Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛)
We study the regularized (2𝑛 − 1)-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of 𝑈(𝑛, 𝑛). The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed, and their equivalence is shown. Some integrable generalization (perturbation) of...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2020 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210761 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛). Anatol Odzijewicz. SIGMA 16 (2020), 087, 23 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859524532772012032 |
|---|---|
| author | Odzijewicz, Anatol |
| author_facet | Odzijewicz, Anatol |
| citation_txt | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛). Anatol Odzijewicz. SIGMA 16 (2020), 087, 23 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the regularized (2𝑛 − 1)-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of 𝑈(𝑛, 𝑛). The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed, and their equivalence is shown. Some integrable generalization (perturbation) of the (2𝑛 − 1)-Kepler problem is proposed.
|
| first_indexed | 2026-03-13T05:46:27Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 087, 23 pages
Perturbed (2n − 1)-Dimensional Kepler Problem
and the Nilpotent Adjoint Orbits of U(n, n)
Anatol ODZIJEWICZ
Department of Mathematics, University of Bia lystok,
Cio lkowskiego 1M, 15-245 Bia lystok, Poland
E-mail: aodzijew@uwb.edu.pl
Received March 11, 2020, in final form September 01, 2020; Published online September 22, 2020
https://doi.org/10.3842/SIGMA.2020.087
Abstract. We study the regularized (2n−1)-Kepler problem and other Hamiltonian systems
which are related to the nilpotent coadjoint orbits of U(n, n). The Kustaanheimo–Stiefel
and Cayley regularization procedures are discussed and their equivalence is shown. Some
integrable generalization (perturbation) of (2n− 1)-Kepler problem is proposed.
Key words: integrable Hamiltonian systems; Kepler problem; nonlinear differential equa-
tions; symplectic geometry; Poisson geometry; Kustaanheimo–Stiefel transformation; celes-
tial mechanics
2020 Mathematics Subject Classification: 53D17; 53D20; 53D22; 70H06
1 Introduction
Kepler problem (not only for historical reasons) is one of the most fundamental subjects of celes-
tial mechanics and quantum mechanics [9, 12, 17, 18, 25, 34]. Such questions as Moser [26] and
Kustaanheimo–Stiefel [15] regularization procedures as well as the relationship between them [14]
are well known for celestial mechanics specialists. Also the questions concerning the quantiza-
tion of the Kepler system and the MIC-Kepler system, which is its natural generalization, are
the subject of many publications, see, e.g., [8, 17, 19, 24, 25, 28, 31]. There are other interesting
generalizations of Kepler and MIC-Kepler problems, for example see [2, 11, 19, 20, 21, 22].
Initially the group U(2, 2), being a natural extension of the Poincaré group, was recognized as
the dynamical group [1, 9, 10, 11, 12, 14, 17] for the three-dimensional Kepler and MIC-Kepler
problems. Consequently, the group U(n, n) plays the same role for higher dimensional case.
Taking this fact into account, in the present paper we study various Hamiltonian systems that
have U(n, n) as a dynamical group. They are related to the adjoint nilpotent orbits of U(n, n)
and could be interpreted as some natural generalizations of Kepler problem.
In Section 2 we investigate the canonically defined vector bundles over the Grassmannian
Gr
(
n,C2n
)
of n-dimensional subspaces of twistor space T =
(
C2n, φ
)
, where φ is hermitian form
on C2n with signature (+ · · ·+︸ ︷︷ ︸
n
− · · ·−︸ ︷︷ ︸
n
).
In Section 3 we consider the Grassmannian Gr0
(
n,C2n
)
of isotropic (with respect to φ) n-
dimensional subspaces of C2n, which as a manifold is diffeomorphic with the unitary group U(n),
and show that T ∗U(n) has the structure of U(n, n)-Hamiltonian space, see Propositions 3.2
and 3.3. In Proposition 3.3, we classify the orbits of U(n, n)-action on T ∗U(n) and specify
the one-to-one correspondence of these orbits with such nilpotent adjoint U(n, n)-orbits whose
elements X satisfy X2 = 0.
In Section 4 we investigate the geometry of the orbit N1,0, which consists of the rank one
nilpotent elements of u(n, n), see Proposition 4.1. We also discuss the equivalent realizations of
the regularized (2n− 1)-dimensional Kepler problem, see Proposition 4.3.
mailto:aodzijew@uwb.edu.pl
https://doi.org/10.3842/SIGMA.2020.087
2 A. Odzijewicz
In Section 5 we show the equivalence of Cayley and Kustaanheimo–Stiefel regularizations in
the context of higher-dimensional Kepler problem, obtaining in this way a natural generalization
of the Kustaanheimo–Stiefel transforms for the arbitrary odd dimension.
Finally, in the last Section 6 we consider some integrable generalization of (2n − 1)-Kepler
problem. For this generalized Kepler problem the Hamiltonian, see formula (6.1), depends on the
positions and momenta through the coordinates of angular momenta and Runge–Lenz vector.
The integrability of this system is proved by the methods developed in [29].
2 Grassmannian Gr
(
n,C2n
)
and related vector bundles
In this section we will study some canonically defined bundles over the Grassmannian Gr
(
n,C2n
)
of n-dimensional complex vector subspaces of C2n. Let us recall that Gr
(
n,C2n
)
is a n2-
dimensional compact complex analytic manifold homogenous with respect to the natural action
of GL(2n,C).
We begin with defining the following complex analytic bundles over Gr
(
n,C2n
)
. Namely, we
consider the bundle πN : N → Gr
(
n,C2n
)
whose fibres consist of nilpotent elements of gl(2n,C).
The total space of this bundle is defined as
N :=
{
(Z, z) ∈ gl(2n,C)×Gr
(
n,C2n
)
: Im(Z) ⊂ z ⊂ Ker(Z)
}
and πN is the projection of N on the second component of the cartesian product. One easily sees
that πN : N → Gr
(
n,C2n
)
is a complex vector bundle of rank n2. The subset pr1(N ) ⊂ gl(2n,C)
consists of such elements Z ∈ gl(2n,C), which satisfy Z2 = 0 and have rank k := dimC Im(Z),
where 1 ≤ k ≤ n.
Next the bundle πP : P → Gr
(
n,C2n
)
is a bundle of idempotents, i.e.,
P :=
{
(p, z) ∈ gl(2n,C)×Gr
(
n,C2n
)
: p2 = p, Im(p) = z
}
,
where πP is the projection of P on the second component of cartesian product. We note that
pr1(P) consists of such idempotents in gl(2n,C) that dimC(Im(p)) = n.
In order to make the structure of πP : P → Gr
(
n,C2n
)
more transparent, we formulate the
following proposition.
Proposition 2.1. The bundle πP : P→Gr
(
n,C2n
)
is an affine bundle with πN : N→Gr
(
n,C2n
)
as the structural vector bundle, i.e., for any z ∈ Gr
(
n,C2n
)
the vector space Nz := π−1
N (z) acts
in a transitive and free way on the fibre Pz := π−1
P (z).
Proof. For p ∈ Pz and Z ∈ Nz we have
(p+ Z)2 = p2 + Z2 + pZ + Zp = p+ Z and (p+ Z)z = pz = z.
This shows that p+ Z ∈ Pz.
For p, p′ ∈ Pz we have
(p′ − p)2 = p′2 + p2 − p′p− pp′ = p′ + p− p− p′ = 0
and dimC Im(p′ − p) ≤ n. Thus, p′ − p =: Z ∈ Nz. Due to the above facts one has free and
transitive action of Nz on Pz. �
We note that for p′, p ∈ Pz the following equalities hold
p′p = p and pp′ = p′.
Perturbed (2n− 1)-Dimensional Kepler Problem 3
Subsequently, using the Cartan–Killing form
gl(2n,C)× gl(2n,C) 3 (Z1,Z2)→ Tr(Z1Z2) ∈ C (2.1)
we will identify the dual space gl(2n,C)∗ with the Lie algebra gl(2n,C).
For any p ∈ pr1(P) one has the open subset
Ωp :=
{
z ∈ Gr
(
n,C2n
)
: z ∩ (1− p)C2n = {0}
}
of the Grassmannian. We define a chart φp : Ωp → (1 − p)gl(2n,C)p ∼= Matn×n(C) in the
following way. The decomposition z ⊕ (1 − p)C2n = C2n defines the projection qz of C2n on
subspace z ⊂ C2n. For projections 1 − qz and 1 − p one has Im(1 − qz) = Im(1 − p). So,
according to Proposition 2.1 there exists Z ∈ (1− p)gl(2n,C)p such that
Z = (1− p)− (1− qz) = qz − p := φp(z). (2.2)
The equality (2.2) defines the chart Ωp 3 z 7→ φp(z) = Z, mentioned above.
In order to find the transition maps φp′ ◦ φ−1
p : φp(Ωp′ ∩ Ωp) → φp′(Ωp′ ∩ Ωp) between the
charts (Ωp, φp) and (Ωp′ , φp′) we observe that for z ∈ Ωp ∩ Ωp′ we have
q′zqz = qz and qzq
′
z = q′z. (2.3)
From (2.2) and (2.3) we obtain
qz = q′zqz = (Z ′ + p′)qz = p′qz + Z ′p′qz, (2.4)
qz = (p+ Z)qz = (p+ Z)pqz = p′(p+ Z)pqz + (1− p′)(p+ Z)pqz
= p′(p+ (1− p)Z)pqz + (1− p′)(p+ (1− p)Z)pqz. (2.5)
Expressions (2.4) and (2.5) give two decompositions of qz on the components from subspaces
p′gl(2n,C)qz and (1− p′)gl(2n,C)qz, which satisfy p′gl(2n,C)qz ∩ (1− p′)gl(2n,C)qz = {0}. So,
we have
p′qz = (a+ cZ)pqz, Z ′p′qz = (b+ dZ)pqz,
where
a := p′p, b := (1− p′)p, c := p′(1− p), d := (1− p′)(1− p).
Observing that p′qz : pC2n → p′C2n and pqz : pC2n → pC2n are isomorphisms of the vector
subspaces we obtain
Z ′ = (b+ dZ)(a+ cZ)−1.
