Curvature as an integrable deformation

Abstract

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the generic anisotropic oscillator Hamiltonian is presented, and its superintegrability for commensurate frequencies is shown. As a second example, an integrable version of the Hénon–Heiles system on the sphere and the hyperbolic plane is introduced. Projective Beltrami coordinates are shown to be helpful in this construction, and further applications of this approach are sketched.¹¹1To appear in “Integrability, Supersymmetry and Coherent States”, A volume in honour of Professor Véronique Hussin. S. Kuru, J. Negro and L.M. Nieto (Eds.), Special volume of the CRM Series in Mathematical Physics (Berlin: Springer, 2019)

Angel Ballesteros, Alfonso Blasco and Francisco J. Herranz

Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain

E-mails: angelb@ubu.es, ablasco@ubu.es, fjherranz@ubu.es

MSC: 37J35, 70H06, 22E60

PACS: 02.30.Ik, 45.20.Jj, 02.20.Sv, 02.40.Ky

Keywords: Integrable systems, Curvature, Sphere, Hyperbolic plane, Integrable perturbations, Oscillator potential, Hénon-Heiles

1 Introduction
2 Geodesic dynamics on the sphere and the hyperboloid
- 2.1 Ambient space coordinates
- 2.2 Geodesic parallel and polar coordinates
3 Beltrami coordinates and projective dynamics
4 Anisotropic oscillators on the Euclidean plane
- 4.1 The $γ = 1$ or 1:1 (isotropic) oscillator
- 4.2 The $γ = 2$ or 2:1 oscillator
5 Anisotropic oscillators on $S^{2}$ and $H^{2}$
6 Integrable Hénon–Heiles systems
- 6.1 An integrable KdV Hénon–Heiles system on the Euclidean plane
7 An integrable KdV Hénon–Heiles system on $S^{2}$ and $H^{2}$
8 Remarks and open problems

section*

1 Introduction

The aim of this contribution is to review some new recent results related to a seemingly elementary issue in the theory of finite-dimensional integrable systems [1, 2, 3, 4, 5], whose solution presents quite a number of interesting features. The problem can explicitly be stated as follows.

Let us consider a certain Liouville integrable natural Hamiltonian system for a particle with unit mass moving on the two-dimensional (2D) Euclidean space endowed with the standard bracket ${q_{i}, p_{j}} = δ_{i j}$ in terms of canonical coordinates and momenta, namely

(1)

where $T$ is the kinetic energy and $V$ is the potential. The Liouville integrability of this system will be provided by a constant of the motion given by a globally defined function $I (p_{1}, p_{2}, q_{1}, q_{2})$ such that ${H, I} = 0$ .

The proposed problem consists in finding a one-parameter integrable deformation of $H$ of the form

H_{κ} = T_{κ} (p_{1}, p_{2}, q_{1}, q_{2}) + V_{κ} (q_{1}, q_{2}), κ \in R,

with integral of the motion given by the smooth and globally defined function $I_{κ} (p_{1}, p_{2}, q_{1}, q_{2})$ (therefore ${H_{κ}, I_{κ}} = 0$ ), and such that the following two conditions hold:

The smooth function $T_{κ}$ is the kinetic energy of a particle on a 2D space whose constant curvature is given by the parameter $κ$ , i.e. the 2D sphere S $^{2}$ will arise in the case $κ > 0$ and the hyperbolic plane $H^{2}$ when $κ < 0$ .
The Euclidean system $H$ given by (1) has to be smoothly recovered in the zero-curvature limit $κ \to 0$ , namely

$H = lim κ \to 0 H_{κ}, I = lim κ \to 0 I_{κ} .$

If these two conditions are fulfilled, we will say that $H_{κ}$ is an integrable curved version of $H$ on the sphere and the hyperbolic space. We stress that within this framework the Gaussian curvature $κ$ of the space enters as a deformation parameter, and the curved system $H_{κ}$ can be thought of as smooth integrable perturbation of the flat one $H$ in terms of the curvature parameter. Therefore, integrable Hamiltonian systems on S $^{2}$ ( $κ > 0$ ), H $^{2}$ ( $κ < 0$ ) and E $^{2}$ ( $κ = 0$ ) will be simultaneously constructed and analysed.

Moreover, it could happen that the initial Hamiltonian $H$ is not only integrable but superintegrable, i.e. another globally defined and functionally independent integral of the motion $K (p_{1}, p_{2}, q_{1}, q_{2})$ does exist such that

{H, I} = {H, K} = 0, {I, K} \neq 0.

In that case we could further impose the existence of the curved (and functionally independent) analogue $K_{κ}$ of the second integral such that

K = lim κ \to 0 K_{κ} .

