic 4-dimensional oscillator,

H0 = 4 = l ( p 2 +p2 +p2 +p2 + q2 + q2 + q2 + q2) (33) w 3

together with the constraint,

1 2 1 ~(Pl + p2 + q2 + q2) = ~(p2 + p42 + q3Z + q2) (34)

where w -- C'Z-ff-E. This is equivalent to requiring that j2 = K 2 i.e., the "square representation" (Baym, 1969) of SO(4). The results so far apply only to the unperturbed system and this approach has the drawback that it provides no easy way to convert the perturbation in Equation (1) into the oscillator coordinates and momenta.

A straightforward way to accomplish the desired transformation is to make use of the Kustaanheimo-Stiefel (1965) coordinates which do allow the perturbation


and H0 to be written in terms of the coordinates and momenta of the 4D isotropic oscillator. The KS transformation was originally designed as a numerical tool to regularize the effect of the Coulomb singularity on the classical dynamics in the vicinity of the nucleus. In the present context it has the distinctly advantageous property of allowing the unperturbed Hamiltonian and the perturbation to be written in terms of the canonical coordinates and momenta of the isotropic 4-dimensional harmonic oscillator. Caution is in order, however, because the coordinates of Kus- taanheimo and Stiefel do not match up directly with the coordinates introduced in (14). Further discussion of this point will be deferred till later. Nevertheless, the KS transformation sets the stage for normalization.

3.1. THE KUSTAANHEIMO-STIEFEL TRANSFORMATION

The KS transformation starts by relating the original coordinates to a set of coordinates in a 4-dimensional space using,

r = Tu (35)

where,

Ul --u2 --u3 u4 )

T = u2 ul - u 4 -u3 (36) U3 U4 Ul U2 U4 --U3 U2 --Ul

and u = (Ul, u2, u3, u4), r : relation,

TtT =t TT =l u 12 .

(Xl, X2, X3,0) and T satisfies the orthogonality

(37)

In Equation (37),

l u 12= ~,,~ + u~ + ~3 ~ + ~4 ~ (38)

The two sets of coordinates are related explicitly by the following,

x l = z = u , ~ - ~ , ~ - ~,~ + u 4 ~

X2 = y = 2(UlU2 - u3u4)

X 3 = X : 2(UlU3 q- uau4).

(39)

(40)

(41)

The dynamical variables can be related by using the momenta Pu conjugate to u,

Pu =t (P1, P2, P3, P4) (42)

for which the following constraint holds,

u l P 4 - u 4 P l --}-- u 3 P 2 - u 2 P 3 = O. (43)

84 DAVID FARRELLY AND I". UZER

Note at this point the difference between the constraint in Equation (43) and that of Equation (34). This confirms that the KS coordinates differ from those introduced in Equation (32). Further, we have

4 4

E Pxidx, : ~ Pidu,. (44) i=1 i=1

and thus,

Px = 1 T P i (45)

where,

Px =t (Pxl, Px2, Pz3,0). (46)

Using (39-41) and (45) the constraint (43) can be expressed in terms of m as,

PC = m = lt4Pl - u lP4 = u 2 P 3 - lt3P2. (47)

The Hamiltonian may now be converted into a system of four anharmonically coupled oscillators by making a transformation to a new time variable s [regularization, or essentially introducing a generalized eccentric anomaly (Stiefel and Schiefel, 1971; Deprit and Williams, 1991)],

dt _ 4r 4 [ U 12 (48) ds

Multiplying through by 4r gives the KS Hamiltonian,

1 2 ~2 8(2 fl2) lu HKs = 4 = ~(Pu + l u 12 ) + - [ 2 (u~ + u~)(u~ + u~)

+ aft 2 [U [2 [(U2 q_ U2)2 q_ (It2 _~_ U~)2] (49)

where,

&2 = - 8 E . (50)

Scaling the coordinates and momenta and introducing the order parameter e yields,

4 1 8e4(2_ flz) 12 (u21 + uZ)(uZ2 + u2) H K S - ~o- 2(e~+lu12)+ oj4 In

4e4fl 2 12 + ~ l u [(u~ + u42) 2 + (u~ + u~)21. (51)

The normal form was obtained in these coordinates using Mathematica as implemented on an IBM RISe 6000. As noted, the KS coordinates are inconvenient in that they do not identify with the coordinates introduced in Equation (32). These


latter coordinates have the advantage of providing a ready connection with the generators of SO(4). Thus it is necessary to relate the two sets of coordinates.

