On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces
We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Eucl...
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| citation_txt | On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces. Claudia Maria Chanu and Giovanni Rastelli. SIGMA 16 (2020), 052, 16 pages |
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| description | We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay-Turbiner-Winternitz, and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We also show how the application of the coupling-constant-metamorphosis technique allows us to obtain new superintegrable Hamiltonians from the previous ones. Moreover, for the Minkowski case, we show the quantum superintegrability of the corresponding quantum Hamiltonian operator. Our results are obtained by applying the theory of extended Hamiltonian systems, which is strictly connected with the geometry of warped manifolds.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 052, 16 pages
On the Extended-Hamiltonian Structure of Certain
Superintegrable Systems on Constant-Curvature
Riemannian and Pseudo-Riemannian Surfaces
Claudia Maria CHANU and Giovanni RASTELLI
Dipartimento di Matematica, Università di Torino, Torino, Italia
E-mail: claudiamaria.chanu@unito.it, giovanni.rastelli@unito.it
Received March 21, 2020, in final form May 20, 2020; Published online June 11, 2020
https://doi.org/10.3842/SIGMA.2020.052
Abstract. We prove the integrability and superintegrability of a family of natural Hamil-
tonians which includes and generalises those studied in some literature, originally defined
on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the
well-known superintegrable system on the Euclidean plane proposed by Tremblay–Turbiner–
Winternitz and they are defined on Minkowski space, as well as on all other 2D manifolds of
constant curvature, Riemannian or pseudo-Riemannian. We show also how the application
of the coupling-constant-metamorphosis technique allows us to obtain new superintegrable
Hamiltonians from the previous ones. Moreover, for the Minkowski case, we show the quan-
tum superintegrability of the corresponding quantum Hamiltonian operator.Our results are
obtained by applying the theory of extended Hamiltonian systems, which is strictly con-
nected with the geometry of warped manifolds.
Key words: extended-Hamiltonian; superintegrable systems; constant curvature
2020 Mathematics Subject Classification: 37J35; 70H33
1 Introduction
In [16] it is proved that the Hamiltonian
H0 = p1p2 − αq2k+1
2 q−2k−31 , (1.1)
α ∈ R, is superintegrable for any k ∈ Q. The proof is obtained by introducing an irregular
bi-Hamiltonian structure, in analogy with the rational Calogero–Moser system.
A partial generalization of this Hamiltonian has been later proved to be again superintegrable
in [2]. This is the Hamiltonian
H = p1p2 − αq2k+1
2 q−2k−31 − β
2
qk2q
−k−2
1 , (1.2)
with k ∈ N, α, β ∈ R. In both the previous articles, the superintegrability of the Hamiltoni-
ans is proved via algebraic methods. In this article, we prove that H is indeed superintegrable
for any k ∈ Q and that its Hamilton–Jacobi equation is separable, implying the integrability
of the system, for any k ∈ R. Moreover, we determine a new class of Hamiltonians which
generalizes H and is again superintegrable. We obtain these results by showing that H can
be written in form of an extended Hamiltonian. We introduced extended Hamiltonian systems
in [4] and in [3, 5, 7, 8] we generalized the idea of extended systems. An extended Hamiltonian is
a Hamiltonian H on a cotangent bundle, with n+ 1 degrees of freedom, which is obtained from
a Hamiltonian L with n degrees of freedom, depending on a rational parameter k and admitting
a non-trivial characteristic and explicitly determined first integral K. When L is polynomial
mailto:claudiamaria.chanu@unito.it
mailto:giovanni.rastelli@unito.it
https://doi.org/10.3842/SIGMA.2020.052
2 C.M. Chanu and G. Rastelli
in the momenta, the same is for H and for K. Moreover, if L is a natural Hamiltonian on the
cotangent bundle of a Riemannian (or pseudo-Riemannian) manifold, i.e., L is a Hamiltonian
of mechanical type, then its extensions (if any) are natural Hamiltonians too. Even if in the
most general setting the Hamiltonians involved in the extension procedure are not necessarily
natural (see [11] for examples of extensions of non-natural Hamiltonians), in this article, how-
ever, we apply the general theory to natural Hamiltonians only, because we are dealing with
the natural Hamiltonian (1.2) and some generalizations of it. The term extension is motivated
by the fact that our procedure increases the degrees of freedom of the Hamiltonian system,
this is the only relation with other existing extension procedures, for example the Eisenhart
extension. Since the characteristic first integral is generated by the recursive application of
a particular operator, our extension procedure is essentially algebraic, but it admits a geomet-
ric characterization in the case of natural Hamiltonians. The class of extended Hamiltonians,
parametrized by k ∈ Q, includes many (but not all, see Remark 3.3) of the known superinte-
grable Hamiltonians admitting a polynomial in the momenta first integral of degree dependent
on any chosen rational number k. Examples which belong to the above class are, for instance,
the isotropic and non-isotropic harmonic oscillators, the three-particle Jacobi–Calogero system,
the Wolfes system, the Tremblay–Turbiner–Winternitz system, the Post–Winternitz system. In
particular, this last system is studied in [9] by applying the technique of coupling-constant-
metamorphosis. We apply here the same technique to obtain from H, and our generalizations
of it, new superintegrable systems. In Section 2 we recall the basic elements of the theory of
extended systems. In Section 3 we show that H is indeed an extended Hamiltonian. In Sec-
tion 4 we apply the theories of extended Hamiltonians and coupling-constant-metamorphosis
to obtain new superintegrable systems. Moreover we show that the same family of extended
Hamiltonians of H includes Hamiltonians on other constant-curvature two-dimensional man-
ifolds: Euclidean planes, spheres, pseudo-spheres, de Sitter and anti-de Sitter spaces. These
Hamiltonians can be considered as generalisations of H to constant-curvature manifolds. In
Section 5, the Laplace–Beltrami quantization of H and its characteristic first integral, together
with their generalisations on flat manifolds, is obtained via the shift-ladder approach proposed
by S. Kuru and J. Negro in [15] and adapted to extended Hamiltonians in [10].