If we succeed in finding such second integral fulfilling

{H_{κ}, I_{κ}} = {H_{κ}, K_{κ}} = 0, {I_{κ}, K_{κ}} \neq 0,

we will say that we have obtained a superintegrable curved generalization of the Euclidean superintegrable Hamiltonian $H$ .

The explicit curvature-dependent description of S $^{2}$ and H $^{2}$ is well-known in the literature and can be found, for instance, in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] (see also references therein) where it has been mainly considered in the classification and description of superintegrable systems on these two spaces. In this contribution we will present several recent works in which this geometric framework has been applied for non-superintegrable systems where the lack of additional symmetries forces to make use of a purely integrable perturbation approach. Moreover, this perturbative viewpoint shows that the uniqueness of this construction is not guaranteed, since in general different $V_{κ}$ integrable potentials (and their associated $I_{κ}$ integrals) having the same $κ \to 0$ limit could exist and be found. As an outstanding example of this plurality, we will present the construction of different integrable curved analogues on S $^{2}$ ( $κ > 0$ ) and H $^{2}$ ( $κ < 0$ ) of some anisotropic oscillators.

The second novel technical aspect to be emphasized in the results here presented is that in some cases projective coordinates turn out to be helpful in order to construct the (super)integrable deformations $H_{κ}$ , since when these coordinates are considered on S $^{2}$ and H $^{2}$ then the curved kinetic energy $T_{κ}$ is expressed as a polynomial in the canonical variables describing the projective phase space. Therefore, some of the examples here presented can be thought of as instances of integrable projective dynamics, in the sense of [29, 30].

The structure of the paper is the following. In the next Section we review the description of the geodesic dynamics on the sphere and the hyperboloid by making use of the above mentioned curvature-dependent formalism. In particular, ambient space coordinates as well as geodesic parallel and geodesic polar coordinates for S $^{2}$ and H $^{2}$ will be introduced. In Section 3 the projective dynamics on the sphere and the hyperboloid in terms of Beltrami coordinates will also be summarized, thus providing a complete set of geometric possibilities for the description of dynamical systems on these curved spaces. In Section 4 we recall the (super)integrability properties of the 2D anisotropic oscillator with arbitrary frequencies and also with commensurate ones, and in Section 5 the explicit construction of the $H_{κ}$ Hamiltonian defining its curved analogue will be presented. Section 6 will be devoted to recall the three integrable versions of the well-known (non-integrable) Hénon–Heiles Hamiltonian. In Section 7 the construction of the curved version on S $^{2}$ and H $^{2}$ of an integrable Hénon–Heiles system related to the KdV hierarchy will be constructed, thus exemplifying the usefulness of the approach here presented for the obtention of new integrable systems on curved spaces. Furthermore, the full Ramani–Dorizzi–Grammaticos series of integrable polynomial potentials will also be generalized to the curved case. Finally, a Section including some remarks and open problems under investigation closes the paper.

2 Geodesic dynamics on the sphere and the hyperboloid

Let us consider the one-parametric family of 3D real Lie algebras ${s o}_{κ} (3) = s p a n {J_{01}, J_{02}, J_{12}}$ with commutation relations given by (in the sequel we follow the curvature-dependent formalism as presented in [31, 32]):

[J_{12}, J_{01}] = J_{02}, [J_{12}, J_{02}] = - J_{01}, [J_{01}, J_{02}] = κ J_{12},

(2)

where $κ$ is a real parameter. The Casimir invariant, coming from the Killing–Cartan form, reads

C = J_{01}^{2} + J_{02}^{2} + κ J_{12}^{2} .

(3)

The family ${s o}_{κ} (3)$ comprises three specific Lie algebras: $s o (3)$ for $κ > 0$ , $s o (2, 1) ≃ {s l}_{2} (R)$ for $κ < 0$ , and $i s o (2) \equiv e (2) = s o (2) \oplus_{S} R^{2}$ for $κ = 0$ . Note that the value of $κ$ can be reduced to ${+ 1, 0, - 1}$ through a rescaling of the Lie algebra generators; therefore setting $κ = 0$ in (2) can be shown to be equivalent to applying an Inönü–Wigner contraction [33].

The involutive automorphism defined by

Θ (J_{01}, J_{02}, J_{12}) = (- J_{01}, - J_{02}, J_{12}),

generates a $Z_{2}$ -grading of ${s o}_{κ} (3)$ in such a manner that $κ$ is a graded contraction parameter [34], and $Θ$ gives rise to the following Cartan decomposition of the Lie algebra:

{s o}_{κ} (3) = h \oplus p, h = s p a n {J_{12}} = s o (2), p = s p a n {J_{01}, J_{02}} .