A solution to this problem is to make a transformation to new coordinates and momenta, equivalent to a rotation in phase space,

(ql -q- P4) (q4 -- Pl) U l - - V~ ' P I - - v '~

u2 -- (q3 + P 2 ) (q2 - P3)

(q2 q- P3) (q3 - P2) U 3 - - V~ ' P 3 - v ~

(q4 + Pl) (ql -- P4) u4 -- v/~ , P4 = v ~ (52)

After this the transformation to an expression in terms of the SO(4) generators is accomplished using Equations (28)-(32).

The normal form through 4th order is (where n = 1/2~0)

"~"[NF = 2n + 1 6 ~ ( n 2 +/32n 2 + $22 - 32S~2 + 4S24 - 32S~4) oJ

+ 16-~(4S~4 -/32S24 - $24 + 4/3S3~4). (53)

Before proceeding to a full study, it is worthwhile noting that three different closed sub-algebras are generated by {$23, S13, SI2}, {S14, Sz4, SI2} and {Sl2, $34}. For the three values of/3 for which Equation (1) is integrable the normal form (53) is expressible entirely in terms of generators of these sub-algebras. In action angle variables it may therefore, in each case, be written such that it contains no angular dependence. These special cases or dynamical symmetry limits are now summarized.

3.2. SYMMETRY LIMITS

(/) /3 = +1/2 In this case the normal form (53) reduces to,

20e4n 3 12e4n. 2 7"/NF = 2n + w----T-- + ----~($12 + 5S24 q- 5S24). (54)

Significantly, the components of a new, non-conventional angular momentum vector, A(A1, A2, A3) - A(S14, $24, S12) generate an SU(2) Lie algebra [see Table I: this A should not be confused with A 3 of Equation (10)]. Equation (54) thus represents a dynamical symmetry limit of the original Hamiltonian. In particular, the reduced phase space is the sphere SU(2) defined by the components of A. The normal form can thus be expressed entirely in terms of the dynamical quantities A 2 and A3.


TABLE I Poisson bracket relations {A, B} between the SO(4) generators

{A, B} $12 $13 &4 5:23 ~24 S34

S12 Sl3 $14 &3 ,924 &4

0 S23 $24 -&3 - & 4 0 -~23 0 ~34 S12 0 -S14 -~24 -~34 0 0 S12 S13 S13 -S12 0 0 $34 -$24 $14 0 -Si2 -$34 0 &3 0 S14 --Sl3 ~24 --~23 0

(ii) 13 = -t-1 This limit corresponds to rotational symmetry and the normal form becomes,

32e4n3 48e4n 2 "J-~NF • 2 n + co----- 7 - + 7 ( S 1 4 -t-- $24 "Jr 8 2 )

or,

(55)

32e4r~ 3 48e4rtA2 7-INF = 2rt + w------ T - + w4 . (56)

In view of the constraint L.A = 0, pure dependence on A 2 is equivalent to pure dependence on L 2. This is expected, given the initial rotational invariance. Again, the reduced phase space is the sphere SU(2) and the energies are degenerate with respect to m.

(iii) /3 = =1:2

In this case the normal form depends only on Lz = S12 and Az = S34 whose mutual Poisson bracket vanishes.

80e4n3 48enn - 2 7-~NF = 2n -t- CO ~ n t- ~ ( 5 S 3 4 - S22). (57)

This again, represents a situation corresponding to degenerate equilibria since the normal form depends only on the action A z and the parameter rn after reduction.

3.3. ACTION-ANGLE VARIABLES AND THE LIMITS S12 = 0

The normal form can also be written in one-dimensional form in terms of action- angle variables. The most useful variables for problems having axial symmetry are based on the canonical conjugate pair (A,, qSAz ). The derivation of these variables is described in the Appendix. In terms of this pair of action-angle variables, the normal form becomes,

"]-~NF = 2 n -- 12A 2 + 18A2z132 + 8n 2 - 2132n 2 - 4rrz 2 - 2132'/7/, 2


-- (8 -- 2/32)v/-A 2 + n 2 - - 2 A z m - 73Z 2

v / - A 2 + n 2 + 2 A z m - m 2 cos(2q~A~). (58)