Note here that a + cZ = (p′qz)(pqz)
−1. In consequence, the inverse (a + cZ)−1 is well defined.
In particular case when Im(1− p) = Im(1− p′) one has Ωp = Ωp′ and, thus
Z ′ = p− p′ + Z.
Let us note that one has another canonical complex vector bundles
E :=
{
(w, z) ∈ C2n ×Gr
(
n,C2n
)
: w ∈ z
}
and
E⊥ :=
{
(ϕ, z) ∈
(
C2n
)∗ ×Gr
(
n,C2n
)
: ϕ|z = 0
}
4 A. Odzijewicz
over Gr
(
n,C2n
)
. The complex linear group GL(2n,C) acts on the above bundles in the following
way
Σg(Z, z) :=
(
gZg−1, σg(z)
)
, (2.6)
Tg(w, z) := (gw, σg(z)), (2.7)
T ∗g (ϕ, z) :=
(
ϕ ◦ g−1, σg(z)
)
, (2.8)
where
σg(z) := gz, (2.9)
for g ∈ GL(2n,C).
The proposition formulated below collects some properties of the above structures which will
be useful in the further considerations.
Proposition 2.2.
(i) One has the canonical isomorphisms
N ∼= E ⊗ E⊥ ∼= T ∗Gr
(
n,C2n
)
(2.10)
of the GL(2n,C)-vector bundles.
(ii) The group GL(2n,C) acts on N , E ⊗ E⊥ and T ∗Gr
(
n,C2n
)
by Σg, Tg ⊗ T ∗g and T ∗σg,
respectively, preserving their vector bundle structures, and isomorphisms from (2.10) are
equivariant with respect to these actions.
(iii) The vector bundle N (and thus the vector bundles E ⊗ E⊥ and T ∗Gr
(
n,C2n
)
) splits into
GL(2n,C)-orbits:
N k := {(Z, z) ∈ N : dimC ImZ = k},
where k = 0, 1, . . . , n.
Proof. (i) For proving the isomorphism N ∼= T ∗Gr
(
n,C2n
)
we note that
Gr
(
n,C2n
) ∼= GL(2n,C)/GL(2n,C)z,
where the subgroup GL(2n,C)z is the stabilizer of z ∈ Gr
(
n,C2n
)
. Thus, one has
TzGr
(
n,C2n
) ∼= gl(2n,C)/gl(2n,C)z.
Using the isomorphism gl(2n,C)∗ ∼= gl(2n,C) defined by the pairing (2.1), we find that
T ∗z Gr
(
n,C2n
) ∼= (gl(2n,C)/gl(2n,C)z)
∗ ∼= (gl(2n,C)z)
⊥,
where (gl(2n,C)z)
⊥ is the annihilator of the subspace gl(2n,C)z ⊂ gl(2n,C). Since Gr
(
n,C2n
)
is a GL(2n,C)-homogenous space, it is enough to prove the above isomorphism for any z0 ∈
Gr
(
n,C2n
)
. So, let us take z0 = {( η0 ) : η ∈ Cn} for which
gl(2n,C)z0 =
{(
A B
0 D
)
∈ Mat2n×2n(C) : A,B,D ∈ Matn×n(C)
}
.
One easily sees that for such z0 we have (gl(2n,C)z0)⊥ ∼= Nz0 .
In order to prove that Ez ⊗ E⊥z ∼= Nz we observe that Z ∈ Ez ⊗ E⊥z ⊂ gl(2n,C) satisfies
ImZ ⊂ z ⊂ KerZ and dimC Ez ⊗ E⊥z = dimCNz.
(ii) The GL(2n,C)-equivariance of the above bundles isomorphisms one easily see from
(2.6)–(2.9).
(iii) By straightforward verification. �
Perturbed (2n− 1)-Dimensional Kepler Problem 5
From Proposition 2.2 we conclude:
Remark 2.3.
(i) The orbit N 0 is the zero section of N → Gr
(
n,C2n
)
so, one can identify it with Gr
(
n,C2n
)
.
(ii) The orbit N n is an open-dense subset of N .
We mention here that N k is the total space of the following GL(2n,C)-homogeneous bundles:
N k
Gr
(
k,C2n
)
,Gr
(
n,C2n
)
Gr
(
n− k,C2n
) ?
�
���
���
H
HHH
HHj
πker π πim
where the bundle projections are defined by
πim(Z, z) = Im(Z), π(Z, z) = z, πker(Z, z) = Ker(Z).
Remark 2.4. In the Penrose twistor theory, see, e.g., [30], which concerns the case n = 2, the
submanifolds π
(
π−1
im (z)
)
and π
(
π−1
ker(z)
)
are called the α-planes and β-planes, respectively.
3 T ∗U(n) as a Hamiltonian U(n, n)-space
Now we will describe some real versions of the structures described in the previous section and
their relation to the structure of the cotangent bundle T ∗U(n) as a U(n, n)-Hamiltonian space.
For this reason we fix a scalar product
〈v, w〉 := v+φw (3.1)
of v, w ∈ C2n, defined by a hermitian matrix φ = φ+ ∈ Mat2n×2n(C) which has signature
(+ · · ·+︸ ︷︷ ︸
n
− · · ·−︸ ︷︷ ︸
n
) and satisfies φ2 = 12n. By “+” in (3.1) and everywhere below we will denote
the hermitian transposition of a matrix. Hence we define the group U(n, n) and Lie algebra
u(n, n) of U(n, n) by
g+φg = φ
and by
X+φ+ φX = 0,
respectively, where by definition g ∈ U(n, n) and X ∈ u(n, n). Since for n = 2 the vector
space C2n provided with scalar product (3.1) is known as twistor space [30], in the subsequent
we will use the same terminology for an arbitrary dimension.
Using scalar product (3.1) we also define on gl(2n,C), Gr
(
n,C2n
)
and N , respectively, the
following involutions
I(Z) := −φZ+φ, (3.2)
⊥(z) := z⊥, (3.3)
Ĩ(Z, z) :=
(
I(Z), z⊥
)
, (3.4)
6 A. Odzijewicz
where z⊥ ⊂ C2n is the orthogonal complement of z ∈ Gr(n,C) with respect to (3.1) and
Z ∈ gl(2n,C). Let us note that (3.2) is an anti-linear map of gl(2n,C) and (3.4) is a fibre-wise
anti-linear map of the bundle πN : N → Gr
(
n,C2n
)
. Hence, taking into account the equivalent
equalities
Im(I(Z)) = (Ker(Z))⊥ and Ker(I(Z)) = (Im(Z))⊥ (3.5)
we obtain the anti-holomorphic bundle isomorphism
N N
Gr
(
n,C2n
)
Gr
(
n,C2n
)
,
? ?
-
-
Ĩ
⊥
which restricts to the isomorphism
N k N k
Gr
(
n,C2n
)
Gr
(
n,C2n
)? ?
-
-
Ĩ
⊥
of the bundle N k → Gr
(
n,C2n
)
. Note here that the bundle N k → Gr
(
n,C2n
)
is not a vector
bundle. All the above isomorphisms are equivariant with respect to the actions of U(n, n) ⊂
GL(2n,C) defined in (2.6) and (2.9).
By πN0 : N0 → Gr0
(
n,C2n
)
we denote the vector bundle over the Grassmannian Gr0
(
n,C2n
)
of complex n-dimensional vector subspaces, which are isotropic with respect to the scalar prod-
uct (3.1), i.e., z ∈ Gr0
(
n,C2n
)
if and only if z = z⊥. The total space N0 of this bundle is defined
as the subset N0 ⊂ N of fixed points of the involution Ĩ : N → N defined in (3.4). Let us note
here that dimR Gr0
(
n,C2n
)
= n2.
We define the map of the vector bundle N0 into the Lie algebra u(n, n) by
pr1(X, z) := X.
The set of values of this map is determined in the following way.
Proposition 3.1. An element X ∈ u(n, n) belongs to pr1(N0) if and only if X2 = 0.
Proof. If X ∈ u(n, n) satisfies X2 = 0 then because of I(X) = X and (3.5) we find that
Im(X) ⊂ Ker(X) = (Im(X))⊥.
From the above and nonsingularity of the scalar product (3.1) we obtain
k := dimC Im(X) ≤ dimC(Im(X))⊥ = 2n− k.
So, 0 ≤ k ≤ n and thus, there exists z ∈ Gr0
(
n,C2n
)
such that
Im(X) ⊂ z ⊂ Ker(X), (3.6)
i.e., X ∈ pr1(N0). Note that Im(X) as an isotropic subspace of
(
C2n, 〈·, ·〉
)
could be extended to
maximal isotropic subspace z, which has dimension n and is contained in (Im(X))⊥.
By definition of N0 any element X ∈ pr1(N0) satisfies (3.6), so one has X2 = 0. �
Perturbed (2n− 1)-Dimensional Kepler Problem 7
Next, taking the decomposition C2n = Cn ⊕ Cn, we will choose the hermitian matrix from
the definition (3.1) in the following diagonal block form
φd =
(
E 0
0 −E
)
, (3.7)
where E and 0 are unit and zero n× n-matrices. Hence, we obtain
〈v, v〉 = η+η − ξ+ξ, (3.8)
for v =
( η
ξ
)
∈ Cn ⊕ Cn.
Now, let us take a set
{
v1 =
( η1
ξ1
)
, . . . , vn =
( ηn
ξn
)}
⊂ C2n of linearly independent vectors
which span z ∈ Gr0
(
n,C2n
)
. Since 〈vk, vl〉 = 0 it follows that
η+
k ηl = ξ+
k ξl (3.9)
for k, l = 1, . . . , n. From (3.9) we see that there exists such Z ∈ U(n) that ηk = Zξk for
k = 1, . . . , n. So, vectors ξ1, . . . , ξn form a basis in Cn.
