Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

0.34 MB·

by  Jose F. Cariñena

Superintegrable systems on 3-dimensional curved spaces:
Eisenhart formalism and separability
Jose F. Carin˜ena†a), Francisco J. Herranz‡b), Manuel F. Ran˜ada†c)
† Departamento de F´ısica Te´orica and IUMA, Facultad de Ciencias
Universidad de Zaragoza, 50009 Zaragoza, Spain
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1 ‡ Departamento de F´ısica, Universidad de Burgos, 09001 Burgos, Spain
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January 23, 2017
n
a
J

0 Abstract
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The Eisenhartgeometricformalism,whichtransformsanEuclideannaturalHamiltonian

]
H =T +V into a geodesic Hamiltonian with one additional degree of freedom, is applied
h
T
p to the four families of quadratically superintegrable systems with multiple separability in
- the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic
h
at HthaemseilftoounriasnysstTemr s(rar=ema,obd,icfi,edd)winithatthhereaed-ddiitmioennsoiofnaaploctuernvteiadlspralceeadairnegsttoudierd=andr+thern.
m U H T U
Secondly,westudy the superintegrabilityofthe fourHamiltonians r = r/µr,where µr is
H H
[ a certain position-dependent mass, that enjoys the same separability as the original system

1 r. All the Hamiltonians here studied describe superintegrable syestems on non-Euclidean
H
v three-dimensional manifolds with a broken spherically symmetry.
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Keywords: Separability. Hamilton–Jacobi equation. Superintegrability. Eisen-
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5 hart lift. Position dependent mass. Geodesic Hamiltonians. Constants of motion.
0 Laplace–Runge–Lenz vector.
.
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AMS classification: 37J35 ; 70H06
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PACS numbers: 02.30.Ik ; 05.45.-a ; 45.20.Jj
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:
v
i
X
r
a
a)E-mail address: [email protected]
b)E-mail address: [email protected]
c)E-mail address: [email protected]
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1 Introduction
It is well known that the harmonic (isotropic) oscillator and the Kepler–Coulomb (KC) problem
are integrable systems admitting additional constants of motion (Demkov–Fradkin tensor [1, 2]
and Laplace–Runge–Lenz vector, respectively). Systems endowed with this property are called
superintegrable. It is also known that if a system is separable (Hamilton–Jacobi (HJ) separable
in the classical case or Schro¨dinger separable in the quantum case), then it is integrable with
integrals of motion of at most second-order in momenta. Thus, if a system admits multisepara-
bility (separability inseveral differentsystems of coordinates) thenitis endowed with‘quadratic
superintegrability’ (superintegrability with linear or quadratic integrals of motion).
Frisetal.studiedin[3]thetwo-dimensional(2D)Euclideansystemsadmittingseparabilityin
more than one coordinate system and they obtained four families of potentials V , r = a,b,c,d,
r
possessingthreefunctionallyindependentintegrals ofmotion (theyweremainlyinterested inthe
quantum 2D Schro¨dinger equation but their results also hold at the classical level). Then other
authors studied similar problems on higher-dimensional Euclidean spaces [4]–[6], on 2D spaces
with a pseudo-Euclidean metric (Drach potentials) [7]–[10], and on curved spaces [11]–[21] (see
[22] for a recent review on superintegrability that includes a long list of references).
The superintegrability property is related with different formalisms and it can be studied by
making use of different approaches, that is, proving that all bounded classical trajectories are
closed, HJ separability, action-angle variables formalism, exact solvability, degenerate quantum
energy levels, complex functions whose Poisson bracket with the Hamiltonian are proportional
to themselves, etc. In this paper, we relate superintegrability with a geometric formalism intro-
duced many years ago by Eisenhart [23].
The theory of general relativity states that the motion of a particle under the action of
gravitational forces is described by a geodesic in the 4D Riemannian spacetime. The Eisenhart
formalism (also known as Eisenhart lift) associates to a system governed by a natural Hamilto-
nian H = T +V (a kinetic term plus a potential) a new geodesic Hamiltonian (so without
T
any potential) with an additional degree of freedom (it is in fact an extended formalism). The
important point is that the solutions of the equations of motion for such a Hamiltonian H come
from geodesics of in an enlarged curved space. That is, it is a geometric formalism introduced
T
with the idea of relating classical nonrelativistic Lagrangian or Hamiltonian mechanics with rel-
ativistic gravitation [24]–[35]. Our idea is that this formalism can also be applied for the study
of superintegrable systems on non-Euclidean spaces.