We denote ${S O}_{κ} (3)$ and $H$ the Lie groups with Lie algebras ${s o}_{κ} (3)$ and $h$ , respectively, and we consider the 2D symmetrical homogeneous space defined by

S_{κ}^{2} = {S O}_{κ} (3) / H, H = S O (2) = ⟨ J_{12} ⟩ .

(4)

This coset space has constant Gaussian curvature equal to $κ$ and is endowed with a metric having positive definite signature. The generator $J_{12}$ leaves a point $O$ invariant, the origin, so generating rotations around $O$ , while $J_{01}$ and $J_{02}$ generate translations which move $O$ along two basic orthogonal geodesics $l_{1}$ and $l_{2}$ .

Therefore $S_{κ}^{2}$ (4) covers the three classical 2D Riemannian spaces of constant curvature:

\begin{matrix} S_{+}^{2} : Sphere & S_{0}^{2} : Euclidean plane & S_{-}^{2} : Hyperbolic % space S^{2} = S O (3) / S O (2) & E^{2} = I S O (2) / S O (2) & H^{2} = S O (2, 1) / S O (2) \end{matrix}

We recall that these three spaces (and their motion groups ${S O}_{κ} (3)$ ) are contained within the family of the so-called 2D orthogonal Cayley–Klein geometries [6, 35, 36], which are parametrized in terms of two graded contraction parameters $κ \equiv κ_{1}$ and $κ_{2}$ [34].

In what follows we describe the metric structure and the geodesic motion on the above spaces in terms of several sets of coordinates that will be used throughout the paper. We stress that all the resulting expressions will have always a smooth and well-defined flat limit (contraction) $κ \to 0$ reducing to the corresponding Euclidean ones.

2.1 Ambient space coordinates

The vector representation of ${s o}_{κ} (3)$ is provided by the following faithful matrix representation $ρ : {s o}_{κ} (3) \to E n d (R^{3})$ [8, 9]

(5)

which satisfies

ρ (J_{i j})^{T} I_{κ} + I_{κ} ρ (J_{i j}) = 0, I_{κ} = d i a g (1, κ, κ) .

(6)

The matrix exponentiation of (5) leads to the following one-parametric subgroups of ${S O}_{κ} (3)$ :

	$e^{α ρ (J_{01})}$			(7)
	$e^{β ρ (J_{02})}$	$= ⎛ ⎜ ⎝ \begin{matrix} C_{κ} (β) & 0 & - κ S_{κ} (β) 0 & 1 & 0 S_{κ} (β) & 0 & C_{κ} (β) \end{matrix} ⎞ ⎟ ⎠,$		(7)

where we have introduced the $κ$ -dependent cosine and sine functions [6, 8]

C_{κ} (x) := \infty \sum l = 0 (- κ)^{l} \frac{x^{2 l}}{(2 l)!} = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} cos \sqrt{κ} x & κ > 0 1 & κ = 0 cosh \sqrt{- κ} x & κ < 0 \end{matrix}

S_{κ} (x) := \infty \sum l = 0 (- κ)^{l} \frac{x^{2 l + 1}}{(2 l + 1)!} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1}{\sqrt{κ}} sin \sqrt{κ} x & κ > 0 x & κ = 0 \frac{1}{\sqrt{- κ}} sinh \sqrt{- κ} x & κ < 0 \end{matrix} .

The $κ$ -tangent function is defined as

T_{κ} (x) := \frac{S_{κ} (x)}{C_{κ} (x)} .

These curvature-dependent trigonometric functions coincide with the circular and hyperbolic ones for $κ = \pm 1$ , while under the contraction $κ = 0$ they reduce to the parabolic functions: $C_{0} (x) = 1$ and $S_{0} (x) = T_{0} (x) = x$ . Some trigonometric relations read [8]

C_{κ}^{2} (x) + κ S_{κ}^{2} (x) = 1, C_{κ} (2 x) = C_{κ}^{2} (x) - κ S_{κ}^{2} (x), S_{κ} (2 x) = 2 S_{κ} (x) C_{κ} (x)

and their derivatives are given by [9]

\frac{d}{d x} C_{κ} (x) = - κ S_{κ} (x), \frac{d}{d x} S_{κ} (x) = C_{κ} (x), \frac{d}{d x} T_{κ} (x) = \frac{1}{C_{κ}^{2} (x)} .