The above considerations show that there is a natural diffeomorphism U(n) ∼= Gr0
(
n,C2n
)
between the unitary group U(n) and the Grassmannian Gr0
(
n,C2n
)
of n-dimensional isotropic
subspaces of
(
C2n, 〈·, ·〉
)
, which is defined in the following way
I0 : U(n) 3 Z 7→ z :=
{(
Zξ
ξ
)
: ξ ∈ Cn
}
∈ Gr0
(
n,C2n
)
. (3.10)
One easily sees that for φd the block matrix elements A,B,C,D ∈ Matn×n(C) of g =
(
A B
C D
)
∈
U(n, n) satisfy
A+A = E + C+C, D+D = E +B+B and D+C = B+A. (3.11)
From (3.10) one finds that U(n, n) acts on U(n) as follows
Z ′ = σg(Z) = (AZ +B)(CZ +D)−1. (3.12)
Subsequently we will need the explicit description of the stabilizer U(n, n)E := {g ∈ U(n, n) :
σg(E) = E} of the group unit E ∈ U(n). Simple considerations shows that g =
(
A B
C D
)
∈
U(n, n)E if and only if
A = 1
2
((
F+
)−1
+ F
)
+H, D = 1
2
((
F+
)−1
+ F
)
−H,
B = 1
2
(
F −
(
F+
)−1)−H, C = 1
2
(
F −
(
F+
)−1)
+H, (3.13)
where F ∈ GL(n,C) and H ∈ Matn×n(C) satisfy the condition
HF+ + FH+ = 0.
Let us take a smooth curve ]−ε, ε[ 3 t 7→ Z(t) ∈ U(n) through the element Z = Z(0). By
Ż := d
dtZ(t)|t=0 ∈ TZU(n) we denote the vector tangent to Z(t) at Z and by τ := Z−1Ż ∈
TEU(n) ∼= u(n) ∼= iH(n), where by H(n), we denote the real vector space of (n× n)-hermitian
matrices. Using the above notation, from (3.12) and (3.11) we obtain
τ ′ = Z ′−1Ż ′ =
(
(AZ +B)(CZ +D)−1
)+(
AŻ(CZ +D)−1
− (AZ +B)(CZ +D)−1CŻ(CZ +D)−1
)
=
(
(AZ +B)(CZ +D)−1
)+
AŻ(CZ +D)−1 − CŻ(CZ +D)−1
8 A. Odzijewicz
=
(
(CZ +D)−1
)+
(Z+
(
A+A− C+C
)
Ż +
(
B+A−D+C
)
Ż)(CZ +D)−1
=
(
(CZ +D)−1
)+
Z+Ż(CZ +D)−1 =
(
(CZ +D)−1
)+
τ(CZ +D)−1. (3.14)
Since Ż = Zτ , we have the isomorphism of vector bundles TU(n) ∼= U(n) × iH(n). It follows
from (3.14) that the covector ρ ∈ T ∗EU(n) ∼= iH(n) transforms in the following way
ρ′ = (CZ +D)ρ(CZ +D)+,
where one identifies the Lie algebra (iH(n), [·, ·]) of U(n) with its dual (iH(n))∗ by the Cartan–
Killing form.
The elements of Lie algebra u(n, n) in the diagonal realization (3.8) of φ are given by matrices
X =
(
α β
β+ δ
)
,
where β ∈ Matn×n(C) and α, δ ∈ iH(n).
Proposition 3.2.
(i) The map I0 : T ∗U(n) ∼= U(n)× iH(n)→ N0 defined by
I0(Z, ρ) :=
((
−ZρZ+ Zρ
(Zρ)+ ρ
)
,
{(
Zξ
ξ
)
: ξ ∈ Cn
})
∈ N0 (3.15)
is a U(n, n)-equivariant
I0 ◦ Λg = Σg ◦ I0
isomorphism of the vector bundles. The action Σg : N0 → N0, g ∈ U(n, n), is a restriction
to U(n, n) and N0 ⊂ N of the action defined in (2.6). The action Λg : U(n) × iH(n) →
U(n)× iH(n) is defined by
Λg(Z, δ) =
(
(AZ +B)(CZ +D)−1, (CZ +D)δ(CZ +D)+
)
, (3.16)
where g =
(
A B
C D
)
.
(ii) The canonical one-form γ0 on T ∗U(n) ∼= U(n)× iH(n) written in the coordinates (Z, δ) ∈
U(n)× iH(n) assumes the form
γ0 = i Tr(ρZ+dZ) (3.17)
and it is invariant with respect to the action (3.16).
(iii) The map J0 : T ∗U(n)→ u(n, n) defined by
J0(Z, ρ) := (pr1 ◦I0)(Z, ρ) =
(
−ZρZ+ Zρ
(Zρ)+ ρ
)
(3.18)
is the momentum map for symplectic form dγ0, i.e., it is a U(n, n)-equivariant
Adg ◦J0 = J0 ◦ Λg (3.19)
Poisson map of symplectic manifold (T ∗U(n),dγ0) into Lie–Poisson space (u(n, n) ∼=
u(n, n)∗, {·, ·}L-P), where
{f, g}L-P(α, δ, β, β+) = Tr
(
α
([
∂f
∂α
,
∂g
∂β
]
+
∂f
∂β
∂g
∂β+
− ∂g
∂β
∂f
∂β+
)
Perturbed (2n− 1)-Dimensional Kepler Problem 9
+ β
(
∂f
∂β+
∂g
∂α
+
∂f
∂δ
∂g
∂β+
− ∂g
∂β+
∂f
∂α
− ∂g
∂δ
∂f
∂β+
)
+ β+
(
∂f
∂α
∂g
∂β
+
∂f
∂β
∂g
∂δ
− ∂g
∂α
∂f
∂β
− ∂g
∂β
∂f
∂δ
)
+ δ
([
∂f
∂δ
,
∂g
∂δ
]
+
∂f
∂β+
∂g
∂β
− ∂g
∂β+
∂f
∂β
))
(3.20)
for f, g ∈ C∞(u(n, n),R).
Proof. (i) From the definition of N0 it follows that (X, z) ∈ N0 if and only if it satisfies (3.6).
Thus, using U(n) ∼= Gr0
(
n,C2n
)
and (3.15) we find that β = Zδ and α = −ZδZ+ for X ∈
pr1(N0). The above shows that I0 : U(n)× iH(n)→ N0 is an isomorphism of vector bundles.
One proves the equivariance property (3.16) by straightforward verification.
(ii) One obtains (3.17) directly from the definition of canonical form γ0 on T ∗U(n) and from
the isomorphism T ∗U(n) ∼= U(n)× iH(n).
(iii) The equivariance property (3.19) and formula (3.20) for Lie–Poisson bracket follow by
straightforward verification. �
Now, we will describe the relation between the Ad(U(n, n))-orbits in pr1(N0) and Λ(U(n, n))-
orbits in T ∗U(n). We present the most important facts in the following proposition.
Proposition 3.3.
(i) Any Λ(U(n, n))-orbit Ok,l in T ∗U(n) = U(n) × iH(n) is uniquely generated from the
element (E, ρk,l) ∈ U(n)× iH(n), where
ρk,l := i diag(1, . . . , 1︸ ︷︷ ︸
k
−1, . . . ,−1︸ ︷︷ ︸
l
0, . . . , 0︸ ︷︷ ︸
n−k−l
) (3.21)
and has structure of a trivial bundle Ok,l → U(n) over U(n), i.e., Ok,l ∼= U(n) × ∆k,l,
where ∆k,l := {Fρk,lF+ : F ∈ GL(n,C)}.
(ii) The momentum map (3.18) gives one-to-one correspondence Ok,l ↔ J0(Ok,l) = Nk,l ⊂
pr1(N0) =
{
X ∈ u(n, n) : X2 = 0
}
between Λ(U(n, n))-orbits in T ∗U(n) and Ad(U(n, n))-
orbits in pr1(N0), where Nk,l = {Adg I0(E, ρk,l) : g ∈ U(n, n)}.
Proof. (i) Since the action of U(n, n) on U(n) is transitive, one can identify any Λ(U(n, n))-
orbit O in T ∗U(n) ∼= U(n)× iH(n) with U(n)×∆, where ∆ is an orbit of U(n, n)E in T ∗EU(n) ∼=
iH(n). The action of g ∈ U(n, n)E , which is defined in (3.13), on (E, ρ) ∈ {E} × iH(n) is given
by
Λg(E, ρ) =
(
E,FρF+
)
, (3.22)
where F ∈ GL(n,C). From (3.22) and Sylvester signature theorem, see [16], follows that ∆ has
form ∆kl := {Fρk,lF+ : F ∈ GL(n,C)}, where ρk,l is defined in (3.21).
(ii) From Proposition 3.1 and point (i) of Proposition 3.2 it follows that any Ad(U(n, n))-orbit
in pr1(N0) has form J0(Ok,l). Since for g ∈ U(n, n)E we have
Adg(J0(E, ρ)) = J0(Λg(E, ρ)) = J0
(
E,FρF+
)
,
the momentum map J0 : T ∗U(n) → pr1(N0) maps Ok,l on the one Ad(U(n, n))-orbit Nk,l ⊂
pr1(N0) only. �
10 A. Odzijewicz
As it follows from general theory, the Ad(U(n, n))-orbit Nk,l is a homogenous symplectic
manifold with the symplectic form ωkl, obtained in a canonical way by Kirillov construction,
see [13]. From point (ii) of Proposition 3.3 we have J−1
0 (Nk,l) = Ok,l. Hence, one can obtain
(Nk,l, ωkl) reducing standard symplectic form dγ0 on T ∗U(n) to the orbit Ok,l. Let us note
here that fibres J−1
0 (X), X ∈ Nk,l, are degeneracy submanifolds for the 2-form dγ0|Ok,l
, so,
Nk,l = Ok,l/∼ and ωkl = dγ0|Ok,l
/∼, where “∼” is an equivalence relation on Ok,l defined by the
submersion J0 : Ok,l → Nk,l.
Ending this section, we mention that in the case when k+ l = n one has Nk,l ∼= Ok,l and the
orbits Ok,l are open subsets of the cotangent bundle T ∗U(n). For symplectic forms ωkl we have
ωkl = dγ0.