One important point is that although the number of superintegrable systems can be con-
sidered as rather limited, they are not, however, isolated ones but, on the contrary, they fre-
quently appear grouped into families; for example, each of the above mentioned 2D potentials
V (r = a,b,c,d), has the structure of a 3D vector space. In this paper we prove that the
r
2D Euclidean potentials V are related, via the Eisenhart formalism, with some superintegrable
r
geodesic Hamiltonian systems on 3D curved spaces, generally of nonconstant curvature and
r
T
with a broken spherically symmetry. Furthermore, natural 3D Hamiltonians, = + , can
r r r
H T U
then be constructed by preserving the same superintegrability and separability properties.
On the other hand, in these last years the interest for the study of systems with a position-
dependentmass(PDM)hasbecomeamatterofgreatinterestandhasattractedalotofattention
of many authors [36]–[50]. It seems therefore natural to enlarge the study of superintegrability
and separability to include systems with a PDM by following the same constructive approach.
Consequently, asanewstepinthisprocedure,wealsoprovethat and admitdeformations,
r r
T H
say = /µ and = /µ , with a PDM µ (λ) depending of a real parameter λ, in such a
r r r r r r r
T T H H
e e
2
way that the latter are superintegrable for all the values of λ (in the domain of the parameter)
and that for λ = 0 they reduce to the previously studied superintegrable Hamiltonians.
We must mention that there exists a certain relationship between the approach presented
in this paper and some previous studies on curved oscillators and KC potentials related to
the so-called Bertrand spacetimes (spherically symmetric and static Lorentzian spacetimes),
firstly introduced by Perlick in [51] and further studied in [52, 53], where generalisations of
superintegrable Hamiltonians fulfilling Bertrand’s theorem [54] on conformally flat spaces have
beenachieved. Westressthatoneofthemaindifferences(inaddition totheuseoftheEisenhart
formalism) is that, in this paper the potentials are not necessary central, that is, our results
mainly concern systems defined on non-conformally flat spaces.
The structure of the paper is as follows. In the next section we establish the main char-
acteristics of the Eisenhart formalism (a rigorous geometrical description can be found in the
Appendix). In Section 3 we briefly review the classification of the four families of quadratic in
the momenta superintegrable Hamiltonians on the Euclidean plane H = T +V (r = a,b,c,d).
r r
InSection 4, theEisenhartapproach isappliedinordertoconstructsuperintegrable3Dgeodesic
Hamiltonians from the previous V . The addition of a potential to leading to superinte-
r r r r
T U T
grable/separableHamiltonians = + isaddressedinSection5. NextinSection 6,aPDM
r r r
H T U
µ is introduced in the 3D geodesic Hamiltonians , by preserving separability, so giving rise to
r r
T
new superintegrable geodesic Hamiltonians = /µ . In Section 7, a separable potential
r r r r
T T U
is added to providing new superintegrable Hamiltonians which constitute the main result
r r
T H
of this paper. We conclude in the last sectionewith some remarks and open problems. e
e e
2 Eisenhart formalism
LetusfirstrecallsomebasicpropertiesrelatingRiemanniangeometrywithLagrangiandynamics
for natural systems.
Suppose a nD manifold M endowed with a Riemannian metric g. If we denote by qi;i =
{
1,...,n , a set of coordinates on M and by g (q) the components of g, the expressions of g and
ij
}
ds2 are given by
g = g (q)dqi dqj, ds2 = g (q)dqidqj.
ij ij
⊗
Then the corresponding equation of the geodesics on M,
1 ∂g ∂g ∂g
q¨i +Γi q˙jq˙k = 0, Γi = gil lj + lk jk , i,j,k = 1,...,n,
jk jk 2 ∂qk ∂qj − ∂ql
(cid:18) (cid:19)
can be obtained as the Euler–Lagrange equations from a Lagrangian L with only a quadratic
kinetic term T and without any potential
g
1
L = T = g (q)vivj.
g ij
2
Conversely,theLagrangianformalismestablishesthatthetrajectories offreemotionofaparticle
in a configuration space Q are (i) the solutions of the equations determined by a pure kinetic
Lagrangian (quadratic kinetic term without potential) and that (ii) these trajectories are just
the geodesics on the space Q. Hence the Lagrangians describing the free motion are also known
as geodesic Lagrangians.