Therefore, under the matrix realization (7), the Lie group ${S O}_{κ} (3)$ becomes a group of isometries of the bilinear form $I_{κ}$ (6),

g^{T} I_{κ} g = I_{κ}, \forall g \in {S O}_{κ} (3),

acting on a 3D linear ambient space $R^{3} = (x_{0}, x_{1}, x_{2})$ through matrix multiplication. The subgroup $e^{γ ρ (J_{12})}$ (7) is the isotropy subgroup of the point $O = (1, 0, 0)$ , which is taken as the origin in the homogeneous space $S_{κ}^{2}$ (4). The orbit of $O$ is contained in the “ $κ$ -sphere” determined by $I_{κ}$ (6):

Σ_{κ} : x_{0}^{2} + κ (x_{1}^{2} + x_{2}^{2}) = 1.

(8)

The connected component of $Σ_{κ}$ is identified with the space $S_{κ}^{2}$ and the action of ${S O}_{κ} (3)$ is transitive on it. The coordinates $(x_{0}, x_{1}, x_{2})$ , satisfying the constraint (8) are called ambient space or Weierstrass coordinates. Notice that for $κ > 0$ we recover the sphere, if $κ < 0$ we find the two-sheeted hyperboloid, and in the flat case with $κ = 0$ we get two Euclidean planes $x_{0} = \pm 1$ with Cartesian coordinates $(x_{1}, x_{2})$ . Since $O = (1, 0, 0)$ , we identify the hyperbolic space $H^{2}$ with the connected component corresponding to the sheet of the hyperboloid with $x_{0} \geq 1$ , and the Euclidean space $E^{2}$ with the plane $x_{0} = + 1$ .

The metric on $S_{κ}^{2}$ comes from the flat ambient metric in $R^{3}$ divided by the curvature $κ$ and restricted to $Σ_{κ}$ :

(d s)_{κ}^{2} = {\frac{1}{κ} (d x_{0}^{2} + κ (d x_{1}^{2} + d x_{2}^{2})) ∣ ∣ ∣}_{Σ_{κ}} = \frac{κ {(x_{1} d x_{1} + x_{2} d x_{2})}^{2}}{1 - κ (x_{1}^{2} + x_{2}^{2})} + d x_{1}^{2} + d x_{2}^{2} .

(9)

Isometry vector fields in ambient coordinates for ${s o}_{κ} (3)$ , fulfilling (2), are directly obtained from the vector representation (5):

J_{01} = κ x_{1} \partial_{0} - x_{0} \partial_{1}, J_{02} = κ x_{2} \partial_{0} - x_{0} \partial_{2}, J_{12} = x_{2} \partial_{1} - x_{1} \partial_{2},

(10)

where $\partial_{μ} = \partial / \partial x_{μ}$ $(μ = 0, 1, 2)$ .

Now we consider the ambient momenta $π_{μ}$ conjugate to $x_{μ}$ fulfilling the canonical Poisson bracket ${x_{μ}, π_{ν}} = δ_{μ ν}$ subjected to the constraint (8). The vector fields (10) give rise to a symplectic realization of ${s o}_{κ} (3)$ in terms of ambient variables by setting $\partial_{μ} \to - π_{μ}$ :

J_{01} = x_{0} π_{1} - κ x_{1} π_{0}, J_{02} = x_{0} π_{2} - κ x_{2} π_{0}, J_{12} = x_{1} π_{2} - x_{2} π_{1},

(11)

which close the Poisson brackets defining the Lie–Poisson algebra ${s o}_{κ} (3)$

{J_{12}, J_{01}} = J_{02}, {J_{12}, J_{02}} = - J_{01}, {J_{01}, J_{02}} = κ J_{12} .

The metric (9) provides the free Lagrangian $L_{κ}$ with ambient velocities ${˙ x}_{μ}$ for a particle with unit mass, so determining geodesic motion on $S_{κ}^{2}$ :

L_{κ} = {\frac{1}{2 κ} ({˙ x}_{0}^{2} + κ ({˙ x}_{1}^{2} + {˙ x}_{2}^{2})) ∣ ∣ ∣}_{Σ_{κ}} = \frac{κ {(x_{1} {˙ x}_{1} + x_{2} {˙ x}_{2})}^{2}}{2 (1 - κ (x_{1}^{2} + x_{2}^{2}))} + \frac{1}{2} ({˙ x}_{1}^{2} + {˙ x}_{2}^{2}) .

(12)

Thus the corresponding momenta $π_{μ} = \partial L_{κ} / \partial {˙ x}_{μ}$ read

π_{0} = {˙ x}_{0} / κ, π_{1} = {˙ x}_{1}, π_{2} = {˙ x}_{2} .

(13)

The time derivative of the constraint (8) provides the relation

Σ_{κ} : x_{0} π_{0} + x_{1} π_{1} + x_{2} π_{2} = 0.