2 Extensions of Hamiltonian systems
Let L
(
qi, pi
)
be any Hamiltonian function with N degrees of freedom, defined on the cotangent
bundle of an N -dimensional manifold with coordinates
(
qi
)
and conjugate momenta (pi).
We say that L admits extensions, if there exists (c, c0) ∈ R2 \ {(0, 0)} such that there exists
a non null solution G
(
qi, pi
)
of
X2
L(G) = −2(cL+ c0)G, (2.1)
where XL is the Hamiltonian vector field of L.
If L admits extensions, then, for any γ(u) solution of the ODE
γ′ + cγ2 + C = 0, (2.2)
depending on the arbitrary constant parameter C, we say that any Hamiltonian H
(
u, qi, pu, pi
)
with N + 1 degrees of freedom of the form
H =
1
2
p2u −
(m
n
)2
γ′L+
(m
n
)2
c0γ
2 +
Ω
γ2
, (m,n) ∈ N \ {0}, Ω ∈ R (2.3)
is an extension of L.
On the Extended-Hamiltonian Structure 3
Extensions of Hamiltonians where introduced in [3] and studied because, when L is a poly-
nomial in the momenta Hamiltonian and in particular when L is a natural Hamiltonian, they
admit a polynomial in the momenta first integral generated via a recursive algorithm; its degree
in (pu, pi) depends on the value of m,n ∈ N \ {0}. For a generic Hamiltonian L, given any
m,n ∈ N \ {0}, let us consider the operator
Um,n = pu +
m
n2
γXL. (2.4)
Proposition 2.1 ([7]). For Ω = 0, the Hamiltonian (2.3) is in involution with the function
Km,n = Umm,n(Gn) =
(
pu +
m
n2
γ(u)XL
)m
(Gn), (2.5)
where Gn is the n-th term of the recursion
G1 = G, Gn+1 = XL(G)Gn +
1
n
GXL(Gn), (2.6)
starting from any solution G of (2.1).
For Ω 6= 0, the recursive construction of a first integral is more complicated: we construct
the following function, depending on two strictly positive integers s, r
K̄2s,r =
(
U2
2s,r + 2Ωγ−2
)s
(Gr), (2.7)
where the operator U2
2s,r is defined according to (2.4) as
U2
2s,r =
(
pu +
2s
r2
γ(u)XL
)2
,
and Gr is, as in (2.5), the r-th term of the recursion (2.6), with G1 = G solution of (2.1). For
Ω = 0 the functions (2.7) reduce to (2.5) and can be computed also when the first of the indices
is odd.
Theorem 2.2 ([8]). For any Ω ∈ R, the Hamiltonian (2.3) satisfies, for m = 2s,{
H, K̄m,n
}
= 0,
for m = 2s+ 1,{
H, K̄2m,2n
}
= 0.
We call K and K̄, of the form (2.5) and (2.7) respectively, characteristic first integrals of the
corresponding extensions. It is proved in [3, 8] that the characteristic first integrals K or K̄ are
functionally independent from H, L, and from any first integral I
(
pi, q
i
)
of L. This means that
the extensions of (maximally) superintegrable Hamiltonians are (maximally) superintegrable
Hamiltonians with one additional degree of freedom (see also [6]). In particular, any extension
of a one-dimensional Hamiltonian is maximally superintegrable.
The explicit expression of the characteristic first integrals is given as follows [8]. For r ≤ m,
we have
U rm,n(Gn) = Pm,n,rGn +Dm,n,rXL(Gn),
with
Gn =
[(n−1)/2]∑
j=0
(
n
2j + 1
)
G2j+1(XLG)n−2j−1(−2)j(cL+ c0)
j ,
4 C.M. Chanu and G. Rastelli
Pm,n,r =
[r/2]∑
j=0
(
r
2j
)(m
n
γ
)2j
pr−2ju (−2)j(cL+ c0)
j ,
Dm,n,r =
1
n
[(r−1)/2]∑
j=0
(
r
2j + 1
)(m
n
γ
)2j+1
pr−2j−1u (−2)j(cL+ c0)
j , m > 1,
where [·] denotes the integer part and D1,n,1 = 1
n2γ.
The expansion of the first integral (2.7) is
K̄2m,n =
m∑
j=0
(
m
j
)(
2Ω
γ2
)j
U
2(m−j)
2m,n (Gn),
with U0
2m,n(Gn) = Gn.