As it is well known, the relativistic theory of gravitation introduced by Einstein in 1915
establishes that the trajectory of a particle under external gravitational forces can be described
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as a geodesic on the 4D spacetime. This, in turn, means that the spatial paths of particles
in the 3D Euclidean space can alternatively be considered as geodesics in a higher-dimensional
non-Euclidean space by introducing a new metric. This was the idea introduced later on by
Eisenhart in 1928/29 in nonrelativistic Lagrangian and Hamiltonian dynamics [23]. Thus the
equations of motion of a particle under the action of a potential force in a nD configuration
space Q can be reformulated as the equations of geodesics in a (n+1)D new configuration space
Q with a new (pseudo-)Riemannian metric constructed by combining the original metric with
the potential defined on Q.
e Moreexplicitly, assumethatwearegivenanaturalLagrangian(quadratickinetictermminus
a potential)
1
L(q,v) = T(q,v) V(q), T(q,v) = g (q)vivj,
ij
− 2
where the coefficients g (q) are symmetric functions of the coordinates and V(q) is a potential.
ij
We can then consider the configuration space Q of the system as a Riemannian space with a
metric determined by the coefficients of the kinetic term
ds2 = g (q)dqidqj.
ij
Since the matrix [g (q)] is invertible the Legendre transformation, p = g (q)vj, leads to the
ij i ij
Hamiltonian function H given by
1
H(q,p) = T(q,p)+V(q), T = gij(q)p p ,
i j
2
where g gjk =δk.
ij i
The Eisenhart formalism (also known as Eisenhart lift) is an extended formalism. The main
idea is introducing a new degree of freedom with a new coordinate, say z, i.e. Q is replaced by
Q = R Q and its corresponding momentum p , in such a way that the new metric dσ2 and
z
×
the new Hamiltonian C∞(T∗Q) are given by
T ∈
e
dz2 1 1
dσ2 = g (q)dqidqej + , = gij(q)p p + V(q)p2, (2.1)
ij V(q) T 2 i j 2 z
so that is homogeneous of degree two in the momenta, and this defines a geodesic Hamil-
T
tonian. As the variable z is cyclic p is a constant of motion and fixing its value p = 1 the
z z
parameter of the integral curves coincides with the arc-length. In this way the motion of a
particle under external forces arising from a potential V is described as a geodesic motion in
an extended configuration space determined by . Although the origin of this formalism is
T
related with properties of relativistic mechanics, this procedure has been studied by making use
of different approaches (see [24]–[35] and references therein). A more detailed geometric study
of the Eisenhart lift is presented in the Appendix.
We must mention that Eisenhart also considered another more extended formalism that
introduces not just one but two additional degrees of freedom; that is, two new variables (Q is
replaced by Q = R2 Q) and two conjugated momenta [23], [32]. This more genera Eisenhart
×
lift is related with the study of time-dependent systems and with problems with external gauge
fields (Lagranegians with terms linear in the velocities). Nevertheless in what follows we study
time-independent natural Hamiltonians without gauge fields and, therefore (see Section 6 of
[23]), we will make use the Eisenhart formalism with only one extra degree of freedom.
4
3 Quadratic superintegrability in the Euclidean plane
Let us denote by V , r = a,b,c,d, the four 2D potentials with separability in two different
r
coordinate systems in the Euclidean plane [3]–[6]. Each resulting potential V is, in fact, a
r
superposition of three potentials
V = k V +k V +k V ,
r 1 1 2 2 3 3
where, hereafter, k ,k ,k are three arbitrary real constants. We remark that, from a mathe-
1 2 3
matical/physical viewpoint, the k -term will be the ‘principal’ potential so that each family will
1
be ‘shortly’ named according to it.