For k = l = 0 the orbit O00
∼= U(n) is the zero section of T ∗U(n) and J0(O00) = N00 = {0}.
4 Regularized (2n − 1)-dimensional Kepler problem
In this section we will describe in detail the various Hamiltonian systems having U(n, n) as
their dynamical group. As we will show in the next section, these systems give the equivalent
description of the regularized (2n− 1)-dimensional Kepler system.
Let us begin by defining U(n, n)-invariant differential one-form
γ+− := i
(
η+dη − ξ+dξ
)
(4.1)
on C2n = Cn ⊕ Cn. The Poisson bracket {f, g}+− and momentum map J+− : C2n → u(n, n)
corresponding to the symplectic form dγ+− are given by
{f, g}+− := i
(
∂f
∂η+
∂g
∂η
− ∂g
∂η+
∂f
∂η
−
(
∂f
∂ξ+
∂g
∂ξ
− ∂g
∂ξ+
∂f
∂ξ
))
and by
J+−(η, ξ) := i
(
−ηη+ ηξ+
−ξη+ ξξ+
)
, (4.2)
respectively, where η, ξ ∈ Cn and f, g ∈ C∞
(
Cn ⊕ Cn
)
. One has the following identity
J+−(η, ξ)2 =
(
η+η − ξ+ξ
)
· J+−(η, ξ)
for this momentum map. Hence, the momentum map J+− maps the space of null-twistors
T 0
+− := I−1
+−(0), where
I+− := η+η − ξ+ξ,
onto the nilpotent coadjoint orbit N1,0 = J0(O1,0) corresponding to k = 1 and l = 0 in sense of
the classification presented in Proposition 3.3. The Hamiltonian flow σt+− : Cn⊕Cn → Cn⊕Cn,
t ∈ R, defined by I+− is given by
σt+−
(
η
ξ
)
:= eit
(
η
ξ
)
. (4.3)
In order to describe fibre bundle structures of N1,0
∼= T 0
+−/U(1) we define the diffeomorphism
Φ: T 0
+−
∼→ S2n−1 × Ċn by
Φ(η, ξ) :=
(
(η+η)−
1
2 η, (ξ+ξ)
1
2 ξ
)
= (η′, ξ′),
Perturbed (2n− 1)-Dimensional Kepler Problem 11
where Ċn := Cn\{0}. Note that U(1) acts on T 0
+− as in (4.3). The inverse diffeomorphism
Φ−1 : S2n−1 × Ċn → T 0
+− is given by
Φ−1(η′, ξ′) =
(
(ξ′+ξ′)
1
4 η′, (ξ′+ξ′)−
1
4 ξ′
)
.
These diffeomorphisms commute with the actions of Hamiltonian flow (4.3) on T 0
+− and on
S2n−1 × Ċn which are defined by (η, ξ) 7→ (λη, λξ) and by (η′, ξ′) 7→ (λη′, λξ′), respectively,
where λ = eit, t ∈ R.
Proposition 4.1.
(i) Nilpotent orbit N1,0 is the total space of the fibre bundle
S2n−1 N1,0
Ċn/U(1)
?
-
over Ċn/U(1) with S2n−1 as a typical fibre. So, this bundle is a bundle of (2n − 1)-
dimensional spheres associated to U(1)-principal bundle Ċn → Ċn/U(1).
(ii) One can also consider N1,0 as the total space of the fibre bundle
Ċn N1,0
CP(n− 1)
?
-
over complex projective space CP(n − 1) which is the base of Hopf U(1)-principal bundle
S2n−1 → S2n−1/U(1) ∼= CP(n− 1).
The total space of the tangent bundle TCP(n− 1)→ CP(n− 1) has the form
TCP(n− 1) ∼=
{
(η′, ξ′) ∈ S2n−1 × Cn : η′+ξ′ = 0
}
/U(1).
So, TCP(n−1)→ CP(n−1) is vector subbundle of the vector bundle S2n−1×Cn
U(1) → S2n−1/U(1) ∼=
CP(n− 1) and its complementary subbundle
E :=
{
(η′, ξ′) ∈ S2n−1 × Cn : ξ′ = sη′, s ∈ C
}
→ CP(n− 1)
is isomorphic to the trivial bundle CP(n− 1)× C.
Summing the above facts we conclude from the point (ii) of Proposition 4.1 that one can
identify N1,0
∼= S2n−1×Cn
U(1) → CP(n− 1) with the vector bundle S2n−1×Cn
U(1) → CP(n− 1) with null
section removed.
To explain the role of U(n, n) as the dynamical group for (2n − 1)-dimensional regularized
Kepler problem we discuss now other description of N1,0 corresponding to the choice of anti-
diagonal
φa := i
(
0 −E
E 0
)
,
realization of twistor form (3.1). For diagonal realization φd see (3.7). Subsequently we will
denote the realizations
(
C2n, φd
)
and
(
C2n, φa
)
of twistor space by T and T̃ , respectively. The
12 A. Odzijewicz
same convention will be assumed for their groups of symmetry, i.e., g =
(
A B
C D
)
∈ U(n, n) iff
g+φdg = φd and g̃ =
(
à B̃
C̃ D̃
)
∈ Ũ(n, n) iff g̃+φag̃ = φa. Hence, for g̃ ∈ Ũ(n, n) and X̃ = ũ(n, n)
one has
Ã+C̃ = C̃+Ã, D̃+B̃ = B̃+D̃, Ã+D̃ = E + C̃+B̃
and
X̃ =
(
α̃ β̃
γ̃ −α̃+
)
,
respectively, where β̃+ = β̃ and γ̃+ = γ̃.
The canonical one-form (4.1) and the momentum map (4.2) for T̃ are given by
γ̃+− = υ+dζ − ζ+dυ
and by
J̃+−(υ, ζ) =
(
υζ+ −υυ+
ζζ+ −ζυ+
)
,
where ( υζ ) ∈ T̃ . The null twistors space is defined as T̃ 0
+− := Ĩ−1
+−(0), where
Ĩ+−(υ, ζ) := i
(
ζ+υ − υ+ζ
)
.
The Hamiltonian flow on C2n generated by Ĩ+− is given by
σ̃t+−
(
υ
ζ
)
= eit
(
υ
ζ
)
∈ T̃ . (4.4)
Both realizations T and T̃ of the twistor space are related by the following unitary transfor-
mation of C2n:(
υ
ζ
)
= C+
(
η
ξ
)
and
(
η
ξ
)
= C
(
υ
ζ
)
,
where
C :=
1√
2
(
E −iE
−iE E
)
, (4.5)
which gives an isomorphism between the U(n, n)-Hamiltonian spaces (T , dγ+−) and
(
T̃ ,dγ̃+−
)
.
Now let us consider H(n)×H(n) with dγ̃0, where
γ̃0 := Tr(Y dX)
and (Y,X) ∈ H(n) × H(n), as a symplectic manifold. We define the symplectic action of
g̃ =
(
à B̃
C̃ D̃
)
on H(n)×H(n) by
σ̃g̃(Y,X) :=
((
ÃY + B̃
)(
C̃Y + D̃
)−1
,
(
C̃Y + D̃
)
X
(
C̃Y + D̃
)+)
. (4.6)
Let us note here that the above action is not defined globally, i.e., the formula (4.6) is valid only
if det
(
C̃Y + D̃
)
6= 0.
Perturbed (2n− 1)-Dimensional Kepler Problem 13
The momentum map J̃0 : H(n)×H(n)→ ũ(n, n) corresponding to dγ̃0 and σ̃g̃ has the form
J̃0(Y,X) =
(
Y X −Y XY
X −XY
)
and it satisfies the equivariance property
J̃ ◦ σ̃g̃ = Adg̃ ◦J̃.
The following diagram
T ∗U(n) u(n, n) T
T̃ũ(n, n)H(n)×H(n)
6 6 6
�
�-
-J0 J+−
J̃+−J̃0
T ∗C AdC C
∪
(4.7)
depicts relationship between the Poisson manifolds defined above. The maps represented by
vertical arrows in (4.7) are defined by (4.5) and by
AdC
(
X̃
)
:= CX̃C+,
T ∗C (Y,X) :=
(
(Y − iE)(−iY + E)−1,
i
2
(−iY + E)X(−iY + E)+
)
, (4.8)
where X̃ ∈ ũ(n, n) and (Y,X) ∈ H(n)×H(n).
Proposition 4.2. All arrows in the diagram (4.7) are the U(n, n)-equivariant Poisson maps.
Proof. By straightforward verification. �
The first component in (4.8), i.e.,
Z = (Y − iE)(−iY + E)−1
is a smooth one-to-one map of H(n) into U(n), which is known as Cayley transform, see, e.g., [6].
Hence, the unitary group U(n) could be considered as a compactification of H(n), Namely, in
order to obtain the full group U(n) one adds to Cayles image of H(n) such unitary matrices Z,
which satisfy the condition det(iZ +E) = 0. One sees this by observing that the inverse Cayley
map is defined by
Y = (Z + iE)(iZ + E)−1,
if det(iZ + E) 6= 0.
Let us define
˙̃O1,0 := H(n)× Cn,1,
where
Cn,1 := {X ∈ H(n) : dim(Im(X)) = 1 and X ≥ 0},
and note that one has
J̃0
( ˙̃O1,0
)
⊂ Ñ1,0 = J̃+−
(
T̃ 0
+−
)
, (4.9)
J0(O1,0) = N1,0 = J+−
(
T 0
+−
)
. (4.10)
14 A. Odzijewicz
Let us recall that positivity X ≥ 0 of X ∈ H(n) means the positivity of its eigenvalues. We
mention here that in [23] the cotangent bundle T ∗Cn,1 of Cn,1 ⊂ H(n) is used as the phase space
of generalized U(1)-Kepler problem.