Remark 2.3. If L is a natural Hamiltonian with a metric and a potential which are alge-
braic (resp. rational) functions of the
(
qi
)
and if L admits an extension associated with a func-
tion G
(
qi, pi
)
which is polynomial in the momenta and algebraic (resp. rational) in the
(
qi
)
, then
the extension of L with γ = (cu)−1 has first integrals which are polynomial in the momenta and
algebraic (resp. rational) in the
(
qi
)
. The first integrals K̄2m,n are polynomial w.r.t.G, L,XLG, γ
and pu. Indeed, by (2.1), it is easy to check that XL(Gn) is a polynomial in G, XLG, L, as well
as Gn.
Remark 2.4. In [5] it is proven that the ODE (2.2) defining γ is a necessary condition in order
to get a characteristic first integral of the form (2.5) or (2.7). According to the value of c and C,
the explicit form of γ(u) is given (up to constant translations of u) by
γ =
−Cu, c = 0,
1
Tκ(cu)
=
Cκ(cu)
Sκ(cu)
, c 6= 0,
where κ = C/c is the ratio of the constant parameters appearing in (2.2) and Tκ, Sκ and Cκ are
the trigonometric tagged functions [19] (see also [7] for a summary of their properties)
Sκ(x) =
sin
√
κx√
κ
, κ > 0,
x, κ = 0,
sinh
√
|κ|x√
|κ|
κ < 0
Cκ(x) =
cos
√
κx, κ > 0,
1, κ = 0,
cosh
√
|κ|x, κ < 0,
Tκ(x) =
Sκ(x)
Cκ(x)
.
Therefore, we have
γ′ =
−C, c = 0,
−c
S2
κ(cu)
, c 6= 0.
Remark 2.5. When c 6= 0 and κ 6= 0, a translation of the variable u allows not so evident
changes in γ and γ′. Indeed, by using standard formulas, we can write for example, for c 6= 0,
γ′(cu) =
4κc
(e
√
−κcu − e−
√
−κcu)2
, γ2(cu) = −κ(e
√
−κcu + e−
√
−κcu)2
(e
√
−κcu − e−
√
−κcu)2
so that, for c = ±1, κ = ±1 we can easily compute, in correspondence with the indicated
translations of u, the expressions of γ′
On the Extended-Hamiltonian Structure 5
Table of γ′ κ = 1 κ = −1
c = 1 − sin−2 u − sinh−2 u
c = −1 sin−2 u sinh−2 u
translation: u→ u+ π/2 u→ u+ iπ/2
c = 1 − cos−2 u cosh−2 u
c = −1 cos−2 u − cosh−2 u
and of γ2
Table of γ2 κ = 1 κ = −1
c = ±1 tan−2 u tanh−2 u
translation: u→ u+ π/2 u→ u+ iπ/2
c = ±1 tan2 u tanh2 u
where the complex translation of the real variable u leaves (2.3) real. The table shows the
remarkable fact that, for both positive and negative values of c, we can always choose positive
or negative values of γ′, by selecting suitable translations and values of κ, i.e., we can obtain
extensions with different signature. This is particularly important since, as pointed out in [5], if L
is a natural Hamiltonian with more than one degree of freedom, then the integrability conditions
of (2.1) require that c satisfies certain conditions (related with the sectional curvatures of the
metric of L), therefore, the sign of c is fixed by L, while κ remains free. Hence, without taking
in account translations in u, the signature of the metric of the extended manifold (i.e., the sign
of γ′ in (2.3)) will be constrained by L. Actually, that sign can be freely chosen.
In order to check if any (N+1)-dimensional Hamiltonian H is an extension of a N -dimensional
Hamiltonian L, we must check if
(i) There exist canonical coordinates (u, pu) such that H can be written as a warped Hamil-
tonian, that is
H =
1
2
p2u + f(u) +
(m
n
)2
α(u)L, m, n ∈ N \ {0}, (2.8)
and the Hamiltonian L is independent of (u, pu). As we see in the next section, these
canonical coordinates can be determined by following an invariant geometrical procedure,
when H is a natural Hamiltonian.
(ii) For some constants c and c0 not both vanishing, the equation (2.1) admits a non null
solution G.
(iii) The functions α and f in (2.8) can be written as in (2.3) for a γ satisfying (2.2).
We remark that if (2.1) has a solution for c 6= 0, then we may assume without loss of generality
c0 = 0, because L is determined up to additive constants.
Examples of the procedure of extension applied on Hamiltonians L which are not of mecha-
nical type, defined possibly on Poisson manifolds, are given in [11]. In the following, we assume
that all Hamiltonian functions are natural and defined on cotangent bundles of Riemannian or
pseudo-Riemannian manifolds.
3 Determination of the extended Hamiltonian structure
In order to show that the natural Hamiltonian (1.2) is an extended Hamiltonian, we have first
to check if it can be written, by a point-transformation of coordinates, in the form of a warped
6 C.M. Chanu and G. Rastelli
Hamiltonian (2.8). We will use the intrinsic characterization of such natural Hamiltonians
making a simple generalization of the result given in [5, Theorem 9]
Let us consider a natural Hamiltonian
H =
1
2
g̃abpapb + Ṽ (3.1)
on a (n + 1)-dimensional Riemannian manifold
(
Q̃, g̃
)
and let X be a conformal Killing vector
of g̃, that is a vector field satisfying
[X, g̃] = LX g̃ = φg̃,
where φ is a function on Q̃, called conformal factor, and [·, ·] denotes the Schouten–Nijenhuis
bracket. We define X[ as the 1-form obtained by lowering the indices of the vector field X with
the metric tensor g̃.