This expression provides ready contact between the special case $12 --- ra -- 0 and the normal form in Equation (53). It is important to establish this connection as an additional and independent check on the validity of the normal form. As described in the Appendix, in the limit m = 0 the perturbed 4D oscillator collapses into a perturbed 2D oscillator. For example, setting m = 0 in Equation (58) yields

"]"[NF = 2n = 12A2z + 18Az2/32 + 8 1 ~ 2 - - 2/32n 2

- 8 ( 1 - ~ ) ( n 2 - A 2 ) c o s ( 2 ( g z ~ ) . (59)

At this point, it is possible to associate A z directly with the SU(2) generator 7rl using Equation (20) and results in the Appendix. In each of the three limits it is easy to show that Equations (21)-(23) are special cases of Equation (53) through 4th order.

4. Degenerate Equilibria in SO(4)

The existence of degenerate equilibria in the reduced phase space SO(4) can, in principle, be translated into a set of restrictions that the normal form must satisfy. This in turn picks out what are presumed to be the integrable limits of the Hamiltonian whose normal form has SO(4) as its reduced phase space. The restrictions are based on the Poisson algebra generated by the SO(4) generators Sij whose Poisson bracket relations are summarized in Table I.

In general the normal form will be an expansion in powers of the SO(4) generators, which can alternatively be expressed in terms of teh components of two angular momenta J and K. Rather than attempting to compose a set of rules for the general case, we focus specifically on the Hamiltonian (2) whose normal form is given by Equation (53). Note that the normal form (53) can be written in terms of four of the six generators, i.e., S14, $24, $34, and SI2. The equations of motion for the six generators are determined as before by evaluation of the Poisson bracket between the generator and the normal form using Table I. The resulting equations of motion are

S12 ~-~ 0

Sl3 ~ (1 - / 3 2 ) 8 1 2 S 2 3 - (4 - f l2)S14834 q- (4f l 2 - 1)S14S34

S14 = (1 - / 3 2 ) S 1 2 S 2 4 - (4 - / 3 2 ) S 1 2 S 2 4 - (4f l 2 - 1)S13S34

5'23 = - ( 1 - / 3 2 ) S 1 2 S 1 3 - (4 - f l2)$24S34 -/- (4/3 2 - 1)$24S34

$24 ~- - ( 1 -/32)S12S14 4- (4 -/32)S12S14 - (4/32 - l)$23S34

$34 ~--- (4 - / 3 2 ) S 1 3 S 1 4 q- (4 - / 3 2 ) $ 2 3 S 2 4 .

(60)

(61)

(62)

(63)

(64)

(65)


These equations are clearly suggestive that 3 --- 4-2, 4-1, and + 1/2 might be special cases and therefore we look for degenerate equilibria. The equations of motion for the six generators are, in the three limits,

(i) 13 = 4-1/2

S12 = 0

S13 = ~S12S23 15

-- --~- S14S34 4 S14 = -3S12S24

$23 = - 3 S 1 2 8 1 3 - ~S24S34

$24 = 3S12S14

$34 = ~-~(S13S14 q- $23S24). (66)

It is convenient to work in the SU(2) phase space associated with A(S14,824, S12) using Equation (54). In this case all points for which m = 0 are fixed points. However, this is only true when m = 0 and, since m occurs as a parameter in the original Hamiltonian, integrability can only be inferred in the limit m -- 0. It does not seem reasonable to impute integrability to the Hamiltonian (2) for all m when degenerate equilibria [or zero characteristic exponents i.e., the modified Deprit conjecture - Miller (1991)], exist only for m = 0. Because the Hamiltonian (2) is known to be integrable for all m when 13 = 4-1/2, this finding demonstrates, by providing a concrete example, that integrability need not have special consequences for the structure of the normal form.

(ii) 3 = 4-1

S12

Sj3

S14

S23 S24

$34

= 0

= 0

= -3S12S24 - 3S13S34

= 0

--: 3S12S14 - 3S23S34

= 3S13S14 q- 3S23S24. (67)

Degenerate equilibria exist for all m and this implies that the underlying Hamilto- nian may be integrable, as is, in fact, the case.