For our purposes we write these four families in terms of Cartesian coordinates (x,y) with
conjugate momenta (p ,p ), in such a manner that the Hamiltonian H reads
x y r
1
H =T +V = (p2 +p2)+V (x,y), r = a,b,c,d. (3.1)
r r 2 x y r
These four types of Hamiltonians determine quadratically superintegrable systems as they are
endowed with three functionally independent constants of motion which are quadratic in the
momenta. Notice that for n = 2, the superintegrability property is, in fact, maximal since
2n 1 =3 is the maximum number of independent integrals.
−
The two first potentials, V and V , represent nonlinear oscillators (harmonic oscillators
a b
with additional terms), meanwhile the two remaining potentials, V and V , correspond to the
c d
superposition of the KC problem with two other terms.
3.1 Family a: Isotropic oscillator
This corresponds to the potential
1 k k
V = k (x2+y2)+ 2 + 3 , (3.2)
a 2 1 x2 y2
which is separable in (i) Cartesian coordinates and (ii) polar ones. The k -potential is just
1
the isotropic oscillator with frequency ω whenever k = ω2 > 0, meanwhile the two remaining
1
potentials are Rosochatius or Winternitz terms (which provide centrifugal barriers when k >
2
0 and k > 0). We recall that the Hamiltonian H is just the 2D version of the so-called
3 a
Smorodinsky–Winternitzsystem [3]whichhasbeenwidely studied(see, e.g., [5,6,15,55,56,57]
and references therein).
Three functionally independent constants of motion are the two 1D energies, I and I ,
a1 a2
along with a third integral I related to the angular momentum; namely,
a3
1 1 k 1 1 k
I = p2 + k x2+ 2 , I = p2 + k y2+ 3 ,
a1 2 x 2 1 x2 a2 2 y 2 1 y2
2 2
y x
I = (xp yp )2+2k +2k .
a3 y x 2 3
− x y
(cid:18) (cid:19) (cid:18) (cid:19)
3.2 Family b: Anisotropic oscillator
The following potential
1 k
V = k (4x2 +y2)+ 2 +k x (3.3)
b 2 1 y2 3
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is separable in (i) Cartesian coordinates and (ii) parabolic ones. The k -potential is just the
1
anisotropic 2 : 1 oscillator provided that k = ω2 > 0 (so with frequencies ω = 2ω and
1 x
ω = ω),thek -potentialisaRosochatius–Winternitz term,andthe(trivial)k -potentialsimply
y 2 3
corresponds to a translation along the x-axis.
Three constants of motion are the two 1D energies, I , I , and a third integral I , related
b1 b2 b3
to one component of the 2D Laplace–Runge–Lenz vector, which are given by
1 1 1 k
I = p2 +2k x2+k x, I = p2 + k y2+ 2 ,
b1 2 x 1 3 b2 2 y 2 1 y2
2k x k y2
I = (xp yp )p k xy2+ 2 3 .
b3 y − x y − 1 y2 − 2
3.3 Family c: Kepler–Coulomb I
The potential given by
k k k x
1 2 3
V = + + (3.4)
c x2+y2 y2 y2 x2+y2
is separable in (i) polar coordinatespand (ii) parabolic onpes. In this case, the k1-term is the KC
potential and the k -term is a Rosochatius–Winternitz potential.
2
One constant of motion is the Hamiltonian itself, that is I = H , and two other integrals,
c1 c
I , and I , read
c2 c3
2k x2 2k x x2+y2
I = (xp yp )2+ 2 + 3 ,
c2 y − x y2 y2
p
k x 2k x k (2x2+y2)
1 2 3
I = (xp yp )p + + + .
c3 y − x y x2+y2 y2 y2 x2+y2
Hence I comes from the angular momentpum, while I is providped by a component of the
c2 c3
Laplace–Runge–Lenz vector.
3.4 Family d: Kepler–Coulomb II
Finally, the fourth potential is given by
1/2 1/2
k x2+y2+x x2+y2 x
1
V = +k +k − , (3.5)
d 2 3
x2+y2 (cid:2)p x2+y2 (cid:3) (cid:2)p x2+y2 (cid:3)
which is separable inp(i) parabolic coorpdinates (τ,σ) and (ii) pa second system of parabolic
coordinates (α,β) obtained from (τ,σ) by a rotation. Thus, we recall that the KC potential
(k -term) can be superposed with two other potentials which are different from the previous
1
ones (3.4) keeping superintegrability.