Theorem 3.1. If on Q̃ there exists a conformal Killing vector field X with conformal factor φ
such that
dX[ ∧X[ = 0, (3.2)
dφ ∧X[ = 0, (3.3)
d‖X‖ ∧X[ = 0, (3.4)
d
(
X
(
Ṽ
)
+ φṼ
)
∧X[ = 0, (3.5)
then, there exist on Q̃ coordinates
(
u, qi
)
such that ∂u coincides up to a rescaling with X and
the natural Hamiltonian (3.1) has the form (2.8). Moreover, if
R̃(X) = aX, a ∈ R, (3.6)
where R̃ is the Ricci tensor of the Riemannian manifold
(
Q̃, g̃
)
, then the function α(u) in (2.8)
satisfies the condition α = −γ′ with γ solution of (2.2).
Proof. Following [5], conditions (3.2), (3.3), (3.4) imply that the metric g̃ab is in the warped
form (2.8) in suitable canonical coordinates (u, qi) such that in these coordinates X is of the
form
X = F (u)∂u
and
g̃ij
(
u, qh
)
= F−2gij
(
qh
)
, φ = 2∂uF.
Condition (3.5) means that X
(
Ṽ
)
+ φṼ is constant on the leaves orthogonal to X; in other
words, X
(
Ṽ
)
+ φṼ is a function of u only. Being X = F (u)∂u, φ = 2F ′ and X
(
Ṽ
)
= F (u)∂uṼ ,
last condition reads as F (u)∂2uiṼ + 2F ′∂iṼ = 0 for all i, that is ∂2uiṼ − ∂u lnF 2∂iṼ = 0. Hence,
∂u
(
ln
(
F 2∂iṼ
))
= 0 and therefore F 2∂iṼ is independent of u, that is ∂iṼ
(
u, qh
)
= ∂iV
(
qh
)
F−2.
It follows that
Ṽ
(
u, qh
)
= V
(
qh
)
F−2 + f(u) = α(u)V
(
qh
)
+ f(u)
and thus it has the form of the potential in (2.8) where V
(
qh
)
is the potential of L. The second
part of the statement is proved in [5]. �
On the Extended-Hamiltonian Structure 7
Remark 3.2. Conditions (3.2), (3.3), (3.4) are the already known intrinsic characterization of
a warped metric, while condition (3.5) intrinsically assures that the scalar part of the Hamilto-
nian is compatible with the warped structure. It is remarkable the fact that an intrinsic condi-
tion – equation 3.6 – characterizes also the subset of the warped metrics which are involved in
extended Hamiltonians. The extended potentials with Ω = 0 are intrinsically characterised by
X
(
Ṽ
)
+ φṼ = 0.
We apply the above statement to the Hamiltonian (1.2) defined on Minkowski plane. First,
we remark that on M2 the Ricci tensor is zero, therefore, the condition (3.6) is verified for any
vector X with a = 0. A conformal Killing vector satisfying the remaining conditions is
X = C
(
q1∂1 + q2∂2
)
, φ = 2C,
where C is any non zero real constant.
The vector X is the gradient of the function 2Cq1q2. Performing the coordinate transforma-
tion
q1 =
1√
2
(
x1 + x2
)
, q2 =
1√
2
(
x1 − x2
)
,
we see that
(
xi
)
are the standard pseudo-Cartesian coordinates such that the geodesic term of
the Hamiltonian H becomes
1
2
(
p21 − p22
)
.
In these coordinates, we have
2q1q2 =
(
x1
)2 − (x2)2,
thus, 2q1q2 = C is the equation verified by points having fixed distance from the origin in the
pseudo-polar coordinates of M2. Hence, a coordinate system (u, χ) such that X is parallel to ∂u
is the pseudo-polar one, with u coinciding with the hyperbolic radius of the hyperbolic circles.
Indeed, by applying the transformations
x1 = u coshχ, x2 = u sinhχ,
u =
√(
x1
)2 − (x2)2, χ = tanh−1
x2
x1
,
q1 =
1√
2
ueχ, q2 =
1√
2
ue−χ,
where χ is the pseudo-angle, we get after simple computations that (1.2) becomes
H =
1
2
(
p2u −
1
u2
p2χ
)
− 1
u2
(
2α
(
e−2χ
)2(k+1)
+ β
(
e−2χ
)k+1
)
.
Finally, we apply a further coordinate transformation, by setting
ψ = (k + 1)χ,
and we get
H =
1
2
p2u −
(k + 1)2
u2
(
1
2
p2ψ + α̃e−4ψ + β̃e−2ψ
)
, (3.7)
8 C.M. Chanu and G. Rastelli
with α̃ = 2
(k+1)2
α, β̃ = β
(k+1)2
, which is an extended Hamiltonian on M2 with
(k + 1)2 = −m
2
n2c
of the one-dimensional Hamiltonian
L =
1
2
p2ψ + V, (3.8)
where
V = α̃e−4ψ + β̃e−2ψ,
and γ = 1
cu satisfying (2.2) for c0 = 0, C = 0, hence with κ = 0, and c satisfying the condition
c = − m2
n2(k + 1)2
,
i.e., c = −η2 with η ∈ Q.