(iii) 13 = + 2


S12

S13

S14

S23

S24 San

= 0

= - S 1 2 S 2 3 + 15S14S34

= - 3 S 1 2 S 2 4 - 15S13S34

= 3S12S13 q- 15S24S34

~- 3S12S14 -- 15S23S34

---~ 0. (68)

Yet again, for arbitrary ra degenerate equilibria exist and, as expected, the original Hamiltonian is integrable in this limit.

Examination of the normal form through order e4 implies that the cases/3 = 4-2, 4-1,4-1/2 (m = 0) correspond to integrable limits. Of course, we must be concerned with the persistence of degeneracy and, therefore, it is necessary to compute the normal form to higher order.

4.1. NORMALIZATION TO HIGH ORDER

Computation of the normal form to high order for this 4-dimensional system is a difficult task. Generation of the normal form with retention of a symbolic "/3" in Equation (51) strains the limits of an IBM RISC 6000 machine running Mathematica. We evaluated the normal form explicitly through 10th order using a symbolic/3, and through higher order only for/3 = 4-2. The maximum order we could achieve is a strong function of the complexity of the original Hamiltonian. Equation (52) is considerably simpler when/3 = 4-2 than in the other two cases. Consequently it is possible to compute the normal form to at least 20th order in this limit. The full (i.e., arbitrary/3) normal form through 10th order contains 4552 terms in the original coordinates and momenta although intermediate expressions may be considerably larger. This number swells considerably when expressed in the Cartesian coordinates of Equation (32), only to shrink to 56 terms when written in terms of the SO(4) generators. It will not be reproduced here. As a check we confirmed that the normal form reduces to the known limits when m = S12 = 0 or /3 = 0 (the QZE).

(/) / 3 = + 2

7-tNF = 2n +

+

+

+

e4(80n 3 - 48nS~ + 240nS~4 ) W 4

~ ( - 1 2 5 7 6 n 5 - + 12480n3S~ 2208nS~)

~8 ( - 125760n3S~ + 37440nS~S~- 62880nS~)

-~2(3774720n 7 -5112576n5S~+ 1831680n3S~ - 162048nS~)


£12 + ~-ff(79269120nsS~4- 51125760n3S~2S~4 + 5495040nS4zS~4)

612 + ~-i~(132115200n3S44 - 25562880nS22S44 + 26423040nS64)

El6 + ~ig(-1465749120n9 + 2504056320n7S~2 - 1347607296n55'42

+ 260175360n3S62) £16

+ ~i8-(-14133888nS82- 52766968320n7S24) El6

+ --~(52585182720nsS~2S24 - 13476072960n3S~2S~4) ~16

+ ~-i-~(780526080nS62S24 - 184684389120n5S44) 616

+ -jd(87641971200n3S22S~4 - 6738036480nS~2S44

- 123122926080n3S64) 616

+ ~7-g(lV528394240nS~zS64 - 13191742080nS84) e2o

+ ~-5-6(655139496960n 11) 620

+ ~-~-6 ( - 1349145768960n9S~2 + 966654535680n7S42) 620

+ ~--~6 ( - 293457733632n5S62)

+ ~o(36237358080n3S82 - 1349968896n5'~ °

+ 36032672332800n9S24) 2O

+ ~2o(-48569247682560nTSZzS24 + 20299745249280n5S42S24) 2O

+ ~-~-6(-2934577336320n3S625'24 + 108712074240nS825'24) 2O

+ ~-62o (216196033996800n7S~4) ~20

+ ~-6(-169992366888960n5S22S44 + 338329048748800n3S42S44) f20

+ ~-~6(- 1467288668160nS6zS'44 + 302674447595520nSS64) ~20

+ " ~ ( - 113328244592640n3S~2S~4)

NORMALIZATION AND THE DETECTION OF INTEGRABILITY

C20 + -~(6766581749760nS~2S64)

~20 + ~ (108098016998400n3S84)

£20 + --~-6(-12142311920640nS22S~4 + 7206534466560nS~°) •

91

(69)

In this limit, through all computed orders the normal form is found to depend only on the squares of the two generators {&z, $34} alias {Lz, A , }; this indicates integrability of the original Hamiltonian.

(ii) /3 = 4-1

7-{NF = 32e4n3 48e4n- 2

2n + co---q-- q- 7 ( S 1 4 -'k ,5'24 -Jr- S24)

e8r~ 3 w8 ($24 + $24)S - 342)(23S24 + 23S224 + 23S324 + 84n2). (70)

In this case the normal form, as expected, is a function only of A = = $24 + $224 + $24 . This agrees with the conjecture that degeneracy must persist to high order if the normal form is to signal underlying integrability. We speculate that this behavior continues to higher and likely all orders of the normal form.