Finally, by introducing (13) in (12) we obtain that the kinetic energy $T_{κ}$ in ambient variables is given by

T_{κ} = {\frac{1}{2} (κ π_{0}^{2} + π_{1}^{2} + π_{2}^{2}) ∣ ∣ ∣}_{Σ_{κ}} = \frac{κ {(x_{1} π_{1} + x_{2} π_{2})}^{2}}{2 (1 - κ (x_{1}^{2} + x_{2}^{2}))} + \frac{1}{2} (π_{1}^{2} + π_{2}^{2}) .

(14)

Notice that the contraction $κ = 0$ is well-defined in the r.h.s. of the equations (9), (12) and (14) yielding the Euclidean expresions

(d s)_{0}^{2} = d x_{1}^{2} + d x_{2}^{2}, L_{0} = \frac{}{1} 2 ({˙ x}_{1}^{2} + {˙ x}_{2}^{2}), T_{0} = \frac{1}{2} (π_{1}^{2} + π_{2}^{2}) .

2.2 Geodesic parallel and polar coordinates

The ambient coordinates (8) can also be parametrized in terms of two intrinsic variables of geodesic type. For our purposes let us consider the so-called geodesic parallel $(x, y)$ and geodesic polar $(r, ϕ)$ coordinates of a point $Q = (x_{0}, x_{1}, x_{2}) \in S_{κ}^{2}$ [7, 9], which are defined through the following action of the one-parametric subgroups (7) on the origin $O = (1, 0, 0)$ :

	$(x_{0}, x_{1}, x_{2})^{T}$	$= exp (x ρ (J_{01})) exp (y ρ (J_{02})) O^{T}$
		$= exp (ϕ ρ (J_{12})) exp (r ρ (J_{01})) O^{T},$

which gives

$x_{0}$	$= C_{κ} (x) C_{κ} (y) = C_{κ} (r),$	(15)
$x_{1}$	$= S_{κ} (x) C_{κ} (y) = S_{κ} (r) cos ϕ,$
$x_{2}$	$= S_{κ} (y) = S_{κ} (r) sin ϕ .$

In this construction, the variable $r$ is the distance between the origin $O$ and the point $Q$ measured along the geodesic $l$ that joins both points, while $ϕ$ is the angle of $l$ with respect to a base geodesic $l_{1}$ (associated with the translation generator $J_{01}$ ). Let $Q_{1}$ be the intersection point of $l_{1}$ with its orthogonal geodesic $l_{2}^{'}$ through $Q$ . Then $x$ is the geodesic distance between $O$ and $Q_{1}$ measured along $l_{1}$ and $y$ is the geodesic distance between $Q_{1}$ and $Q$ measured along $l_{2}^{'}$ . On $E^{2}$ with $κ = 0$ , the relations (15) lead to $x_{0} = 1$ and $(x_{1}, x_{2}) = (x, y) = (r cos ϕ, r sin ϕ)$ so reducing to Cartesian and polar coordinates.

These coordinates are shown Figure 1 for $S^{2}$ and $H^{2}$ . In these pictures, $l_{2}$ is the base geodesic orthogonal to $l_{1}$ through $O$ , so related to $J_{02}$ , and $Q_{2}$ is the intersection point of $l_{2}$ with its orthogonal geodesic $l_{1}^{'}$ through $Q$ .

We substitute (15) in the ambient metric (9) and in the free Lagrangian (12), finding that

	$(d s)_{κ}^{2}$	$= C_{κ}^{2} (y) d x^{2} + d y^{2} = d r^{2} + S_{κ}^{2} (r) d ϕ^{2},$
	$L_{κ}$	$= \frac{1}{2} (C_{κ}^{2} (y) {˙ x}^{2} + ˙ y^{2}) = \frac{1}{2} ({˙ r}^{2} + S_{κ}^{2} (r) ˙ ϕ^{2}) .$

Now, we denote $(p_{x}, p_{y})$ and $(p_{r}, p_{ϕ})$ the conjugate momenta of the coordinates $(x, y)$ and $(r, ϕ)$ , respectively, and the free Hamiltonian (kinetic energy) turns out to be

T_{κ} = \frac{1}{2} (\frac{p_{x}^{2}}{C_{κ}^{2} (y)} + p_{y}^{2}) = \frac{1}{2} (p_{r}^{2} + \frac{p_{ϕ}^{2}}{S_{κ}^{2} (r)}) .