Taking into account the properties of Poisson maps presented in the diagram (4.7), as well
as the relations (4.10) and (4.9), one obtains the following morphisms of the reduced U(n, n)-
Hamiltonian spaces
O1,0/∼ N1,0 T 0
+−/∼
T̃ 0
+−/∼,Ñ1,0
˙̃O1,0/∼
6 6 6
�
�-
-
J0/∼ J+−/∼
J̃0/∼ J̃+−/∼⊂
T ∗C /∼ AdC /∼ C/∼
∪
(4.11)
which are symplectic isomorphisms, except for
T ∗C /∼ : ˙̃O1,0/∼ ↪→ O1,0/∼ and J̃0/∼ : ˙̃O1,0/∼ ↪→ Ñ1,0,
which are one-to-one symplectic maps only. The equivalence relations ∼ in (4.11) are defined
by the degeneracy leaves of the restrictions of respective symplectic forms defined on manifolds
which appear on the left- and right-hand sides of the diagram (4.7).
For any X =
(
a b
b+ d
)
∈ u(n, n) one defines the linear function
LX
(
α β
β+ δ
)
:= Tr
((
a b
b+ d
)(
α β
β+ δ
))
(4.12)
on the Lie–Poisson space (u(n, n), {·, ·}L-P), where the Lie–Poisson bracket {·, ·}L-P is defined
in (3.20). These functions satisfy
{LX1 , LX2}L-P = L[X1,X2].
In particular cases when X++ = i
(
E 0
0 E
)
and X+− = i
(
E 0
0 −E
)
one obtains(
LX++ ◦ J+−
)
(η, ξ) = η+η − ξ+ξ = I+−, (4.13)(
LX+− ◦ J+−
)
(η, ξ) = η+η + ξ+ξ =: I++, (4.14)(
LX+− ◦ J0
)
(Z, ρ) = −2i Tr ρ =: I0, (4.15)(
LX++ ◦ J0
)
(Z, ρ) = 0. (4.16)
Rewriting the above formula in the anti-diagonal realization, where X̃++ = CX++C+ = X++
and X̃+− =
(
0 −E
E 0
)
= CX+−C+ we find(
LX̃++
◦ J̃+−
)
(υ, ζ) = i
(
υζ+ − ζυ+
)
, (4.17)(
LX̃+−
◦ J̃+−
)
(υ, ζ) = υ+υ + ζ+ζ =: Ĩ++, (4.18)(
LX̃+−
◦ J̃0
)
(Y,X) = Tr
(
X
(
E + Y 2
))
=: Ĩ0, (4.19)(
LX̃++
◦ J̃0
)
(Y,X) = 0. (4.20)
The functions I++, I0, and Ĩ++, Ĩ0 are invariants of the Hamiltonian flows presented in (4.3)
and (4.4), respectively. Note that these flows are generated by X++ = X̃++ ∈ u(n, n) ∩ ũ(n, n).
So, the reduced functions I++/∼, I0/∼, Ĩ++/∼ and Ĩ0/∼ defined by (4.14), (4.15), (4.18)
and (4.19), respectively, could be considered as Hamiltonians on the reduced symplectic mani-
folds T 0
+−/∼, O1,0/∼, T̃ 0
+−/∼ and ˙̃O1,0/∼. The function LX+− : u(n, n)→ R, see definition (4.12),
as well as the function LX̃+−
: ũ(n, n) → R, after restriction to N1,0 ⊂ u(n, n) and to Ñ1,0 ⊂
ũ(n, n) give Hamiltonians on N1,0 and on Ñ1,0, respectively. Taking into account the symplectic
manifolds morphisms mentioned in the diagram (4.11) we conclude
Perturbed (2n− 1)-Dimensional Kepler Problem 15
Proposition 4.3.
(i) The U(n, n)-Hamiltonian systems:
(
T 0
+−/∼, I++/∼
)
,
(
T̃ 0
+−/∼, Ĩ++/∼
)
, (O1,0/∼, I0/∼),
(N1,0, LX+−) and
(
Ñ1,0, LX̃+−
)
are mutually isomorphic.
(ii) The Hamiltonian system
( ˙̃O1,0/∼, Ĩ0/∼
)
possesses two extensions (regularizations) to
U(n, n)-Hamiltonian systems given by the injective symplectomorphisms T ∗C /∼ : ˙̃O1,0/∼ ↪→
O1,0/∼ and J̃0/∼ : ˙̃O1,0/∼ ↪→ Ñ1,0, respectively.
Since the Hamiltonian LX+− and, thus Hamiltonians I++/∼, I0/∼, Ĩ++/∼ and Ĩ0/∼ are
defined by the element X+− of the Lie algebra u(n, n) one can consider U(n, n) as a dynamical
group for all systems mentioned in (i) of Proposition 4.3. As a matter of fact we can treat all of
them as various realizations of the same Hamiltonian system. See also [10, 11, 12] for U(n, n)
as the dynamical group of MIC-Kepler system.
The easiest way to find the symmetry groups of these systems, and thus, their integrals of
motion, is to consider the case
(
T 0
+−/∼, I++/∼
)
. In this case the symmetry group is the subgroup
of U(n, n), which preserve the canonical form γ+−, defined in (4.1), and the Hamiltonian I++,
i.e., it is U(n, n)∩U(2n) ∼= U(n)×U(n). So, the corresponding integrals of motion one obtains
by restricting the matrix functions
I+
(
η+, ξ+, η, ξ
)
:= ηη+ and I−
(
η+, ξ+, η, ξ
)
:= ξξ+
to T 0
+−. Let us note that {I++, I+}+− = {I++, I−}+− = 0.
The integrals of motion M : H(n)×H(n)→ H(n) and R : H(n)×H(n)→ H(n) for Hamil-
tonian system (H(n)×H(n), Ĩ0) have the following matrix forms
M := i[X,Y ] and R := X + Y XY. (4.21)
Reducing them to
( ˙̃O1,0/∼, Ĩ0/∼
)
we obtain their correspondence
I+ ◦ (C/∼) ◦ Kreg =
1
2
(R−M) and I− ◦ (C/∼) ◦ Kreg =
1
2
(R+M) (4.22)
to the integrals of motion I+ and I−, where Kreg : ˙̃O1,0/∼ → T̃ 0
+−/∼ is defined by
Kreg :=
(
J̃+−/∼
)−1 ◦
(
J̃0/∼
)
(4.23)
The Hamilton equations defined by Ĩ0 are
d
dt
Y = E + Y 2,
d
dt
X = −(XY − Y X), (4.24)
i.e., they could be classified as a matrix Riccati type equations. In order to obtain their solution
we note that after passing to
(
T 0
+−/∼, I++/∼
)
they asssume the form of a linear equations which
are solved by
σt++
(
η
ξ
)
=
(
eitE 0
0 e−itE
)(
η
ξ
)
,
i.e., the Hamiltonian flow σt++ is one-parameter subgroup of U(n, n) generated by X+− ∈ u(n, n).
Therefore, going through the symplectic manifold isomorphisms presented in (4.11), we obtain
the solution
Y (t) = (Y cosh t− iE sinh t)(iY sinh t+ E cosh t)−1,
X(t) = (iY sinh t+ E cosh t)X(iY sinh t+ E cosh t)+
16 A. Odzijewicz
of (4.24) by specifying the transformation formula (4.6) to the one-parameter subgroup σ̃t+− =
C+
(
eitE 0
0 e−itE
)
C of the group Ũ(n, n).
Ending this section let us mention the papers [8, 24, 25, 28, 31], where Kepler and MIC-Kepler
problems were considered on the classical and quantum levels. Let us also mention some inter-
esting generalizations of these problems [2, 19, 20, 21, 22] based on the theory of Jordan algebras.
5 Cayley and Kustaanheimo–Stiefel transformations
In this section we discuss two regularizations of the Hamiltonian system
( ˙̃O1,0/∼, Ĩ0
)
which were
mentioned in the point (ii) of Proposition 4.3. At first we will show that the regularization
Kreg : ˙̃O1,0/∼ → T̃ 0
+−/∼, defined in (4.23), could be interpreted as a generalization for arbitrary
dimension of Kustaanheimo–Stiefel regularization, which was introduced in [15] for the case
n = 2. Then we will discuss shortly the regularization Creg : ˙̃O/∼ → T 0
+−/∼ defined by Cayley
transformation
Creg := (J+−/∼)−1 ◦ (J0/∼) ◦ (T ∗C/∼).
We will also show the equivalence of the both considered regularizations.
Comparing the values
J̃0(X,Y ) =
(
Y X −Y XY
X −XY
)
=
(
vζ+ −vv+
ζζ+ −ζv+
)
= J̃+−(v, ζ)
of momentum maps J̃0 and J̃+− we find that (Y,X) ∈ J̃−1
0
(
J̃+−(v, ζ)
)
iff
X = ζζ+, (5.1)
v = Y ζ. (5.2)
Let us define ˙̃T 0
+− :=
{
( vζ ) ∈ T̃ 0
+− : ζ 6= 0
}
and observe that the surjective submersion
R : ˙̃O1,0 → ˙̃T 0
+−/∼ defined by
R(Y,X) :=
[(
Y ζ
ζ
)]
,
where X = ζζ+ and
[(
Y ζ
ζ
)]
:=
{
λ
(
Y ζ
ζ
)
: λ ∈ U(1)
}
, satisfies
R∗γ̃+−| ˙̃T 0
+−
= γ̃0| ˙̃O1,0
, Ĩ++ ◦ R = Ĩ0.
We also observe that the fibres R−1
([(
Y ζ
ζ
)])
, where
[(
Y ζ
ζ
)]
∈ ˙̃T 0
+−/∼ are the degeneracy
leaves of dγ̃0/ ˙̃O1,0
, so, one can identify the quotient map R̃ : ˙̃O1,0/∼ → ˙̃T
0
+−/∼ with the map
Kreg : ˙̃O1,0/∼ → ˙̃T 0
+−/∼ defined in (4.23).