(iii) /3 = ± 1/2

20~4n3 12~4n - 2 7"/NF = 2n + w----- T - + ~ ( m + 5S~4 q- 5S224)

e 8 - 6n~--g(23m 4 + 230m2n 2 + 131n 4)

E 8 + 30~--g[2- (31S24 - 31S224 + 8S24)Trt 2

2 2 2n2)]. (71) -- 131($24 + $24)(S14 -I- $24 -~

Unlike the previous two examples, when fl = -4-1/2 for non-zero m the original Hamiltonian (1), while integrable, is non-separable. Curiously, the normal form at orders higher than 4th contains terms in S~4 which, however, vanish in the limit m = 0. This emphasizes that while normalization might provide a clue that a system is integrable in a certain limit, there is no guarantee that the normal form need contain an indicator of integrability. At any rate, the exact dynamical symmetry that appeared in 4th order has been broken at 10th order. The normal form does not depend only on the generators {$14, $24, $12} and thus the reduced phase space is not the sphere SU(2). Stated differently, it does not seem possible to eliminate all angular dependence from the normal form when expressed in action- angle variables. Incidentally, this result differs from the conclusion of Alhassid et


al. (1987) who indicated that fl = 1/2 is dynamical symmetry limit independent of re.

5. A Related Example

Having established that normalization can identify certain integrable, separable limits of Equation (2), it is natural to examine other cases where potential integrability exists. As an example we consider the following Hamiltonian which represents a modification of Equation (2),

1 2 1 f12Z2 ] m 2 = - - + - - ( 7 2 ) n + + [p2 + + (z3 + a zP2)] T 2p2

This Hamiltonian was constructed with ease of application in mind rather than because it corresponds to any particular physical situation. Odd powers of p were omitted because their presence complicates regularization and the construction of the normal form which are not the main points here. The normal form was constructed as before and to 6th order is the following,

~ N F = 2n + 1 6 9 ( n 2 + f12n2 q- nS22 - fl2322 q- 4S24 - fl2324)

£4n 2 q- 16--~(4S~4 - f12S224 - S24 q- 4fl2S24)

20c~c6&4n ( - 3 s 2 - 3s~4 + 4s324) (73) q- w6

20°~e6'S'34 n / . - ,-,2 q- ~ I,°D14al + 6S24al - S~4a I -- 3 m 2 + 3 a i m 2 + 3n 2 + a ln2) .

All terms in the 6th order part of the normal form depend explicitly on $34 and this immediately rules out the possibility that fl --- + l , or/3 = 4-1/2 correspond to integrable limits. The only remaining possibility is to require that the normal form depend only on $34 and S12 and this may be accomplished by choosing al --- 1/2 and fl -- 4-2. This result caused us to question whether the Hamiltonian (72) might be integrable in this limit; it turns out that it is not only integrable but also separable in the parabolic coordinates of Equation (4) when al = 1/2 and /3 ----- 4-2. Evidently, normalization has again identified a new non-trivial integrable limit of a perturbed Kepler Hamiltonian.

6. Conclusions

We have presented a detailed analysis of the application of a conjecture due to A. Deprit and co-workers to a perturbed Kepler problem that is of considerable theoretical and experimental interest in atomic and molecular physics. The Deprit conjecture was found to hold up in cases where the reduced phase space turned


out to be SU(2) or SO(4). However, while particular structures exhibited by the normal form corresponded in all cases to integrable limits, not all integrable limits give rise to these structures. Thus, normalization is a valuable weapon in the arsenal of methods to detect integrability, particularly of perturbed Keplerian systems. In common with other methods (e.g., the Painlev6 approach) it is capable of uncovering integrable limits but, like them, it seems, cannot be guaranteed to detect all possible integrable cases.