One constant of motion is again the Hamiltonian itself, I = H , meanwhile two other
d1 d
integrals, I , and I , turn out to be
d2 d3
k x k y x2+y2 x 1/2 k y x2+y2+x 1/2
1 2 3
I = (xp yp )p + − + ,
d2 y x y
− x2+y2 − (cid:2)p x2+y2 (cid:3) (cid:2)p x2+y2 (cid:3)
p k y k x px2+y2 x 1/2 k x px2+y2+x 1/2
1 2 3
I = (xp yp )p − + .
d3 y x x
− − x2+y2 − (cid:2)p x2+y2 (cid:3) (cid:2)p x2+y2 (cid:3)
p p p
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These are related to both components of the 2D Laplace–Runge–Lenz vector.
Obviously, when k = k = 0 both KC I and II families reduce to the common KC k -
2 3 1
potential. Nevertheless, throughout the paper we shall deal with the three generic k -terms so
i
describing two essential different families of superintegrable Hamiltonians.
4 Geodesic Hamiltonians endowed with multiple separability
T
on 3D curved spaces
Letusconsiderthe2DEuclideanHamiltonianH (3.1)withoneofthesuperintegrablepotentials
r
V given in the above section. By applying the Eisenhart lift (2.1) with g = δ , q = x,q = y,
r ij ij 1 2
we obtain a new 3D Riemannian metric and associated free Hamiltonian defined by
dz2 1
dσ2 = dx2+dy2+ , = p2 +p2 +V (x,y)p2 , r = a,b,c,d, (4.1)
r V (x,y) Tr 2 x y r z
r
(cid:0) (cid:1)
where (x,y,z) are Cartesian coordinates and (p ,p ,p ) their conjugate momenta.
x y z
In what follows we study the separability of the corresponding HJ equation for each of
the four types of geodesic Hamiltonians . We stress that the separability of is, in fact,
r r
T T
provided by the separability of H , that is, if H is separable in the coordinates (q ,q ), we
r r 1 2
assume that is separable in the coordinates (q ,q ,z). We advance that we shall obtain four
r 1 2
T
independentintegralsforeach inanexplicitform(threeofthembeingmutuallyininvolution).
r
T
Consequently, willdetermineasuperintegrablesystembutnotamaximalsuperintegrableone,
r
T
since an additional fifth constant of motion would be necessary to get the maximum number of
2n 1= 5integrals(correspondington = 3degreesoffreedom). Inthissense, canberegarded
r
− T
as either a minimally superintegrable Hamiltonian [4] or a quasi-maximally superintegrable
one [57].
At this point we mention that the idea of obtaining a new superintegrable (n+1)D Hamilto-
nian starting with a simpler and previously known superintegrable Hamiltonian with n degrees
of freedom is a matter that has beenanalyzed by some authors (see e.g. [58, 59, 60]) butmaking
use of other approaches different to the Eisenhart formalism presented in this paper.
4.1 Geodesic Hamiltonian from isotropic oscillator
a
T
We construct theHamiltonian (4.1) with thepotential V (3.2). Sincetheinitial 2D Hamilto-
a a
T
nianH isseparableinCartesian(x,y)andpolarvariables(r,φ),wenowanalysetheseparability
a
of the new 3D system in Cartesian (x,y,z) and cylindrical (r,φ,z) coordinates.