In order to obtain explicitly a local expression for R̃−1 let us take the map S : Ω → ˙̃O1,0
defined by
S
(
v+, ζ+, v, ζ
)
:=
(
Y
(
υ+, ζ+, υ, ζ
)
, ζζ+
)
(5.3)
on an open U(1)-invariant subset Ω ⊂ T̃ 0
+−, where the map Y : Ω→ H(n) fulfills the conditions
Y
(
υ+, ζ+, υ, ζ
)
ζ = υ, (5.4)
Perturbed (2n− 1)-Dimensional Kepler Problem 17
and
Y
(
(λυ)+, (λζ)+, λυ, λζ
)
= Y
(
υ+, ζ+, υ, ζ
)
(5.5)
for λ ∈ U(1). From (5.4) and (5.5) we see that S is a local section of R, i.e., R◦S = idΩ. Thus
one can choose S
(
υ+, ζ+, υ, ζ
)
∈ R−1([( υζ )]) as a representative of the degeneracy leaf
R−1
[(
υ
ζ
)]
=
{(
Y (υ+, ζ+, υ, ζ) + Y ′, ζζ+
)
: Y ′ ∈ H(n) and Y ′ζ = 0
}
of the differential closed form dγ̃0| ˙̃O1,0
. Let “∼” be the equivalence relation on ˙̃O1,0 defined by
these leaves, then identifying the quotient manifold R−1(Ω/∼)/∼, with respect to this equiv-
alence, with the local section S(Ω) we obtain the following local diffeomorphism S : Ω/∼
∼→
S(Ω) ∼= R−1(Ω/∼)/∼.
In next examples we will present two local sections S : Ω→ ˙̃O1,0 of R : ˙̃O1,0 → ˙̃T 0
+−/∼.
Example 5.1. Let us take Ω = ˙̃T 0
+− and define Y : Ω→ H(n) as follows
Y
(
υ+, ζ+, υ, ζ
)
:=
1
ζ+ζ
[
ζυ+ + υζ+ − 1
2
(
υ+ζ + ζ+υ
)
E
]
. (5.6)
One easily checks that the map Y : Ω → H(n) defined in (5.6) satisfies the conditions (5.4)
and (5.5), so, it defines by (5.3) a local section of R.
Example 5.2. In this example we assume Ω :=
{
( vζ ) ∈ T̃ 0
+− : v+ζ 6= 0
}
and define Y : Ω →
H(n) by
Y
(
υ+, ζ+, υ, ζ
)
=
υυ+
υ+ζ
.
The meaning of the first example will be explained at the end of this section. The second
example illustrates another possibility to define a local diffeomorphism S : Ω/∼
∼→ S(Ω) ∼=
R−1(Ω/∼)/∼.
Having in mind a physical interpretations of the discussed Hamiltonian systems, we will
consider the case n = 2 in details. Expanding (Y,X) ∈ H(2)×H(2) in Pauli matrices σ0 := ( 1 0
0 1 ),
σ1 := ( 0 1
1 0 ), σ2 :=
(
0 i
−i 0
)
and σ3 :=
(
1 0
0 −1
)
, i.e.,
Y = y0σ0 + ~y · ~σ and X = x0σ0 + ~x · ~σ, (5.7)
where ~σ = (σ1, σ2, σ3), we find that
1
2
γ̃0 = y0dx0 + ~y · d~x.
In this case we assume that Ω = ˙̃T 0
+− and define S : ˙̃T 0
+− →
˙̃O1,0 taking Y : ˙̃T 0
+− → H(n)
such as in (5.6). We see from (5.3) and (5.6) that (Y,X) ∈ S
( ˙̃T 0
+−
)
iff Tr(Y ) = 2y0 = 0 and
detX = x02 − ~x2 = 0, Tr(X) = 2x0 > 0. From the above it follows that S
( ˙̃T 0
+−
) ∼= R3 × Ṙ3,
where Ṙ3 = R3 \ {0}, and the canonical form γ̃0 after restriction to S
( ˙̃T 0
+−
)
is given by
γ̃0|S( ˙̃T 0
+−)
= 2~y · d~x = 2ykdx
k.
Using the identity
σkσl + σlσk = 2δkl (5.8)
18 A. Odzijewicz
valid for Pauli matrices σk, k = 1, 2, 3, we find that the Hamiltonian H0 := 1
2 Ĩ0, defined in (4.19),
after restriction to S
( ˙̃T 0
+−
)
assumes the following form
H0 = Ĩ0|S( ˙̃T 0
+−)
= ‖~x‖
(
1 + ‖~y‖2
)
on R3 × Ṙ3. Let us note that ‖~x‖ = x0 = 1
2ζ
+ζ > 0.
Summing up the above facts we state that the Hamiltonian system
(
H(2) × H(2),dγ̃0, Ĩ0
)
after reduction to
(
R3× Ṙ3, 2d~y∧d~x,H0
)
is exactly the 3-dimensional Kepler system written in
the “fictitious time” s which is related to the real time t via the rescaling
ds
dt
=
1
‖~x‖
.
For an exhaustive description of the regularized Kepler problem we address to original papers
of Moser [26] and of Kustaanheimo and Stiefel [15] as well as to [14], where the relationship
between Moser and Kustaanheimo–Stiefel regularization was established.
In order to express (~y, ~x) ∈ R3 × Ṙ3 by ( υζ ) ∈ T̃ 0
+− we put Y = ~y · ~σ = ykσk into (5.2)
and multiply this equation by ζ+σl. Then, using (5.8) and (5.1) we obtain the one-to-one map
defined by
~y =
1
ζ+ζ
1
2
(
υ+~σζ + ζ+~συ
)
, ~x =
1
2
ζ+~σζ, (5.9)
of ˙̃T 0
+−/∼ onto R3×Ṙ3. This map is known in literature of celestial mechanics as Kustaanheimo–
Stiefel transformation, see [14, 15]. There are possible some variations of (5.9) naturally pre-
sented in quaternion language, see equation (15) in [7]. This quaternionic approach does not
extend to an arbitrary dimension, where symplectic geometry methods are effective only.
Therefore, having in mind the case n = 2, it is reasonable to interpret:
i) the Hamiltonian systems
(
T 0
+−/∼, I++
)
,
(
T̃ 0
+−/∼, Ĩ++
)
,
(
O1,0/∼, I0
)
,
(
Ñ1,0, LX̃+−
)
and(
N1,0, LX+−
)
as the various equivalent realizations of the regularized (2n− 1)-dimensional
Kepler problem;
ii) the map S : ˙̃T 0
+− → S
( ˙̃T 0
+−
)
, where Y : ˙̃T 0
+− → H(n) is given by (5.6), as Kustaanheimo–
Stiefel transformation for the (2n− 1)-dimensional Kepler problem.
Finally let us briefly discuss the regularization of
( ˙̃O1,0/∼, Ĩ0
)
given by Creg : ˙̃O1,0/∼ → T 0
+−/∼
which we will call Cayley regularization of the (2n− 1)-dimensional Kepler problem. From the
commutativity of the diagram (4.11) we conclude that
Creg = (C/ ∼) ◦ Kreg.
Therefore, the Kustaanheimo–Stiefel regularization is equivalent to the Cayley regularization of
the (2n− 1)-dimensional Kepler problem.
By Proposition 4.3 the (2n − 1)-Kepler system
( ˙̃O1,0, Ĩ0/∼
)
is extended to (regularized by)
arbitrary U(n, n)-Hamiltonian system occurred in the diagram (4.11). In accordance with ter-
minology assumed here, the extension of
( ˙̃O1,0, Ĩ0/∼
)
to a U(n, n)-Hamiltonian system from the
upper row of the diagram (4.11) is called the Cayley regularization, whereas the extension to
the one from the lower row is the Kustaanheimo–Stiefel regularization. The justification of this
nomenclature follows from the appearance in (4.11) the maps (4.8) and (5.3).
The benefit of using the various isomorphic realizations of the same U(n, n)-Hamiltonian
system is based on the possibility to admit different physical interpretations for them. For
Perturbed (2n− 1)-Dimensional Kepler Problem 19
example, if n = 2 one can consider the symplectic manifold O1,0/∼ as the phase space of
massless scalar particle in the conformally compactified Minkowski space M1,3
∼= U(2), see [27].
The realizations T̃ 0
+−/∼ and T 0
+−/∼ play the crucial role in the twistor theory [30] of R. Penrose.
In the papers [3, 4] a method of linearization of the regularized Kepler problem based on the
Clifford algebra C(2, n + 1) of the Lie group SO(2, n + 1) was proposed. The Spin(2, n + 2)-
invariant symplectic structure ω on an ideal V ⊂ C(2, n+ 1) of the Clifford algebra C(2, n+ 1)
is fixed. Then, using the momentum map J : V → sl(2, n + 1) on this auxiliary Sp(2, n + 2)-
symplectic manifold (V, ω), the Marsden–Weinstein reduction procedure to the Ad∗(Spin(2, n+
2))-orbits O = ι(T ∗Sn) is applied. The inverse (KS)−1 of Kustaanheimo–Stiefel map is defined
by (KS)−1 = l ◦ π, where symplectomorphism l is defined as the one making the diagram
T ∗
(
Rn\{0}
)
T+Sn O
J−1(O)/ ∼
?
- -
�
�
�
���
π ι
l
J/ ∼
⊂
commutative, see [4, 5], where π is Moser regularization [26] map and ι is the momentum map
for Moser phase space T+Sn = T ∗Sn\{null section}. Comparing the above approach with ours,
we conclude that the construction of Kustaanheimo–Stiefel map presented in [4, 5] combines the
symplectic geometry with Clifford algebras theory and is obtained in an implicit way. In our
case we use the Poisson geometry methods only and obtain the explicit formulas, see (5.3), (5.6),
for Kustaanheimo–Stiefel map. Both approaches intersect in the case n = 2.
Although here we have considered the odd-dimensional Kepler problem only, the even-
dimensional case is none the less important. Since one can obtain the planar Kepler problem
from the spatial one by some reduction procedure [32, 33], the question arise: is it possible in
general case? Another interesting question concerns the Kepler problem of positive energy. But,
these are the tasks for a next paper.
6 An integrable generalization of (2n − 1)-dimensional
Kepler problem
We present here an integrable Hamiltonian system which will be a natural generalization (pre-
turbation) of regularized (2n− 1)-dimensional Kepler problem discussed in Section 4.