Acknowledgements

We would like to thank Professor Andr6 Deprit, Professor Sebastian Ferrer, Dr. Herbert Smew, Professor John Wood, Professor Raymond Flannery and Jefferey Humpherys for valuable discussions and comments on the manuscript, and P. Edgar Raines and J. P. Skelton for technical assistance. One of us (DF) would like to thank A. Deprit, S. Ferrer and the Grupo de Mec,Snica Espacial, Universidad de Zaragoza, Spain, for the hospitality extended to him during a visit to Zaragoza in June 1993, and the members of the Institute for Theoretical Atomic and Molecular Physics at the Harvard-Smithsonian Observatory for their hospitality. This work was supported, in part, by NATO, the American Chemical Society, the USDA, and by the National Science Foundation through a grant for the Institute for Theoretical Atomic and Molecular Physics at Harvard University and the Harvard-Smithsonian Astrophysical Observatory.

Appendix

This appendix describes the conversion of the normal form into action-angle variables and the limit m = 0 (Boiteux, 1973; Iwai, 1981, 1982a,b; Krantzman et al., 1992; Farrelly et al., 1992).

The original Hamiltonian (2) was converted into a perturbed 4D isotropic oscillator (51). Considering only the unperturbed part of Equation (51) and referring to the oscillators as 1, 2, 3 and 4 we make the transformation to Cartesian action-angle variables.

qi ---- V ~ / sin qSi; Pi = ~ sin c~i (A.1)

which yields,

4 7-/0 -- -- 2n = I1 + h + / 3 + /4 . (A.2)

oJ

In order to observe the collapse of the system into a 2D oscillator when m = 0 it is first convenient to perform the following sequence of transformations,


(Ia + h ) I1 - - 2 ' C t = C a + C b

12 -- (Ia -- Ib) 2 ' ¢2 = Ca -- ¢b

+ IV) 13 -- , C3 = C c + C d

2

1 4 - (Ic--Id)

2 ' C4 = Cc - Cd. (m.3)

In these variables (43) requires that la = Ic = n. Here n is the principal action or quantum number (Born, 1925). The final transformation to action-angle variables is,

(Cra + CA,) Ib = m + A z , Cb = 2

(Cm -- CA,) Id = m - A z , Cd -- 2 (A.4)

which converts the Hamiltonian 7-/0 into,

2 7-10 - fl - 2n (A.5)

where f l = ~ . In the limit that m = 0, Equation (A.4) shows that Ib = --Id =

A z and the oscillators 1 and 4 become identical to each other as do oscillators 2 and 3 in Equation (A.2). Relabelling them "u" and "v" and using Equation (A.2) yields,

7-lo = 2 n = 2Iu + 2I~. (A.6)

Interpreting L, and Iv as Cartesian action-angle variables then allows Equation (A.6) to be written as a 2D isotropic oscillator.

R e f e r e n c e s

Alhassid, Y., Hinds, E. A., and Meschede, D.: 1987, Phys. Rev. Let. 59, 945. Arnold, V. I.: 1985 Dynamical Systems III, Springer-Verlag, New York, NY. Baym, G.: 1969, Quantum Mechanics, Benjamin-Cummings, Menlo Park, CA. Baumann, G. and Nonnenmacher, T. F.: 1992, Phys. Rev. A 46, 2682. Bliimel, R., Kappler, C., Quint, W. and Walther, H.: 1989, Phys. Rev. A 40, 808. Boiteux, M.: 1973, Physica 65, 381. Born, M.: 1925, Mechanics of the Atom, republished by E Ungar, New York, NY, 1960. Translation

by J. W. Fisher. Coffey, S. L., Deprit, A., Miller, B. and Williams, C. A.: 1987, Annals N. Y. Academy of Sciences 497,

22. Coffey, S. L., Deprit, A., Deprit, E. and Healy, L. C. A.: 1990, Science 247, 833. Cushman, R.: 1984, 'Normal Form for Vectorfields with Periodic Flow', in S. Sternberg (ed.),

Differential Geometric Methods in Mathematical Physics, D. Reidel Publ. Co., Dordrecht.