(16)

According to (15) and avoiding singularities in (16), we find that the domain of the geodesic coordinates on $S^{2}$ and $H^{2}$ reads (always $ϕ \in [0, 2 π)$ )

	$S^{2} (κ > 0) :$		$- \frac{π}{\sqrt{κ}} < x \leq \frac{π}{\sqrt{κ}}, - \frac{π}{2 \sqrt{κ}} < y < \frac{π}{2 \sqrt{κ}}, 0 < r < \frac{π}{\sqrt{κ}} .$
	$H^{2} (κ < 0) :$		$- \infty < x < \infty, - \infty < y < \infty, 0 < r < \infty .$		(17)

Figure 1: Ambient $(x_{0}, x_{1}, x_{2})$ , geodesic parallel $(x, y)$ and geodesic polar $(r, ϕ)$ coordinates of a point $Q$ on the sphere $S^{2}$ with $κ = + 1$ and on the hyperbolic space $H^{2}$ with $κ = - 1$ and $x_{0} \geq 1$ . The origin of the space in ambient coordinates is $O = (1, 0, 0)$ . Note that on $S^{2}$ , $O_{1} = (0, 1, 0)$ and $O_{2} = (0, 0, 1)$

3 Beltrami coordinates and projective dynamics

The quotients $(x_{1} / x_{0}, x_{2} / x_{0}) \equiv (q_{1}, q_{2})$ of the ambient coordinates (8) are just the Beltrami coordinates of projective geometry for the sphere and the hyperbolic plane. They are obtained by applying the central stereographic projection with pole $(0, 0, 0) \in R^{3}$ of a point $Q = (x_{0}, x_{1}, x_{2})$ onto the projective plane with $x_{0} = 1$ and coordinates $(q_{1}, q_{2})$ :

(x_{0}, x_{1}, x_{2}) \in Σ_{κ} \to (0, 0, 0) + μ (1, q_{1}, q_{2}) \in Σ_{κ},

giving rise to the expressions

	$x_{0}$	$= μ = \frac{1}{\sqrt{1 + κ (q_{1}^{2} + q_{2}^{2})}}, x_{i} = μ q_{i} = \frac{q_{i}}{\sqrt{1 + κ (q_{1}^{2} + q_{2}^{2})}},$		(18)
	$q_{i}$	$= \frac{x_{i}}{x_{0}}, q_{1}^{2} + q_{2}^{2} = \frac{1 - x_{0}^{2}}{κ x_{0}^{2}}, i = 1, 2.$		(18)

Thus the origin $O = (1, 0, 0) \in Σ_{κ}$ goes to the origin $(q_{1}, q_{2}) = (0, 0)$ in the projective space $S_{κ}^{2}$ .

The domain of $(q_{1}, q_{2})$ depends on the value of the curvature $κ$ . We write $κ$ in terms of the radius $R$ of the space as $κ = \pm 1 / R^{2}$ and we find that in the sphere $S^{2}$ with $κ = 1 / R^{2} > 0$ , $q_{i} \in (- \infty, + \infty)$ . The points in the equator in $Σ_{κ}$ with $x_{0} = 0$ ( $x_{1}^{2} + x_{2}^{2} = R^{2}$ ) go to infinity, so that the projection (18) is well-defined for the hemisphere with $x_{0} > 0$ . In the hyperbolic or Lobachevski space $H^{2}$ with $κ = - 1 / R^{2} < 0$ and $x_{0} \geq 1$ it is satisfied that

q_{1}^{2} + q_{2}^{2} = \frac{x_{0}^{2} - 1}{| κ | x_{0}^{2}} < R^{2},

which is the Poincaré disk in Beltrami coordinates and

q_{i} \in (- 1 / \sqrt{| κ |}, + 1 / \sqrt{| κ |}) = (- R, + R) .

The points at the infinity in $H^{2}$ $(x_{0} \to \infty)$ are mapped onto to the circle $q_{1}^{2} + q_{2}^{2} = R^{2}$ . Finally, in the Euclidean plane $E^{2}$ , with $κ = 0$ ( $R \to \infty$ ), the Beltrami coordinates are just the Cartesian ones $x_{i} = q_{i} \in (- \infty, + \infty)$ .

By introducing (18) in the ambient metric (9) and in the free Lagrangian (12) we obtain that

(d s)_{κ}^{2} = \frac{(1 + κ q^{2}) d q^{2} - κ (q \cdot d q)^{2}}{(1 + κ q^{2})^{2}}, L_{κ} = \frac{(1 + κ q^{2}) {˙ q}^{2} - κ (q \cdot ˙ q)^{2}}{2 (1 + κ q^{2})^{2}},

(19)

where $q = (q_{1}, q_{2})$ and hereafter we shall use the following notation for any 2-vectors $a = (a_{1}, a_{2})$ and $b = (b_{1}, b_{2})$ :

a^{2} = a_{1}^{2} + a_{2}^{2}, a \cdot b = a_{1} b_{1} + a_{2} b_{2} .