a
T
4.1.1 Cartesian separability
The HJ equation takes the form
2 2 2
∂W ∂W ∂W
+ +V (x,y) = 2E,
a
∂x ∂y ∂z
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
so that if we assume that W can be written as W = W (x) + W (y) + W (z), then we can
x y z
perform a separation of variables which leads to the following one-variable expressions
1 k 1 k
(W′)2 = α, (W′)2 α k x2+ 2 = β+E, (W′)2 α k y2+ 3 = β+E,
z − x − 2 1 x2 y − 2 1 y2 −
(cid:18) (cid:19) (cid:18) (cid:19)
7
where α and β denote two constants associated with separability. Each one of these expressions
determines a constant of motion; so the following functions
1 k 1 k
K = p , K = p2 + k x2+ 2 p2, K = p2 + k y2+ 3 p2, (4.2)
a1 z a2 x 2 1 x2 z a3 y 2 1 y2 z
(cid:18) (cid:19) (cid:18) (cid:19)
are three functionally independent constants of motion,
dK dK dK = 0,
a1 a2 a3
∧ ∧ 6
satisfying the following properties
1
K ,K = 0, K ,K = 0, K ,K = 0, = K +K .
a1 a2 a1 a3 a2 a3 a a2 a3
{ } { } { } T 2
(cid:0) (cid:1)
4.1.2 Cylindrical separability
We introduce the usual polar coordinates, x = rcosφ and y = rsinφ, finding that the free
Hamiltonian T reads
a
1 p2 1 k k
= p2+ φ +V p2 , V = k r2+ 2 + 3 .
Ta 2 r r2 a z! a 2 1 r2cos2φ r2sin2φ
The HJ equation turns out to be
2 2 2
∂W 1 ∂W ∂W
+ +V (r,φ) = 2E.
∂r r2 ∂φ a ∂z
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
If we suppose that W =W (r)+W (φ)+W (z), then we can perform a separation of variables;
r φ z
we first obtain (W′)2 = γ and next
z −
1 k k
r2(W′)2 2r2E k r4γ = (W′)2+γ 2 + 3 = δ,
r − − 2 1 − φ cos2φ sin2φ
(cid:18) (cid:19)
where γ and δ are two constants. Hence we obtain the following constants of motion
k k 1
J = p K , J = p2 + 2 + 3 p2, J = r2p2+ k r4p2 2r2 , (4.3)
a1 z ≡ a1 a2 φ cos2φ sin2φ z a3 r 2 1 z − Ta
(cid:18) (cid:19)
such that J ,J = 0 and J +J = 0.
a1 a2 a2 a3
{ }
We summarize the above results in the following statement.
Proposition 1. The 3D geodesic Hamiltonian
1 1 k k
= p2 +p2 +V p2 , V = k (x2+y2)+ 2 + 3 , (4.4)
Ta 2 x y a z a 2 1 x2 y2
(cid:0) (cid:1)
is HJ separable in Cartesian (x,y,z) and cylindrical (r,φ,z) coordinates. This determines a su-
perintegrable system endowed with four independent constants of motion given by K ,K ,K
a1 a2 a3
(4.2) and J (4.3). Furthermore, K ,K ,K are mutually in involution and = 1 K +
a2 a1 a2 a3 Ta 2 a2
K .
a3
(cid:0)
(cid:1)
8
4.2 Geodesic Hamiltonian from anisotropic oscillator
b
T
Let be the 3D free Hamiltonian (4.1) with V (3.3). Since V is separable in Cartesian and
b b b
T
parabolic (a,b) coordinates, we study the separability of in Cartesian (x,y,z) and parabolic-
b
T
cylindrical (a,b,z) coordinates.
4.2.1 Cartesian separability
The HJ equation yields
2 2 2
∂W ∂W ∂W
+ +V (x,y) = 2E,
b
∂x ∂y ∂z
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
sothatifweassumethatW can bewrittenasW = W (x)+W (y)+W (z)thenwecan perform
x y z
a separation of variables obtaining the one-variable expressions
1 k
(W′)2 = α, (W′)2 α 2k x2+k x = β +E, (W′)2 α k y2+ 2 = β+E,
z − x − 1 3 y − 2 1 y2 −
(cid:18) (cid:19)
(cid:0) (cid:1)
whereαandβ aretwoconstants. Eachoneoftheseexpressionsdeterminesaconstantofmotion,
namely,
1 k
K = p , K = p2 + 2k x2+k x p2, K = p2 + k y2+ 2 p2, (4.5)
b1 z b2 x 1 3 z b3 y 2 1 y2 z
(cid:18) (cid:19)
(cid:0) (cid:1)
which, moreover, are functionally independent
dK dK dK = 0,
b1 b2 b3
∧ ∧ 6
and they satisfy the following properties
1
K ,K = 0, K ,K = 0, K ,K = 0, = K +K .