Darboux, G.: 1901, 'Sur un probl6me de m6canique', Arch. Need. (ii) 6, 371. Deprit, A.: 1991, Celest. Mech. 51, 361. Deprit, A. and Elipe, A.: 1991, Celest. Mech. 51, 227. Deprit, A., Elipe, A. and Ferrer, S.: 1994, 'Linearization: Laplace vs. Stiefel', Celest. Mech. 58,

151-201. Deprit, A. and Ferrer, S.: 1991, Phys. Lett. A 148, 412. Deprit, A. and Miller, B. R.: 1988, Annals N. Y. Academy of Sciences 536, 101. Deprit, A. and Williams, C. A.: 1991, Celest. Mech. 51, 271. Edmonds, A. R. and Pullen, R. A.: 1979, 'Semiclassical Treatment of the Quadratic Zeeman Effect:

Classical Orbits', Imperical College preprint ICTP (79-80) (unpublished). Farrelly, D.: 1986, J. Chem. Phys, 85, 2119. Farrelly, D., Uzer, T., Raines, P. E., Skelton, J. P. and Milligan, J. A.: 1992, Phys. Rev. A 45, 4738. Farrelly, D. and Howard, J. E.: 1993, Phys. Rev. A 48, 851. Ferrer, S. and Miller, B. R.: 1992, Celest. Mech. 53, 3. Ford, J., Stoddard, S. D. and Turner, J. S.: 1973, Prog. Theor. Phys. 50, 1547. Ganesan, K. and Lakshmanan, M.: 1989, Phys. Rev. Lett. 62, 232. Ganesan, K. and Lakshmanan, M.: 1990, Phys. Rev. A 42, 3940. Ganesan, K. and Lakshamanan, M.: 1992, Phys. Rev. A 45, 1548. Ganesan, K. and Lakshamanan, M.: 1993, Phys. Rev. A 48, 964. Ghikas, D.: 1990, Phys. Lett. A 137, 183. Grozdanov, T. P. and Rackovic, H. J.: 1990, J. Phys. B 23, 3531. Gutzwiller, M. C.: 1990, Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York,

NY. Hietarinta, J.: 1987, Phys. Rep. 147, 87. Hietarinta, J.: 1988, Annals N. Y. Academy of Sciences 536, 33. Howard, J. E. and Farrelly, D.: 1993, Phys. Lett. A 178, 62. Iwai, T.: 1981, J. Math. Phys. 22, 1628. Iwai, T.: 1982a, J. Math. Phys. 23, 1088. Iwai, T.: 1982b, J. Math. Phys. 23, 1093. Kibler, M. and Negadi, T.: 1983, Lett. alNuovo Cimento 37, 225. Krantzman, K. D., Milligan, J. A. and Farrelly, D.: 1992, Phys. Rev. A 45, 3093. Kustaanheimo, P. and Stiefel, E.: 1965, J. rein. Angew. Math. 218, 204. Martens, C. C. and Ezra, G. S.: 1987, J. Chem. Phys. 87, 284. Miller, B. R.: 1991, Celest. Mech. 51, 361. Milligan, J. A. and Farrelly, D.: 1993, 'Atomic Analogs of Local and Normal Modes: The Hydrogen

Atom in a Generalized van der Waals Potential', Phys. Rev. A 47, 3137. Paul, W.: 1990, Rev. Mod. Phys. 62, 531. Raines, P. E. and Uzer, T.: 1992, Comput. Phys. Commun. 70, 569. Sahm, D. K., Weaver, R. V. and Uzer, T.: 1990, J. Opt. Soc. Am. B 7, 1865. Sahm, D. K. and Uzer, T.: 1989, Chem. Phys. Lett. 163, 5. Stiefel, E. and Scheifele, G.: 1971, Linear and Regular Celestial Mechanics, Springer-Verlag, New

York, NY. Tabor, M.: 1988, Annals N. Y. Academy of Sc&nces 536, 43. Toda, M.: 1970, Prog, Theor. Phys. Suppl. 45, 174. Uzer, T., Farrelly, D., Milligan, J. A., Raines, P. E. and Skelton, J. P.: 1991, Science 242, 41. van der Meer, J.-C. and Cushman, R.: 1986, J. Appl. Math. andPhys. 37, 402. van Moerbecke, P.: 1976, Invent. Math. 37, 45. Whittaker, E. T.: 1944, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Dover

Publications, New York, NY. Wolfram, S.: 1988, Mathematica. A System for Doing Mathematics by Computer, Addison-Wesley,

Redwood City, CA. Yoshida, H.: 1984, 'Integrability of Generalized Toda Lattice Systems and Singularities in the Corn-

pies t-Plane', in M. Jimbo and T. Miwa (eds.), Nonlinear Integrable Systems-Classical Theory and Quantum Theory. World Scientific, Singapore.

Normalization and the detection of integrability: The generalized Van Der Waals potential

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