As a second step, we have to find a non-trivial solution G(pψ, ψ) of the equation
X2
LG = −2cLG = 2η2LG. (3.9)
If we assume
G = g(ψ)pψ,
by expanding equation (3.9) we obtain(
g′′ − η2g
)
p3ψ +
[
β̃
(
6g′ −
(
4 + 2η2
)
g
)
e−2ψ + α̃
(
12g′ −
(
16 + 2η2
)
g
)
e−4ψ
]
pψ = 0.
The coefficients of p3ψ and pψ vanish identically if and only if the following conditions hold
g = a1e
−ηψ + a2e
ηψ,
g = a3
(
β̃e2ψ + 2α̃
) η2
12
− 1
3 e(
η2
6
+ 4
3
)ψ,
where ai are arbitrary real constants. A common solution g exists for
η = 2, a1 = 0, a2 = a3.
Indeed, we get the non-trivial solution of (3.9)
G = e2ψpψ, c = −4,
that corresponds to the extended Hamiltonian
H =
1
2
p2u −
m2
4n2u2
L, c = −4.
Hence, by solving
m
n
= 2(k + 1), (3.10)
(for any k ∈ Q, the equation admits positive integer solutions m, n) we can write the Hamilto-
nian (3.7) as an extended Hamiltonian.
As immediate corollaries of the fact that the Hamiltonian (1.2) is an extension of (3.8), we
get additional information about (1.2)
On the Extended-Hamiltonian Structure 9
(i) the function L is a quadratic first-integral of H, in accordance with the results of [2];
(ii) as a consequence of (i), the Hamilton–Jacobi equation associated with H is separable in
a Stäckel coordinates system for any k ∈ R. It is evident from (3.7) that this coordinate
system is (u, ψ) (or, equivalently, (u, χ)), the pseudo-polar coordinates of M2. The sepa-
rability of the Hamiltonian (1.2) was not noticed until now, while the separability of (1.1)
with β = 0, k ∈ R was found in [16] in the same pseudo-polar coordinates.
Remark 3.3. In [16] it is proved the superintegrability of other two Hamiltonians in M2,
H1 = 2p1p2 +
qd2√
q1
, d = p, or d =
1− 2p
2
, p ∈ N,
with the quadratic in the momenta first integral
I1 = 2p1(q2p2 − p1q1) +
qd+1
2√
q1
,
and
H2 = 2p1p2 + q1q
d
2 , d =
1− p
p
, or d =
1 + 2p
1− 2p
, p ∈ N,
with first integral
I2 = p21 +
qd+1
2
d+ 1
.
These Hamiltonians cannot be written as extensions by using point transformations of coordi-
nates. Indeed, due to the form of the extended Hamiltonians, in dimension two it is necessary
that the extended Hamiltonian is diagonalized in the coordinates adapted to the warped product
characterizing the extension. Necessarily, also the first integral L, quadratic in the momenta,
must be diagonalized in these coordinates, since the Hamiltonian is Stäckel separable. It is easy
to check that I1 and I2 are the unique non-trivial quadratic first integrals of the Hamiltonians H1
and H2 respectively, and it is impossible to find coordinates, real or complex, simultaneously di-
agonalizing H1 and I1, or H2 and I2. It is possible to see that the Killing tensors associated to I1
and I2 have an eigenvalue with algebraic multiplicity two, but with one-dimensional eigenspace
(in the case of indefinite metric there exist symmetric tensors which do not admit a basis of
eigenvectors). Therefore, the Killing tensors cannot determine separable coordinates [1].
4 Generalizations
Since the theory of extended Hamiltonians can be applied to Hamiltonian (1.2), which is the
extension of (3.8) as it is proved in Section 3, we can obtain from (1.2) some new families of
extended Hamiltonians, which are maximally superintegrable systems on M2 and other two-
dimensional Riemannian manifolds of constant curvature.
(i) The first evident generalization is that, from (3.10), we have that (1.2) is an extended
Hamiltonian for any
k =
m
2n
− 1,
and, therefore, for any rational number k ∈ Q instead of k ∈ N. In [16] the superintegrability
of (3.7) with β = 0 was proved for k ∈ Q.
10 C.M. Chanu and G. Rastelli
From Section 2, it is straightforward to find another generalization of the Hamiltonian (1.2)
just by adding to it the scalar
Ωu2 = 2Ωq1q2, Ω ∈ R,
so that we get a family of superintegrable potentials on M2 depending on three real parameters
(α, β,Ω), in analogy with the Tremblay–Turbiner–Winternitz potential on E2. Indeed, the gen-
eral theory of extended Hamiltonians resumed in Section 2 assures the existence of a constant
of motion of the form (2.7) for these new Hamiltonians.
(ii) We can try to find more general scalar potentials V (ψ) in L = 1
2p
2
ψ + V allowing the
existence of functions G solutions of (3.9), where c = −η2 in order to stay on a Lorentzian
manifold, without restrictions on η ∈ R.
We start by looking for G = G(ψ). By solving equation (3.9) for both G and V as in the
previous section, we find
G = C1e
ηψ − C2e
−ηψ, (4.1)
V = C3
(
C1e
ηψ + C2e
−ηψ)−2, (4.2)
where C1, C2, C3 are arbitrary parameters (with the obvious constraints C3 6= 0, C1 and C2
not both zero). These general expressions of G and V were obtained already in [3]. Instead, by
looking for G of the form G = g(ψ)pψ, we obtain
g = C1e
ηψ + C2e
−ηψ, (4.3)
V = (C1e
ηψ + C2e
−ηψ)−2
(
C3 + C4
(
C1e
ηψ − C2e
−ηψ)) =
(
C3 +
C4
η
g′
)
g−2, (4.4)
where C1, C2, C3, C4 are arbitrary parameters (with the obvious constraints C1, C2 not both
zero, as well as C3, C4).