b1 b2 b1 b3 b2 b3 b b2 b3
{ } { } { } T 2
(cid:0) (cid:1)
4.2.2 Parabolic-cylindrical separability
If we introduce the parabolic coordinates defined by
1
x = τ2 σ2 , y = τσ, (4.6)
2 −
(cid:0) (cid:1)
the Hamiltonian (4.1) and the potential V (3.3) become
b b
T
1 p2 +p2 1 k 1 1 k
= τ σ +V p2 , V = 1(τ6 +σ6)+k + + 3(τ4 σ4) ,
Tb 2 τ2+σ2 b z b τ2+σ2 2 2 τ2 σ2 2 −
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:21)
and the HJ equation adopts the form
2 2 2
1 ∂W ∂W ∂W
+ +V (τ,σ) = 2E,
τ2+σ2 ∂τ ∂σ b ∂z
" #
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
so that if we assume that W is of the form W = W (τ)+ W (σ) +W (z) we can perform a
τ σ z
separation of variables obtaining first (W′)2 = γ and then
z −
k k k k k k
(W′)2 2τ2E + (W′)2 2σ2E =γ 1 τ6+ 2 + 3 τ4 +γ 1 σ6+ 2 3 σ4 ,
τ − σ − 2 τ2 2 2 σ2 − 2
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
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providing three integrals
k k k
J = p K , J = p2 + 1 τ6+ 2 + 3 τ4 p2 2τ2 ,
b1 z ≡ b1 b2 τ 2 τ2 2 z − Tb
(cid:18) (cid:19)
k k k
J = p2 + 1 σ6+ 2 3 σ4 p2 2σ2 , (4.7)
b3 σ 2 σ2 − 2 z − Tb
(cid:18) (cid:19)
such that J ,J = 0 and J +J = 0.
b1 b2 b2 b3
{ }
The following proposition summarizes these results.
Proposition 2. The 3D geodesic Hamiltonian given by
1 1 k
= p2 +p2 +V p2 V = k (4x2 +y2)+ 2 +k x, (4.8)
Tb 2 x y b z b 2 1 y2 3
(cid:0) (cid:1)
is HJ separable in Cartesian (x,y,z) and parabolic-cylindrical (τ,σ,z) coordinates. It represents
a superintegrable system endowed with four independent constants of motion corresponding to
K ,K ,K (4.5) and J (4.7). Moreover, K ,K ,K are mutually in involution and =
b1 b2 b3 b2 b1 b2 b3 b
T
1 K +K .
2 b2 b3
(cid:0) (cid:1)
4.3 Geodesic Hamiltonian from Kepler–Coulomb I
c
T
Now we consider (4.1) with V (3.4). Recall that V is separable in polar and parabolic
c c c
T
coordinates, so that we analyse the separability of in cylindrical and parabolic-cylindrical
c
T
coordinates.
4.3.1 Cylindrical separability
In the variables (r,φ,z), the geodesic Hamiltonian is expressed as
c
T
1 p2 k k k cosφ
= p2+ φ +V p2 , V = 1 + 2 + 3 .
Tc 2 r r2 c z! c r r2sin2φ r2sin2φ
The HJ equation reads
2 2 2
∂W 1 ∂W ∂W
+ +V (r,φ) = 2E.
∂r r2 ∂φ c ∂z
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Hence if we assume that W is of the form W = W (r) + W (φ) + W (z) we can perform a
r φ z
separation of variables, finding first that (W′)2 = γ and then
z −
k k cosφ
r2(W′)2 2r2E k γr = (W′)2+γ 2 + 3 = δ,
r − − 1 − φ sin2φ sin2φ
(cid:18) (cid:19)
so that the following functions
k k cosφ
K = p , K = p2 + 2 + 3 p2,
c1 z c2 φ sin2φ sin2φ z
(cid:16) (cid:17)
K = r2p2+k rp2 2r2 , (4.9)
c3 r 1 z − Tc
are constants of motion such that K ,K = 0 and K +K = 0.
c1 c2 c2 c3
{ }
10