The Beltrami momenta $p = (p_{1}, p_{2})$ conjugate to the coordinates $q$ , such that ${q_{i}, p_{j}} = δ_{i j}$ , come from $p_{i} = \partial L_{κ} / \partial {˙ q}_{i}$

p_{i} = \frac{(1 + κ q^{2}) {˙ q}_{i} - κ (q \cdot ˙ q) q_{i}}{(1 + κ q^{2})^{2}}, {˙ q}_{i} = (1 + κ q^{2}) (p_{i} + κ (q \cdot p) q_{i}) .

(20)

And by inserting these expressions into $L_{κ}$ (19) we get the free Hamiltonian

T_{κ} = \frac{1}{2} (1 + κ q^{2}) (p^{2} + κ (q \cdot p)^{2}) .

(21)

By introducing (18) and (20) in (13) we obtain the ambient momenta written in terms of the Beltrami variables, $π_{μ} (q, p)$ , and from this result a symplectic realization of the Lie–Poisson generators (11) in these variables is directly found. These expressions are displayed in Table 1. Notice that the kinetic energy (21) can also be recovered by computing the symplectic realization of the Casimir (3) of ${s o}_{κ} (3)$ in Beltrami variables as $T_{κ} \equiv \frac{1}{2} C$ . Likewise the ambient momenta $π_{μ}$ and symplectic realization of the Lie–Poisson generators $J_{μ ν}$ can be computed in the geodesic variables introduced in Section 2.2, and these are also presented in Table 1.

We recall that a similar procedure can be performed with Poincaré coordinates [37] which come from the stereographic projection with pole $(- 1, 0, 0)$ . The resulting expressions can be found in [32].

	Beltrami $(q, p)$	Geodesic parallel $(x, y, p_{x}, p_{y})$	Geodesic polar $(r, ϕ, p_{r}, p_{ϕ})$
$x_{0}$	$\frac{1}{(1 + κ q^{2})^{1 / 2}}$	$C_{κ} (x) C_{κ} (y)$	$C_{κ} (r)$
$x_{1}$	$\frac{q_{1}}{(1 + κ q^{2})^{1 / 2}}$	$S_{κ} (x) C_{κ} (y)$	$S_{κ} (r) cos ϕ$
$x_{2}$	$\frac{q_{2}}{(1 + κ q^{2})^{1 / 2}}$	$S_{κ} (y)$	$S_{κ} (r) sin ϕ$
$π_{0}$	$- \sqrt{1 + κ q^{2}} (q \cdot p)$	$- \frac{S_{κ} (x)}{C_{κ} (y)} p_{x} - C_{κ} (x) S_{κ} (y) p_{y}$	$- S_{κ} (r) p_{r}$
$π_{1}$	$\sqrt{1 + κ q^{2}} p_{1}$	$\frac{C_{κ} (x)}{C_{κ} (y)} p_{x} - κ S_{κ} (x) S_{κ} (y) p_{y}$	$C_{κ} (r) cos ϕ p_{r} - \frac{sin ϕ}{S_{κ} (r)} p_{ϕ}$
$π_{2}$	$\sqrt{1 + κ q^{2}} p_{2}$	$C_{κ} (y) p_{y}$	$C_{κ} (r) sin ϕ p_{r} + \frac{cos ϕ}{S_{κ} (r)} p_{ϕ}$
$T_{κ}$	$\frac{1}{2} (1 + κ q^{2}) (p^{2} + κ (q \cdot p)^{2})$	$\frac{1}{2} (\frac{p_{x}^{2}}{C_{κ}^{2} (y)} + p_{y}^{2})$	$\frac{1}{2} (p_{r}^{2} + \frac{p_{ϕ}^{2}}{S_{κ}^{2} (r)})$
$J_{01}$	$p_{1} + κ (q \cdot p) q_{1}$	$p_{x}$	$cos ϕ p_{r} - \frac{sin ϕ}{T_{κ} (r)} p_{ϕ}$
$J_{02}$	$p_{2} + κ (q \cdot p) q_{2}$	$C_{κ} (x) p_{y} + κ S_{κ} (x) T_{κ} (y) p_{x}$	$sin ϕ p_{r} + \frac{cos ϕ}{T_{κ} (r)} p_{ϕ}$
$J_{12}$	$q_{1} p_{2} - q_{2} p_{1}$	$S_{κ} (x) p_{y} - C_{κ} (x) T_{κ} (y) p_{x}$	$p_{ϕ}$

Table 1: Expressions for the ambient variables

(x_{μ}, π_{μ})

, free Hamiltonian

T_{κ}

and symplectic realization of the Lie–Poisson generators

J_{μ ν}

{s o}_{κ} (3)

in terms of Beltrami, geodesic parallel and geodesic polar canonical variables. The specific expressions for

S^{2}

H^{2}

and

E^{2}

correspond to set

κ > 0

κ < 0

and

κ = 0

, respectively

4 Anisotropic oscillators on the Euclidean plane

To start with, let us consider the Hamiltonian determining the anisotropic oscillator with unit mass and frequencies $ω_{x}$ and $ω_{y}$ on the Euclidean plane in Cartesian coordinates $(x, y) \in R^{2}$ and conjugate momenta $(p_{x}, p_{y})$ :

H = \frac{1}{2} (p_{x}^{2} + p_{y}^{2}) + \frac{1}{2} (ω_{x}^{2} x^{2} + ω_{y}^{2} y^{2}) .