It is immediate to see that (4.2) is a particular case of (4.4), obtained for C4 = 0, while the
functions G and g (given in (4.1) and (4.3), respectively) are equal up to the sign of C2. By
setting
η = 2, C1 = 1, C2 = 0, C3 = α̃, C4 = β̃,
in (4.3), (4.4), we exactly find the potential of the Hamiltonian (3.8), with the corresponding
function G solution of (3.9). In general, the change of coordinates
q1 =
u√
2
e
η
k
ψ, q2 =
u√
2
e−
η
k
ψ,
transforms the extension
H =
1
2
p2u − k2
1
η2u2
(
1
2
p2ψ + V
)
+ η4Ωu2, k ∈ Q,
into
H = p1p2 −
4k̃2
2η2q1q2
(
C1z
k̃ + C2z
−k̃)−2(C3 + C4
(
C1z
k̃ − C2z
−k̃))+ 2η4Ωq1q2,
with
z =
q1
q2
, k̃ =
k
2
.
On the Extended-Hamiltonian Structure 11
It is now evident that, since η is now an inessential multiplicative parameter of the potential,
the extensions with a general η ∈ R are Hamiltonians essentially equivalent to those obtained
for η = 2.
We can write the functions (4.3) and (4.4) in a simpler form, focusing on essential parameters
(with η fixed equal to 2). According to the sign of C1C2 we can write (4.3) as
g = a0e
2εψ, ε = ±1, V = αe−2εψ + βe−4εψ, C1C2 = 0,
where a0 6= 0 is C1 or C2, ε = 1 if C2 = 0 and ε = −1 if C1 = 0, α = C4εa
−1
0 , β = C3a
−2
0 ,
g = A cosh(2ψ + ψ0), V =
α+ β sinh(2ψ + ψ0)
cosh2(2ψ + ψ0)
, C1C2 > 0, (4.5)
g = A sinh(2ψ + ψ0), V =
α+ β cosh(2ψ + ψ0)
sinh2(2ψ + ψ0)
, C1C2 < 0, (4.6)
where C1 = Aeψ0 , C2 = |C1C2|
C1C2
Ae−ψ0 , ψ0 = 1
2 ln |C1/C2|, α = C3/A
2, β = C4/A.
The form (4.6) appeared already in [8] and it is mapped to (4.5) by an imaginary translation
of iπ/2. Moreover, the transformation
q1 =
u√
2
e
2
k
ψ, q2 =
u√
2
e−
2
k
ψ,
maps the extended Hamiltonians
1
2
p2u − k2
1
4u2
(
1
2
p2ψ + V
)
+ 16Ωu2, k ∈ Q,
into
H = p1p2 −
k̃2
2q1q2
(
C1z
k̃ + C2z
−k̃)−2(C3 + C4
(
C1z
k̃ − C2z
−k̃))+ 32Ωq1q2.
Remark 4.1. The potentials V described in (4.4) are always algebraic functions of q1, q2,
provided η ∈ Q (which in this case is absorbed by the rational k).
The generalized extended Hamiltonian obtained from (1.2) is therefore
H̃ =
1
2
p2u −
m2
η2n2u2
(
1
2
p2ψ + V
)
+ η4Ωu2, (4.7)
where V is (4.4) and m
n = η(k + 1), c = −η2.
Remark 4.2. The Hamiltonian (4.7) becomes (3.7) when C2 = 0, C1 = 1, C3 = α̃, C4 = β̃.
When C2 = 1, C1 = 0, the potentials correspond up to the exchange between q1 and q2. For C1
and C2 both 6= 0, we obtain new superintegrable potentials. Of the four parameters Ci only two
are essential, the remaining two being reducible to a single multiplicative factor.
One could search for other superintegrable potentials V by assuming different expressions for
G = G(ψ, pψ), but the resulting differential equations are much less easy to solve than for G
constant or linear in the momentum pψ.
(iii) From the Hamiltonian (4.7) we can obtain another family of superintegrable Hamiltonians
depending on a rational parameter k by applying to it the coupling-constant metamorphosis
(CCM) as described in [9, 12, 14, 18].
12 C.M. Chanu and G. Rastelli
The CCM transforms integrable or superintegrable systems in new integrable or superinte-
grable ones, by mapping first integrals in first integrals. It is characterized by the following
theorem
Let us consider a Hamiltonian H = Ĥ−ẼU in canonical coordinates
(
xi, pi
)
, where Ĥ
(
xi, pi
)
is independent of the arbitrary parameter Ẽ and U
(
xi
)
, with an integral of the motion K
(depending on Ẽ). If we define the CCM of H and K as H ′ = U−1
(
Ĥ − E
)
and K ′ = K|Ẽ=H̃
then K ′ is an integral of the motion for H ′.