Therefore, assuming for z ∈ C and l ∈ Z the convention
zl :=
{
zl for l ≥ 0,
z̄−l for l < 0
we define the following Hamiltonian
H = h0
(
|η1|2, . . . , |ηn|2, |ξ1|2, . . . , |ξn|2
)
+ g0
(
|η1|2, . . . , |ηn|2, |ξ1|2, . . . , |ξn|2
)
×
(
ηk11 · · · η
kn
n ξl11 · · · ξ
ln
n + η−k11 · · · η−knn ξ−l11 · · · ξ−lnn
)
, (6.1)
on the symplectic manifold
(
C2n, dγ+−
)
, where h0 and g0 are arbitrary smooth functions of 2n
real variables and k1, . . . kn, l1, . . . , ln ∈ Z. Let us note here that taking in (6.1) h0 = I++
and g0 = 0 we obtain (2n − 1)-dimensional regularized Kepler Hamiltonian on T 0
+−/∼. We
see from (6.1) that H is a radical generalization of I++. Nevertheless, as we will show in the
subsequent, the Hamiltonian system
(
T 0
+−/∼, H/∼
)
is still integrable in quadratures.
20 A. Odzijewicz
For this reason, according to [29], we define, for r = 1, . . . , 2n, the functions
Ir :=
n∑
j=1
ρr,j |ηj |2 −
n∑
j=1
ρr,n+j |ξj |2, (6.2)
ψr :=
n∑
j=1
κj,rφj +
n∑
j=1
κn+j,rφn+j ,
where ηj = |ηj |eiφj , ξj = |ξj |eiφn+j . By definition the real 2n × 2n matrix [ρr,s] is invertible
and the matrix [κr,s] is its inverse. The functions (I1, . . . , I2n, ψ1, . . . , ψ2n) form a system of
coordinates on the open subset
Ω2n :=
{
(η, ξ) ∈ Cn ⊕ Cn : |η1| 6= 0, . . . , |ηn| 6= 0, |ξ1| 6= 0, . . . , |ξn| 6= 0
}
of C2n. They are a canonical coordinates for symplectic form dγ+−, i.e., their Poisson brackets
satisfy
{Ir, Is} = 0, {Ir, ψs} = δrs, {ψr, ψs} = 0.
What is more, one easily checks that for r = 2, . . . , 2n one has {H, Ir} = 0 if and only if
n∑
j=1
(ρr,jlj + ρr,n+jkj) = δr1. (6.3)
So, the Hamiltonian system on
(
Ω2n, dγ+−
)
given by the Hamiltonian (6.1) is integrable and
H, I2, . . . , I2n−1 are its functionally independent integrals of motion in involution. Considering
(I2, . . . , I2n) as the components
J(η+, ξ+, η, ξ) =
I2(η+, ξ+, η, ξ)
...
I2n(η+, ξ+, η, ξ)
(6.4)
of the momentum map J : Ω2n → R2n−1, where one identifies R2n−1 with the dual space to the
Lie algebra of (2n − 1)-dimensional torus T2n−1 := U(1)× · · · × U(1)︸ ︷︷ ︸
2n−1
, we can apply Marsden–
Weinstein reduction procedure to
(
Ω2n, dγ+−, H
)
. In this way we reduce the above Hamiltonian
system to J−1(c2, . . . , c2n)/T2n−1 ∼= ]a, b[×S1 with ωred = dI1∧dψ1 as a symplectic form, where
(I1, ψ1) ∈ ]a, b[ × S1, and the Hamiltonian (6.1) after the reduction to J−1(c2, . . . , c2n)/T2n−1
assumes the following form
Hred = H0(I1, c2, . . . , c2n) + 2
√
G0(I1, c2, . . . , c2n) cosψ1,
where H0(I1, I2, . . . , I2n) and G0(I1, I2, . . . , I2n) are defined as the superpositions of the functions
h0
(
|η1|2, . . . , |ηn|2, |ξ1|2, . . . , |ξn|2
)
and (
g0
(
|η1|2, . . . , |ηn|2, |ξ1|2, . . . , |ξn|2
))2|η1|2|k1| · · · |ηn|2|kn||ξ1|2|l1| · · · |ξn|2|ln|
with the map inverse to the map defined in (6.2). For the explicit expression for a and b see [29].
Perturbed (2n− 1)-Dimensional Kepler Problem 21
The Hamilton equations defined by Hred in the canonical coordinates (I1, ψ1) have form
dI1
dt
= 2
√
G0(I1, c2, . . . , c2n) sinψ1, (6.5)
dψ1
dt
=
∂H0
∂I1
(I1, c2, . . . , c2n) +
∂G0
∂I1
(I1, c2, . . . , c2n) cosψ1.
From (6.5) and E := Hred(I1(t), ψ1(t), c2, . . . , c2n) = const, where E is the total energy of the
system, we obtain(
dI1
dt
)2
= 4G0(I1, c2, . . . , c2n)− (E −H0(I1, c2, . . . , c2n))2. (6.6)
Separating variables in (6.6) we integrate it by quadratures. Next, using integrals of motion
I2, . . . , I2n, we integrate our initial system defined on
(
C2n, dγ+−
)
by the Hamiltonian (6.1).
A detailed description of this method of integration can be found in [29, Section 2].
Now let us assume that the last two of integrals of motion I2, . . . , I2n−1, I2n are given by
I2n−1 := I++ = η+η + ξ+ξ, I2n := I+− = η+η − ξ+ξ.
Hence, from (6.3), we obtain the conditions
k1 + · · ·+ kn = 0 and l1 + · · ·+ ln = 0
on the exponents k1, . . . , kn, l1, . . . , ln ∈ Z, which guarantee integrability of the Hamiltonian
system
(
C2n, dγ+−, H
)
. Because I+− is one of the integrals of motion, we find that the reduced
system
(
T 0
+−/∼, H/∼
)
is also integrable. So, using the symplectomorphism Creg◦Kreg : ˙̃O1,0/∼ →
T 0
+−/∼, see diagram (4.11), we obtain an integrable Hamiltonian system on ˙̃O1,0/∼ with Hamil-
tonian (H/∼) ◦ Creg.
In the particular case, if k ∈ {k1, . . . , kn} and l ∈ {l1, . . . , ln} then −k ∈ {k1, . . . , kn} and
−l ∈ {l1, . . . , ln}, the Hamiltonian (6.1) depends on the matrix elements of I+ and I− only. So,
in this case we obtain from (4.22) that the Hamiltonian (H/∼) ◦ Creg could be defined as the
reduction H̃/∼ to ˙̃O1,0/∼ of the Hamiltonian
H̃ = h0
(
N−11, . . . , N
−
nn, N
+
11, . . . , N
+
nn
)
+ g0
(
N−11, . . . , N
−
nn, N
+
11, . . . , N
+
nn
)
×
[(
N−i1j1
)ki1 · · · (N−irjr)kir (N+
a1b1
)la1 · · · (N+
asbs
)las + h.c.
]
(6.7)
on H(n)×H(n), where N±kl := 1
2(Rkl±Mkl), R and M depend on (Y,X) by (4.21). The subsets of
exponents {ki1 , . . . , kir} ⊂ {k1, . . . , kn} and {la1 , . . . , las} ⊂ {l1, . . . , ln} satisfy kim = −kjm > 0
for m = 1, 2, . . . , r and lam = −lbm > 0 for m = 1, 2, . . . , s.
Ending this section, we write the Hamiltonian (6.7) in the more explicit form for the case
n = 2. In this case the integrals of motion M and R can be written in terms of Pauli matrices
M = M0E + ~M · ~σ and R = R0E + ~R · ~σ,
where ~M and ~R are angular momentum and Runge–Lenz vector, respectively. Using the linear
relation
|η1|2
|η2|2
|ξ1|2
|ξ2|2
=
1
2
1 1 −1 −1
1 −1 −1 1
1 1 1 1
1 −1 1 −1
R0
R3
M0
M3
22 A. Odzijewicz
and defining M+ := M1 + iM2 and M− := M1 − iM2 we write this Hamiltonian as follows
H̃ = h̃0(R0, R3,M0,M3) + g̃0(R0, R3,M0,M3)
×
(
(Rσ −Mσ)k(Rσ′ +Mσ′)
l + (R−σ −M−σ)k(R−σ′ +M−σ′)
l
)
, (6.8)
where σ, σ′ = +,−, k, l ∈ N ∪ {0} and h̃0, g̃0 are arbitrary smooth functions. Let us note that
R0 = 1
2I0. Note also that equation M0 = −η+η + ξ+ξ = 0 leads to the reduced system T 0
+−/∼.
In order to represent this Hamiltonian in the canonical coordinates (~y, ~x) ∈ R3×Ṙ3, see (5.7),
we note that
~M = 2~x× ~y, (6.9)
~R =
(
1− ~y2
)
~x+ 2~y(~x · ~y). (6.10)
After substituting (6.9), (6.10) and M0 = 0 and R0 = ||~x||
(
1 + (~y)2
)
into (6.8) we reduce the
Hamiltonian H̃ to the phase space
(
S(Ω) ∼= R3 × Ṙ3, 2d~y ∧ d~x
)
.
As it follows from the general method presented above, the Hamiltonian system on T 0
+−/ ∼
described by the Hamiltonian (6.8) for n = 2, is integrable in quadratures, see equation (6.6).
The third integral of motion complementary to I3 = I++ and I4 = I+− is the following
I2 = ρ2,1|η1|2 + ρ2,2|η2|2 − ρ2,3|ξ1|2 − ρ2,4|ξ2|2,
where the resonance condition
(ρ2,1 − ρ2,2)l + (ρ2,3 − ρ2,4)k = 0
is subjected to be fulfilled. In the Section IV of the paper [5], where a perturbed Kepler problem
(the hydrogen atom interacting with the constant electric and magnetic fields) is considered,
the authors, using the normalization procedure, obtain an integrable approximation of the per-
turbed Kepler Hamiltonian investigated by them. See also [11] for MIC-Kepler problem. This
approximated system could be treated as a special subcase of (6.8), what follows from the fact
that (6.8) is the general Hamiltonian, which has three Manley–Rowe type integrals of motion
given by (6.4). The quantum version of the Hamiltonian system (6.1), as well as its integration
by quantum reduction method, can be found in [29]. Some methods of integration of a quantum
perturbed Kepler system can be found in [5]. All these questions for the integrable generalized
(2n− 1)-Kepler problem defined by the Hamiltonian (6.1) will be a subject of the next paper.