(22)

Clearly, this Hamiltonian is always integrable due to its separability in Cartesian coordinates so that it Poisson-commutes with the (quadratic in the momenta) integrals of motion

I_{x} = \frac{1}{2} p_{x}^{2} + \frac{1}{2} ω_{x}^{2} x^{2}, I_{y} = \frac{1}{2} p_{y}^{2} + \frac{1}{2} ω_{y}^{2} y^{2},

which are not independent since

H = I_{x} + I_{y} .

Furthermore, it is also well-known that for commensurate frequencies $ω_{x} : ω_{y}$ the Hamiltonian (22) provides a superintegrable system [38, 39, 40], in such a manner that an “additional” (in general higher-order in the momenta) integral of motion does exist.

The (super)integrability properties of the commensurate oscillator will be sketched by following the approach given in [41, 42], which is based on a classical factorization formalism (see [43, 44, 45, 46, 47] and references therein). If we denote

ω_{x} = γ ω_{y}, ω_{y} = ω, γ \in R^{+} / {0},

(23)

then $H$ (22) can be written in terms of the parameter $γ$ and frequency $ω$ as

H = \frac{1}{2} (p_{x}^{2} + p_{y}^{2}) + \frac{ω^{2}}{2} ((γ x)^{2} + y^{2}) .

(24)

Next we introduce new canonical variables

ξ = γ x, p_{ξ} = p_{x} / γ, ξ \in R,

(25)

giving rise to

H = \frac{1}{2} p_{y}^{2} + \frac{ω^{2}}{2} y^{2} + γ^{2} (\frac{1}{2} p_{ξ}^{2} + \frac{ω^{2}}{2 γ^{2}} ξ^{2}) .

(26)

Therefore we obtain two 1D Hamiltonians $H^{ξ}$ and $H^{y}$ given by

H^{ξ} = \frac{1}{2} p_{ξ}^{2} + \frac{ω^{2}}{2 γ^{2}} ξ^{2}, H^{y} = \frac{1}{2} p_{y}^{2} + \frac{ω^{2}}{2} y^{2}, H = H^{y} + γ^{2} H^{ξ},

(27)

which are two integrals of the motion for $H$ . The 1D Hamiltonian $H^{ξ}$ (27) can then be factorized in terms of “ladder functions” $B^{\pm}$ as

H^{ξ} = B^{+} B^{-}, B^{\pm} = \mp \frac{i}{\sqrt{2}} p_{ξ} + \frac{1}{\sqrt{2}} \frac{ω}{γ} ξ,

(28)

fulfilling

{H^{ξ}, B^{\pm}} = \mp i \frac{ω}{γ} B^{\pm}, {B^{-}, B^{+}} = - i \frac{ω}{γ} .

The remaining 1D Hamiltonian $H^{y}$ (27) can also be factorized through “shift functions” $A^{\pm}$ in the form

H^{y} = A^{+} A^{-}, A^{\pm} = \mp \frac{i}{\sqrt{2}} p_{y} - \frac{ω}{\sqrt{2}} y,

(29)

so that

{H^{y}, A^{\pm}} = \pm i ω A^{\pm}, {A^{-}, A^{+}} = i ω .

Notice that the sets of functions $(H^{ξ}, B^{\pm}, 1)$ and $(H^{y}, A^{\pm}, 1)$ span a Poisson–Lie algebra isomorphic to the harmonic oscillator Lie algebra $h_{4}$ . Hence, the 2D Hamiltonian (26) can finally be expressed in terms of the above ladder and shift functions as

H = A^{+} A^{-} + γ^{2} B^{+} B^{-}, {H, B^{\pm}} = \mp i γ ω B^{\pm}, {H, A^{\pm}} = \pm i ω A^{\pm} .

The remarkable fact now is that if we consider a rational value for $γ$ ,

γ = \frac{ω_{x}}{ω_{y}} = \frac{m}{n}, m, n \in N^{*},

(30)

we obtain two additional complex constants of the motion $X^{\pm}$ for $H$ (26)

X^{\pm} = (B^{\pm})^{n} (A^{\pm}