In our case, the configuration manifold of the resulting system is again M2. Indeed, from (4.7),
by applying the CCM based on Ẽ = −η4Ω, and recalling that k + 1 = m
n , we get
H ′ =
1
2u2
p2u −
m2
η2n2u4
(
1
2
p2ψ + V
)
− E
u2
,
where V is given by (4.4). By performing the rescaling u2 = 2v, we obtain
H ′ =
1
2
p2v −
m2
4η2n2v2
(
1
2
p2ψ + V
)
− E
2v
, (4.8)
or, in coordinates
q1 =
v√
2
e
η
k
ψ, q2 =
v√
2
e−
η
k
ψ,
H ′ = p1p2 −
4k̃2
2η2q1q2
(
C1z
k̃ + C2z
−k̃)−2(C3 + C4
(
C1z
k̃ − C2z
−k̃))− E
2
√
2q1q2
,
with
z =
q1
q2
, k̃ =
m
4n
.
The configuration manifold of H ′, is again M2. The form of the characteristic first integral
of (4.8) can be found in [9].
(iv) The choice c = −η2, for η = 2, and C = κ = 0 in Section 3 has been made in order to
obtain the Hamiltonian (1.2) in M2 as an extension of (3.8). However, from Section 1 we know
that for different values of c and C we can obtain extensions of the same L of (3.8) which are
defined on manifolds different from M2, but which still belong to the same family of extensions
of L and, therefore, which can be seen as generalizations of the Hamiltonian (1.2) in these new
manifolds. A first possible generalization is the following: we consider the expressions (4.3), (4.4)
for G and V and we allow that η takes imaginary values. Such values of η imply the appearance of
trigonometric functions in G and V , which remain real provided C1 = C̄2 and C4 are imaginary.
In this way we see that the extension
H0 =
1
2
p2u +
m2
|η|2n2u2
L+ η4u2Ω (4.9)
is now defined in the Euclidean plane, since c = |η|2 > 0. It follows that H0 is nothing but the
Tremblay–Turbiner–Winternitz Hamiltonian discussed in [8].
Moreover, we can consider values of κ different from 0, according to the freedom of choice of
the parameter C in (2.2). Consequently, for η real or imaginary (i.e., for c < 0 or c > 0), we
have the possible extensions of L
H1 =
1
2
p2u +
(k + 1)2c
sin2 cu
L+ Ω tan2 cu, κ = 1, (4.10)
On the Extended-Hamiltonian Structure 13
H2 =
1
2
p2u +
(k + 1)2c
sinh2 cu
L+ Ω tanh2 cu, κ = −1, (4.11)
corresponding, for c > 0, to Hamiltonians on the sphere S2 and on the pseudosphere H2, respec-
tively, and, for c < 0, to Hamiltonians defined on the de Sitter space dS2 and on the anti-de Sitter
space AdS2, respectively. We recall that, in this case, the value of c is related to the value of
the curvature of the configuration manifold of H1, H2.
In this way, we obtained some kind of generalizations of the Hamiltonian (1.2) in manifolds
of constant positive and negative curvature, on Riemannian and pseudo-Riemannian manifolds,
since they are all obtained as extensions of the same one-dimensional Hamiltonian.
Finally, we consider the case c = 0, which cannot be deduced from (4.3), (4.4), since we
assumed there that c0 = 0 in (2.1). For the case c = 0, c0 6= 0, we have from [8] either
g = (a1ψ + a2), V =
c0
4a21
g2 + c1g + c2,
or
g = a2, V = c0ψ
2 + c1ψ + c2.
Therefore, we have that the function L (3.8) is not included in this case.
The Hamiltonians (4.10), (4.11), defined on conformally flat Riemannian manifolds, can be
again transformed into new superintegrable systems by applying to them the CCM.
The results of above are summarized in the following statement
Theorem 4.3. The Hamiltonians (4.7), (4.8), (4.9), (4.10), (4.11) are superintegrable with
a first integral polynomial in the momenta computed through the procedure described in Section 2.
Remark 4.4. The Hamiltonian
H̃ = p1p2 − αq2k+1
2 q−2k−31 − β
2
qk2q
−k−2
1 + 2Ωq1q2, k ∈ R,
as well as any Hamiltonian if the form
H = p1p2 +
1
q1q2
F
(
q1
q2
)
+G
(
q1q2
)
,
remain separable in pseudo-polar coordinates for any k ∈ R, and therefore it is Liouville inte-
grable.
Remark 4.5. All the Hamiltonians of Theorem 4.3 correspond essentially to the Tremblay–
Turbiner–Winternitz system, its generalizations on constant-curvature manifolds and its CCM
transformed Hamiltonians. This becomes evident by comparing the Hamiltonians of above with
those given in [8, 13, 17] and references therein.
5 Quantization
We consider the Hamiltonian (4.7) with V given by (4.4), thus including the Hamiltonian (1.2).