Acknowledgements
Author would like to express his sincere gratitude for all the anonymous referees for their com-
ments and remarks which improved the paper and made it more readable.
References
[1] Barut A.O., Kleinert H., Transition probabilities of the hydrogen atom from noncompact dynamical groups,
Phys. Rev. 156 (1967), 1541–1545.
[2] Bouarroudj S., Meng G., The classical dynamic symmetry for the U(1)-Kepler problems, J. Geom. Phys.
124 (2018), 1–15, arXiv:1509.08263.
[3] Cordani B., On the generalisation of the Kustaanheimo-Stiefel transformations, J. Phys. A: Math. Gen. 22
(1989), 2441–2446.
[4] Cordani B., Reina C., Spinor regularization of the n-dimensional Kepler problem, Lett. Math. Phys. 13
(1987), 79–82.
https://doi.org/10.1103/PhysRev.156.1541
https://doi.org/10.1016/j.geomphys.2017.10.012
https://arxiv.org/abs/1509.08263
https://doi.org/10.1088/0305-4470/22/13/036
https://doi.org/10.1007/BF00570771
Perturbed (2n− 1)-Dimensional Kepler Problem 23
[5] Efstathiou K., Sadovskii D.A., Normalization and global analysis of perturbations of the hydrogen atom,
Rev. Mod. Phys. 82 (2010), 2099–2154.
[6] Faraut J., Korányi A., Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon
Press, Oxford University Press, New York, 1994.
[7] Ferrer S., Crespo F., Alternative angle-based approach to the KS-map. An interpretation through symmetry
and reduction, J. Geom. Mech. 10 (2018), 359–372, arXiv:1711.08530.
[8] Horowski M., Odzijewicz A., Geometry of the Kepler system in coherent states approach, Ann. Inst.
H. Poincaré Phys. Théor. 59 (1993), 69–89.
[9] Iwai T., The geometry of the SU(2) Kepler problem, J. Geom. Phys. 7 (1990), 507–535.
[10] Iwai T., A dynamical group SU(2, 2) and its use in the MIC-Kepler problem, J. Phys. A: Math. Gen. 26
(1993), 609–630.
[11] Iwai T., Matsumoto S., Poisson mechanics for perturbed MIC-Kepler problems at both positive and negative
energies, J. Phys. A: Math. Theor. 45 (2012), 365203, 34 pages.
[12] Iwai T., Uwano Y., The four-dimensional conformal Kepler problem reduces to the three-dimensional Kepler
problem with a centrifugal potential and Dirac’s monopole field. Classical theory, J. Math. Phys. 27 (1986),
1523–1529.
[13] Kirillov A.A., Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften,
Vol. 220, Springer-Verlag, Berlin – New York, 1976.
[14] Kummer M., On the regularization of the Kepler problem, Comm. Math. Phys. 84 (1982), 133–152.
[15] Kustaanheimo P., Stiefel E., Perturbation theory of Kepler motion based on spinor regularization, J. Reine
Angew. Math. 218 (1965), 204–219.
[16] Lang S., Algebra, 3rd ed., Graduate Texts in Mathematics, Vol. 211, Springer-Verlag, New York, 2002.
[17] Malkin I.A., Man’ko V.I., Symmetry of the hydrogen atom, JETP Lett. 2 (1965), 146–148.
[18] McIntosh H.V., Cisneros A., Degeneracy in the presence of a magnetic monopole, J. Math. Phys. 11 (1970),
896–916.
[19] Meng G., Generalized MICZ-Kepler problems and unitary highest weight modules, II, J. Lond. Math. Soc.
81 (2010), 663–678, arXiv:0704.2936.
[20] Meng G., Euclidean Jordan algebras, hidden actions, and J-Kepler problems, J. Math. Phys. 52 (2011),
112104, 35 pages, arXiv:0911.2977.
[21] Meng G., Generalized Kepler problems. I. Without magnetic charges, J. Math. Phys. 54 (2013), 012109,
25 pages, arXiv:1104.2585.
[22] Meng G., The universal Kepler problem, J. Geom. Symmetry Phys. 36 (2014), 47–57, arXiv:1011.6609.
[23] Meng G., On the trajectories of U(1)-Kepler problems, in Geometry, Integrability and Quantization XVI,
Avangard Prima, Sofia, 2015, 219–230.
[24] Mladenov I., Tsanov V., Geometric quantization of the multidimensional Kepler problem, J. Geom. Phys.
2 (1985), 17–24.
[25] Mladenov I.M., Tsanov V.V., Geometric quantisation of the MIC-Kepler problem, J. Phys. A: Math. Gen.
20 (1987), 5865–5871.
[26] Moser J., Regularization of Kepler’s problem and the averaging method on a manifold, Comm. Pure Appl.
Math. 23 (1970), 609–636.
[27] Odzijewicz A., A conformal holomorphic field theory, Comm. Math. Phys. 107 (1986), 561–575.
[28] Odzijewicz A., Świȩtochowski M., Coherent states map for MIC-Kepler system, J. Math. Phys. 38 (1997),
5010–5030.
[29] Odzijewicz A., Wawreniuk E., Classical and quantum Kummer shape algebras, J. Phys. A: Math. Theor.
49 (2016), 265202, 33 pages, arXiv:1512.09279.
[30] Penrose R., Twistor algebra, J. Math. Phys. 8 (1967), 345–366.
[31] Simms D.J., Bohr–Sommerfeld orbits and quantizable symplectic manifolds, Proc. Cambridge Philos. Soc.
73 (1973), 489–491.
[32] Stiefel E.L., Scheifele G., Linear and regular celestial mechanics, Grundlehren der mathematischen Wis-
senschaften, Vol. 174, Springer-Verlag, Berlin – Heidelberg, 1971.
[33] Zhao L., Kustaanheimo–Stiefel regularization and the quadrupolar conjugacy, Regul. Chaotic Dyn. 20 (2015),
19–36, arXiv:1308.2314.
[34] Zwanzinger D., Exactly soluble nonrelativistic model of particles with both electric and magnetic charges,
Phys. Rev. 176 (1968), 1480–1488.
https://doi.org/10.1103/RevModPhys.82.2099
https://doi.org/10.3934/jgm.2018013
https://arxiv.org/abs/1711.08530
https://doi.org/10.1016/0393-0440(90)90004-M
https://doi.org/10.1088/0305-4470/26/3/021
https://doi.org/10.1088/1751-8113/45/36/365203
https://doi.org/10.1063/1.527112
https://doi.org/10.1007/978-3-642-66243-0
https://doi.org/10.1007/BF01208375
https://doi.org/10.1515/crll.1965.218.204
https://doi.org/10.1515/crll.1965.218.204
https://doi.org/10.1007/978-1-4613-0041-0
https://doi.org/10.1063/1.1665227
https://doi.org/10.1112/jlms/jdq019
https://arxiv.org/abs/0704.2936
https://doi.org/10.1063/1.3659283
https://arxiv.org/abs/0911.2977
https://doi.org/10.1063/1.4775343
https://arxiv.org/abs/1104.2585
https://doi.org/10.7546/jgsp-36-2014-47-57
https://arxiv.org/abs/1011.6609
https://doi.org/10.7546/giq-16-2015-219-230
https://doi.org/10.1016/0393-0440(85)90016-6
https://doi.org/10.1088/0305-4470/20/17/020
https://doi.org/10.1002/cpa.3160230406
https://doi.org/10.1002/cpa.3160230406
https://doi.org/10.1007/BF01205486
https://doi.org/10.1063/1.531930
https://doi.org/10.1088/1751-8113/49/26/265202
https://arxiv.org/abs/1512.09279
https://doi.org/10.1063/1.1705200
https://doi.org/10.1017/s0305004100077070
https://doi.org/10.1134/S1560354715010025
https://arxiv.org/abs/1308.2314
https://doi.org/10.1103/PhysRev.176.1480
1 Introduction
2 Grassmannian Gr(to.n,C2n)to. and related vector bundles
3 T*U(n) as a Hamiltonian U(n,n)-space
4 Regularized (2n-1)-dimensional Kepler problem
5 Cayley and Kustaanheimo–Stiefel transformations
6 An integrable generalization of (2n-1)-dimensional Kepler problem
References
|
| id | nasplib_isofts_kiev_ua-123456789-210761 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T05:46:27Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Odzijewicz, Anatol 2025-12-17T14:29:58Z 2020 Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛). Anatol Odzijewicz. SIGMA 16 (2020), 087, 23 pages 1815-0659 2020 Mathematics Subject Classification: 53D17; 53D20; 53D22; 70H06 arXiv:1806.05912 https://nasplib.isofts.kiev.ua/handle/123456789/210761 https://doi.org/10.3842/SIGMA.2020.087 We study the regularized (2𝑛 − 1)-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of 𝑈(𝑛, 𝑛). The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed, and their equivalence is shown. Some integrable generalization (perturbation) of the (2𝑛 − 1)-Kepler problem is proposed. The author would like to express his sincere gratitude to all the anonymous referees for their comments and remarks, which improved the paper and made it more readable. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛) Article published earlier |
| spellingShingle | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛) Odzijewicz, Anatol |
| title | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛) |
| title_full | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛) |
| title_fullStr | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛) |
| title_full_unstemmed | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛) |
| title_short | Perturbed (2𝑛 − 1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of 𝑈(𝑛, 𝑛) |
| title_sort | perturbed (2𝑛 − 1)-dimensional kepler problem and the nilpotent adjoint orbits of 𝑈(𝑛, 𝑛) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210761 |
| work_keys_str_mv | AT odzijewiczanatol perturbed2n1dimensionalkeplerproblemandthenilpotentadjointorbitsofunn |