We indicate how to quantize it together with its characteristic first integral corresponding to
any rational parameter, proving the quantum superintegrability of this two-dimensional system
with a potential involving three arbitrary parameters. The quantization is made possible by
applying the method described in [10], where the Kuru–Negro quantization introduced in [15],
based on factorization in shift and ladder operators, is adapted to extended Hamiltonians on flat
14 C.M. Chanu and G. Rastelli
manifolds. Indeed, in order to apply the method for H defined on a two-dimensional manifold,
it is necessary to find functions of the form
F± = ±F ′(ψ)pψ + F (ψ)f +
1
f
c1, f =
√
2
(
η2L+ c0
)
,
solutions of
X2
LF
± = fF±. (5.1)
The functions F± are known as ladder functions of L and they are determined by solutions F (ψ)
of the system
F ′′ − η2F = 0,
V ′F ′ + 2η2V F + c1 = 0,
equivalent to (5.1) and corresponding to (20) and (21) of [10], for a suitable constant c1. We
have in our case
F = g′, c1 = −C4η
3,
where g is given in (4.3). The construction of the quantum Hamiltonian and of the characteristic
symmetry operator is then explicitely described in [10, Theorem 10]. To the function L we
associate the quantum operator
L̂ = −~2
2
∂2ψψ + V,
which can be extended to the quantum Hamiltonian
Ĥ = −~2
2
(
∂2uu +
1
u
∂u −
k2
η2u2
∂2ψψ
)
− k2
η2u2
V +
Ωη4
2
u2, k =
m
n
, m, n ∈ N \ {0},
which commutes with L̂. The operator H̃ coincides with the Laplace–Beltrami operator of the
metric
ds2 = du2 − η2u2
k2
dψ2,
which, for η2 = 1, is the metric of the Minkowski plane in pseudo-polar coordinates (u, kψ), plus
a scalar term. Then, the operator
X̂k
ε,E =
(
Ĝ+
ε
)2n ◦ (Â1,1
kε
)2m ◦ (D̂+
E
)m
,
is such that
Ĥ
(
X̂k
ε,EfE
)
= EX̂k
ε,E(fE),
for all multiplicatively separated eigenfunctions fE = φME (u)χλ(ψ) of Ĥ such that χλ is an
eigenfunction of L̂ with eigenvalue λ, and
M = k2λ, ε =
√
|λ|, η2λ ≤ 0,
where φME (u) is (not restrictively) any eigenfunction of the “radial” operator
ĤM = −~2
2
(
∂2uu +
1
u
∂u
)
− M
η2u2
,
On the Extended-Hamiltonian Structure 15
corresponding to the eigenvalue E, and where
Â1,1
kε = ∂u −
1
~
(
η2
√
Ωu+
kε
√
2
2ηu
)
,
D̂±E =
~√
2
u∂u +
~√
2
±
(
− E
2η2
√
Ω
+
√
Ω
2
η2u2
)
,
Ĝ±ε = η2g∂ψ ± εη
√
2
~
g′ +
2η2C4
~
(
~η ± 2
√
2ε
) .
The operator X̂k
ε,E is therefore a symmetry of Ĥ, called “warped symmetry” in [10], corre-
sponding to the characteristic first integral of the classical extended Hamiltonian H.
6 Conclusions
In this article we show that the theory of extended Hamiltonians allows not only to prove the
superintegrability of the Hamiltonian (1.2) for any non-zero rational k, but also gives imme-
diate generalizations of H, both on flat and curved manifolds, which are still superintegrable
Hamiltonians and include the superposition of a harmonic oscillator term.
Moreover, the application of the coupling-constant-metamorphosis produces further superin-
tegrable Hamiltonians, when adapted to the structure of extended Hamiltonians.
The possible generalizations are not exhausted by the examples given in the present paper.
For example, the methods employed in [6] allow to extend n-dimensional superintegrable Hamil-
tonians into (n+ 1)-dimensional ones, again superintegrable. The application of those methods
to the present case has not been considered yet.
The procedure of Laplace–Beltrami quantization of extended Hamiltonians and their associ-
ated first integrals of high-degree developed in [10] on flat manifolds is applied to the present case,
so the quantum superintegrability of (1.2) and its generalisations on flat manifolds is proved.
The problem of the quantum superintegrability of (1.2) was left open in [2]. The quantization of
extended Hamiltonians on curved manifolds, such as (4.10) and (4.11), is still an open problem.
The search for extended Hamiltonian structures is limited for the moment to point-coordinate
transformations. A possible future direction of research is the search for extended Hamiltonian
structures under more general canonical transformations, for example concerning the systems
described in Remark 3.3.
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1 Introduction
2 Extensions of Hamiltonian systems
3 Determination of the extended Hamiltonian structure
4 Generalizations
5 Quantization
6 Conclusions
References
|
| id | nasplib_isofts_kiev_ua-123456789-210698 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:31Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chanu, Claudia Maria Rastelli, Giovanni 2025-12-15T15:24:00Z 2020 On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces. Claudia Maria Chanu and Giovanni Rastelli. SIGMA 16 (2020), 052, 16 pages 1815-0659 2020 Mathematics Subject Classification: 37J35; 70H33 arXiv:2001.08613 https://nasplib.isofts.kiev.ua/handle/123456789/210698 https://doi.org/10.3842/SIGMA.2020.052 We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay-Turbiner-Winternitz, and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We also show how the application of the coupling-constant-metamorphosis technique allows us to obtain new superintegrable Hamiltonians from the previous ones. Moreover, for the Minkowski case, we show the quantum superintegrability of the corresponding quantum Hamiltonian operator. Our results are obtained by applying the theory of extended Hamiltonian systems, which is strictly connected with the geometry of warped manifolds. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces Article published earlier |
| spellingShingle | On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces Chanu, Claudia Maria Rastelli, Giovanni |
| title | On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces |
| title_full | On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces |
| title_fullStr | On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces |
| title_full_unstemmed | On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces |
| title_short | On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces |
| title_sort | on the extended-hamiltonian structure of certain superintegrable systems on constant-curvature riemannian and pseudo-riemannian surfaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210698 |
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